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A theory of elementary couplings I
ANNALS
OF PHYSICS:
73, 18O-209 (1972)
A Theory
of Elementary MARCEL
Department
of Physics,
Syracuse
Couplings
I*
WELLNER
University,
Syracuse,
New
York
13210
Received June 30, 1971
Some recent attempts at constructing a detailed dynamical theory (“compensation theory”) of strong interactions are revised, extended, and systematized. It is hoped that this article, which aims at being self-contained, may serve as an introduction to the subject.
1. INTRODUCTION The problem considered in this article is the relatively old one of finding a workable Lagrangian for the hadrons. What precise form the strong interactions ought to assume in a Lagrangian theory has been a matter of debate since the early days of the pseudoscalar and pseudovector schemes for the pi-nucleon coupling. Eventually, the feeling has prevailed that such a debate will remain in some sense a sterile one until we know how to extract sufficiently reliable numbers from the formal Lagrangian for comparison with experiment. Now such a mathematical technique, which so clearly exists in the case of electrodynamics, is very effectively hidden when one deals with the strong couplings. One thesis proposed here is that the methods of electrodynamics (i.e., Born series and Feynman diagrams) will also be effective and accurate in the strong interactions when applied to well-chosen processes. If such be the case, it becomes meaningful, and indeed necessary, to ask for the detailed structure of the strong-interaction Lagrangian. This involves first of all the question of which particles to represent by a field operator for use in such a Lagrangian (these particles might be called elementary), and which other particles (resonances and bound states) must be content with a pole in a scattering matrix whose labels are the elementary particles only. The set of elementary hadrons adopted here, namely, the spin-half baryons and the pseudoscalar mesons, will be shown in the subsequent pages to be exceedingly likely candidates for the role. * Research supported in part by the National Science Foundation.
180 Copyright All rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
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Once this particle list has been settled upon, there remains to decide on the spin structure of the couplings, and, finally, one must determine their dependence on the internal quantum numbers. It is in the latter determination that this study will make use of a new systematic set of consistency requirements, to be called compensation relations. Compensation, like renormalization which it resembles and supplements, is designed to ensure the convergence (i.e., the cutoff independence) of all scattering-matrix elements term by term in a perturbation series. More specifically, the main ingredient of the method is the recognition that the weak and strong parts of the Lagrangian exert restrictive influences on each other through those transition amplitudes which they jointly determine, and that, of the two, the weak Lagrangian is the one more clearly related to the group SU(3), not through any symmetry, but through a Lie algebra as suggested by Gell-Mann [I]. In fact, no SU(3) symmetry, exact or approximate, will be assumed in the present paper. The need for compensation, as distinct from renormalization, arises as follows: Some Feynman diagrams (such as the bubble formed by a baryon-antibaryon pair mediating the decay of a pseudoscalar meson into a lepton pair) are well known to be divergent, i.e., dependent on a cutoff in the momentum-space integration. Such diagrams cannot be renormalized in the context of a current-current weak interaction. Indeed, any renormalization term would have to be first-order in the strong coupling, so that no appropriate Lagrangian counterterm can be devised. (The first-order term in the Lagrangian constitutes the basic strong interaction itself.) Instead, one can arrange for the mutual cancellation of infinities arising from diagrams of the same topology, but with different internal baryons. The latter’s strong couplings provide the needed parameters. This compensation program was initially undertaken [2] on the rather speculative basis of a pair of excellent numerical agreements with observed rates of decay, namely, for the charged and neutral pi mesons. (The latter decay rate has fulfilled its motivational purpose, but its currently accepted value is no longer very close to our theoretical results.) Further numerical results and a much better understanding of the weak currents responsible for the compensation relations emerged rather haphazardly in later published [3] and unpublished work, and in the interest of a coherent presentation this article will attempt to start ab initio. We note that the contents of Refs. [2, 31 are thereby in many cases considerably modified. The paper is organized in such a way that the reader may obtain the general philosophy, the specific assumptions, and a bare statement of the results from Sections 2-8. The detailed derivations and calculations are Ieft to the subsequent Sections 9-12. The Appendices A-F are quoted in the text where needed. It is hoped that the reader will tolerate a number of anticipatory statements, which this subdivision of material necessitates.
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2. PERTURBATION METHOD AND STRONG INTERACTIONS To many students of the strong interactions, the notion of a perturbative treatment is little more than an historical museum piece. The present work will nevertheless demonstrate, it is hoped, that there is a large class of physical processes which open a promising field of application for precisely this technique: a Born series expansion in powers of the strong coupling constants. The processes selected for study are, first, the subject matter of this introductory article, namely, the decays of a pseudoscalar meson into an arbitrary number of leptons, photons, or weak intermediate bosons (W bosons); and, second, the nonleptonic decays of a baryon into another baryon and a single meson-these processes being left for later discussion. The reason for selecting these particular amplitudes (most of which, incidentally, are virtual) is one of practical feasibility: they may well be the simplest cases in which compensation theory is applicable. Many other types of amplitude can, no doubt, be studied by this method. Consider first the meson decays. Two important danger signals for the breakdown of the Born series seem absent: (1) N o intermediate resonances appear to be available; (2) The effective strong coupling is quite possibly of moderate size. This latter point requires some explanation. We may assume that the behavior of the first Born term is indicative of that of the higher ones. In the first term, the amplitudes basically consist of three factors: a weak coupling constant G; a strong coupling constant g, whose magnitude must be similar to the pi-nucleon coupling, i.e., g2/4n E 15; and a coefficient reflecting the dynamics of the situation. Now this latter coefficient consists of a divergent term, which gets cancelled exactly (by design), and a finite term, over which one has no control. Such finite parts, as will turn out, almost cancel each other too, so that, for example, they combine to introduce the factor1 ln(Z2/AE) in the decay r -+ p + v. This factor may optimistically be viewed as a reduction, by an order of magnitude or so, of the effective strong coupling. Such a view can eventually be justified, of course, only through a detailed calculation of the subsequent Born approximation. Consider next the nonleptonic baryon decays. Here intermediate resonances, such as vector mesons and decuplet baryons, are likely to prove important, and hence one should not expect the Born series in terms of spin-half baryons and pseudoscalar mesons to be a good approximation to the physics. However, one should stil1 require the Born series to exist term by term, i.e., each separate order should be cutoff independent. There are, quite apart from mathematical neatness, two good reasons for this. First, all the well-established analyticity properties of the hadronic processes (with the exception of Regge behavior) are obtained theoretically by looking at the individual Born terms. Second, there are some very 1The particlesymbolsstandfor the masses.
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promising approximation techniques, such as the Pad6 method [4], which are based on a reordering of the Born series. Thus, the latter ought at least to have well-defined terms even if it diverges as a whole. The fact that these terms exist to all orders in an important class of models, thanks to renormalization, is, of course, nontrivial and fortunate. To assume that they continue to exist under the application of a weak-interaction “probe” is what the compensation postulate amounts to, and, when stated in this fashion, might not appear unreasonable. To conclude this discussion on the relevance of the perturbation method, one may classify the numerical results of the present study under three headings: (i) (ii) (iii)
Masses and mixing angles; Strong coupling constants; Decay lifetimes,
the order being one of increasing dependence on a convergent Born series. Before discussing these three categories of results it is well to note that in all the following material, only the first Born approximation (in the strong coupling) is ever made use of, or indeed computed. A systematic investigation of the next term, while presenting no difficulty of principle, actually requires far more extensive calculations, in which renormalization must now be enforced as well as compensation. The full “compensation postulate,” according to which the Born series is well defined term by term even under a weak-interaction probe, largely represents an act of faith in the mathematical consistency of such a requirement separately to all orders. The fact that a similar “miracle” has turned up in renormalization theory ought to encourage us to proceed likewise in compensation theory without waiting for a general proof of compensability. Thus, our immediate aim will be the relatively modest one of establishing consistency for the set of first Born approximations. Returning to classes (i)-(iii) of results, and beginning with (ii): this simply records the allowed values of the strong coupling constants g, subject to the compensation requirements, and using a weak-coupling scheme which has somehow been guessed or postulated a priori. These g’s are nothing but expansion parameters. In the first Born approximation they are identical with the physical coupling constants, defined from the residue of the baryon pole in a physical meson-baryon scattering process. Numerical evidence will be presented in Section 8 showing that this relation [expansion parameter = pole residue] appears to persist with great accuracy even when the g’s are large. (This is in the only accurately verifiable case so far of the pi-nucleon coupling.) To summarize the reliability of class (ii), the g’s are Born expansion parameters; compensation theory furnishes exact information concerning their relative sizes (i.e., the Born series may be as divergent as it pleases); their mathematical relation to the physical coupling
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constants is obscure, however, but numerically extremely close where a precise comparison can be made. We next come to the results of class (i). These arise through overdetermination of the g’s by the compensation relations. Since the g’s are thereby eliminated, the question of their exact interpretation does not arise. Thus, class (i) is made of results which are theoretically
exact in the strict sense.
Finally, to obtain class (iii), one simply inserts the results (ii) into the first Born term for various decay amplitudes. Hence, the numerical reliability depends here more clearly than elsewhere on a good convergence of the Born series. Besides the reasons already given for believing such to be the case in certain processes, there is the further one of numerical agreements with experiment which, at least in one case, appears too close to be coincidental. 3. THE ELEMENTARY PARTICLES
Since we are considering, at least in principle, all orders in the strong interaction, it seems reasonable to choose the latter in such a way that, when taken by itself, i.e., without perturbations from weaker interactions, it be renormalizable. This consideration very likely restricts the choice of hadrons to spins zero and one half. Quark theories will not be considered, although it is conceivable that one such theory could lend itself to the compensation technique. The usual ones will not, however. In the case of a single quark triplet with spin one half, the coupling must be to mesons because renormalizability appears to rule out the four-quark interaction. Then there are not enough parameters to satisfy all the compensation relations which arise. In conclusion, the minimal set of hadrons must comprise, first, the spin-half baryons, amounting to eight particles, and augmented (by necessity, as will be shown) with a ninth baryon L having all the quantum numbers of the lambda, except for an opposite parity. Thus the isospin wave functions are N = ($3
ur =
50 ( -v-- 19 (3.1)
A, and L.
C =
One will also need the following definite-parity
mixtures:
Am = A cos w + iybL sin w, LW = L cos w -
iybA sin w,
(3.2)
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I
with a mixing angle w to be determined later. The mixed wavefunction with some increase in generality over the usual case w = 0, as the in Cabibbo’s strangeness-changing currents for the weak interaction, in such a current being py”(l + ir5) L’P. Second, one has the set of nine pseudoscalar mesons with isospin
4~ is defined, one occurring a typical term functions
(3.3)
7, and H (R”
=
K”*;
K--
=
K+*;
7T- =
,+*).
Here KC is the charge conjugate of K, and H denotes a more massive analog of the q-perhaps the X0. All considerations in this first article will be completely insensitive to any mixtures that may occur between 7 and H, of for that matter to the existence of the H. In addition to the above-mentioned hadrons, one requires nine scalar analogs (3.4)
of the mesons. No strong coupling is assumed for these, at least to first order, so that they should not necessarily be identified with any observed scalar mesons, and indeed should perhaps not be called mesons at all at this stage. The need for them arises if one wishes to construct a chiral algebra of currents and simultaneously to satisfy the conserved vector current hypothesis. This will be discussed later in more detail; the scalar particles will play no role in this first article. Next, one assumes the existence of an unspecified number of W bosons (with vector wavefunctions W”), which mediate the weak-and perhaps superweakinteractions. The detailed properties of the W bosons will not be required at present. Finally, one must consider the photon field Afi, but it, too, will largely play a spectator role. Our notation has &{r“,r”>
= g@” = diag(l, -1, -1, -1)
y5 = y”y1y2y3.
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4. MASS RELATIONS AND MIXING ANGLE Some results of class (i), involving the baryon masses and the mixing angle W, are recorded here for reference. These results are exact in the sense that they do not depend in any way on the convergence of the strong-interaction Born series. All mass shifts of electromagnetic or weak origin are, however, ignored, as is consistent with this lowest-order study in the weaker couplings. The equations below are to be derived piecemeal throughout this and the subsequent articles, by eliminating the strong coupling constants from overdetermined sets of compensation relations. The particles’ symbols stand for their masses when no ambiguity is likely. The results are (l/N) + Cl/@ + UP) = 6/V + Q, (4.1) or, equivalently,
Z((s - N)2 = NE(2Z - N - S); also (3/A) + u/9
= (2/W + (2/E),
(4.2)
and fTL = z’2.
(4.3)
The “average” mass il is defined as A = A cos2 w - L sin2 w.
(4.4)
The minus sign in front of L is related to the negative L/A relative parity. The results (4.1)-(4.3) depend in part on the consideration of nonleptonic baryon decays, to be discussed in the next installment (II) of this article. Specifically, (4.1), which is verified to great accuracy (see below), will be shown there to follow from the statement [see Section 12 and Eq. (11.13) of the present article] that the strong ?rZA coupling vanishes. Another result, anticipated from II and used here, is that the strong interactions of the II and L baryons occur through the wave function /lo [Eq. (3.2)], but not through L”. This statement will be referred to as the no-nonet theorem. For the details the reader is referred to Section 6 of this article. Equation (4.2) corresponds to the Gell-Mann-Okubo mass relation [5] 3A+Z=2N+29,
(4.5)
and indeed reduces to it in the limit w ---f 0 and for small mass differences; Eq. (4.2), however, is exact and does not depend for its derivation on any approximate SU(3) symmetry [see Eq. (11.18) later in this article]. Equation (4.3) follows from Eq. (11.19); see later. After some algebra, one obtains from (4.1)-(4.3) L - Z = [2Z/(N
+ E)](2Z
- N - E).
(4.6)
A THEORY OF ELEMENTARY
COUPLINGS I
187
There are three adjustable masses, say N, 8, and /.I. An excellent fit to Z is obtained by choosing as an input the observed [6] neutral masses (in MeV/?) N=
940;
8 = 1315.
(4.7)
Then (4.1) gives i?Y= 1195
(4.8)
(between the observed [6] Co = 1193 and Z- = 1197); Eq. (4.6) gives L = 1340.
(4.9)
(A possible candidate has occasionally been seen [7] near 1330 MeV/c2.) Then, from (4.3), A = 1066. (4.10) Taking from experiment /.l = 1115.6, one then gets from (4.4) sin2 w = (A - ir>/(d + L) = 0.0202 or sin w = 0.1421 (positive without loss of generality), so that CIJ= 8.17”.
(4.11)
5. THE STRONG INTERACTION The strong-interaction Lagrangian density Z&,ng conserves isospin and strangeness, and is invariant under the separate C, P, and T transformations. Invariance under SU(3) is not assumed, even approximately. We consider the Yukawa coupling because of its renormalizability:
where the & , & stand for the baryon octet fields n, p, etc., or for the ninth field iy5L having the same parity; the & are the pseudoscalar meson nonet fields. The indices 01,p, y label the nonet members. The expression shown below represents the most general trilinear Hermitian PStronB built out of the above-mentioned nonets, and which satisfies the stated conservation laws. The present listing will serve as a definition for the various g’s. Numerically, for the pi-nucleon coupling constant [S], we take g$N/4n = 14.85 f 0.25. (5.2)
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The spin dependence shown in (5.1) is suppressed for brevity in the listing below. Thus, d*C stands for ify5Z, L*Z stands for (iy5L) y5C, etc. One has 9 strong = x7 + -% + -E”, + dz;, , (5.3) where _Ep,= 7cc’k,,N+~N + (g,,L*C
+ (gz,A*C
+ H.C.)
+ H.C.) - ig&2* x Z + g&?d],
=%c= [N+k,d + gd + givs * Z)K + H.C.1 + [E+(gmA + gd + g,zT - z) KC+ KC.], q
= 7)[gNN+N + &J*d + g,L*L + (g,,x’.t*L + H.C.) + g,C* . C + g,E+E],
9H = H[h,N+N + hJ*A
(5.4) (5.5) (5.6)
+ hLL*L
+ (hA,d*L + H.C.) + hEC* * C + h,E+E].
(5.7)
In these expressions, “H.C.” means Hermitian conjugate, and 7 stands for the Pauli matrices. All the g’s are real without loss of generality (this is shown in Appendix E), the fields being defined with suitable phases. The explicit, particleby-particle expressions for Zn and 5pK are given in Appendix A for the reader’s use in calculations and as his check on this article’s isospin notation. The behavior of arong under Gell-Mann’s &Y transformation [9] is outlined in Appendix B; in Appendix C we list the values that would be assigned to the couplings if one did assume SU(3) invariance [lo]. The complete renormalizable strong interaction should also include terms of the form &3,~+~~,~,~8
(5.8)
*
Renormalization theory requires that the constants h at least contain even powers of the g’s. Such “$4 terms” play no role in the present study.
6. RESULTS FOR THE STRONG COUPLING
CONSTANTS
The strong Lagrangian of the preceding section contains 23 adjustable coupling constants. In the course of the following sections, they will be reduced to 8 adjustable parameters. Two results of the subsequent article will be used: Eq. (4.1) will be freely applied to simplify some expressions, and the no-nonet theorem will be extensively drawn upon. This theorem states that one must consider only an octet (rather than a nonet)
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I
of baryon wavefunctions in the strong-interaction Lagrangian. Specifically, claim that AU, but not L”, occurs in (5.3). This immediately implies gEA : gzL = g,, : g,, = g,
we
: g,, = cos w : sin W,
g, : g,, : g, = h, : h,, : hL = co9 co : cos w sin w : sin2 W.
(6.1)
This theorem is linked to SU(3) in that w is the corresponding mixing angle used in the SU(3) currents. The theorem is also strongly hinted at by the present paper’s compensation relations, which then admit the existence of a special scalar current density .F, Eq. (D.lO). [This will be commented on in Section 11 in connection with(10.17),(11.3)and(11.25);seelater.] One finds gEA gNN
= pz/lN,
-
&L
=
GE = -p,
0,
g,
= p-z/E,
(6.2)
&,7/j = q[(E - if)@]
cos w,
gm = qW
- Al/d]
cm w,
gNL = q[(E - if)/A]
sin w,
gEL = q[(N - li)/?i]
sin w,
gN, gN
= q[(z - E)/d3
z],
g_F= r.zJB,
= r-W,
g‘q = ucos2 w, [hN = r’Z/N,
g, = dW - z)/1’\/5Jzl,
gL = 24sin2 w,
g,, = u sin w cos 0,
etc.: see gN , etc., with I -+ r’, u -+ u’.]
(6.3) (6.4) (6.5) (6.6)
(6.7)
In Eqs. (6.3), ii is given by (4.4). Thus, only the quantities p, q, r, r’, U, u’, g, , and hz are still undetermined relative to each other. Numerically, from (5.2) and (6.2): &-/4n
= 9.18 f 0.15,
gQ4lr
= 7.57 f 0.13,
(6.8)
the UnCertaintieS ariSing from gNN . It iS worth noting that, in these reSults, sU(3) symmetry is badly broken, or indeed absent altogether (cf. Appendix C). Experimentally, precise information is difficult to obtain and its interpretation [ll] may be affected by including a (l/2)- particle, our L (1340), in the analysis. There are two possible arguments leading to g,, = gzL = 0. One of them is “numerological” and invokes the excellent mass relation (4.1). A purely deductive argument, presented in Section 12, consists of taking the weak current ,$ [Eq. (D. 16)] as the limit of a somewhat more general one. The experimental situation as regards g,, appears somewhat confused [I 11, some authors favoring a very large value.
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7. THE WEAKER
INTERACTIONS
The combined electromagnetic, weak, and possibly superweak Lagrangian density for the hadrons is postulated to be of the form (assumed Hermitian) (7.1) (p = four-vector index). The W, are intermediate bosons, among which is the photon; the J, are appropriate current densities, to be described presently; and the GOBare the corresponding coupling constants, not further specified in this work. For the purpose of this article, only the baryonic currents are relevant. They consist of vector (V) and axial-vector (A) terms of the form 09 : cLv%
(7.2)
and (4 : hJ”(-ir3
$5 9
(7.3)
where K and X are nonet labels. Seventeen linearly independent currents are considered. As explained in later sections, this amounts to as many currents as could be found, subject to the requirements that (a) the electromagnetic (V) current be included; (b) the V - A generators of W(3) be included, somewhat like in Cabibbo’s theory [12]; (c) all compensation relations be mutually compatible; (d) the set of all currents, or, better, all “charges”, constitute a Lie algebra (or an Abelian set of Lie algebras), in the spirit of Gell-Mann’s prescription [l]. The list of currents follows. (i) SU(3) algebra: Eight Y - A currents 8+, PO, P-, X+, X0, Y”, Y-, Y, respectively transforming under SU(3) like the particle octet n+, no, W-, K+, K”, KO, R-, 7).
These currents are F combinations and, thus, their F’ parts are the generators of W(3). No D combinations were found compatible with the requirements (a)-(d) above. (ii) number. (iii)
Universal scalar: A single V current S?, expressing the total baryon SU(3) scalar: A single V - A isosinglet current2, dp.
a No confusion with a Lagrangian density should arise.
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191
COUPLINGS I
(iv) SU(2) algebra: Three V + A currents, Y’+, PO, 9’-. They form an isovector identical to g+, PO, 9- except for the sign of the A term. (v) Completion to SO(4) algebra: Three V + A currents, f+, &O, f-, forming an isovector. Of the seventeen currents, they are the only C&violating ones. Together with (iv), they make up the algebra of SO(4). The CP violation is chosen, for the sake of simplicity, to be maximal in a sense to be explained in Section 12. (vi) SO(4) scalar: A single V + A isosinglet current ?Y’, identical to ??I except for the sign of the A term. These currents may also be classified according to their properties under the R transformation, as exhibited in Appendix B. This makes, for example, a distinctLoon between ?Y and L?. The .A-like wavefunctions (see Section 3) occur as follows: Only /lW in (i); d and L in (ii); only LW in (iii); none in (iv); only L in (v); none in (vi). These assignments are not arbitrary and will be justified in Section 12. It is worth noting that the @ current cannot occur in the axial-vector form, as it leads to a wrong overdetermination of the K-meson couplings, see Appendix F. A particle-by-particle list of the seventeen currents is given in Appendix D. To summarize, the detailed form (Appendix D) of the weaker currents is determined primarily from the requirements (a)-(d) above, but also from certain simplicity considerations which, not being fundamental, could easily be relaxed if necessary. In particular, the amount of CP violation in 3 is chosen maximal.
8. LEPTONIC COUPLING
For the purpose of calculating fermion coupling 9
semileptonic
=
AND PI-MESON
DECAY RATES
meson lifetimes we assume an effective fourcos 0 + Yu+ sin 0) 1-1~ + H.C.
c2 (CPU’
(8.1)
(G = four-vector index), after Cabibbo [12]. Here 1 is the leptonic current: z-*0 = hd~“(l
+ iy5) h9 + (e -
p, ye -
G);
(8.2)
we take 1131 G = 1.435 the normalization
1O-4serg cm3;
x
(8.3)
of 8+ and Z+ is as in Appendix D: 90+ = +ZEy,(l xi+ = -v3
+ ir5)p + . . . ,
Y&(1
+
iy5)p
+ ... .
(8.4)
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For the Cabibbo angle we take [14] I9 = 12”.
(8.5)
Equation (8.1) differs from Cabibbo’s phenomenological coupling in that it has only pure V - A currents and no D-type SU(3) currents. Thus it is clear that strong-interaction corrections will be needed in order to calculate semileptonic decays from (8.1); however, that aspect of the present theory is as yet unexplored. The greater simplicity of (8.1) over the phenomenological coupling not only befits a “basic” Lagrangian, but is also forced on the theory by the fact that the latter has no room for the D currents of W(3) (see the previous section) if current algebras are to be valid. The decay of a meson into a lepton pair is calculated here to first order in both the strong and the weak couplings. For the charged pi meson, the dimensionless partial rate for decay into p + v is then, from the bubble diagram (Section 9), -=- 1 W&1
gLN 2Gm,cos 477 [ Gw2
0
1
(8.6)
i
where Z2 ma* ~4~~) = In -7 + -&- (& NLI
+ &
- -&)
The rnZ term adds only about 0.35 % to the ln(Z2/N@ masses of Section 4, one gets from (8.6) I/(m&*
+ O&t;).
(8.7)
term. Using the baryon
= (1.85 & 0.05 f 0.07) x lo-16,
(8.8)
as against the experimental [6] (1.85 f 0.02) x lo-la. About 80 % of the first uncertainty in (8.8) is due to gNN, the rest to 0. The second uncertainty in (8.8) arises from allowing N and 9 to range independently between the neutral and charged values, keeping Z constrained by Eq. (4.1). Turning next to the decay r” + 2y, as computed from the triangle diagram (Section 9) [15], one finds the dimensionless rate -=-
1
(mTLo
d(nd = (
(8.9)
(8.10)
A THEORY OF ELEMENTARY
COUPLINGS I
193
The term in rn$ amounts only to a 0.26 % correction to the amplitude. Numerically, (8.9) gives l/(ms-),,, = (2.48 f 0.05 f 0.05) x 1O-s (8.11) [corresponding to TV,-,= (1.96 & 0.04 f 0.04) x IO-l6 set], as compared to the experimental [6] l/(m-r)& (5.46 f 1.1) x 1O-s. The uncertainties in (8.11) are from g,, and from the electromagnetic mass shifts in N and E. This discrepancy, which was not apparent [2] from older Z-Odata, would indicate that the perturbation expansion for this type of process is only moderately good (by a factor of 0.7 in the amplitude), in contrast to the n* case. In agreement with the heuristic argument of Section 2, we note that the “finite-part cancellation” l/W - l/S2 (no case) is indeed far less perfect that in ln(Z2/NE) (T+ case), so that even in terms of the present formalism some such discrepancy is perhaps to be expected.
9. FEYNMAN DIAGRAMS:
DIVERGENT
AND FINITE PARTS
For the purpose of this study we assume that all the weaker (i.e., electromagnetic, weak, or superweak) vertices involve a W boson (which may be a photon). In order to arrive at the set of Feynman diagrams to be considered for compensation, we go through the following sequence of increasing specializations: (a) Any process, real of virtual, is considered only to lowest contributing order in the weaker coupling constants. Thus, apart from kinematic factors, all diagrams reduce to those whose external lines are just hadrons and W bosons. (b) There is, by choice, only one external hadron line, namely, a pseudoscalar meson. (c) Although the compensation postulate requires the convergence of these processes to each separate order in the strong coupling constants, only the first order will be considered here. (d) Since only the closed loops must be watched for divergences, and since, with the present restrictions, the only closed loops which can occur are baryon loops (parity conservation prohibits strong three-meson vertices), one has to consider just the set of single baryon loops with one strong and one or more weaker vertices. (e) Counting momentum factors in the Feynman diagrams, one may disregard the case of loops with four or more weaker vertices. (f) The case of three weaker vertices is shown to be harmless by y-matrix counting.
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WELLNER
Thus, in the end, one gets just the bubble diagram with an A-type weak vertex, and the triangle diagram with one A and one V. These processes are shown in Fig. 1 and will be designated by the symbols M(ab)A and M(abc) VA [or M(abc) AV]. Their degree of divergence is logarithmic in the cutoff mass &‘. Internal L baryons are treated dynamically just like the other baryons, the iy5L wavefunction being used throughout in the Lagrangian and current listings. However, since the mass term rnLEL changes sign under L * iy5L, one must formally use a negative mass, -m, , for the iy5L “particle”. The divergent terms of the bubble and triangle diagrams are, up to a constant factor, M(ub)A ---f (a + b)k” In d2, (k = meson momentum), M(ubc) VA + (2~ + b - c)g”” In M2, M(ubc)A V --f (-2~
P-1)
+ b - c)gUvIn A2
(CL,v = vector indices of the currents AU, v’). We note that, for the triangle, these divergent terms change sign upon reversal of the internal arrows. It will ordinarily be more convenient to look at current combinations V f A. Equations (9.1) then give ,rise to the mass dependence
~(aW’
f A) - ztt(a+ b),
M(ubc)( V - A)( V - A) -M(ubc)(V+
A)(V+
A) -b
- c,
(9.2)
M(ubc)( V - A)( V + A) - M(ubc)( V + A)( F’ - A) - a, overall factors being ignored. A single, universal cutoff &’ is used for all Feynman diagrams. For use in Section 8, one, also needs the finite parts of the diagrams M(ub)A and M(ubc)VV for a = b and a = b = c, respectively. One has, for M(uu)A:
Tr1 d4PY-P+a_ p2
= -4iuk”
u2
Y5y(;(;
tkj8k)_+nau
YGy5)
d4p s (p2 - a”)[(~ + kj2 - a2i (9.3)
= (2~r)~ uku [In M2 - In u2 - 1 + -$ where k is the incoming meson momentum.
+ o((k2)2)],
A THEORY OF ELEMENTARY
Similarly,
195
COUPLINGS I
for M(uaa) VV:
Tr & Y. (P+ r) + a y.r+a s (p + r)2 - a2 ‘” r2 - a2
7"
Y. (r - 4)+ a (r
-
-27r2i ‘k2 = - a E”YKAP& [l + ij-$ + 4(k2)2)] 3
q)2
-
p2
y5
(9.4)
wherep and q are the outgoing vector current (or photon) momenta, with k = p + q being the incoming meson momentum. This diagram, being automatically convergent, does not fall within the class (9.1).
10. PROCESSES To BE COMPENSATED For a given process, i.e., for given currents or particles as external lines, the first step in the “compensation calculus” is to determine all the possible particle labels (baryon labels in this article) which are allowed for the internal lines. This is done here for all processes which may diverge. The notation is that of Fig. 1.
M(ab)A
M(cbc)VA [or M(obc)AV]
FIG. 1. The divergent diagrams considered in this article. A pseudoscalar meson M is strongly coupled to a baryon loop (internal masses a, b or a, b, c). One or two W bosons go out, coupled to Y or A current densities. The symbols M(ab)A and M(abc)VA [or M(abc)A V] will be used in order to refer to these processes.
One significant short cut consists of simulating isospin conservation by attaching “spurions” to the W bosons. This procedure is correct as long as W bosons do not occur internally. Thus, as far as the incoming meson is concerned, only one member of each isospin multiplet needs be considered. Also, if the incoming meson is given, certain outgoing currents (in the bubble case) or combinations of currents (in the triangle case) will yield no constraint because the amplitude changes isospin and thus vanishes identically. This consideration will be used to omit many processes;
196
WELLNER
in addition, if two processes are related by charge conjugation, only one of them will be listed. Triangular graphs having .G?as one of the outgoing currents need never be listed: the clockwise and counterclockwise loops always cancel each other term by term. Sometimes the internal baryon labels are unimportant and will be indicated by dots. This happens when, in a triangle, both currents are of the I’- A type (or both V + A) and the meson is coupled to two baryons of the same mass. Eq. (9.2) then shows that there is no divergent contribution. In the end, one is left with the following set, which is understood to include also possible replacements of 7 by H: Bubbles 7r+@lp, ‘Pz+,
Z-P,
E-Eo)B+,
n+(id.)P’+, 3i-+(np, 8-D;
Z-L,
L.Z+)$+,
K+(Ap, Lp, Zap, Z-n, EO,Z’+, E-P, E-A, E-L&f+, EOEO, E-E- g, ~PP, nn, 1
v(id.)W, ,l(AA, AL, LA, LL)8.
(10.1) (10.2) (10.3) (10.4) (10.5) (10.6) (10.7)
Triangles
In this subsection the small Roman numerals refer to the classes (see Section 7) from which the external currents are selected. (9, (9: T+( ***p%P+ + ?T+(.. ‘).LP+P, 7r+(-)YP+ + 7r+(**-)8+Y, 7r+( ..,, n,Z-A, nEL)Y”.X+ + 7r+( . ..) E-Z-A, E-Z-L, K+(pAp, pLp, p.?YOp,nEn, + K+(z-x-n, BOSOM+,
(10.8) (10.9) EOA,z’, soLz+)sf-+xo,
Z+E”2J’+)9’X+ S-E-ZO, Z-E-A,
K+@Ap, pLp, pZ”p, n2-n)CVZ+ + K+@OsOz+, E-E-EO, B-E-A,
&E-L)s+p,
E-s-L)zfg
,
7j( * **)P+LP- + 7]( ***)8-P’+, rl(***W~, rl(..., PAL, p~A)c%?-SC+ + 77(..., g-AL,
E-LA)X+Y-;
(10.10) (10.11) (10.12) (10.13) (10.14) (10.15)
A THEORY
OF ELEMENTARY
COUPLINGS
I
197
(i), (iii):: K+(flE-A, L&A, M-L, LB-L)mYX+ + K+(AAp, ALP, LAP, LLp)X+cY;
(10.16)
(i), (iv): r+(pnp, BiPZ+, ~“E-Eo)~o.P‘+ + r+(nnp, Z-Z-.Zo, 5-E-E”)8’+Po, r+(pnp, E”~-Eo)~~‘+
+ z-+(nnp, E-E-EO)B’+SY,
K+(pAp, pLp, pZ”p, d-n, .Z+EOZ+)B’OX+ + Kf(z-z-n, sOEOz+, E-&zO, E-E-A,
E-&L)x+$PO
(10.17) (10.18) (10.19)
[cf. (lO.ll)]; (9, 69: r+@np, E”E-E”; Z+LZ+)P”$+ + ir+(nnp, E-E-E”; Z-Z-L)$+P”, rr+(pnp, E”E-Eo)CYf+ K+(nEn, +
+ n+(nnp, E-B-E”)y+CY,
pZOp, pAp, pLp; LE-Z”, Z”E-L)$OX+ E-~-L, E-E-z0 , E”Eoz+; LiPp, zoLp)x+~o,
Kt(E-E-A,
q(pnn, E”B-E-)f-9
(10.20) (10.21)
+ v(npp, E-E”Eo)P+$-;
(10.22) (10.23)
+ r+(nnp, E-E-S”)~+W,
(10.24)
(i), (vi): r+@np, E”E-So)WB+
K+(pAp, pLp, pZ”p, nZ-n)WX+ + Ki-(~O~O~t, E-E-20, E-&A,
&-~)$“+q’
(10.25)
+ ~(id.)YY;
(10.26)
[cf. (10.12)], r](nnn, ppp, E”EoEo, E-E-E-)CY’CY
(iii), (iii): rl(..., AAL, ALA, LAL, LLA)S’mC?;
(10.27)
(iii), (v): x+(LZ-L,
LEA)
2’9’
+ ,+(LAZ+,
LLZ+)$+P’;
(10.28)
(iv), (iv): 7T+(.. .p’ogd’t
+ n+(. . .)y+p’o;
(10.29)
+ 7rIT+(.‘.,z-z-L)~+9J’o;
(10.30)
(iv), (9: 7+(‘.‘,
z+Lz+)P$+
(iv), (vi): n+(. . .)gy’$za’++ n+(. . .)gwy’;
(10.31)
198
WELLNER
(v>, 09: 7r+(.-*)f+&o
+ n-+(...)fo2+,
v(.-.>$‘f-
+ q(***W-y+;
(10.32) (10.33)
+ n-+(‘.‘)w$+;
(10.34)
(9, (vi): 7r’(-.-)f’W
(vi), (vi): (10.35)
q( ***)5?PTv’.
11. COMPENSATION RELATIONS In this section, the results (9.2) are written out for every process listed in Section 10. The particle labels, unless they are subscripts, now denote masses. Pi Relations
From (lO.l), (10.2), or (10.17) NgNN + 2-h.z + &.m
= 0;
(11.1)
From,(10.3) or (10.20), Ng,r, - EgEE = 0;
(11.2)
it is noteworthy that the CP violating loops [after the semicolon in (10.3) and (10.20)] make no net divergent contribution to (11.2). From (10.28), g,, cos 0 - gzA sin w = 0.
(11.3)
Processes (lO.lO), (10.18), (10.21), (10.24), and (10.30) are automatically finite. The relations (1 1.1)--(11.3) are equivalent to (6.2). Equation (11.3) shows the interdependence of the no -nonet theorem (6.1) and the existence of the current density J.Z’. K Relations
In these, it will be convenient to define t?hiA
=
gNA
cos
w
+
gNL
sin
w3
EEA = g, cos w + ggL sin w, rNA
=
r,,
= Ag,
&,A
cos w - LgNL sin w, cos w - LgaL sin w.
(11.4)
A THEORY
OF ELEMENTARY
COUPLINGS
-199
I
Then, from (10.4), NgNn -- E&,
+ r,,
- TEA + d/YN + Z)g,
+ 2/3(E + Z)g,
= 0.
(11.5)
TO study (10.11) and (10.12) it is simpler to consider first the formal “electric charge” combination Y” + (l/ ~‘3) g/, or, symbolically, the process (10.11) + (l/1/3) x (10.12). This yields
~/3rN, + %w,, I’~N&, - ,1/3%l- (N - E)g, - (9 - xZ)gEz = 0.
(11.6)
From (10.12) alone, mNA - &
- rNA -t rEA f A&N
+ 4/5&T - Z)gsz = 0.
(11.7)
+ (8 + 2Z)g,
(11.8)
- z)g,
From. (10.19) + (l/d/3) (10.25), d--NgN/, - x’~@m
+ (N + 2Z)g,
= o.
From (10.25) alone,
From (10.22), d/3N&,
+ dE&,
Z(gNL - gd
- Ng,
+ &m
sin w + 43 Lb
= 0,
+ gNz) = 0.
(11.10) (11.11)
Of these two equations, the first is the real part, the second the imaginary part of the full compensation relation. The semicolons in (10.22) are meant to show how the generating processes correspondingly split up. The remaining K process, (10.16), is automatically finite-a strong suggestion for the existence of the current 3’. The information in (11.5)-(11 .l 1) may be extracted as follows. First note that (11.9) is equivalent to (11.5) + (11.7), and (11.8) to d/5 x (11.5) + (11.6). Thus, it is enough to consider the independent set (11.5), (11.8)-(11.12). From (11.8)(1 l.lO), the values of gNA , & , g,, and g, are obtained up to some factor q’: STNA= E(--2218
+ NE + NZ) q’,
&I
+ NE + EZ) q’,
= N(--2zN
g,, = 2/3 Nc”(E
- ,i?) q’,
g,
- N) q’.
= v/5 NE@
(11.12)
200
WELLNER
[In a sequel to this article, and starting from the statement gzn = 0, the equation (11.13)
ghln + EEA - 43ggN.z + z/3&% = 0 will be derived, whence (4.1).] An alternative formulation is readily obtained Eq. (6.1). From (11.4),
from the no-nonet
theorem,
where /r = A cos2 w - L sin2 w.
(11.15)
Then, from (11.5): (N+
J~NA
-
(s
+
/r>g,,
+
dJ(N
+
z)&.,Z
+
g/3@
From (11.16), (11.8) and (11.9) one finds, up to a factor ?TNL~ = gN,
C?(E
-
4lif
= dz - @/(d3
z),
q(N
-
+
z)gsz
=
0.
(11.16)
q,
&A
=
ir>/&
g,
= 4w - q/(1/3
(11.17) 2).
The compatibility of (11.12) and (11.17) is most easily expressed by substituting (11.17) in (ll.lO), yielding (3/A) + (l/a
= (2/N
(11.18)
+ (2/Q,
i.e., Eq. (4.2). Turning next to Eq. (11.1 l), one obtains after substituting
(11.17) with
(L/Z)@ - N) - (C/A)@ - iv) = 0,
q #
0,
(11.19)
whence (4.3) if S # N. Eta Relations From (10.5) or (10.6), N&q - E& = 0,
(11.20)
AgA sin2 w + (L - (l)gA, sin w cos w - LgL cos’ w = 0.
(11.21)
whence (6.5). From (10.7),
Processes (10.15), (10.23), (10.26), and (10.27) are automatically finite. Relation (11.21) is automatic under the no-nonet theorem (6.1).
A THEORY OF ELEMENTARY
12. LAMBDA-LIKE
WAVEFUNCTIONS
201
COUPLINGS I
AND THE VANISHING
OF gzA
The detailed choice and form of the seventeen currents of Section 7 are only partly constrained by the requirement of Lie algebras. One other guide has been the purely heuristic idea that, if simple integers are to characterize a particle model, these integers should be found in the weaker currents rather than in the strong couplings. Then also, ultimately, the numerical results which are yielded through the compensation conditions should agree with experiment. It appears that, in the framework of this study, one can afford to be exceedingly particular as to the accuracy of such results if they are of class (i), Section 2. This observation is to be kept in mind throughout the present section. Our task here will be to show why those terms which, in the weaker currents, contain the wavefunctions (1 or L have the special coefficients assigned to them as described at the end of Section 7. Consider first the SU(3) currents, see (i), Section 7, and formulas (D.l)-(D.9). The form of these currents is entirely traditional, and the occurrence of rl” instead of II is actually an increase in generality. The form of 2 comes about as follows. There are two independent SU(3) scalars, Lw*Lw and the baryon number, both of which commute with the SU(3) octet currents. Although the baryon number 9 is acceptable in the V form, it cannot occur in the V - A form (see Appendix F). This leaves only 5? to be tried out, which leads to consistent relations. An additional piece of theoretical evidence for 2, independent of (6.1), is provided by the the automatically finite nature of (10.16). We next turn to the occurrence of &2i(I:*L - L*Z) in y, see (v), Section 7, and formula (D.16). Consider a more general form f2i(C*LA
- LA*Z),
(12.1)
where LA = eiX(L cos X - A sin h),
(12.2)
for some angles x and h. (We still assume no parity mixing of the wavefunctions, i.e., L stands for iy5L.) Some term of the form (12.1) is required to complete a Lie algebra. The phase x is chosen to be zero (a “maximal” CP violation) on the numerical evidence of formula (4.1), which in turn depends on (11.10) and on the (anticipated) (11.13). The presence of a nonzero x in (12.2) would permit nonzero contributions from (11 .I 1) to “leak” into (11. IO), and thereby simultaneously destroy the accuracy of (4.1) and the simplicity of the K-coupling results. Nevertheless, a very small but nonzero x has the interesting consequence of forcing gZL = 0, a fact to be discussed presently.
202
WELLNER
For general h, compensation Gh
relation (11.11) reads
- gd 03s h - %h - sEAI sin h + pCX(L cos h sin w + (1 sin h cos w)(gsZ + gNZ) = 0
(12.3)
L%h - EEA)+ vWtk~ + gd tan 0 = F%%v~- hII) - 1/Wg8z + khdl tan A.
(12.4)
or, with (11.14),
Inserting (11.17), and taking q # 0, B # N, (Z2 - &)
tan w = (Z” + Ll)
tan X.
(12.5)
Even under the rather weak assumption that the L mass be of the order of the other baryons’, we see that h < w, the angle w already being quite small owing to (11.18). Hence X = 0 is fairly compelling if any simplicity is to be retained in the theory. Next, generalizing slightly from x = 0 to x very small, we observe that processes (10.3) and (10.20) both acquire g ZL terms but are no longer equivalent. Thus, we get gZr. = 0 as well as (11.2). Hence also gZA = 0 by the no-nonet theorem. To summarize, the form’ of $ adopted here is the one which simultaneously yields the strongest and simplest results. That is to say, f is maximally CP violating as a (continuous) limit x --f 0 of the nonmaximal case. It is, of course, to be expected that even such simplicities may not survive in future compensation models. However, in this initial test of the theory it seems an essential strategy to keep the weaker interactions as simple as possible.
APPENDIX
A: EXPLICIT
gT AND -EK
The particle-by-particle expressions for 6p, and ZK [Eqs. (5.4) and (5.91 are listed here for convenience. The coupling constants are chosen in the real representation, as explained in Appendix E. gr = gNN[Ghr+p*n + G!?r-n*p + 7p(p*p - n*n)] + {g&r-(A*z+
+ z-*(1) + r+(A*E
+ 27+*/l>
+ T”(A*zo + z”*(1)] + g&l - L]} + g~&+(z~*z- z+*zo) + 7r-(z-*.P - ,P*#x+) + 7ryz+*z+ - z-*2-)] + d/zr+EO*,F + dTT-s-*E“ + nO(~O.*~O- E-*3-)], gEE[
(A-1)
A THEORY
OF
ELEMENTARY
COUPLINGS
203
I
9K = {g,,[K+p*A + K%*A + K-A*p + RoA*n] + gNL[A + L]} + g,[K+(p*zo + d/Zn*E) + KO( z/zp*z+ - n*q + K-(P*p + l/zz-*n) + KO( lLE+*p - D*n)] + {ggA[-Koso*A + R-E-*A - KOA*Eo + K+A*E-] + &LW - LII + g,,[~~o(~o*~o + 2/zE-*z-) + R-(qZEO*z+ _ s-*zo) - KO(~O*~O + d\/zz-*8-) + K+( d\/z,Z+*E” - ,ZO*E-)].
APPENDIX
(A.2)
B: THE R TRANSFORMATION
A great deal of work is spared, mostly in the nonleptonic decay calculations to be presented at a later date, by the use of Gell-Mann’s R-transformation [9]. One observes that -YStrong(Section 5; Appendix A) is invariant under the purely formal substitutions (no physical invariance being implied) /l-A,
rl t--) 79
L+-+L, P4-3 D, z+tt -z-, nt,EO, p-E-, -
&?,A,
gNL*&YL
3
gNA
&2l-
H-H, n-0e-b 770, rr+ tt -?r 9 K”e, -K”, K+ +-+ i?, 8ZA 3
g‘?,t+g,,
gN.Z+-+--8E2,
gNN
-
hz+-+
gA+-+gA,
9
gL++gL,
-hiI?, -Km
h,
gz++gz 9
+‘h,
,
k-b, 3
h.z++h.z,
hNt)hE.
gN-gS,
The seventeen currents of Section 7 transform as follows under R: “F-type”
transformation: 9+ t) @-, CYM-Y, Jf-+ t) -SF, @‘+ tt @‘- >
@Ott w t+ xot, 9’0 t)
-90; -w; 20; -9’0.
“D type” transformation: la’t,, %‘-
-3-y
9++t,, $0 t-) $0.
[The sign behavior for given isospin is what suggests the F, D nomenclature.]
204
WELLNER
APPENDIX
C: THE W(3)
INVARIANT STRONG COUPLING
If the strong interactions were SU(3) invariant, there would be no mixing between ~4 and iy5L, since a positive mass for L implies a negative mass for iy5L. There could, however, be an 7, H mixing. In the SU(3)-invariant case, and without any mixing, the 23 coupling constants in L%trong , Section 5, would depend as follows [lo] on five parameters, say d, f, gAL , hA , and hL :
gNN= d-kf, gNA
= - -&
lb= Cd + 3f),
gm = z
gm=d--f, gN
-
&
g,=
=
gSL
=
d, &L
1
-d+f, C--d + 3f),
-(d+f),
g, = - --& (d + 3f),
(F-d + 3f), g, = - $
gNL
g,,=
;?f,
=
g.z = -& g/IL
hN = h, = hz = h,
4
arbitrary, arbitrary,
h,, = gL = 0, h, arbitrary.
APPENDIX
D: EXPLICIT CURRENTS
The particle-by-particle expressions for the seventeen currents of Section 7 are listed below. Spin variables are suppressed. Thus, for example, n*p stands for Zr“(l 4~ iy5)p, L stands for iy5L, etc. Attention is called to the occurrence of the various A-like wavefunctions (1, L, (lW, and Lw, as outlined in Section 7. v - A currents [16]. P. 1) 03.2) (D.3) (D.4)
A THEORY
OF
ELEMENTARY
COUPLINGS
205
I
(D.5)
03.6)
(D.7) w3)
(D.9) (D.10)
37 = G(N+N
+ E+B + x* * z + /1*/l + L*L).
(D.11)
(the factor 1/g is just for later convenience). V + A currents Y+, PO, 9”-: f+
= 1/Z n*p -
go=p*p
p-
-
=
z/Zp$
see 9+, 80, 8-;
d/z E--*9O (T) 2i(Z-*L
n*n--o*EO+
-
q/z
E-*E-
EOo*E--
(f,
(D.12) (D.13)
- L*z+), 2i(c”*L
- L*ZO),
(D.14)
2i(Z+*L
- L*Z-)
(D.15)
$8 = N+TN - 3+75 ($‘) 2i(c*L
- L*q],
(D.16)
(‘,
W : see CV.
(D.17)
V current (Electric)
(D.18) Concerning 9, we note the two possible signs of the ZL terms. Both choices lead to the same Lie algebra, as well as being inconsequential to the rest of this paper. It is interesting that this sign is changed by a CP-transformation, so that the algebra is mapped outside of itself by that transformation. Thus, any interaction using 3 would appear to be CP nonconserving. The wavefunction L in (D.16) is to be understood as eiXL (x real), where x -+ 0 after all compensation calculations (see Section 12).
206
APPENDIX
WELLNER
E: REAL REPRESENTATION
FOR THE STRONG COUPLING
CONSTANTS
All the coupling constants in Eqs. (5.4)--(5.7) either must be real, or may be taken real without loss of generality. I. The Hermiticity
of =.YStrongrequires that the following be real:
gNN9ax ,gEs in (5.4); gN, gA , gL , a , gE in (5.6); and hi , hA , hL , hz , hs in (5.7).
II. Next, turning to LZK [Eq. (5.5)], we assume at frrst that all its g’s are complex: gNL = I gNL I ei92,
(E. 1)
g,L = I sk I ei*5, Time-reversal (T) invariance requires two relations between these six phases. Indeed, the most general T transformation is of the form K + K*&
B+y&e o-* iX,r,
N + yaN *.@N,
9 (1 -+ y”/l*eixA,
C --f y°C*eix2
iy5L -+ y0(iy5L)* eiXL,
(t -+ -t being understood), for some phase angles xK ,..., xr. under T then implies =
gNA&K+XA-XN’
(E.2)
Invariance of ,EpK
, etc.,
gsA
so that, module 2a, XK + XA -
XN
+ 24,
=
0,
XK
+
XL
-
XN
+
242
=
0,
XK
+
XZ
-
XN
+
29%
=
0,
-XK
+
?hl
-
x.R
+
244
=
0,
-XK
+
XL
-
XS
+
2+5
=
0,
-XK
+
xr:
-
XS
+
294 =
0.
,
(E-3)
Therefore, 41 - & - +4 + & = multiple +1 - & L +* + & = multiple
of n=, of 7r.
We now redefine the wavefunctions’ phases as follows: K --f Kei”K,
N + Nei*N,
(1 -+ fle”“,
L + LeidL,
(E.4)
A THEORY OF ELEMENTARY
COUPLINGS I
207
for six $‘s to be adjusted. Inserting in -YK, we find the modifications g,, -, / g,, 1 e~(~l+~K+hl--IIN), etc. One way of obtaining a real representation therefore consits of choosing the #‘s equal to half the corresponding x’s, which have been seen to exist. After the redefinitions (E.4) have been made, all the new g’s in SK are real. III. We finally consider gzA , gzL [Eq. (5.4)], g,, [Eq. (5.6)], and hnL [Eq. (5.7)], under the assumption that the redefinitions (E.4) have already been carried out, so that the 4’s are themselves multiples of 7~. Consider the remaining most general T transformation for n and L: the difference of the first two Eqs. (E.3) gives xn - xr. = multiple
of 27r.
Making use of the fact that the T invariance of, e.g., z * Nt7N implies phase angle for the T transformation of 7r, we get, under T, x . (g,mA*C
+ H.C.) -
z . (g&‘l*C
a zero
+ H.C.),
so that gzA = gz,, . The reality of g,, , g,, , and hllL is proved in analogous fashion. APPENDIX
F: ABSENCE OF g'a
We examine the consequence of assuming that a weak boson is coupled to the baryon number current (D.l l), but changed to the axial vector form aA . Since a(=gr) is automatically allowed, we consider, for convenience, a current L$?“+~ . The process K+(nZ--n, pz”p, pAp, pLp, AS-A,
Lz-L,
+ K+(AAp, LLp, S”Eo,E+, 8-&.Z”,
rE”E-.Zo, Z+E”.Z+)S?,,
E-E-A, 8-E-L,
X+
Z”,Zop, Z-Z-n)X+9v+,
,
(F.1) with the help of (6.1), gives rise to the compensation 0’ -. &!,A
+ (s - &m
+ d/3(N - Z)g,
relation
- d/3@ - Z)gsz = o.
(F.2)
Inserting (11.17), (N - ii>@ - jT,//i
= (N - Z)(E - Z)/Z
(F.3)
or (NE - Z?f)(Z - i!f) = 0. 595/73/I-14
(F-4)
208
WELLNER
One possibility, Z = A, taken with (11. I8), yields 2/z = (l/N) + (l/B). This, apart from being poorly satisfied in Nature, also implies N = E if taken together with (4.1). The other possibility, NE = Z& again taken with (11.18), gives
leading, likewise, to poor numerical agreement with facts and to N = E from (4.1).
ACKNOWLEDGMENTS The author would like to thank J. A. Cronin for comments on the possible importance of a negative-parity lambda, and J. Weinberg for a conversation on Lie algebras. A considerable portion of this research was carried out in the stimulating environment of the Cavendish Laboratory at Cambridge University, and the hospitality of R. J. Eden there is gratefully acknowledged. Last but not least, the long-standing confidence of the author’s colleagues at Syracuse University deserves thankful mention.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
8.
9. 10. 11.
M. GELL-MANN, Physics 1 (1964), 63. M. WELLNER, Phys. Rev. 168 (1968), 1855. M. WELLNER, Phys. Rev. 184 (1969), 1782; D3 (1971), 2295. For example, G. A. BAKER in “Advances in Theoretical Physics,” Vol. 1, p. 1, Wiley Interscience, New York, 1965); J. Z~NN-JUSTM, Phys. Rep. C 1 (1971), 55. M. GELL-MANN, Phys. Rev. 125 (1962), 1067; S. OKUBO, Progr. Theor. Phys. 27 (1962), 949. Particle Data Group, Rev. Mod. Phys. 42 (1970), 87. G. BOZ~KI, E. FENYVES, T. GBMESY, E. GOMBOSI, S. KRASZNOVSZKY, E. NAGY, N. P. BOGACHEV, Yu. A. BUDAGOV, V. B. VINOGRADOV, A. G. VOLODKO, V. P. DZHELEPOV, V. G. IVANOV, V. S. KLADNITSKY, S. V. KLIMENKO, Yu. F. LOMAKIN, Yu. P. MEREKOV, I. PATOEKA, V. B. FLIAGIN, AND P. V. SHLYAPNIKOV, Phys. Lett. B 28 (1968), 360. J. HAMILTON AND W. S. WOOLCOCK, Rev. Mod. Phys. 35 (1963), 737; V. K. SAMARANAYAKE AND W. S. WOOLCOCK, Phys. Rev. Lett. 15 (1965), 936; M. H. MACGREGOR AND R. A. ARNDT, Phys. Rev. B139 (1965), 362. M. GELL-MANN, Ref. [5]. Way,” pp. 51, 52, Benjamin, For example, M. GELL-MANN AND Y. NE’EMAN, “The Eightfold New York, 1964. C. H. CHAN AND L. L. SMALLEY, Phys. Rev. D 2 (1970), 2635; C. H. CHAN AND F. T. MEIERE, Phys. Lett. B28 (1968), 125, and Phys. Rev. L&t. 20 (1969), 568; D. CLINE, R. LAUMANN, AND J. Mapp, Phys. Lett. B3J (1971), 606.
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I
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12. N. CABIBBO, Phys. Rev. Left. 10 (1963), 531. 13. For example, R. E. MARSHAK, RLGUDDIN, AND C. P. RYAN, ‘Theory of Weak Interactions in Particle Physics,” p. 39, Wiley Interscience, New York, 1969. 14. Reference [13], p. 403. 15. The internal-nucleon contribution was already calculated in this way by J. STEINBERGER, Phys. Rev. 76 (1949), 1180. 16. For example, P. A. CARRUTHERS, “Introduction to Unitary Symmetry,” Wiley Interscience, New York, 1966.
OF PHYSICS:
73, 18O-209 (1972)
A Theory
of Elementary MARCEL
Department
of Physics,
Syracuse
Couplings
I*
WELLNER
University,
Syracuse,
New
York
13210
Received June 30, 1971
Some recent attempts at constructing a detailed dynamical theory (“compensation theory”) of strong interactions are revised, extended, and systematized. It is hoped that this article, which aims at being self-contained, may serve as an introduction to the subject.
1. INTRODUCTION The problem considered in this article is the relatively old one of finding a workable Lagrangian for the hadrons. What precise form the strong interactions ought to assume in a Lagrangian theory has been a matter of debate since the early days of the pseudoscalar and pseudovector schemes for the pi-nucleon coupling. Eventually, the feeling has prevailed that such a debate will remain in some sense a sterile one until we know how to extract sufficiently reliable numbers from the formal Lagrangian for comparison with experiment. Now such a mathematical technique, which so clearly exists in the case of electrodynamics, is very effectively hidden when one deals with the strong couplings. One thesis proposed here is that the methods of electrodynamics (i.e., Born series and Feynman diagrams) will also be effective and accurate in the strong interactions when applied to well-chosen processes. If such be the case, it becomes meaningful, and indeed necessary, to ask for the detailed structure of the strong-interaction Lagrangian. This involves first of all the question of which particles to represent by a field operator for use in such a Lagrangian (these particles might be called elementary), and which other particles (resonances and bound states) must be content with a pole in a scattering matrix whose labels are the elementary particles only. The set of elementary hadrons adopted here, namely, the spin-half baryons and the pseudoscalar mesons, will be shown in the subsequent pages to be exceedingly likely candidates for the role. * Research supported in part by the National Science Foundation.
180 Copyright All rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
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Once this particle list has been settled upon, there remains to decide on the spin structure of the couplings, and, finally, one must determine their dependence on the internal quantum numbers. It is in the latter determination that this study will make use of a new systematic set of consistency requirements, to be called compensation relations. Compensation, like renormalization which it resembles and supplements, is designed to ensure the convergence (i.e., the cutoff independence) of all scattering-matrix elements term by term in a perturbation series. More specifically, the main ingredient of the method is the recognition that the weak and strong parts of the Lagrangian exert restrictive influences on each other through those transition amplitudes which they jointly determine, and that, of the two, the weak Lagrangian is the one more clearly related to the group SU(3), not through any symmetry, but through a Lie algebra as suggested by Gell-Mann [I]. In fact, no SU(3) symmetry, exact or approximate, will be assumed in the present paper. The need for compensation, as distinct from renormalization, arises as follows: Some Feynman diagrams (such as the bubble formed by a baryon-antibaryon pair mediating the decay of a pseudoscalar meson into a lepton pair) are well known to be divergent, i.e., dependent on a cutoff in the momentum-space integration. Such diagrams cannot be renormalized in the context of a current-current weak interaction. Indeed, any renormalization term would have to be first-order in the strong coupling, so that no appropriate Lagrangian counterterm can be devised. (The first-order term in the Lagrangian constitutes the basic strong interaction itself.) Instead, one can arrange for the mutual cancellation of infinities arising from diagrams of the same topology, but with different internal baryons. The latter’s strong couplings provide the needed parameters. This compensation program was initially undertaken [2] on the rather speculative basis of a pair of excellent numerical agreements with observed rates of decay, namely, for the charged and neutral pi mesons. (The latter decay rate has fulfilled its motivational purpose, but its currently accepted value is no longer very close to our theoretical results.) Further numerical results and a much better understanding of the weak currents responsible for the compensation relations emerged rather haphazardly in later published [3] and unpublished work, and in the interest of a coherent presentation this article will attempt to start ab initio. We note that the contents of Refs. [2, 31 are thereby in many cases considerably modified. The paper is organized in such a way that the reader may obtain the general philosophy, the specific assumptions, and a bare statement of the results from Sections 2-8. The detailed derivations and calculations are Ieft to the subsequent Sections 9-12. The Appendices A-F are quoted in the text where needed. It is hoped that the reader will tolerate a number of anticipatory statements, which this subdivision of material necessitates.
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2. PERTURBATION METHOD AND STRONG INTERACTIONS To many students of the strong interactions, the notion of a perturbative treatment is little more than an historical museum piece. The present work will nevertheless demonstrate, it is hoped, that there is a large class of physical processes which open a promising field of application for precisely this technique: a Born series expansion in powers of the strong coupling constants. The processes selected for study are, first, the subject matter of this introductory article, namely, the decays of a pseudoscalar meson into an arbitrary number of leptons, photons, or weak intermediate bosons (W bosons); and, second, the nonleptonic decays of a baryon into another baryon and a single meson-these processes being left for later discussion. The reason for selecting these particular amplitudes (most of which, incidentally, are virtual) is one of practical feasibility: they may well be the simplest cases in which compensation theory is applicable. Many other types of amplitude can, no doubt, be studied by this method. Consider first the meson decays. Two important danger signals for the breakdown of the Born series seem absent: (1) N o intermediate resonances appear to be available; (2) The effective strong coupling is quite possibly of moderate size. This latter point requires some explanation. We may assume that the behavior of the first Born term is indicative of that of the higher ones. In the first term, the amplitudes basically consist of three factors: a weak coupling constant G; a strong coupling constant g, whose magnitude must be similar to the pi-nucleon coupling, i.e., g2/4n E 15; and a coefficient reflecting the dynamics of the situation. Now this latter coefficient consists of a divergent term, which gets cancelled exactly (by design), and a finite term, over which one has no control. Such finite parts, as will turn out, almost cancel each other too, so that, for example, they combine to introduce the factor1 ln(Z2/AE) in the decay r -+ p + v. This factor may optimistically be viewed as a reduction, by an order of magnitude or so, of the effective strong coupling. Such a view can eventually be justified, of course, only through a detailed calculation of the subsequent Born approximation. Consider next the nonleptonic baryon decays. Here intermediate resonances, such as vector mesons and decuplet baryons, are likely to prove important, and hence one should not expect the Born series in terms of spin-half baryons and pseudoscalar mesons to be a good approximation to the physics. However, one should stil1 require the Born series to exist term by term, i.e., each separate order should be cutoff independent. There are, quite apart from mathematical neatness, two good reasons for this. First, all the well-established analyticity properties of the hadronic processes (with the exception of Regge behavior) are obtained theoretically by looking at the individual Born terms. Second, there are some very 1The particlesymbolsstandfor the masses.
A THEORY
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183
promising approximation techniques, such as the Pad6 method [4], which are based on a reordering of the Born series. Thus, the latter ought at least to have well-defined terms even if it diverges as a whole. The fact that these terms exist to all orders in an important class of models, thanks to renormalization, is, of course, nontrivial and fortunate. To assume that they continue to exist under the application of a weak-interaction “probe” is what the compensation postulate amounts to, and, when stated in this fashion, might not appear unreasonable. To conclude this discussion on the relevance of the perturbation method, one may classify the numerical results of the present study under three headings: (i) (ii) (iii)
Masses and mixing angles; Strong coupling constants; Decay lifetimes,
the order being one of increasing dependence on a convergent Born series. Before discussing these three categories of results it is well to note that in all the following material, only the first Born approximation (in the strong coupling) is ever made use of, or indeed computed. A systematic investigation of the next term, while presenting no difficulty of principle, actually requires far more extensive calculations, in which renormalization must now be enforced as well as compensation. The full “compensation postulate,” according to which the Born series is well defined term by term even under a weak-interaction probe, largely represents an act of faith in the mathematical consistency of such a requirement separately to all orders. The fact that a similar “miracle” has turned up in renormalization theory ought to encourage us to proceed likewise in compensation theory without waiting for a general proof of compensability. Thus, our immediate aim will be the relatively modest one of establishing consistency for the set of first Born approximations. Returning to classes (i)-(iii) of results, and beginning with (ii): this simply records the allowed values of the strong coupling constants g, subject to the compensation requirements, and using a weak-coupling scheme which has somehow been guessed or postulated a priori. These g’s are nothing but expansion parameters. In the first Born approximation they are identical with the physical coupling constants, defined from the residue of the baryon pole in a physical meson-baryon scattering process. Numerical evidence will be presented in Section 8 showing that this relation [expansion parameter = pole residue] appears to persist with great accuracy even when the g’s are large. (This is in the only accurately verifiable case so far of the pi-nucleon coupling.) To summarize the reliability of class (ii), the g’s are Born expansion parameters; compensation theory furnishes exact information concerning their relative sizes (i.e., the Born series may be as divergent as it pleases); their mathematical relation to the physical coupling
184
WELLNER
constants is obscure, however, but numerically extremely close where a precise comparison can be made. We next come to the results of class (i). These arise through overdetermination of the g’s by the compensation relations. Since the g’s are thereby eliminated, the question of their exact interpretation does not arise. Thus, class (i) is made of results which are theoretically
exact in the strict sense.
Finally, to obtain class (iii), one simply inserts the results (ii) into the first Born term for various decay amplitudes. Hence, the numerical reliability depends here more clearly than elsewhere on a good convergence of the Born series. Besides the reasons already given for believing such to be the case in certain processes, there is the further one of numerical agreements with experiment which, at least in one case, appears too close to be coincidental. 3. THE ELEMENTARY PARTICLES
Since we are considering, at least in principle, all orders in the strong interaction, it seems reasonable to choose the latter in such a way that, when taken by itself, i.e., without perturbations from weaker interactions, it be renormalizable. This consideration very likely restricts the choice of hadrons to spins zero and one half. Quark theories will not be considered, although it is conceivable that one such theory could lend itself to the compensation technique. The usual ones will not, however. In the case of a single quark triplet with spin one half, the coupling must be to mesons because renormalizability appears to rule out the four-quark interaction. Then there are not enough parameters to satisfy all the compensation relations which arise. In conclusion, the minimal set of hadrons must comprise, first, the spin-half baryons, amounting to eight particles, and augmented (by necessity, as will be shown) with a ninth baryon L having all the quantum numbers of the lambda, except for an opposite parity. Thus the isospin wave functions are N = ($3
ur =
50 ( -v-- 19 (3.1)
A, and L.
C =
One will also need the following definite-parity
mixtures:
Am = A cos w + iybL sin w, LW = L cos w -
iybA sin w,
(3.2)
A THEORY
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185
I
with a mixing angle w to be determined later. The mixed wavefunction with some increase in generality over the usual case w = 0, as the in Cabibbo’s strangeness-changing currents for the weak interaction, in such a current being py”(l + ir5) L’P. Second, one has the set of nine pseudoscalar mesons with isospin
4~ is defined, one occurring a typical term functions
(3.3)
7, and H (R”
=
K”*;
K--
=
K+*;
7T- =
,+*).
Here KC is the charge conjugate of K, and H denotes a more massive analog of the q-perhaps the X0. All considerations in this first article will be completely insensitive to any mixtures that may occur between 7 and H, of for that matter to the existence of the H. In addition to the above-mentioned hadrons, one requires nine scalar analogs (3.4)
of the mesons. No strong coupling is assumed for these, at least to first order, so that they should not necessarily be identified with any observed scalar mesons, and indeed should perhaps not be called mesons at all at this stage. The need for them arises if one wishes to construct a chiral algebra of currents and simultaneously to satisfy the conserved vector current hypothesis. This will be discussed later in more detail; the scalar particles will play no role in this first article. Next, one assumes the existence of an unspecified number of W bosons (with vector wavefunctions W”), which mediate the weak-and perhaps superweakinteractions. The detailed properties of the W bosons will not be required at present. Finally, one must consider the photon field Afi, but it, too, will largely play a spectator role. Our notation has &{r“,r”>
= g@” = diag(l, -1, -1, -1)
y5 = y”y1y2y3.
186
WELLNER
4. MASS RELATIONS AND MIXING ANGLE Some results of class (i), involving the baryon masses and the mixing angle W, are recorded here for reference. These results are exact in the sense that they do not depend in any way on the convergence of the strong-interaction Born series. All mass shifts of electromagnetic or weak origin are, however, ignored, as is consistent with this lowest-order study in the weaker couplings. The equations below are to be derived piecemeal throughout this and the subsequent articles, by eliminating the strong coupling constants from overdetermined sets of compensation relations. The particles’ symbols stand for their masses when no ambiguity is likely. The results are (l/N) + Cl/@ + UP) = 6/V + Q, (4.1) or, equivalently,
Z((s - N)2 = NE(2Z - N - S); also (3/A) + u/9
= (2/W + (2/E),
(4.2)
and fTL = z’2.
(4.3)
The “average” mass il is defined as A = A cos2 w - L sin2 w.
(4.4)
The minus sign in front of L is related to the negative L/A relative parity. The results (4.1)-(4.3) depend in part on the consideration of nonleptonic baryon decays, to be discussed in the next installment (II) of this article. Specifically, (4.1), which is verified to great accuracy (see below), will be shown there to follow from the statement [see Section 12 and Eq. (11.13) of the present article] that the strong ?rZA coupling vanishes. Another result, anticipated from II and used here, is that the strong interactions of the II and L baryons occur through the wave function /lo [Eq. (3.2)], but not through L”. This statement will be referred to as the no-nonet theorem. For the details the reader is referred to Section 6 of this article. Equation (4.2) corresponds to the Gell-Mann-Okubo mass relation [5] 3A+Z=2N+29,
(4.5)
and indeed reduces to it in the limit w ---f 0 and for small mass differences; Eq. (4.2), however, is exact and does not depend for its derivation on any approximate SU(3) symmetry [see Eq. (11.18) later in this article]. Equation (4.3) follows from Eq. (11.19); see later. After some algebra, one obtains from (4.1)-(4.3) L - Z = [2Z/(N
+ E)](2Z
- N - E).
(4.6)
A THEORY OF ELEMENTARY
COUPLINGS I
187
There are three adjustable masses, say N, 8, and /.I. An excellent fit to Z is obtained by choosing as an input the observed [6] neutral masses (in MeV/?) N=
940;
8 = 1315.
(4.7)
Then (4.1) gives i?Y= 1195
(4.8)
(between the observed [6] Co = 1193 and Z- = 1197); Eq. (4.6) gives L = 1340.
(4.9)
(A possible candidate has occasionally been seen [7] near 1330 MeV/c2.) Then, from (4.3), A = 1066. (4.10) Taking from experiment /.l = 1115.6, one then gets from (4.4) sin2 w = (A - ir>/(d + L) = 0.0202 or sin w = 0.1421 (positive without loss of generality), so that CIJ= 8.17”.
(4.11)
5. THE STRONG INTERACTION The strong-interaction Lagrangian density Z&,ng conserves isospin and strangeness, and is invariant under the separate C, P, and T transformations. Invariance under SU(3) is not assumed, even approximately. We consider the Yukawa coupling because of its renormalizability:
where the & , & stand for the baryon octet fields n, p, etc., or for the ninth field iy5L having the same parity; the & are the pseudoscalar meson nonet fields. The indices 01,p, y label the nonet members. The expression shown below represents the most general trilinear Hermitian PStronB built out of the above-mentioned nonets, and which satisfies the stated conservation laws. The present listing will serve as a definition for the various g’s. Numerically, for the pi-nucleon coupling constant [S], we take g$N/4n = 14.85 f 0.25. (5.2)
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WELLNER
The spin dependence shown in (5.1) is suppressed for brevity in the listing below. Thus, d*C stands for ify5Z, L*Z stands for (iy5L) y5C, etc. One has 9 strong = x7 + -% + -E”, + dz;, , (5.3) where _Ep,= 7cc’k,,N+~N + (g,,L*C
+ (gz,A*C
+ H.C.)
+ H.C.) - ig&2* x Z + g&?d],
=%c= [N+k,d + gd + givs * Z)K + H.C.1 + [E+(gmA + gd + g,zT - z) KC+ KC.], q
= 7)[gNN+N + &J*d + g,L*L + (g,,x’.t*L + H.C.) + g,C* . C + g,E+E],
9H = H[h,N+N + hJ*A
(5.4) (5.5) (5.6)
+ hLL*L
+ (hA,d*L + H.C.) + hEC* * C + h,E+E].
(5.7)
In these expressions, “H.C.” means Hermitian conjugate, and 7 stands for the Pauli matrices. All the g’s are real without loss of generality (this is shown in Appendix E), the fields being defined with suitable phases. The explicit, particleby-particle expressions for Zn and 5pK are given in Appendix A for the reader’s use in calculations and as his check on this article’s isospin notation. The behavior of arong under Gell-Mann’s &Y transformation [9] is outlined in Appendix B; in Appendix C we list the values that would be assigned to the couplings if one did assume SU(3) invariance [lo]. The complete renormalizable strong interaction should also include terms of the form &3,~+~~,~,~8
(5.8)
*
Renormalization theory requires that the constants h at least contain even powers of the g’s. Such “$4 terms” play no role in the present study.
6. RESULTS FOR THE STRONG COUPLING
CONSTANTS
The strong Lagrangian of the preceding section contains 23 adjustable coupling constants. In the course of the following sections, they will be reduced to 8 adjustable parameters. Two results of the subsequent article will be used: Eq. (4.1) will be freely applied to simplify some expressions, and the no-nonet theorem will be extensively drawn upon. This theorem states that one must consider only an octet (rather than a nonet)
A THEORY
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189
I
of baryon wavefunctions in the strong-interaction Lagrangian. Specifically, claim that AU, but not L”, occurs in (5.3). This immediately implies gEA : gzL = g,, : g,, = g,
we
: g,, = cos w : sin W,
g, : g,, : g, = h, : h,, : hL = co9 co : cos w sin w : sin2 W.
(6.1)
This theorem is linked to SU(3) in that w is the corresponding mixing angle used in the SU(3) currents. The theorem is also strongly hinted at by the present paper’s compensation relations, which then admit the existence of a special scalar current density .F, Eq. (D.lO). [This will be commented on in Section 11 in connection with(10.17),(11.3)and(11.25);seelater.] One finds gEA gNN
= pz/lN,
-
&L
=
GE = -p,
0,
g,
= p-z/E,
(6.2)
&,7/j = q[(E - if)@]
cos w,
gm = qW
- Al/d]
cm w,
gNL = q[(E - if)/A]
sin w,
gEL = q[(N - li)/?i]
sin w,
gN, gN
= q[(z - E)/d3
z],
g_F= r.zJB,
= r-W,
g‘q = ucos2 w, [hN = r’Z/N,
g, = dW - z)/1’\/5Jzl,
gL = 24sin2 w,
g,, = u sin w cos 0,
etc.: see gN , etc., with I -+ r’, u -+ u’.]
(6.3) (6.4) (6.5) (6.6)
(6.7)
In Eqs. (6.3), ii is given by (4.4). Thus, only the quantities p, q, r, r’, U, u’, g, , and hz are still undetermined relative to each other. Numerically, from (5.2) and (6.2): &-/4n
= 9.18 f 0.15,
gQ4lr
= 7.57 f 0.13,
(6.8)
the UnCertaintieS ariSing from gNN . It iS worth noting that, in these reSults, sU(3) symmetry is badly broken, or indeed absent altogether (cf. Appendix C). Experimentally, precise information is difficult to obtain and its interpretation [ll] may be affected by including a (l/2)- particle, our L (1340), in the analysis. There are two possible arguments leading to g,, = gzL = 0. One of them is “numerological” and invokes the excellent mass relation (4.1). A purely deductive argument, presented in Section 12, consists of taking the weak current ,$ [Eq. (D. 16)] as the limit of a somewhat more general one. The experimental situation as regards g,, appears somewhat confused [I 11, some authors favoring a very large value.
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WELLNER
7. THE WEAKER
INTERACTIONS
The combined electromagnetic, weak, and possibly superweak Lagrangian density for the hadrons is postulated to be of the form (assumed Hermitian) (7.1) (p = four-vector index). The W, are intermediate bosons, among which is the photon; the J, are appropriate current densities, to be described presently; and the GOBare the corresponding coupling constants, not further specified in this work. For the purpose of this article, only the baryonic currents are relevant. They consist of vector (V) and axial-vector (A) terms of the form 09 : cLv%
(7.2)
and (4 : hJ”(-ir3
$5 9
(7.3)
where K and X are nonet labels. Seventeen linearly independent currents are considered. As explained in later sections, this amounts to as many currents as could be found, subject to the requirements that (a) the electromagnetic (V) current be included; (b) the V - A generators of W(3) be included, somewhat like in Cabibbo’s theory [12]; (c) all compensation relations be mutually compatible; (d) the set of all currents, or, better, all “charges”, constitute a Lie algebra (or an Abelian set of Lie algebras), in the spirit of Gell-Mann’s prescription [l]. The list of currents follows. (i) SU(3) algebra: Eight Y - A currents 8+, PO, P-, X+, X0, Y”, Y-, Y, respectively transforming under SU(3) like the particle octet n+, no, W-, K+, K”, KO, R-, 7).
These currents are F combinations and, thus, their F’ parts are the generators of W(3). No D combinations were found compatible with the requirements (a)-(d) above. (ii) number. (iii)
Universal scalar: A single V current S?, expressing the total baryon SU(3) scalar: A single V - A isosinglet current2, dp.
a No confusion with a Lagrangian density should arise.
A THEORY OF ELEMENTARY
191
COUPLINGS I
(iv) SU(2) algebra: Three V + A currents, Y’+, PO, 9’-. They form an isovector identical to g+, PO, 9- except for the sign of the A term. (v) Completion to SO(4) algebra: Three V + A currents, f+, &O, f-, forming an isovector. Of the seventeen currents, they are the only C&violating ones. Together with (iv), they make up the algebra of SO(4). The CP violation is chosen, for the sake of simplicity, to be maximal in a sense to be explained in Section 12. (vi) SO(4) scalar: A single V + A isosinglet current ?Y’, identical to ??I except for the sign of the A term. These currents may also be classified according to their properties under the R transformation, as exhibited in Appendix B. This makes, for example, a distinctLoon between ?Y and L?. The .A-like wavefunctions (see Section 3) occur as follows: Only /lW in (i); d and L in (ii); only LW in (iii); none in (iv); only L in (v); none in (vi). These assignments are not arbitrary and will be justified in Section 12. It is worth noting that the @ current cannot occur in the axial-vector form, as it leads to a wrong overdetermination of the K-meson couplings, see Appendix F. A particle-by-particle list of the seventeen currents is given in Appendix D. To summarize, the detailed form (Appendix D) of the weaker currents is determined primarily from the requirements (a)-(d) above, but also from certain simplicity considerations which, not being fundamental, could easily be relaxed if necessary. In particular, the amount of CP violation in 3 is chosen maximal.
8. LEPTONIC COUPLING
For the purpose of calculating fermion coupling 9
semileptonic
=
AND PI-MESON
DECAY RATES
meson lifetimes we assume an effective fourcos 0 + Yu+ sin 0) 1-1~ + H.C.
c2 (CPU’
(8.1)
(G = four-vector index), after Cabibbo [12]. Here 1 is the leptonic current: z-*0 = hd~“(l
+ iy5) h9 + (e -
p, ye -
G);
(8.2)
we take 1131 G = 1.435 the normalization
1O-4serg cm3;
x
(8.3)
of 8+ and Z+ is as in Appendix D: 90+ = +ZEy,(l xi+ = -v3
+ ir5)p + . . . ,
Y&(1
+
iy5)p
+ ... .
(8.4)
192
WELLNER
For the Cabibbo angle we take [14] I9 = 12”.
(8.5)
Equation (8.1) differs from Cabibbo’s phenomenological coupling in that it has only pure V - A currents and no D-type SU(3) currents. Thus it is clear that strong-interaction corrections will be needed in order to calculate semileptonic decays from (8.1); however, that aspect of the present theory is as yet unexplored. The greater simplicity of (8.1) over the phenomenological coupling not only befits a “basic” Lagrangian, but is also forced on the theory by the fact that the latter has no room for the D currents of W(3) (see the previous section) if current algebras are to be valid. The decay of a meson into a lepton pair is calculated here to first order in both the strong and the weak couplings. For the charged pi meson, the dimensionless partial rate for decay into p + v is then, from the bubble diagram (Section 9), -=- 1 W&1
gLN 2Gm,cos 477 [ Gw2
0
1
(8.6)
i
where Z2 ma* ~4~~) = In -7 + -&- (& NLI
+ &
- -&)
The rnZ term adds only about 0.35 % to the ln(Z2/N@ masses of Section 4, one gets from (8.6) I/(m&*
+ O&t;).
(8.7)
term. Using the baryon
= (1.85 & 0.05 f 0.07) x lo-16,
(8.8)
as against the experimental [6] (1.85 f 0.02) x lo-la. About 80 % of the first uncertainty in (8.8) is due to gNN, the rest to 0. The second uncertainty in (8.8) arises from allowing N and 9 to range independently between the neutral and charged values, keeping Z constrained by Eq. (4.1). Turning next to the decay r” + 2y, as computed from the triangle diagram (Section 9) [15], one finds the dimensionless rate -=-
1
(mTLo
d(nd = (
(8.9)
(8.10)
A THEORY OF ELEMENTARY
COUPLINGS I
193
The term in rn$ amounts only to a 0.26 % correction to the amplitude. Numerically, (8.9) gives l/(ms-),,, = (2.48 f 0.05 f 0.05) x 1O-s (8.11) [corresponding to TV,-,= (1.96 & 0.04 f 0.04) x IO-l6 set], as compared to the experimental [6] l/(m-r)& (5.46 f 1.1) x 1O-s. The uncertainties in (8.11) are from g,, and from the electromagnetic mass shifts in N and E. This discrepancy, which was not apparent [2] from older Z-Odata, would indicate that the perturbation expansion for this type of process is only moderately good (by a factor of 0.7 in the amplitude), in contrast to the n* case. In agreement with the heuristic argument of Section 2, we note that the “finite-part cancellation” l/W - l/S2 (no case) is indeed far less perfect that in ln(Z2/NE) (T+ case), so that even in terms of the present formalism some such discrepancy is perhaps to be expected.
9. FEYNMAN DIAGRAMS:
DIVERGENT
AND FINITE PARTS
For the purpose of this study we assume that all the weaker (i.e., electromagnetic, weak, or superweak) vertices involve a W boson (which may be a photon). In order to arrive at the set of Feynman diagrams to be considered for compensation, we go through the following sequence of increasing specializations: (a) Any process, real of virtual, is considered only to lowest contributing order in the weaker coupling constants. Thus, apart from kinematic factors, all diagrams reduce to those whose external lines are just hadrons and W bosons. (b) There is, by choice, only one external hadron line, namely, a pseudoscalar meson. (c) Although the compensation postulate requires the convergence of these processes to each separate order in the strong coupling constants, only the first order will be considered here. (d) Since only the closed loops must be watched for divergences, and since, with the present restrictions, the only closed loops which can occur are baryon loops (parity conservation prohibits strong three-meson vertices), one has to consider just the set of single baryon loops with one strong and one or more weaker vertices. (e) Counting momentum factors in the Feynman diagrams, one may disregard the case of loops with four or more weaker vertices. (f) The case of three weaker vertices is shown to be harmless by y-matrix counting.
194
WELLNER
Thus, in the end, one gets just the bubble diagram with an A-type weak vertex, and the triangle diagram with one A and one V. These processes are shown in Fig. 1 and will be designated by the symbols M(ab)A and M(abc) VA [or M(abc) AV]. Their degree of divergence is logarithmic in the cutoff mass &‘. Internal L baryons are treated dynamically just like the other baryons, the iy5L wavefunction being used throughout in the Lagrangian and current listings. However, since the mass term rnLEL changes sign under L * iy5L, one must formally use a negative mass, -m, , for the iy5L “particle”. The divergent terms of the bubble and triangle diagrams are, up to a constant factor, M(ub)A ---f (a + b)k” In d2, (k = meson momentum), M(ubc) VA + (2~ + b - c)g”” In M2, M(ubc)A V --f (-2~
P-1)
+ b - c)gUvIn A2
(CL,v = vector indices of the currents AU, v’). We note that, for the triangle, these divergent terms change sign upon reversal of the internal arrows. It will ordinarily be more convenient to look at current combinations V f A. Equations (9.1) then give ,rise to the mass dependence
~(aW’
f A) - ztt(a+ b),
M(ubc)( V - A)( V - A) -M(ubc)(V+
A)(V+
A) -b
- c,
(9.2)
M(ubc)( V - A)( V + A) - M(ubc)( V + A)( F’ - A) - a, overall factors being ignored. A single, universal cutoff &’ is used for all Feynman diagrams. For use in Section 8, one, also needs the finite parts of the diagrams M(ub)A and M(ubc)VV for a = b and a = b = c, respectively. One has, for M(uu)A:
Tr1 d4PY-P+a_ p2
= -4iuk”
u2
Y5y(;(;
tkj8k)_+nau
YGy5)
d4p s (p2 - a”)[(~ + kj2 - a2i (9.3)
= (2~r)~ uku [In M2 - In u2 - 1 + -$ where k is the incoming meson momentum.
+ o((k2)2)],
A THEORY OF ELEMENTARY
Similarly,
195
COUPLINGS I
for M(uaa) VV:
Tr & Y. (P+ r) + a y.r+a s (p + r)2 - a2 ‘” r2 - a2
7"
Y. (r - 4)+ a (r
-
-27r2i ‘k2 = - a E”YKAP& [l + ij-$ + 4(k2)2)] 3
q)2
-
p2
y5
(9.4)
wherep and q are the outgoing vector current (or photon) momenta, with k = p + q being the incoming meson momentum. This diagram, being automatically convergent, does not fall within the class (9.1).
10. PROCESSES To BE COMPENSATED For a given process, i.e., for given currents or particles as external lines, the first step in the “compensation calculus” is to determine all the possible particle labels (baryon labels in this article) which are allowed for the internal lines. This is done here for all processes which may diverge. The notation is that of Fig. 1.
M(ab)A
M(cbc)VA [or M(obc)AV]
FIG. 1. The divergent diagrams considered in this article. A pseudoscalar meson M is strongly coupled to a baryon loop (internal masses a, b or a, b, c). One or two W bosons go out, coupled to Y or A current densities. The symbols M(ab)A and M(abc)VA [or M(abc)A V] will be used in order to refer to these processes.
One significant short cut consists of simulating isospin conservation by attaching “spurions” to the W bosons. This procedure is correct as long as W bosons do not occur internally. Thus, as far as the incoming meson is concerned, only one member of each isospin multiplet needs be considered. Also, if the incoming meson is given, certain outgoing currents (in the bubble case) or combinations of currents (in the triangle case) will yield no constraint because the amplitude changes isospin and thus vanishes identically. This consideration will be used to omit many processes;
196
WELLNER
in addition, if two processes are related by charge conjugation, only one of them will be listed. Triangular graphs having .G?as one of the outgoing currents need never be listed: the clockwise and counterclockwise loops always cancel each other term by term. Sometimes the internal baryon labels are unimportant and will be indicated by dots. This happens when, in a triangle, both currents are of the I’- A type (or both V + A) and the meson is coupled to two baryons of the same mass. Eq. (9.2) then shows that there is no divergent contribution. In the end, one is left with the following set, which is understood to include also possible replacements of 7 by H: Bubbles 7r+@lp, ‘Pz+,
Z-P,
E-Eo)B+,
n+(id.)P’+, 3i-+(np, 8-D;
Z-L,
L.Z+)$+,
K+(Ap, Lp, Zap, Z-n, EO,Z’+, E-P, E-A, E-L&f+, EOEO, E-E- g, ~PP, nn, 1
v(id.)W, ,l(AA, AL, LA, LL)8.
(10.1) (10.2) (10.3) (10.4) (10.5) (10.6) (10.7)
Triangles
In this subsection the small Roman numerals refer to the classes (see Section 7) from which the external currents are selected. (9, (9: T+( ***p%P+ + ?T+(.. ‘).LP+P, 7r+(-)YP+ + 7r+(**-)8+Y, 7r+( ..,, n,Z-A, nEL)Y”.X+ + 7r+( . ..) E-Z-A, E-Z-L, K+(pAp, pLp, p.?YOp,nEn, + K+(z-x-n, BOSOM+,
(10.8) (10.9) EOA,z’, soLz+)sf-+xo,
Z+E”2J’+)9’X+ S-E-ZO, Z-E-A,
K+@Ap, pLp, pZ”p, n2-n)CVZ+ + K+@OsOz+, E-E-EO, B-E-A,
&E-L)s+p,
E-s-L)zfg
,
7j( * **)P+LP- + 7]( ***)8-P’+, rl(***W~, rl(..., PAL, p~A)c%?-SC+ + 77(..., g-AL,
E-LA)X+Y-;
(10.10) (10.11) (10.12) (10.13) (10.14) (10.15)
A THEORY
OF ELEMENTARY
COUPLINGS
I
197
(i), (iii):: K+(flE-A, L&A, M-L, LB-L)mYX+ + K+(AAp, ALP, LAP, LLp)X+cY;
(10.16)
(i), (iv): r+(pnp, BiPZ+, ~“E-Eo)~o.P‘+ + r+(nnp, Z-Z-.Zo, 5-E-E”)8’+Po, r+(pnp, E”~-Eo)~~‘+
+ z-+(nnp, E-E-EO)B’+SY,
K+(pAp, pLp, pZ”p, d-n, .Z+EOZ+)B’OX+ + Kf(z-z-n, sOEOz+, E-&zO, E-E-A,
E-&L)x+$PO
(10.17) (10.18) (10.19)
[cf. (lO.ll)]; (9, 69: r+@np, E”E-E”; Z+LZ+)P”$+ + ir+(nnp, E-E-E”; Z-Z-L)$+P”, rr+(pnp, E”E-Eo)CYf+ K+(nEn, +
+ n+(nnp, E-B-E”)y+CY,
pZOp, pAp, pLp; LE-Z”, Z”E-L)$OX+ E-~-L, E-E-z0 , E”Eoz+; LiPp, zoLp)x+~o,
Kt(E-E-A,
q(pnn, E”B-E-)f-9
(10.20) (10.21)
+ v(npp, E-E”Eo)P+$-;
(10.22) (10.23)
+ r+(nnp, E-E-S”)~+W,
(10.24)
(i), (vi): r+@np, E”E-So)WB+
K+(pAp, pLp, pZ”p, nZ-n)WX+ + Ki-(~O~O~t, E-E-20, E-&A,
&-~)$“+q’
(10.25)
+ ~(id.)YY;
(10.26)
[cf. (10.12)], r](nnn, ppp, E”EoEo, E-E-E-)CY’CY
(iii), (iii): rl(..., AAL, ALA, LAL, LLA)S’mC?;
(10.27)
(iii), (v): x+(LZ-L,
LEA)
2’9’
+ ,+(LAZ+,
LLZ+)$+P’;
(10.28)
(iv), (iv): 7T+(.. .p’ogd’t
+ n+(. . .)y+p’o;
(10.29)
+ 7rIT+(.‘.,z-z-L)~+9J’o;
(10.30)
(iv), (9: 7+(‘.‘,
z+Lz+)P$+
(iv), (vi): n+(. . .)gy’$za’++ n+(. . .)gwy’;
(10.31)
198
WELLNER
(v>, 09: 7r+(.-*)f+&o
+ n-+(...)fo2+,
v(.-.>$‘f-
+ q(***W-y+;
(10.32) (10.33)
+ n-+(‘.‘)w$+;
(10.34)
(9, (vi): 7r’(-.-)f’W
(vi), (vi): (10.35)
q( ***)5?PTv’.
11. COMPENSATION RELATIONS In this section, the results (9.2) are written out for every process listed in Section 10. The particle labels, unless they are subscripts, now denote masses. Pi Relations
From (lO.l), (10.2), or (10.17) NgNN + 2-h.z + &.m
= 0;
(11.1)
From,(10.3) or (10.20), Ng,r, - EgEE = 0;
(11.2)
it is noteworthy that the CP violating loops [after the semicolon in (10.3) and (10.20)] make no net divergent contribution to (11.2). From (10.28), g,, cos 0 - gzA sin w = 0.
(11.3)
Processes (lO.lO), (10.18), (10.21), (10.24), and (10.30) are automatically finite. The relations (1 1.1)--(11.3) are equivalent to (6.2). Equation (11.3) shows the interdependence of the no -nonet theorem (6.1) and the existence of the current density J.Z’. K Relations
In these, it will be convenient to define t?hiA
=
gNA
cos
w
+
gNL
sin
w3
EEA = g, cos w + ggL sin w, rNA
=
r,,
= Ag,
&,A
cos w - LgNL sin w, cos w - LgaL sin w.
(11.4)
A THEORY
OF ELEMENTARY
COUPLINGS
-199
I
Then, from (10.4), NgNn -- E&,
+ r,,
- TEA + d/YN + Z)g,
+ 2/3(E + Z)g,
= 0.
(11.5)
TO study (10.11) and (10.12) it is simpler to consider first the formal “electric charge” combination Y” + (l/ ~‘3) g/, or, symbolically, the process (10.11) + (l/1/3) x (10.12). This yields
~/3rN, + %w,, I’~N&, - ,1/3%l- (N - E)g, - (9 - xZ)gEz = 0.
(11.6)
From (10.12) alone, mNA - &
- rNA -t rEA f A&N
+ 4/5&T - Z)gsz = 0.
(11.7)
+ (8 + 2Z)g,
(11.8)
- z)g,
From. (10.19) + (l/d/3) (10.25), d--NgN/, - x’~@m
+ (N + 2Z)g,
= o.
From (10.25) alone,
From (10.22), d/3N&,
+ dE&,
Z(gNL - gd
- Ng,
+ &m
sin w + 43 Lb
= 0,
+ gNz) = 0.
(11.10) (11.11)
Of these two equations, the first is the real part, the second the imaginary part of the full compensation relation. The semicolons in (10.22) are meant to show how the generating processes correspondingly split up. The remaining K process, (10.16), is automatically finite-a strong suggestion for the existence of the current 3’. The information in (11.5)-(11 .l 1) may be extracted as follows. First note that (11.9) is equivalent to (11.5) + (11.7), and (11.8) to d/5 x (11.5) + (11.6). Thus, it is enough to consider the independent set (11.5), (11.8)-(11.12). From (11.8)(1 l.lO), the values of gNA , & , g,, and g, are obtained up to some factor q’: STNA= E(--2218
+ NE + NZ) q’,
&I
+ NE + EZ) q’,
= N(--2zN
g,, = 2/3 Nc”(E
- ,i?) q’,
g,
- N) q’.
= v/5 NE@
(11.12)
200
WELLNER
[In a sequel to this article, and starting from the statement gzn = 0, the equation (11.13)
ghln + EEA - 43ggN.z + z/3&% = 0 will be derived, whence (4.1).] An alternative formulation is readily obtained Eq. (6.1). From (11.4),
from the no-nonet
theorem,
where /r = A cos2 w - L sin2 w.
(11.15)
Then, from (11.5): (N+
J~NA
-
(s
+
/r>g,,
+
dJ(N
+
z)&.,Z
+
g/3@
From (11.16), (11.8) and (11.9) one finds, up to a factor ?TNL~ = gN,
C?(E
-
4lif
= dz - @/(d3
z),
q(N
-
+
z)gsz
=
0.
(11.16)
q,
&A
=
ir>/&
g,
= 4w - q/(1/3
(11.17) 2).
The compatibility of (11.12) and (11.17) is most easily expressed by substituting (11.17) in (ll.lO), yielding (3/A) + (l/a
= (2/N
(11.18)
+ (2/Q,
i.e., Eq. (4.2). Turning next to Eq. (11.1 l), one obtains after substituting
(11.17) with
(L/Z)@ - N) - (C/A)@ - iv) = 0,
q #
0,
(11.19)
whence (4.3) if S # N. Eta Relations From (10.5) or (10.6), N&q - E& = 0,
(11.20)
AgA sin2 w + (L - (l)gA, sin w cos w - LgL cos’ w = 0.
(11.21)
whence (6.5). From (10.7),
Processes (10.15), (10.23), (10.26), and (10.27) are automatically finite. Relation (11.21) is automatic under the no-nonet theorem (6.1).
A THEORY OF ELEMENTARY
12. LAMBDA-LIKE
WAVEFUNCTIONS
201
COUPLINGS I
AND THE VANISHING
OF gzA
The detailed choice and form of the seventeen currents of Section 7 are only partly constrained by the requirement of Lie algebras. One other guide has been the purely heuristic idea that, if simple integers are to characterize a particle model, these integers should be found in the weaker currents rather than in the strong couplings. Then also, ultimately, the numerical results which are yielded through the compensation conditions should agree with experiment. It appears that, in the framework of this study, one can afford to be exceedingly particular as to the accuracy of such results if they are of class (i), Section 2. This observation is to be kept in mind throughout the present section. Our task here will be to show why those terms which, in the weaker currents, contain the wavefunctions (1 or L have the special coefficients assigned to them as described at the end of Section 7. Consider first the SU(3) currents, see (i), Section 7, and formulas (D.l)-(D.9). The form of these currents is entirely traditional, and the occurrence of rl” instead of II is actually an increase in generality. The form of 2 comes about as follows. There are two independent SU(3) scalars, Lw*Lw and the baryon number, both of which commute with the SU(3) octet currents. Although the baryon number 9 is acceptable in the V form, it cannot occur in the V - A form (see Appendix F). This leaves only 5? to be tried out, which leads to consistent relations. An additional piece of theoretical evidence for 2, independent of (6.1), is provided by the the automatically finite nature of (10.16). We next turn to the occurrence of &2i(I:*L - L*Z) in y, see (v), Section 7, and formula (D.16). Consider a more general form f2i(C*LA
- LA*Z),
(12.1)
where LA = eiX(L cos X - A sin h),
(12.2)
for some angles x and h. (We still assume no parity mixing of the wavefunctions, i.e., L stands for iy5L.) Some term of the form (12.1) is required to complete a Lie algebra. The phase x is chosen to be zero (a “maximal” CP violation) on the numerical evidence of formula (4.1), which in turn depends on (11.10) and on the (anticipated) (11.13). The presence of a nonzero x in (12.2) would permit nonzero contributions from (11 .I 1) to “leak” into (11. IO), and thereby simultaneously destroy the accuracy of (4.1) and the simplicity of the K-coupling results. Nevertheless, a very small but nonzero x has the interesting consequence of forcing gZL = 0, a fact to be discussed presently.
202
WELLNER
For general h, compensation Gh
relation (11.11) reads
- gd 03s h - %h - sEAI sin h + pCX(L cos h sin w + (1 sin h cos w)(gsZ + gNZ) = 0
(12.3)
L%h - EEA)+ vWtk~ + gd tan 0 = F%%v~- hII) - 1/Wg8z + khdl tan A.
(12.4)
or, with (11.14),
Inserting (11.17), and taking q # 0, B # N, (Z2 - &)
tan w = (Z” + Ll)
tan X.
(12.5)
Even under the rather weak assumption that the L mass be of the order of the other baryons’, we see that h < w, the angle w already being quite small owing to (11.18). Hence X = 0 is fairly compelling if any simplicity is to be retained in the theory. Next, generalizing slightly from x = 0 to x very small, we observe that processes (10.3) and (10.20) both acquire g ZL terms but are no longer equivalent. Thus, we get gZr. = 0 as well as (11.2). Hence also gZA = 0 by the no-nonet theorem. To summarize, the form’ of $ adopted here is the one which simultaneously yields the strongest and simplest results. That is to say, f is maximally CP violating as a (continuous) limit x --f 0 of the nonmaximal case. It is, of course, to be expected that even such simplicities may not survive in future compensation models. However, in this initial test of the theory it seems an essential strategy to keep the weaker interactions as simple as possible.
APPENDIX
A: EXPLICIT
gT AND -EK
The particle-by-particle expressions for 6p, and ZK [Eqs. (5.4) and (5.91 are listed here for convenience. The coupling constants are chosen in the real representation, as explained in Appendix E. gr = gNN[Ghr+p*n + G!?r-n*p + 7p(p*p - n*n)] + {g&r-(A*z+
+ z-*(1) + r+(A*E
+ 27+*/l>
+ T”(A*zo + z”*(1)] + g&l - L]} + g~&+(z~*z- z+*zo) + 7r-(z-*.P - ,P*#x+) + 7ryz+*z+ - z-*2-)] + d/zr+EO*,F + dTT-s-*E“ + nO(~O.*~O- E-*3-)], gEE[
(A-1)
A THEORY
OF
ELEMENTARY
COUPLINGS
203
I
9K = {g,,[K+p*A + K%*A + K-A*p + RoA*n] + gNL[A + L]} + g,[K+(p*zo + d/Zn*E) + KO( z/zp*z+ - n*q + K-(P*p + l/zz-*n) + KO( lLE+*p - D*n)] + {ggA[-Koso*A + R-E-*A - KOA*Eo + K+A*E-] + &LW - LII + g,,[~~o(~o*~o + 2/zE-*z-) + R-(qZEO*z+ _ s-*zo) - KO(~O*~O + d\/zz-*8-) + K+( d\/z,Z+*E” - ,ZO*E-)].
APPENDIX
(A.2)
B: THE R TRANSFORMATION
A great deal of work is spared, mostly in the nonleptonic decay calculations to be presented at a later date, by the use of Gell-Mann’s R-transformation [9]. One observes that -YStrong(Section 5; Appendix A) is invariant under the purely formal substitutions (no physical invariance being implied) /l-A,
rl t--) 79
L+-+L, P4-3 D, z+tt -z-, nt,EO, p-E-, -
&?,A,
gNL*&YL
3
gNA
&2l-
H-H, n-0e-b 770, rr+ tt -?r 9 K”e, -K”, K+ +-+ i?, 8ZA 3
g‘?,t+g,,
gN.Z+-+--8E2,
gNN
-
hz+-+
gA+-+gA,
9
gL++gL,
-hiI?, -Km
h,
gz++gz 9
+‘h,
,
k-b, 3
h.z++h.z,
hNt)hE.
gN-gS,
The seventeen currents of Section 7 transform as follows under R: “F-type”
transformation: 9+ t) @-, CYM-Y, Jf-+ t) -SF, @‘+ tt @‘- >
@Ott w t+ xot, 9’0 t)
-90; -w; 20; -9’0.
“D type” transformation: la’t,, %‘-
-3-y
9++t,, $0 t-) $0.
[The sign behavior for given isospin is what suggests the F, D nomenclature.]
204
WELLNER
APPENDIX
C: THE W(3)
INVARIANT STRONG COUPLING
If the strong interactions were SU(3) invariant, there would be no mixing between ~4 and iy5L, since a positive mass for L implies a negative mass for iy5L. There could, however, be an 7, H mixing. In the SU(3)-invariant case, and without any mixing, the 23 coupling constants in L%trong , Section 5, would depend as follows [lo] on five parameters, say d, f, gAL , hA , and hL :
gNN= d-kf, gNA
= - -&
lb= Cd + 3f),
gm = z
gm=d--f, gN
-
&
g,=
=
gSL
=
d, &L
1
-d+f, C--d + 3f),
-(d+f),
g, = - --& (d + 3f),
(F-d + 3f), g, = - $
gNL
g,,=
;?f,
=
g.z = -& g/IL
hN = h, = hz = h,
4
arbitrary, arbitrary,
h,, = gL = 0, h, arbitrary.
APPENDIX
D: EXPLICIT CURRENTS
The particle-by-particle expressions for the seventeen currents of Section 7 are listed below. Spin variables are suppressed. Thus, for example, n*p stands for Zr“(l 4~ iy5)p, L stands for iy5L, etc. Attention is called to the occurrence of the various A-like wavefunctions (1, L, (lW, and Lw, as outlined in Section 7. v - A currents [16]. P. 1) 03.2) (D.3) (D.4)
A THEORY
OF
ELEMENTARY
COUPLINGS
205
I
(D.5)
03.6)
(D.7) w3)
(D.9) (D.10)
37 = G(N+N
+ E+B + x* * z + /1*/l + L*L).
(D.11)
(the factor 1/g is just for later convenience). V + A currents Y+, PO, 9”-: f+
= 1/Z n*p -
go=p*p
p-
-
=
z/Zp$
see 9+, 80, 8-;
d/z E--*9O (T) 2i(Z-*L
n*n--o*EO+
-
q/z
E-*E-
EOo*E--
(f,
(D.12) (D.13)
- L*z+), 2i(c”*L
- L*ZO),
(D.14)
2i(Z+*L
- L*Z-)
(D.15)
$8 = N+TN - 3+75 ($‘) 2i(c*L
- L*q],
(D.16)
(‘,
W : see CV.
(D.17)
V current (Electric)
(D.18) Concerning 9, we note the two possible signs of the ZL terms. Both choices lead to the same Lie algebra, as well as being inconsequential to the rest of this paper. It is interesting that this sign is changed by a CP-transformation, so that the algebra is mapped outside of itself by that transformation. Thus, any interaction using 3 would appear to be CP nonconserving. The wavefunction L in (D.16) is to be understood as eiXL (x real), where x -+ 0 after all compensation calculations (see Section 12).
206
APPENDIX
WELLNER
E: REAL REPRESENTATION
FOR THE STRONG COUPLING
CONSTANTS
All the coupling constants in Eqs. (5.4)--(5.7) either must be real, or may be taken real without loss of generality. I. The Hermiticity
of =.YStrongrequires that the following be real:
gNN9ax ,gEs in (5.4); gN, gA , gL , a , gE in (5.6); and hi , hA , hL , hz , hs in (5.7).
II. Next, turning to LZK [Eq. (5.5)], we assume at frrst that all its g’s are complex: gNL = I gNL I ei92,
(E. 1)
g,L = I sk I ei*5, Time-reversal (T) invariance requires two relations between these six phases. Indeed, the most general T transformation is of the form K + K*&
B+y&e o-* iX,r,
N + yaN *.@N,
9 (1 -+ y”/l*eixA,
C --f y°C*eix2
iy5L -+ y0(iy5L)* eiXL,
(t -+ -t being understood), for some phase angles xK ,..., xr. under T then implies =
gNA&K+XA-XN’
(E.2)
Invariance of ,EpK
, etc.,
gsA
so that, module 2a, XK + XA -
XN
+ 24,
=
0,
XK
+
XL
-
XN
+
242
=
0,
XK
+
XZ
-
XN
+
29%
=
0,
-XK
+
?hl
-
x.R
+
244
=
0,
-XK
+
XL
-
XS
+
2+5
=
0,
-XK
+
xr:
-
XS
+
294 =
0.
,
(E-3)
Therefore, 41 - & - +4 + & = multiple +1 - & L +* + & = multiple
of n=, of 7r.
We now redefine the wavefunctions’ phases as follows: K --f Kei”K,
N + Nei*N,
(1 -+ fle”“,
L + LeidL,
(E.4)
A THEORY OF ELEMENTARY
COUPLINGS I
207
for six $‘s to be adjusted. Inserting in -YK, we find the modifications g,, -, / g,, 1 e~(~l+~K+hl--IIN), etc. One way of obtaining a real representation therefore consits of choosing the #‘s equal to half the corresponding x’s, which have been seen to exist. After the redefinitions (E.4) have been made, all the new g’s in SK are real. III. We finally consider gzA , gzL [Eq. (5.4)], g,, [Eq. (5.6)], and hnL [Eq. (5.7)], under the assumption that the redefinitions (E.4) have already been carried out, so that the 4’s are themselves multiples of 7~. Consider the remaining most general T transformation for n and L: the difference of the first two Eqs. (E.3) gives xn - xr. = multiple
of 27r.
Making use of the fact that the T invariance of, e.g., z * Nt7N implies phase angle for the T transformation of 7r, we get, under T, x . (g,mA*C
+ H.C.) -
z . (g&‘l*C
a zero
+ H.C.),
so that gzA = gz,, . The reality of g,, , g,, , and hllL is proved in analogous fashion. APPENDIX
F: ABSENCE OF g'a
We examine the consequence of assuming that a weak boson is coupled to the baryon number current (D.l l), but changed to the axial vector form aA . Since a(=gr) is automatically allowed, we consider, for convenience, a current L$?“+~ . The process K+(nZ--n, pz”p, pAp, pLp, AS-A,
Lz-L,
+ K+(AAp, LLp, S”Eo,E+, 8-&.Z”,
rE”E-.Zo, Z+E”.Z+)S?,,
E-E-A, 8-E-L,
X+
Z”,Zop, Z-Z-n)X+9v+,
,
(F.1) with the help of (6.1), gives rise to the compensation 0’ -. &!,A
+ (s - &m
+ d/3(N - Z)g,
relation
- d/3@ - Z)gsz = o.
(F.2)
Inserting (11.17), (N - ii>@ - jT,//i
= (N - Z)(E - Z)/Z
(F.3)
or (NE - Z?f)(Z - i!f) = 0. 595/73/I-14
(F-4)
208
WELLNER
One possibility, Z = A, taken with (11. I8), yields 2/z = (l/N) + (l/B). This, apart from being poorly satisfied in Nature, also implies N = E if taken together with (4.1). The other possibility, NE = Z& again taken with (11.18), gives
leading, likewise, to poor numerical agreement with facts and to N = E from (4.1).
ACKNOWLEDGMENTS The author would like to thank J. A. Cronin for comments on the possible importance of a negative-parity lambda, and J. Weinberg for a conversation on Lie algebras. A considerable portion of this research was carried out in the stimulating environment of the Cavendish Laboratory at Cambridge University, and the hospitality of R. J. Eden there is gratefully acknowledged. Last but not least, the long-standing confidence of the author’s colleagues at Syracuse University deserves thankful mention.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
8.
9. 10. 11.
M. GELL-MANN, Physics 1 (1964), 63. M. WELLNER, Phys. Rev. 168 (1968), 1855. M. WELLNER, Phys. Rev. 184 (1969), 1782; D3 (1971), 2295. For example, G. A. BAKER in “Advances in Theoretical Physics,” Vol. 1, p. 1, Wiley Interscience, New York, 1965); J. Z~NN-JUSTM, Phys. Rep. C 1 (1971), 55. M. GELL-MANN, Phys. Rev. 125 (1962), 1067; S. OKUBO, Progr. Theor. Phys. 27 (1962), 949. Particle Data Group, Rev. Mod. Phys. 42 (1970), 87. G. BOZ~KI, E. FENYVES, T. GBMESY, E. GOMBOSI, S. KRASZNOVSZKY, E. NAGY, N. P. BOGACHEV, Yu. A. BUDAGOV, V. B. VINOGRADOV, A. G. VOLODKO, V. P. DZHELEPOV, V. G. IVANOV, V. S. KLADNITSKY, S. V. KLIMENKO, Yu. F. LOMAKIN, Yu. P. MEREKOV, I. PATOEKA, V. B. FLIAGIN, AND P. V. SHLYAPNIKOV, Phys. Lett. B 28 (1968), 360. J. HAMILTON AND W. S. WOOLCOCK, Rev. Mod. Phys. 35 (1963), 737; V. K. SAMARANAYAKE AND W. S. WOOLCOCK, Phys. Rev. Lett. 15 (1965), 936; M. H. MACGREGOR AND R. A. ARNDT, Phys. Rev. B139 (1965), 362. M. GELL-MANN, Ref. [5]. Way,” pp. 51, 52, Benjamin, For example, M. GELL-MANN AND Y. NE’EMAN, “The Eightfold New York, 1964. C. H. CHAN AND L. L. SMALLEY, Phys. Rev. D 2 (1970), 2635; C. H. CHAN AND F. T. MEIERE, Phys. Lett. B28 (1968), 125, and Phys. Rev. L&t. 20 (1969), 568; D. CLINE, R. LAUMANN, AND J. Mapp, Phys. Lett. B3J (1971), 606.
A THEORY
OF
ELEMENTARY
COUPLINGS
I
209
12. N. CABIBBO, Phys. Rev. Left. 10 (1963), 531. 13. For example, R. E. MARSHAK, RLGUDDIN, AND C. P. RYAN, ‘Theory of Weak Interactions in Particle Physics,” p. 39, Wiley Interscience, New York, 1969. 14. Reference [13], p. 403. 15. The internal-nucleon contribution was already calculated in this way by J. STEINBERGER, Phys. Rev. 76 (1949), 1180. 16. For example, P. A. CARRUTHERS, “Introduction to Unitary Symmetry,” Wiley Interscience, New York, 1966.