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The Mixed-Isospin Vector Current Correlator in Chiral Perturbation Theory and QCD Sum Rules


The Mixed-Isospin Vector Current Correlator in Chiral Perturbation Theory and QCD Sum Rules ADP-95-20/T179 hep-ph/9504237
Kim Maltman
Department of Mathematics and Statistics, York University, 4700 Keele St., North York, Ontario, Canada M3J 1P3

(April 10, 1995)

Abstract
The mixed-isospin vector current correlator, h0jT (V V ! )j0i is evaluated using both QCD sum rules and Chiral Perturbation Theory (ChPT) to one-loop order. The sum rule treatment is a modi cation of previous analyses necessitated by the observation that those analyses produce forms of the correlator that fail to be dominated, near q 2 = 0, by the most nearby singularities. Inclusion of contributions associated with the meson rectify this problem. The resulting sum rule t provides evidence for a signi cant direct ! ! coupling contribution in e+ e? ! + ? . It is also pointed out that results for the q 2 -dependence of the correlator cannot be used to provide information about the (o -shell) q 2 -dependence of the o -diagonal element of the vector meson propagator unless a very speci c choice of interpolating elds for the vector mesons is made. The results for the value of the correlator near q 2 = 0 in ChPT are shown to be more than an order of magnitude smaller than those extracted from the sum rule analysis and the reasons why this suggests slow

1

convergence of the chiral series for the correlator given. 11.55.Hx, 12.39.Fe, 14.40.Cs, 24.85.+p

Typeset using REVTEX 2

I. INTRODUCTION
Non-electromagnetic isosopin breaking is well-established in many strongly interacting systems (e.g., splittings in the hadron spectrum, binding energy di erences in mirror nuclei, asymmetries in polarized np scattering, binding energies and level splittings of light hypernuclei 1]). In few-body systems, an important source of this breaking has been thought to be the mixing of isoscalar and isovector mesons appearing in meson exchange diagrams. In particular, the bulk of the non-Coulombic contributions to the charge symmetry breaking nn-pp scattering length di erence and to the A=3 binding energy di erence, and of the np asymmetry at 183 MeV, can be explained 2,3] using the value of ? ! mixing extracted from an analysis of e+e? ! + ? in the ? ! interference region 4,5] . The plausibility of this explanation (which employs the observed mixing, measured at q2 = m2 , unchanged in ! the spacelike region q2 < 0) has, however, recently been called into question by Goldman, Henderson and Thomas 6] who pointed out that, in the context of a particular model, the relevant ? ! mixing matrix element has signi cant q2-dependence. Subsequently, various authors, employing various computational and/or model framewords, have showed that the presence of such q2-dependence appears to be a common feature of isospin-breaking in both meson-propagator- and current-correlator matrix elements 7{16] . In the present paper we will concentrate on the isospin-breaking vector current correlator (q2) = i where
Z

d4x eiq:xh0jT (V (x) V ! (0))j0i ;

(1.1)

V = (u u ? d d)=2 ; V ! = (u u + d d)=6 :

(1.2)

This correlator was rst analyzed using QCD sum rules in Ref. 17] , and the analysis updated by the authors of Ref. 12] who, in particular, stressed the q2-dependence of the correlator implicit in the results of this analysis. As will be shown below, a worrisome feature of the resulting t is that the phenomenological representation of the correlator near q2 = 0 3

is not dominated by the most nearby singularities, suggesting that some ingredient may be missing from the form chosen for this representation. This missing ingredient is identi ed below and it is shown that a reanalysis of the correlator, which includes it, recti es the problem. The resulting correlator still displays a very strong q2-dependence, and, in addition, provides evidence for the presence of signi cant direct ! ! coupling in e+e? ! + ?. The behavior of the resulting correlator near q2 = 0 is then compared with that obtained from ChPT to one-loop. The latter is found to be more than an order of magnitude smaller than the former, the reason why this suggests the likelihood of a slow convergence of the chiral series for the correlator explained. The paper is organized as follows. In Section II, those features of the behavior of quantum eld theories under eld rede nitions relevant to attempts in the literature to relate meson propagators and current correlators are discussed, and it is explained why the freedom of eld rede nition implies that (1) one cannot obtain o -shell information about the o -diagonal element of the vector meson propagator from the o -diagonal element of the vector current correlator without making speci c choices for the vector meson interpolating elds, and (2) if one writes the o -diagonal element of the vector meson propagator as
! (q 2 ) ! (q 2)

( ?(g ? q q =q2) (q2 ? m2)(q 2)? m2 ) q
!

2

!

(1.3)

cannot, in general, be q2-independent. In Section III we return to the QCD sum rule analysis of the vector current correlator, rst explaining why certain features of the existing analyses suggest the need for a modi ed analysis, and then performing this analysis. The results both correct the apparently unphysical features of the previous analyses and provide evidence for non-negligible direct ! ! + ? contributions to e+e? ! + ? in the ? ! interference region. In Section IV, the correlator is computed to one-loop in ChPT, and the results compared to those obtained from the sum rule analysis. Implications for the discrepancy between the results of the two approaches are discussed there. Finally, in Section V, a brief summary of the main results of the paper is given. 4

II. CONSEQUENCES OF THE FREEDOM OF FIELD REDEFINITION
Let us begin by clarifying the relation (or lack thereof) between the vector-mesonpropagator and vector-current-correlator matrices. The former is an, in general, o -shell Green function, which we may think of as being associated with some low-energy e ective Lagrangian, Leff , in which the vector meson degrees of freedom have been made explicit. As is well-known 18{20] , the form of such a Lagrangian is not unique: if and are two possible eld choices describing a given particle, related by = F ( ), with F (0) = 1, then Leff ] and L0eff ] Leff F ( )] produce exactly the same experimental observables 18] . However, while the S-matrix elements of the two theories are identical, this is not true of the general o -shell Green functions. One is free to make eld rede nitions of the form above (as is done, e.g., in order to obtain the canonical form of the e ective Lagrangian for ChPT 19{21] ) without changing the physical consequences of the theory; the Green functions, however, are not in general invariant under such eld rede nitions. Useful pedagogical illustrations of this general principle, for pion Compton scattering and the linear -model, are given in Ref. 22] and Chapter IV of Ref. 23] , respectively. In the case of interest to us, what this means is that, when we make a rede nition of the , ! elds in Leff ; !], we generate a new e ective Lagrangian, L0eff 0; !0], the Green functions of which are, in general, di erent from those of Leff (though when we piece such Green functions together to form S-matrix elements, these di erences produce no net e ect). The o -shell behavior of the vector meson propagator is thus dependent on the particular choice of elds used to represent the vector mesons (the choice of \interpolating eld"). It is not a physical observable. In contrast, the R vector current correlators ab (q2) = i d4x eiq:xh0jT (V a(x) V b(0))j0i are, in fact, physical objects, independent of interpolating eld choice. The spectral functions for 33 and 88 are, for example, accessible from a combination of ? ! ? 0 and e+e? ! ; data, and that for 38 could in principle be obtained from a careful analysis of the deviation of the ratio of the di erential decay rates for ? ! ? 0 and e+e? ! + ? from that predicted by isospin symmetry. As such there can be no general (i.e. valid for all choices of 5

interpolating eld) relation between the correlator and propagator matrices. This point is the source of some confusion in Ref. 12] where an attempt is made to obtain the o -shell propagator based on an analysis of the correlator. Before proceeding to the reanalysis of the correlator, let us be more precise about the problems with the interpretation of the results of Ref. 12], in the light of the above comments. The authors begin by writing a general form for the spectral function of the correlator: Im (q2) = A0 Im
! (q 2) + A

1 Im

0

!

0

+

(2.1)

where the superscripts on the RHS should, for the moment, be taken only as labelling the region of spectral strength, and where + refers to all other contributions (we return to this below). Eqn. (2.1) is, of course, completely general. The authors of Ref. 12], however, then identify A0 with m2m2 =g g! , where g ;! are the vector meson decay constants, de ned ! by
2 h0jV ;! j ; !i m ;! ; g
;!

(2.2)

and ! with the o -diagonal element of the vector meson propagator. This amounts to assuming that the isospin-unmixed I = 1 state, (0), couples only to V , and the isospin-unmixed I = 0 ! state, !(0), only to V ! , the isospin-breaking contribution to of Eqn. (2.2) from the ; ! region then resulting solely from the (0)-!(0) mixing in the meson propagator. In this interpretation, xing the imaginary part of the correlator in the , ! region (via the sum rule analysis) allows one to obtain the isospin-breaking parameters of the imaginary part of the vector meson propagator, and, via a dispersion relation, the behavior of the o -diagonal element of the propagator o -shell. However, as explained above, such a possibility is excluded on general grounds. The problem with going from A0 and ! of Eqn. (2.2) to the interpretation of these quantities in Ref. 12] is that, not one, but three sources of isospin breaking exist in the contributions to from the , ! region: that due to (0)-!(0) mixing (discussed above), that due to the direct coupling of V to !(0), and that 6

due to the direct coupling of V ! to (0). The same I = 1 strong operator which gives rise to non-zero (0)-!(0) mixing will also necessarily give rise to the latter two couplings. These couplings would be described by new isospin breaking parameters, ( )! and (!) ,
2 h0jV ! j (0)i m! g 2 h0jV j!i m g
!

(!)

( )!

(2.3)

where (!) , ( )! are, in general, q2-dependent, and also interpolating- eld-dependent o -shell. Thus, o -shell, the -! region contribution to depends not only on the (interpolating- eld-choice-dependent) isospin-breaking parameters of the o -diagonal element of the vector meson propagator, but also on the (interpolating- eld-choice-dependent) isospin-breaking parameters (!) , ( )!. The total contribution is independent of the interpolating eld choice, but the individual contributions are not. One is, of course, free to choose a convenient set of , ! interpolating elds and work with these, provided one calculates contributions to S-matrix elements. Since, to O(md ? mu) Eqn. (2.2) remains valid when we replace and ! with (0) and !(0), the elds g V (0)c m2 g (2.4) !(0)c m!2 V ! ;
!

satisfyh0j (0)cj (0)i = terpolating elds for others) one obtains

and h0j!(0)cj!(0)i = , and hence serve as possible choices of in(0) and ! (0). With this choice of interpolating elds (and not with (q2) =

m2 m2 (c) ! 2 ! (q ) (2.5) g g! where (c) ! (q2) is the o -diagonal element of the vector meson propagator for the interpolating eld choice above. If one simultaneously evaluates, e.g. NN , NN! vertex form factors using the same interpolating elds, one could, of course, piece the resulting vertices and propagators together to obtain contributions to NN scattering S-matrix elements which are independent of eld choice. For a general choice of interpolating eld, however, neither
7

nor ! is proportional to ! . Given the existence of QCD sum rules and ChPT methods, which are rather e cient at handling current-current correlators and vector current vertex functions, the choice (2.4) for vector meson interpolating elds would appear to be a convenient and sensible one. With this choice, Eqn. (2.1) provides the basis of a spectral representation of (c) ! , but for other choices of the vector meson interpolating elds this is not the case. Note that the above discussion also clari es one ongoing point of debate in the literature, namely that concerning the q2-dependence of the quantity ! (q2) appearing in Eqn. (1.3). De ning ^ (q2) by (q2) (g ? q q =q2) ^ (q2) ; (2.6)

the absence of massless singularities implies that ^ (0) = 0 14] . This in turn implies, with
(c) ! (q 2) (c) ! (q 2)

?(g ? q q =q2)

(c) ! (q 2)

;

(2.7)

= 0, and hence ! (0) = 0. Since this is true for one choice of the vector meson interpolating elds, it is incumbent upon those advocating
! (q 2)

=

! (m2 ) !

(2.8)

to explicitly demonstrate the existence of an interpolating eld choice for the vector mesons for which Eqn. (2.8) is valid; the relation cannot be true in general.

III. THE QCD SUM RULE ANALYSIS OF

(Q2) REVISITED

With the above discussion in mind, let us turn to the sum rule analysis of the vector correlator, rst brie y reviewing the treatment and results of Refs. 12,17] . The sum rule approach consists of writing an operator product expansion (OPE) representation for the correlator, valid in the region of validity of perturbative QCD, and a second, phenomenological, representation in terms of hadronic parameters, and then Borel transforming both. The Borel transform serves to extend the ranges of validity of both representations and, in 8

addition, to (1) emphasize the operators of lowest dimension in the OPE representation and (2) give higher weight to the parameters of the lowest lying resonances in the phenomenological representation. One then matches the transformed representations in order to make predictions for the relevant hadronic parameters. The OPE for the correlator of interest was performed long ago 17] . Truncating the expansion at operators of dimension six, one nds that, de ning (q2) by (q q ? q2g ) (q2) ; one has
OP E (Q2)

(3.1)
#

c c 1 3 = 12 ?c0 log(Q2) + Q12 + Q24 + 2c6 Q

"

(3.2)

where Q2 = ?q2 and
EM c0 = 16 3 c1 = 23 2 (m2 ? m2 ) d u " ! md + mu md ? mu 2f 2m2 1 + c2 = m + m 2 + # md ? mu d u " c3 = ? 224 shqqi0]2 8 EM2) ? 81 s(

#

(3.3)

with hddi0 =huui0 ? 1. Taking for the phenomenological representation (in the narrow resonance approximation) Im
phen (q 2)

= 12 f (q2 ? m2) ? f! (q2 ? m2 ) + f (q2 ? m2 ) ? f! (q2 ? m2 ) + ! !
0 0 0 0

h

i

192
0 0

EM

2

;

(3.4)

(where f , f! , f , f! may be thought of as the parameters to be determined from the sum rule analysis) one nds, upon Borel transformation and matching, 1 hf exp(?m2=M 2 ) ? f exp(?m2 =M 2 ) + f exp(?m2 =M 2 ) ? f exp(?m2 =M 2)i ! ! ! ! M2 c c c EM + 16 3 exp(?s0=M 2 ) = c0 + M12 + M24 + M36 ; (3.5)
0 0 0 0

9

where M is the Borel mass. As pointed out in Ref. 12] , to O( m2; m02), where m2 = m2 ? m2, m02 = m2 ? m2 , Eqn. (3.5) can be rewritten in terms of the parameters , , 0 ! ! and 0, where
0 0

with m2

m2 = m4 f + f! 2 = (f! ? f ) m2 0 2 f + f! ! 0= m m04 2 0 = (f! ? f ) m02 0 (m2 + m2 )=2 and m02 (m2 + m2 )=2, as ! ! ! m2 m2 ? exp(?m2=M 2 ) + 0 m02 m02 ? 0 exp(?m02=M 2) M2 M2 M2 c c c EM + 16 3 exp(?s0=M 2 ) = c0 + M12 + M24 + M36 :
0 0 0 0 0 0

!

(3.6)

(3.7)

If c0?3 were precisely known, Eqn. (3.5) or Eqn. (3.7) could, in principle, be used to determine the parameters , , 0, 0. There are, however, some uncertainties in the values of the ci, associated with the imprecision in our knowledge of the values of the four-quark condensates and of the isospin-breaking ratio of the huui0 and hddi0 condensates. The authors of Ref. 12] (which updates Ref. 17] ) consider a range of possibilities for these quantities, and also take for r (md ? mu)=(md + mu) the value r = 0:28, obtained from an analysis of pseudoscalar isomultiplet splittings 24] employing Dashen's theorem 25] for the electromagnetic contributions to these splittings. The last ingredient of the analysis of Ref. 12] is the imposition of an external constraint on the hadronic parameter , based on the observed interference in the -! interference region in e+e? ! + ?. This constrained value, = 1:13 10?3 , is based on (1) the assumed connection between the correlator and the propagator (presumably valid for the essentially on-shell value of the mixing, though not elsewhere) and (2) the assumption that direct !(0) ! contributions to e+e? ! + ? can be neglected (see Ref. 26] for a discussion of these issues). There appears to be no particularly good reason for the latter assumption, and, indeed, it would seem appropriate to allow 10

to be t by the sum rule analysis as a test of this assumption (as will be done below), but let us follow the analysis of Ref. 12] for the moment. Using the sum rule, Eqn. (3.7), and imposing the constraint = 1:13 10?3, as discussed above, the authors of Ref. 12] solve for , 0 and 0 for four di erent input sets fcig. Using the expression (3.4) for Im phen (q2) and the fact that (q2) satis es an unsubtracted dispersion relation, one may show that, to rst order in m2 and m02, 1 Re (0) = 12 (1 ? ) + 0 (1 ? 0)] : (3.8)

Using the values of the parameters obtained in Ref. 12] , one nds that the ratios of the contributions to Re (0) from the 0-!0 region to those from the -! region are 1:8, 0:8, 0:3 and 0:8 for input sets I, II, III, IV, respectively. The failure of the results to be dominated by the nearby ( , !) singularities suggests that the phenomenological form employed for the spectral function may well be incomplete, either in missing low-lying contributions or in failing to include the e ect of even more distant singularities. If we consider Eqns. (3.4) and (3.8) for a moment an interesting possibility becomes evident. If one had all isospin-breaking e ects generated solely by (0)-!(0) mixing, and if the physical vector mesons were a simple rotation of the isospin-pure basis (not in general true when the wavefunction renormalization matrix of the system is non-diagonal), we would have f = f! for f , f! as written in Eqn. (3.4). While the assumptions required to arrive at this conclusion are certainly not satis ed in general, this nonetheless indicates that there should be signi cant cancellation between the and ! contributions to the correlator. Thus, a single isolated resonance, even with a coupling much smaller than that of the or !, could in fact contribute signi cantly to . This suggests that the contribution to Im , neglected in Ref. 12], may well be non-negligible. In fact we can make a rough estimate of the expected size of f (where f is de ned by adding a contribution 12 f (q2 ? m2 ) to Im phen(q2) in Eqn. (3.4)) as follows. is known to be not quite pure ss. If, e.g., we take the Particle Data Group (PDG) 27] value for the octet-singlet mixing angle, = 390 (quadratic t), ' (0) ? !(0), where (0) is the pure ss state and = :065 rad is the deviation of from ideal mixing. The 11

contribution of the pole term to due to mixing in the propagator should then be of order ? times that associated with the ! pole, i.e. ' 0:065 f! ' 0:065 f . There will, of course, also, in general, be isospin-breaking contributions from direct couplings to the current vertices, not just from mixing in the propagator, but the above discussion shows that f ' (0:05 ? 0:10) f ;! should be a reasonable expectation. As we will see below, this (rather crude) estimate is indeed borne out by the sum rule analysis. Let us, therefore, add a term 12 f (q2 ? m2 ) to Im phen (q2) on the RHS of Eqn. (3.4), and perform a reanalysis of that equation. We will follow Ref. 12] in choosing the range of input values for the fcig, with, however, the following modi cations. First, the small c1 term dropped in Ref. 12] will be retained, though, as pointed out there, it in fact has little e ect on the nal results. The numerical value is obtained by using (md +mu)(1GeV) = 12:5 2:5MeV from Ref. 28] and the updated value of r discussed below. The main modi cation to the input is in the parameter r. There is now considerable evidence that Dashen's theorem is signi cantly violated 29{31], Refs. 30,31] in particular suggesting that (m2 + ? m2 0 )EM ' 1:9 (m2 + ? m2 0 )expt K K (3.9)

(where the factor 1:9 on the RHS of Eqn. (3.9) is absent in Dashen's theorem). Using Eqn. (3.9) in place of Dashen's theorem for the electromagnetic contribution to the kaon mass splitting produces a rescaling of r by 1:22. The resulting change in the ci is essentially to rescale the values of c2 in Ref. 12] by this same factor. In assessing the e ect of the uncertainties in the values of the fcig for a given input set, the input errors on c2 have also been rescaled by this factor of 1:22. Finally, since the masses of all the resonances appearing above, including the 0 and !0, are known, we may take these as input and use the sum rule to extract the isospin-breaking parameters, ffk g, where i = 1 5 correspond to , !, 0, !0 and , respectively. Note that, in taking this approach, we are abandoning the constraint on employed in Ref. 12]. If the direct !(0) ! + ? coupling is, indeed, negligible in e+e? ! + ?, this will manifest itself by the value of resulting from the sum rule analysis being near 1:13 10?3 . 12

The analysis of the modi ed version of the sum rule, (3.5), proceeds as follows. First, from the terms of O(M 0), c0 = EM =16 3. One may check that, as in Ref. 12] , the analysis is very insensitive to the value of the EM threshold parameter, s0. We will, therefore, quote all results below for the value, s0 = 1:8 GeV, employed in a number of the results quoted in Ref. 12] . Second, again as in Ref. 12] , we impose the local duality relation
Z1

0

ds Im

phen (s) =

O(

EM ; mq )

2

(3.10)

(which is equivalent to matching the coe cients of the O(1=M 2 ) terms in Eqn. (3.5)). With the index k = 1; ; 5 labelling , !, 0, !0 and , respectively, as above, this relation is
X
k

(?1)k+1fk = c0s0 + c1 :

(3.11)

(Note that the ci tabulated in Ref. 12] have had the appropriate factors of m2 required to leave the remaining coe cient dimensionless factored out of them. Thus, e.g., c1 in Eqn. (3.11) is m2 times that tabulated in Ref. 12] .) The remaining four relations required to obtain a solution for the ve unknowns, ffk g, are obtained by acting on Eqn. (3.5) with @n (?1)n @(1=M 2)n for n = 1; ; 4. One may check that the results are not sensitive to using precisely the PDG values for the 0 and !0 masses. Indeed, shifting either mass by 50 MeV induces changes of < 4% in , < 2:5% in , < 5% in 0 and < 20% in 0. The resulting d changes in the correlator itself are even smaller: e.g. (0) and dq2 (0) are changed by < 2% by the above mass shifts. In Table 1, the results of the modi ed sum rule analysis are displayed for the input sets I, III, IV of Ref. 12] , modi ed as described above. The errors shown in the table correspond to the uncertainties in the input parameters, c2 and c3, (those quoted in Ref. 12] in the case of c3 and the rescaled version thereof in the case of c2). The stability of the analysis is illustrated, for input set IV, in Figs. 1-5, which display the parameters , , 0, 0, f as a function of the Borel mass, M , in the range 1-10 GeV (the choice of the rst four parameters, rather than corresponding fk values, is made in order to facilitate comparison with Ref. 12] ). Set I generates results of comparable stability, while the results of set III 13

are even more stable than those of set IV. In all three cases a wide stability window exists in the Borel mass for all ve output parameters. This stability window, moreover, occurs without the necessity of using unphysical values for the the average of the 0 and !0 masses. As noted previously in Ref. 12] , results for input set II are considerably less stable than for the other sets: in fact, no stability window exists anywhere in the range M = 1 and M = 10 GeV, apart from for the very lower edge of the error band for the magnitude of c3, for which values input set II is very close to the upper end of the corresponding error band for input set I. The instability of the analysis for input set II is illustrated (for the central values of c2 and c3) in Fig. 6, where the parameter, f , is plotted as a function of the Borel mass, M . As a result of this instability, results corresponding to input set II are not quoted in the table; for most of the input range (i.e. for larger values of the magnitude of c3) the input set appears, from the sum rule analysis, to be unphysical. A number of features are evident from the results of the above analysis. First, from Table 1, we see that the magnitude of di ers signi cantly from that which would be expected from the analysis of e+ e? ! + ?, neglecting !(0) ! + ? contributions, suggesting that the latter are, indeed, not negligible. It should be stressed that the errors quoted in the table correspond to varying c2 and c3 separately within the range of quoted errors, and taking the maximum variation of the resulting output. One can obtain even lower values of , i.e. closer to that expected if one can indeed neglect !(0) ! + ? contributions to e+e? ! + ?, by letting c2 lie at the bottom of its error band and, simultaneously, the magnitude of c3 lie at the top of its error band in set I. However, such a combination (which produces = 1:43 10?3 ) is quite unstable, the values of 0, e.g., varying by more than 20% between M = 3 and 5 GeV. A similar result, = 1:48 10?3 , can be obtained from set II for the central value of c2 and the lower edge of the error band for the magnitude of c3, with comparable (' 20% over the range M = 3 to 5 GeV) instability. All other portions of the set II error band are even more unstable. Thus it appears very clear that the value = 1:13 10?3 is excluded by the sum rule analysis. The second observation is that the inclusion of the pole term in the phenomenological representation of the correlator 14

recti es the problem of the strength of the distant singularities. This can be seen from the relative size of and 0 in Table 1, but is more evident in Table 2, where the output values for the parameters ffk g are tabulated, for the central values of the input parameters fcig, for input sets I, III, IV. The ratios of f to f! are 0:062, 0:068 and 0:066 for sets I, III and IV, respectively. This is in (better than should be expected) agreement with the rough estimate given above, con rming the physical plausibility of the solutions obtained. Moreover, f and f! are now a factor of 40-60 smaller than f and f! . The structure of the resulting contributions to the correlator near q2 = 0 is shown in Table 3, where the , !, and also the 0, !0 contributions have been combined. Note that the individual and ! contributions are a factor of ' 13 larger than the contribution, but the cancellation between them is such that the contribution is approximately twice as large as their sum. The 0-!0 region contribution is then less than 10% of the contribution. The more distant singularities, thus, have only a small e ect, justifying, a posteriori, the neglect of yet more distant singularities in the phenomenological side of the sum rule analysis. Given that the results satisfy all the above tests for being physically sensible and stable, it appears that the resulting values for the correlator and its slope with respect to q2 at q2 = 0 should be taken as good estimates, within the uncertainties resulting from those in the input parameters. The fact that, due to cancellation between the otherwise dominant and ! contributions, the contribution is actually dominant, no doubt accounts for the unphysical behavior of the spectral distribution of the correlator obtained in the absence of the term. Note that, despite the signi cant changes in the t, as compared to Ref. 12] , the slope of the correlator remains large in the present results.
0 0

IV. THE CORRELATOR TO ONE-LOOP ORDER IN CHPT
The starting point for the computation of the mixed-isospin correlator, e ective chiral Lagrangian of Ref. 21], 1 1 Leff = 4 f 2Tr(D D y) + 2 f 2Tr B0M ( + y)] + L1 Tr(D D y)]2 15 (q2), is the

+ L2Tr(D D + L4Tr(D D

y)Tr(D y)Tr 2B

D
0M (

y) + L

3 Tr(D

yD

D

yD

)
yD yD

+ y)] + L5Tr 2B0(M + yM )D
y + FL D

] ] (4.1)

+ L6 Tr 2B0M ( + y)]]2 + L7 Tr 2B0M ( ? y)]]2
2 + L8Tr 4B0 (M M + M yM y)] ? iL9Tr F R D D 2 + L10Tr yF R F L ] + H1Tr F R F L + F L F L ] + H2 Tr 4B0 M 2] :

In Eqn. (4.1), B0 is a mass scale related to the value of the quark condensate in the chiral limit, = exp(i~ ~ =f ) (with ~ the usual SU (3) Gell-Mann matrices and ~ the octet of pseudoscalar (pseudo-) Goldstone boson elds), f is a dimensionful constant, equal to f in leading order, M is the current quark mass matrix, and D is the covariant derivative

D

= @ ? i(v + a ) + i (v ? a ):

(4.2)

In Eqn. (4.1) the external pseudoscalar sources (which occur in the most general form of Leff ) have already been set to zero, and the external scalar source to 2B0 times the current quark mass matrix, since we are interested here only in the vector current correlator and, therefore, require only the external vector sources. For this same reason we may drop the external axial vector sources from the expression for the covariant derivative in Eqn. (4.2). The left and right external source eld strength tensors, F L;R, then both reduce to F L;R = @ v ? @ v ? i v ; v ], where v = 2a va, with va the octet of external SU (3) vector elds. In principle, Eqn. (4.1) should be supplemented by terms involving Tr(F ) in order to treat the case at hand, since the current V ! contains both octet and singlet pieces. However, to one-loop order, the additional terms do not contribute to the isospin-mixed correlator p R (the correlator is identical to 38 (q2)=3 3, with 38 (q2) = i d4xh0jT (V 3(x)V 8(0))j0i, to this order), so we will not explicitly display these terms. The unrenormalized higher order coe cients L1; ; L10 and H1 ; H2 appearing in Eqn. (4.1) contain divergent pieces which cancel those of the one-loop graphs involving vertices from the rst two terms in Leff above, and also nite, renormalization-scale-dependent pieces, Lr . Expressions for the divergent i pieces of the Li, Hi, relevant to one-loop calculations, may be found in Ref. 21]. 16

Contributions to the correlator resulting from Eqn. (4.1) are of two types, corresponding to the two types of contribution to the low-energy representation of the product of currents, V V ! : (1) those terms arising from the product of the low-energy representations of the individual currents, V and V ! (obtained from the terms in Leff linear in the va , a = 0; ; 8), and (2) contact terms (generated by the terms in Leff quadratic in the va ). To leading order (i.e. keeping only the rst two terms in Leff ) the correlator vanishes. This is because it is isospin-breaking and the only isospin-breaking at leading order lies in the term involving the quark mass matrix, which does not contain the external vector sources, and hence does not contribute to the correlator in zero-loop graphs. The leading contributions to the correlator are, therefore, next-to-leading order in the usual chiral counting. As such, the contributions consist of one-loop contact and non-contact graphs (where the current vertices are obtained from the rst term in the e ective Lagrangian above) and meson- eldindependent contact terms from the remainder of Leff . These latter contributions, which would in general produce terms involving the Lr , may be easily shown to vanish for the case i at hand. Thus only the contact and non-contact graphs mentioned above contribute. It is straightforward to demonstrate then that, to one-loop order, the O(md ? mu) expression for the correlator is (q2) = where
Z1 h i JP (q2) = ? 161 2 0 dx log 1 ? x(1 ? x)q2=m2 : P

1 log(m2 0 =m2 + ) + 4m2 0 ? 1 J 0 (q2) ? 4m2 + ? 1 J + (q2) K K K K 12 48 2 3q2 3 K 3q2 3 K

"

!

!

#

(4.3)

(4.4)

For our purposes we will not need the general expression for J (which is quoted in Appendix A of Ref. 21] ), but only the behavior near q2 = 0, which is given by

q2 1 q4 JP (q2) = 961 2 m2 + 960 2 m4 + P P

:

(4.5)

In Eqn. (4.3), m2 0;K+ are the leading-order expressions for the kaon squared-masses, m2 0 = K K B0(ms + md) and m2 + = B0(ms + mu) and terms have been kept only to O(md ? mu). As K 17

such, we must also expand all terms occuring there to the same order. Doing so, and making the expansion of Eqn. (4.5) for the loop integrals JK0;K+ , we obtain, for the behavior of the correlator in the vicinity of q2 = 0, (q2) = 1 (m2 0 ? m2 + ) K K 12 48 2m2 K
!

2 1 + 10q 2 + mK

!

;

(4.6)

where m2 is the average of the K + and K 0 squared masses. Thus, 12 (0) = (m2 0 ? K K m2 + )=48 2 m2 , where the kaon mass di erence is that due to the strong isospin-breaking, K K i.e., with the electromagnetic contribution removed. Using Eqn. (3.9) for the electromagnetic contribution, we nd that the RHS of this expression is 5:5 10?5 , to be compared with the results of the sum rule analysis, ' 1 10?3 . The one-loop ChPT result is a factor of ' 20 smaller than the sum rule result. The discrepancy between the one-loop ChPT and sum rule analyses for the correlator near q2 = 0 should actually not come as a complete surprise. Indeed, when the leadingorder contribution to a physical quantity (order 2 in the chiral expansion) vanishes, as it does here, one has no obvious scale to use in judging whether or not the next-to-leadingorder contribution obtained is abnormally small, i.e., whether or not the resulting oneloop expression is likely to represent a well-converged approximation to the whole chiral expansion. In fact, the structure of the expression, (4.3), above for the correlator, (q2), is such as to suggest that it is unlikely to be well-converged. The reason for this statement is that Eqn. (4.3) is independendent of the low-energy constants (LEC's), Lr , and results i purely from one-loop graphs involving internal kaon loops. Such loops, for the non-contact graphs, are well-known to be suppressed in size (the coe cient of q2 in the leading term of JK in Eqn. (4.5), e.g., is a factor of m2 =m2 smaller than for the corresponding loop K integral, J ) and, moreover, in the case at hand, i.e. the correlator (q2), those terms in which this suppression would be lifted by the presence of the m2 =q2 factor in the coe cient K multiplying JK (q2) cancel, since the expression for the correlator involves the di erence of the K + and K 0 loop contributions. The correlator, of course, has a cut beginning at q2 = 4m2 , associated with intermediate states, but such intermediate states do not enter 18

until two-loop order in the chiral expansion. Since the relevant loop integral is intrinsically much larger than its kaonic counterpart, it is likely that the two-loop contributions will not be negligible, despite being higher order in the chiral expansion. Other examples of slow convergence of the chiral series when the leading contribution vanishes and the next-to-leading order contribution results purely from loop graphs (i.e. is independent of the fourth-order LEC's, Lr ) are, in fact, already known. One is the process i ! 0 0, whose amplitude, to one-loop order, receives contributions only from loop graphs (though in this case, loop graphs with internal lines). The one-loop expression 32,33] deviates from the experimental amplitude 34] even very close to threshold, and one nds that extending the calculation to two-loop order (sixth order in the chiral expansion) produces corrections to the one-loop result of order 30% 35] , which corrections bring the amplitude into agreement with experiment. Even more closely similar to the case at hand is the process ! 0 . The one-loop amplitude again has no leading term and no contributions from the fourth order LEC's, but here, although there are loop contributions, these contributions are suppressed by a factor (md ? mu). The K loop contributions are naturally small, as noted above. The result is that the one-loop prediction for the partial rate 36] is a factor of ' 170 smaller than observed experimentally 27] . It is worth considering the process ! 0 in somewhat more detail since, not only does it closely parallel the case at hand, but the physical origin of the smallness of the one-loop result for this process is well-understood. The source of the problem lies in the fact that the dominant contribution to the amplitude is known to be due to vector meson exchange 37,38] . As is well-known 20,39,40] , it is possible to make standard eld choice for the various meson resonances and write an e ective chiral Lagrangian which includes both these resonances and the octet of pseudoscalar (pseudo-) Goldstone bosons. One may then integrate out the (heavy) resonance elds to obtain an e ective Lagrangian of the form Leff for the pseudoscalars alone. The resonance contributions to the LEC's are then determined by the coupling parameters of the original, extended Lagrangian (which are xed by experimental data). The e ect of the resonances (for the initial eld choices used) 19

then lies solely in their contributions to the Lr 39,40] . Those Lr to which the vector i i and axial-vector mesons can contribute (Lr , i = 1; 2; 3; 9; 10) are known to be essentially i saturated by these contributions 39,40] . Thus, the absence of any Lr -dependence in the i one-loop amplitude implies the absence of the e ects of vector meson exchange (for the given initial choice of vector meson interpolating elds) and if, as seems to be the case for ! 0 , vector meson exchange is the dominant contribution, the one-loop amplitude can be expected to be a poor representation of the full chiral series. The vector meson contributions, in this case, rst appear as tree-level contributions arising from the O(p6) part of the e ective Lagrangian, not included in Eqn. (4.1) above (the general form of the O(p6) part of the e ective Lagrangian is given in Ref. 41] ). Thus, only by including these contributions (which requires, for consistency, a full two-loop-order calculation) can one hope to obtain a well-converged approximation to the full chiral series for the amplitude. The ! 0 discussion above can obviously be transferred directly to the case of the mixed-isospin vector current correlator under consideration here. We expect signi cant contributions to the correlator from the vector meson resonances and, for a particular choice of vector meson interpolating elds, these contributions are completely absent from the oneloop result. As a consequence, we can expect signi cant contributions from the tree-level O(p6) terms in which such contributions reside. As already discussed, the O(p6) loop contributions (arising from two-loop graphs with lowest-order vertices and one-loop graphs with a single O(p4) vertex) may also be signi cant. A two-loop calculation is, therefore, almost certainly required in order to obtain convergence of the chiral series for the mixed-isospin correlator. Similar statements hold for the related correlator, 38 (q2), for which work on the two-loop calculation is in progress 42] . Note that, in the latter case, only a single combination of the O(p6) LEC's enters the two-loop result. This combination, which in the notation of Ref. 43] (where the analogous 33 and 88 correlators are computed to two-loop order), ? is written Q0( ) ? 3L(9?1) ( ) ? 3L(10 1) ( ), with the renormalization scale, is in principle obtainable from experimental data using the chiral sum rules of Ref. 43] (Eqns. (97) and (98) therein). In the case at hand, one further O(p4) LEC and one further O(p6) LEC will 20

be present at the two-loop level, but the sum rule analysis above, in combination with a full two-loop evaluation of (q2), would provide a useful constraint on these parameters, albeit it with the ' 20 ? 30% errors displayed in Table III and associated with the uncertainties in the values of the input parameters fcig which determine the correlator near q2 = 0. A similar sum rule analysis of the correlator 38 would constrain the combination ? Q0( ) ? 3L(9?1)( ) ? 3L(10 1)( ), mentioned above. It is interesting to note that the relation between the sum rule and ChPT results for the mixed-isospin vector correlator is e ectively the reverse of what occurs in the mixed-isospin axial correlator case. In the latter case, the ChPT 16] and sum rule 13] results for the value of that piece of the correlator proportional to q q at q2 = 0 are comparable, but the ChPT result for the slope of the correlator with q2 is more than an order of magnitude larger than that obtained from a sum rule analysis analogous to that employed above for the vector correlator case 16] . The source of the discrepancy, in the axial correlator case, is that the sum rule result for the slope has the incorrect chiral behavior, being in fact missing its leading contribution in the chiral expansion. This problem with the sum rule treatment is easily exposed using chiral methods, but is completely non-obvious without them. In the present case, since we do not know what portion of the vector meson masses survives in the chiral limit, we cannot make as precise statements about the required chiral behavior of the vector correlator. There is, however, no obvious problem with the form of the sum rule result above. The sum rule result, moreover, provides clear evidence to indicate that the chiral series for the vector correlator is indeed, as suggested by analogy to the known behavior of the ! 0 process, slowly converging. The sum rule result, in this case, should also provide useful input for the two-loop analysis in ChPT. The two examples clearly indicate the advantages of applying both methods, within their common range of validity, in any given physical process.

21

V. SUMMARY OF RESULTS
The basic results of the paper are as follows. We have demonstrated that (1) in making a sum rule analysis of the mixed-isospin vector current correlator, it is necessary to include the pole term in the phenomenological form of the representation of the correlator, and that, when one does so, the spectral structure of the correlator becomes physically sensible; (2) the expression for the correlator away from q2 = m2 has no general interpretation ! as the o -diagonal element of a vector-meson propagator except for a particular vector meson interpolating eld choice; (3) the freedom of eld rede nition shows that the isospinbreaking factor ! (q2), which occurs in the numerator of the expression, (1.3), for the o -diagonal element of the vector meson propagator, cannot, in general, be taken to be independent of q2; (4) the behavior of the correlator near q2 = m2 suggests that the direct ! !(0) ! + ? contribution to e+ e? ! + ? is not negligible in the -! interference region; (5) the discrepancy between the behavior of the correlator near q2 = 0 as obtained from the sum rule analysis and from ChPT to one-loop indicates a slow convergence of the chiral series for the correlator and, in consequence, the necessity of a two-loop calculation of this quantity in ChPT. The sum rule result for the correlator near q2 = 0 can then be used, in such a calculation, to constrain the O(p6) LEC's of ChPT.

ACKNOWLEDGMENTS
The hospitality of the Department of Physics and Mathematical Physics of the University of Adelaide and the continuing nancial support of the Natural Sciences and Research Engineering Council of Canada are gratefully acknowledged. The author also wishes to thank Tony Williams, Tony Thomas, Terry Goldman, and Jerry Stephenson for numerous useful discussions on the issues of charge symmetry breaking in few-body systems.

22

REFERENCES
Current address: Department of Physics and Mathematical Physics, University of Adelaide, Adelaide, South Australia 5005, Australia 1] G. A. Miller, B. M. K. Nefkens and I. Slaus, Phys. Rep. 194, 1 (1990). 2] P. C. McNamee, M. D. Scadron and S. A. Coon, Nucl. Phys. A249, 483 (1975); S. A. Coon, M. D. Scadron and P. C. McNamee, Nucl. Phys. A287, 381 (1977); J. L. Friar and B. F. Gibson, Phys. Rev. C17, 1752 (1978) R. A. Brandenburg, S. A. Coon and P. U. Sauer, Nucl. Phys. A294, 305 (1978); S. A. Coon and R. C. Barrett, Phys. Rev. C36, 2189 (1987); P. G. Blunden and M. J. Iqbal, Phys. Lett. B198, 14 (1987); Y. Wu, S. Ishikawa and T. Sasakawa, Phys. Rev. Lett. 64, 1875 (1990). 3] G. A. Miller, A. W. Thomas and A. G. Williams, Phys. Rev. Lett. 56, 2567 (1986); Phys. Rev. C36, 1956 (1987). 4] L. M. Barkov et al., Nucl. Phys. B256, 365 (1985); see also S. A. Coon and R. C. Barrett, Phys. Rev. C36, 2189 (1987) and references cited therein. 5] H. B. O'Connell, B. C. Pearce, A. W. Thomas and A. G. Williams, \Rho-Omega Mixing and the Pion Electromagnetic Form-Factor", preprint ADP-95-15/T176, hepph/9503332, 1995. 6] T. Goldman, J. A. Henderson and A. W. Thomas, Few-Body Systems 12, 123 (1992). 7] K. Maltman, Phys. Lett. B313, 203 (1993). 8] G. Krein, A. W. Thomas, and A. G. Williams, Phys. Lett. B317, 293 (1993). 9] J. Piekarewicz and A. G. Williams, Phys. Rev. C47, R2461 (1993). 10] J. Piekarewicz, Phys. Rev. C48, 1555 (1993). 11] K. Maltman and T. Goldman, Nucl. Phys. A572, 682 (1994). 23

12] T. Hatsuda, E. M. Henley, T. Meissner and G. Krein, Phys. Rev. C49, 452 (1994). 13] C. -T. Chan, E. M. Henley and T. Meissner, Phys. Lett. B343, 7 (1995). 14] H. B. O'Connell, B. C. Pearce, A. W. Thomas and A. G. Williams, Phys. Lett. B336, 1 (1994). 15] K. L. Mitchell, P. C. Tandy, C. D. Roberts and R. T. Cahill, Phys. Lett. B335, 282 (1994). 16] K. Maltman, \Comparison of Chiral Perturbation Theory and QCD Sum Rule Results for Pseudoscalar Isoscalar-Isovector Mixing", preprint ADP-95-2-T169, January 1995, to appear Phys. Lett. B. 17] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B147, 519 (1979). 18] R. Haag, Phys. Rev. 112, 669 (1958). 19] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177, 2239 (1969). 20] C. G. Callen, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177, 2247 (1969). 21] J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465 22] S. Scherer and H. W. Fearing, \Compton Scattering by a Pion and O -Shell E ects", TRIUMF preprint TRI-PP-94-65, hep-ph/9408312, 1994. 23] J. F. Donoghue, E. Golowich and B. Holstein, \Dynamics of the Standard Model", Cambridge University Press, New York, N.Y., 1992. 24] J. Gasser and H. Leutwyler, Phys. Rep. 87C, 77 (1982). 25] R. Dashen, Phys. Rev. 183, 1245 (1969). 26] H. B. O'Connell, B. C. Pearce, A. W. Thomas and A. G. Williams, hep-ph/9501251, to appear in \Trends in Particle and Nuclear Physics", ed. W. -Y. Pauchy Hwang (Plenum Press). 24

27] Particle Data Group, \Review of Particle Properties", Phys. Rev. D45, 1 (1992). 28] J. Bijnens, J. Prades and E. de Rafael, \Light Quark Masses from QCD", Nordita preprint PP-94/62 N,P, hep-ph/9411285, Nov. 1994. 29] K. Maltman and D. Kotchan, Mod. Phys. Lett. A5, 2457 (1990). 30] J.F. Donoghue, B.R. Holstein and D. Wyler, Phys. Rev. Lett. 69, 3444 (1992); Phys. Rev. D47, 2089 (1993). 31] J. Bijnens, Phys. Lett. B306, 343 (1993). 32] J. Bijnens and F. Cornet, Nucl. Phys. B296, 557 (1988). 33] J. F. Donoghue, B. R. Holstein and Y. C. Lin, Phys. Rev. D37, 2423 (1988). 34] H. Marsiske et al. (Crystal Ball Collaboration), Phys. Rev. D41, 3324 (1990). 35] S. Bellucci, J. Gasser and M. E. Sainio, Nucl. Phys. B423, 80 (1994). 36] G. Ecker, Nucl. Phys. B (Proc. Suppl.) 7A, 78 (1989). 37] T. P. Cheng, Phys. Rev. 162, 1734 (1967). 38] L. J. Ametller, J. Bijnens, A. Bramon and F. Cornet, Phys. Lett. B276, 185 (1992). 39] G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B321, 311 (1989). 40] J. F. Donoghue, C. Ramirez and G. Valencia, Phys. Rev. D39, 1947 (1989). 41] H. W. Fearing and S. Scherer, \Extension of the Chiral Perturbation Theory Meson Lagrangian to Order p6 ", TRIUMF preprint TRI-PP-94-68, hep-ph/9408346, 1994. 42] K. Maltman, work in progress. 43] E. Golowich and J. Kambour, \Two-Loop Analysis of Vector Current Propagators in Chiral Perturbation Theory", Univ. of Mass. preprint UMHEP-414, hep-ph/9501318, 1995. 25

FIGURES

FIG. 1. Dependence of on the Borel mass, M , for modi ed input set IV.

FIG. 2. Dependence of on the Borel mass, M , for modi ed input set IV.

26

FIG. 3. Dependence of 0 on the Borel mass, M , for modi ed input set IV.

FIG. 4. Dependence of 0 on the Borel mass, M , for modi ed input set IV.

27

FIG. 5. Dependence of f on the Borel mass, M , for modi ed input set IV.

FIG. 6. Dependence of f on the Borel mass, M , for modi ed input set II.

28

TABLES
TABLE I. Sum rule t for the parameters , , 0 , 0 and f Input Set I Set III Set IV 103 2.18 0.39 3.10 0.39 2.59 0.39 1.49 0.06 1.62 0.02 1.55 0.04
0

105

0

f

103

-2.63 0.79 -4.57 0.69 -3.47 0.61

-5.84 0.12 -5.72 0.01 -5.78 0.04

2.30 0.52 3.57 0.52 2.86 0.45

TABLE II. Sum rule t for the isospin-breaking parameters ffk g. Values are quoted for the central values of the input parameters fci g. The units are GeV2. Input Set I Set III Set IV
f

102 3.53 5.00 4.18

f!

102

f

103 2.30 3.57 2.86

f

0

104

f!

0

104

3.73 5.30 4.42

5.34 9.32 7.06

8.45 14.6 11.1

TABLE III. Behavior of the correlator near q 2 = 0. Contributions to 12 (0) from the -! , and 0-! 0 regions are quoted for central values of the input parameters fcig for each input set,
d while the e ect of the uncertainties in these values is displayed explicitly for 12 (0) and 12 dq2 (0). d All entries are in units of 10?3, except for 12 dq2 (0), which is in units of 10?3 GeV?2 .

Input Set I Set III Set IV

-! -1.06 -1.92 -1.43 2.21 3.44 2.75

0 -! 0

12 (0) 0.96 0.14 1.22 0.14 1.08 0.14

d 12 dq2 (0)

-0.18 -0.31 -0.24

3.88 0.61 5.10 0.60 4.43 0.61

29


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