Pion transitions and models of chiral symmetry

University of Massachusetts Amherst From the SelectedWorks of John Donoghue 1989 Pion transitions and models of chiral symmetry John Donoghue, University of Massachusetts - Amherst Barry Holstein, University of Massachusetts - Amherst Available at: https://works.bepress.com/john_donoghue/ 91/ PHYSICAL REVIEW D VOLUME 40, NUMBER 7 1 OCTOBER 1989 Pion transitions and models of chiral symmetry John Department of Physics and F. Donoghue Astronomy, and Barry Uniuersity R. Holstein of Massachusetts, Amherst, Massachusetts 01003 (Received 15 June 1989) We describe a set of pion-decay and scattering amplitudes which are described by only two lowenergy parameters in the effective chiral Lagrangian of QCD. After a phenomenological analysis of the data, we demonstrate how the effective-Lagrangian framework correlates the many predictions of these reactions which have been made in the literature using a variety of models with chiral symmetry. A comparison with the data then also determines which model represents @CD. Not surprisingly, the winner is a form of vector dominance. I. INTRODUCTION One of the common ways to probe quantum chromodynamics (QCD) is by means of very-high-energy scattering. In this case, because of asymptotic freedom the coupling constant o., is small enough that a perturbative treatment of hard scattering is possible. There remains, however, dependence upon noncalculable soft physics such as structure functions, intrinsic pT distributions, and fragmentation models, which must be determined pheAt high energies then, QCD leads to nomenologically. relations between scattering processes, parametrized by e, and empirically determined structure functions. ' There exists an analogous program to test QCD at very low energies. Here one cannot use perturbation theory in a„but rather one exploits the symmetries and anomalies of the theory. The symmetries, especially chiral symmetry, predict the forms of possible reactions at low energy. There remains, however, dependence on noncalculable soft physics, such as the pion-decay constant F and coefficients in an effective chiral Lagrangian, which must be determined experimentally. At low energies then, QCD leads to relations between scattering and decay processes, parametrized by F and empirically determined low-energy constants. In this paper we explore a self-contained set of lowenergy reactions: ~7T 77, ~chevy, 7T f ~/AT +~eve+e, m'~yy . At one level, our motivation is to see how well this lowenergy program, chiral perturbation theory in QCD, is working. At a deeper level, however, we are interested in the structure of the chiral Lagrangian that parametrizes low-energy QCD. There exist in the literature many differing predictions for these reactions within a variety of theories, all chirally symmetric. We show how these simply represent different assumptions for the physics which determines the chiral Lagrangian. By comparison with experiment, we demonstrate that nature selects one version of the physics as being superior. In the next section then, we present a brief overview of of Gasser and techniques the chiral perturbative Leutwyler, while in Sec. III we confront speci6c general predictions of chiral symmetry with experimental results. In Sec. IV we examine theoretical prediction for chiral expansion parameters within various chiral models and demonstrate how previous (model-dependent) predictions can be understood. Finally, in Sec. V our findings are summarized. II. FORMALISM In the limit that the u , d-, s-qua-rk masses vanish, QCD is known to possess an exact global SU(3)L X SU(3)it chiral symmetry: -a qL = I.qL, AJP& qit —Rqtt 8 qL ~exp i+ qR ~exp / A, 8 g Here q refers to the three-component — column vector Q (2) S Chiral SU(3)I and I, ; are the Gell-Mann matrices. X SU(3)tt invariance is dynamically broken to SU(3) i, and Goldstone's theorem requires the existence of eight Goldstone bosons which are identified with n, E, and g (Ref. 4). The classical axial U(l) transformation „ q —+e i Oy~ (3) q is, however, not a symmetry, and leads to the well-known QCD anomaly. The inclusion of quark mass introduces a small explicit breaking of the chiral symmetry, which can be accounted for via a perturbative expansion in the energy. In this paper all of the processes that we study only involve chiral SU(2). Nevertheless, in order to make contact with other work in the field, we shall use the language of chiral SU(3). This involves no loss of accuracy or generality, as the effects of K or g can be absorbed 2378 1989 The American Physical Society PION TRANSITIONS AND MODELS OF CHIRAL SYMMETRY in the low-energy constants and do not modify predictions of chiral SU(2) (Ref. 7). Since @CD possesses this approximate chiral symmetry, the symmetry must be manifested somehow in the interactions of the (pseudo-)Goldstone bosons, and this point has been carefully exploited in recent studies. The strictures that arise from chiral invariance are most succinctly described in terms of the nonlinear order parameter 8 l U=exp m' g j-= A, . (j)i (4) Including the effects of quark mass, the simplest grangian consistent with chiral, Lorentz, and U(1) gauge invariance is then La- '= md 0 0 0 0, Q= 0 —' —, o m, 0 o The first piece of L' ' contains the meson kinetic energy contribution and is chiral invariant. The second term transforms as (3L, 3+ )+(31,3z ) under chiral rotations and describes the breaking of chiral invariance by the with quark masses. Comparing masses yields the normalization 2BO m m„+md +md TrD P U D "Dt+ F2 4 Trm( U+ U") —4'Fpv F"',— (6) where D„U= B„U— +ie[Q, U]A„ is the covariant derivative and ao=,„ 7m~ ao=— m~ 16' 32~F (10) which are roughly borne out experimentally. However, loop diagrams arising from L' ' produce effects of higher m"), and contain divergences. These order, O(p, m infinites can be eliminated by being absorbed into renorchiral couplings of order 4. malizing phenomenological The most general such Lagrangian has been given by Gasser and Leutwyler: p, +" Here F„,F„are external FLpv ~ =g FLv FI., R P p (9) The last piece of L' ' is simply the free photon Lagrang1an. Even at this level the theory has predictive power the tree-level evaluation of L' ' yields [at O(p, m )] the familiar Weinberg scattering lengths for ~-m scattering, L' '=Li[Tr(D UD"U )] +Lz(TrD„UD, U ) +L3Tr(D„UD"U ) +L4Tr(D„UD"Ut)Trm(U+U ) +L~TrD„UD" U mU + Um+L6[Trm(U+ U )] +L7[Trm(U —U )] +L8Tr(mUmU+mU mU "UD Ut+F U D "U)+L,OTr(F„„UF"' U )+L»TrF„F""+L,2Trm iL9Tr(F„+— via meson 6m„ m„+ md +4m, 2m ~ 2m~ = experimental — +L UR U— L' =2Bo 0 1 where P are the pseudoscalar fields and F =94 MeV is the pion-decay constant. Under SU(3)I SU(3)z the matrix U is defined to transform as F2 m„o m 2379 ii yP tensors defined field-strength —r)g pI- ii —i [FLp & FLv ii ] where y is Euler's dimensionality, and constant, e=d —4 ) represents r, = — „, r, = '„r,=o, r, =-,', r =-'„r6 = — „~„r,=o, r, = ', , the —, (12) P S —, &4& ~ Y8~ (15) and the covariant derivative is defined as D =8„— i[V The coefficients L [A„ I . . . , L, 2 are ] i „. — (13) arbitrary and unphysical (bare) inasmuch as they can be used to absorb divergent loop contributions from the lowest-order chiral Lagrangian L ' '. The physical (renormalized) couplings are found to be L,"(p, ) =L, + r. — 2 +ln4~+ —y l 327T 1 p are constants chosen to cancel the divergences. In addition to the above terms, the effect of the anomaThis is contained in the Wessly must be included. Zumino-Witten action at order E (Ref. 10): Swzw= fd x + fd (14) with e'~"' Tr(L;Lq x LkL. A„J"+fd"xe" ~B„A & Tp (16) 2380 JOHN L„=B„UU~, J"=e" F. DONOGHUE AND BARRY R. HOLSTEIN R„=U B„U, F ~Tr(QL L L&+QR R R&), (17) . T13=Tr(Q L&+Q R&+ ,'QU—QU L13+ ,'QU—QUR&) . We have given the gauge dependence only for the U(1) photon field. The full non-Abelian anomaly is much more complicated and is not needed for our purposes. The first term in Eq. (16) in this action is written as the of a five-dimensional integral space, whose fourdimensional boundary is our usual space-time. Even though the physics is determined entirely by the fourdimensional boundary, this construction gives a remarkably simple form for the anomaly. The one-loop renormalization of the anomaly action has recently been carried out. Although there is no modification at all to the coefficients given above, there exist induced new terms at order E . Such terms are themselves purely four dimensional in character and do not modify the anomalous Ward identities. Since we are in this paper working to order E we will not consider them further. Gasser and Leutwyler have analyzed an entire set of electroweak reactions involving pions and kaons, + + + + 0 + + 7T ~& e " v~~ K+~m e+v„K+~m. p+v„, K ~m e+v, Ep~m- p+v„, etc. , (18) , and have shown that such reactions are completely determined in terms of only three of the fourth-order constants: L5, L9, L ]p For example, the charged-pion electromagnetic form factor is calculated as & ( ) 2 — 1+q F2 2 L9—32~2F2 ] +— ln The experimental value p determines m)=( and the measured constants The MeV —5. 2+0. 3)X10 (25) fact that such "-10, is satisfying and is consistent chiral coefficients are small, with the theoretthat chiral ical observation perturbation theory represents an expansion in momentum with a parameter A, which sets the scale of said expansion, given by' L, A-4+F —1 GeV . c' 'c' 'c' ' (26) of terms in L' ', L' ', L' ', etc. , That is, coefficients should be of order . —1 A (27) A which is consistent with, e.g. , 0. 015 F 1 (28) (1100 MeV) m It is this feature which guarantees the success of chiral perturbation theory at low energies: s (&A . In fact, the analysis made by Gasser and Leutwyler successfully related all the reactions quoted in Eq. (18) in terms of the three parameters L5, L9, L",p, and the physical particle masses. Thus far our discussion has constituted a brief review of the Gasser-Leutwyler procedure. The remainder of our paper will involve analysis of previous theoretical attempts to calculate the chiral parameters in chirally symmetric models. We shall first focus our discussion on certain higher-order properties of the pion, and then provide a more general overview. III. (20) (21) (22) L &p L", (pa= L5(p=m„)=(2. 2+0. 5) X 10 (19) Similarly the experimental value for the axial structure constant hz in radiative pion decay, n+ ~e+v, y (Ref. 13), '=(8. 7+0. 2) X10 (24) 2 mg can then be used in order to produce an empirical value for the chiral parameter L9. L9(p=m„)=(6. 9+0. 2) X10 = 1.22+0. 01 provides the size of L 5: of the pion charge radius' (r )'" '=0 439+0 030 fm h~" exPt (23) SU(3) breaking in the pion, kaon decay PHENOMENOLOGICAL UPDATE Before discussing specific theoretical models, we will first describe the analysis of various pionic propertiesthe electromagnetic form factor, radiative weak decay, and the polarizability using the e6'ective Lagrangian framework, and compare the predictions of chiral symmetry with experiment. The electromagnetic form factor, defined as — +(p )I&™~ (p ))=f (q')(p, +p, )„, f is characterized in terms of the form factor (q to lowest order is written in the (on-shell) form f (q )=1+ '(r —, )q +. (29) ) which (30) Radiative pion beta decay, ~+~e+v, y, is described by the weak electromagnetic m. atrix element' PION TRANSITIONS AND MODELS OF CHIRAL SYMMETRY M„,(qp)= fd x e'~ 2381 "(O~T[V™(x )J","(0)]~sr+(p)) q)„2 "(p — — =V2F„(p —q), (2p — . —&2F g +h~[(p q)' q)— „q. — m'2 (1+ (r )q )+&2F„(p q), q„6t(r ) 61 . g—(p . — q, q. ) . r~— (g, q' q— ) q]+hvep. @ q (31) from Here the first piece is the pion pole contribution, while h z, r~, and h z are so-called direct or structure-dependent factors associated with axial-vector and polar-vector compounds of the weak current. We have retained terms proporwhich has recently been meational to q which vanish for physical photons in order to include also ~ ~e+v, sured at SIN (Ref. 17). Ordinarily such structure dependence is difficult to measure, being dwarfed by inner bremsallowing experistrahlung. However, in this case the nonradiative ~+ e v, process is strongly helicity suppressed' mental access to the direct emission terms. The pion polarizability is given in terms of the Compton-scattering ampli- e+e, ~ tude A„(p„q, , q2)= fd x ' (rr+(p2)~T[V' (x)V™(0)]~1r+(p,)) e (P 1 + gpv+ q2 P 1 )Tp(P2 P2+q (p, —q2 ) —m 3 ~ r tr )(q lgpv 1 Tp(P ) 'qlpqlv+q2gpv T„(p,p )=(p +p, )„(1+-,'(r„')q2) ' )(p', —p', ) ' +q„—, (p, —q, ) —m „ 'q2pq2v)+2Vrr(ql expt (33) & Amp(yn~ym)=e, E2 .— + CO +e, Xq, [1 ——,'(r )(q, +q2)] IC02A~ (34) e2X Here a is the fine-structure constant while aE, pM are the electric and magnetic pion polarizabilities, respectively, defined such that f d x H;„, o- 2t(aEE +pMB — ) . ) qlv'q2p q2gpv reveals CX the That aE must be the negative of p~ is also clear from the wherein the framework, effective-Lagrangian order chiral contribution must have the form ) o-(E B) . lowest- (37) Before discussing predictions for these basic pion properties, we note that experimental values exist for each, (32) (Ref. 12), =0.46+0. 02 (Ref. 21), =2. 3+0. 6 (Ref. 17), (38) aE+I3~~'""'=(l. 4+3. 1) X 10 fm aE ~'"~'=(6. 8+1.4) X 10 (Ref. fm (Ref. 22), 23), with which to compare the theoretical analysis. It is also important to note that these viue experimental quantities are predicted in terms of just two of the chiral 'parameters L9 and Llo. (r ) ~' e'=, F 12L9 =32m hv theo ~ (36) m~ )+ V (35) with Eq. (32) immediately Comparison chiral-symmetry requirement F,F" Tr(QUQU h~ expt is the pion electromagnetic vertex. Connection with the polarizability is made by use of the nonrelativistic expansion L,fr-e ql P 1 )T (P2~P2+q2 (r ) ~'" "=(0.44+0. 01)fm where &E 1 4a h~ =32m (L ~ +Lr L" a +P I F (L + L r '""=0 ) (39) r The effects of pion loops have been worked out for these processes in Ref. 7. We do not include these in our analysis for two reasons. (1) When regularized as in Gasser and Leutwyler's work, the corrections induced by loops in these processes are found to be small. In particular they are smaller both than the experimental uncertree-level tainty and than the magnitude — coefficients L9, L Io. A corollary of thisof is the that the scale dependence is not important either. (2) The theories which we are exploring yield predictions for the tree-level coefficients. Given these comments made in case (1) we JOHN 2382 F. DONOGHUE AND BARRY R. HOLSTEIN in chiral theories has already been noted. The between r„lb~, (r ) and between az, h~/hz required in chiral theories can also easily be shown to be results of current-algebra —PCAC (partial conservation of axial-vector current) limits. We have, correction — q) (p „(p,q)= 1 d x e'~ ~'"(O~T[B"A v'2F— p P +v'2F m 7f 7T e.g. ) (x)V„' (0)]~a+(p)) —(O~A„(0)~m+(p)) M— 17 — aE+P~ feel that the use of the coefFicients at the tree level is most illuminating. Before discussing specific theoretical models for L9, L&0, we first examine the relationships between these experimental parameters which follow strictly from chiral the reason for the vanishing of symmetry. One of these P )P ( TP, (p —q, p) (40) expt which yields r„=v'2F„,'&r') (41) —. Also lim A p& 0 „(p„q„q~ ) (q„p, )+M „(qz,p, )) 2F [M„, (42) which requires 8~2m F2 h v a Smm The predictions quoted in Eq. (39) then follow from the use of the SU(2) relation 1 4v'2~ (49) fm This disturbing discrepancy represents a serious violation of the chiral prediction and we urge an experimental Indeed the exeffort to remeasure the polarizability. istence of such a large violation would suggest a significant breaking of chiral symmetry required by the validity of QCD. Incidentally, loops are of no help here. Inclusion of final-state mm and KE effects in yymm yield (43) v'ZF. =2. 8 X 10 F (4~F m 3 2 1 m ) t m 2 (44) F 1 which relates the vector structure constant in radiative pion decay to the ~ ~yy amplitude given by the anomaly. The strictures (45) 2 K F t 2 m~ (50) where t =(q, +q~) (51) transfer and, for x & 0, I/2 i2 F(x) = —4 arcsinh x is the momentum then must arise in any chirally invariant theory. Previous chiral evaluations which independently calculate hz and the polarizability or r~ and the charge radius are redun- dant. On the experimental az = —P~ side, besides the requirement which is well satisfied, we note that' expt 877 2 3 (46) 2 —+ —1— 12m (47) fm is more than a factor of 2 larger than (48) I ~ ~ (54) consistent with the current-algebra —PCAC constraint. Higher-order contributions are to proportional and yield more complex angular depent/(4vrF ) dence not seen in the data. Having examined the experimental state of relations between pionic properties required by chiral syrnrnetry, «1 However, ' + so that —+ 0 8am F hy&~0 =2. 6 . a'"~'=(6. 8+1.4) X 10 m a~ — with expt (52) Note that, in the limit of small t, m =2. 3+0. 6 is in good agreement 4 PION TRANSITIONS AND MODELS OF CHIRAL SYMMETRY 40 to touch base with previous calculations we are prepared of the absolute size of such effects, which is done in the next section. We shall proceed by relating the various theories to the corresponding low-energy parameters L, ,'"'. This shows that many of the model calculations of individual reactions are redundant, being merely the recalculation of the same low-energy parameter in different reactions and allows us to understand in a simple fashion the classes of predictions which appear in the literature. In addition we can compare the results with the phenomenological analysis of the previous section in order to select the most realistic of these theoretical models. IV. PREDICTIONS AT ORDER E' trX X — (trX X) 16 4 (55) where X=o+iw (56) m ' and defining .1 X =(u +s)exp i v rm =(— u +s) U L = —,'(u+s) TrB„UB"U 3 ~ 4 s4 )s diagram, a con- which yields, due to the scalar-exchange tribution to L' ', 4m'S (Tra Va~V')'. (62) In such a model then L& is nonvanishing but. there is no contribution to L2 or L3, and this pattern, were it to be confirmed experimentally, would act as a confirmation of this particular theoretical picture. The linear o model also predicts, however, L9=L&o=0 at the tree level in contradiction to the expermental results of Eq. (38). Of course, our presentation here is simplified. A realistic discussion requires also the inclusion of pion and scalar loop effects and one can also rule out the full o. model's predictions for L&, L2, L3, from the amplitudes in mm. scattering. ' However, the main point which we wish to make is that any such chirally symmetric model can be completely characterized in terms of its order-4 chiral exthe predicted values of these pansion coefficients coefficients provide a complete and accurate representation of the predictions of the theory. The theories applied to various rare pion transitions can be characterized in the following five basic groups. (i) Chiral quark model. In this approach one considers the pion to be a simple qq bound state and evaluates its properties in a simple single quark loop approximation. Calculations have yielded — (r') = 4 3 2F2 ~ E ~M a 8 2 F2 (63) 1 4~'&2F. + '(B„sB"s—2p —, (ii) Chiral baryon field theory. In this model one takes the pion as a fundamental field which is coupled to the nucleon in a chirally invariant fashion. Calculations in this case have given 3 s gg ) (59) The theory then contains massless pions described by the nonlinear representation U together with scalar mesons of mass 2 2 (58) we determine vs (TrB„UB"U L;„,= — 1/2 (57) A I h~=hv= The vacuum state is (60) =v we see that the form of L' ' is as reIdentifying At an quired. energy scale E &&m„such that creation of s particles is not permitted, the scalars still affect the theory through virtual processes. To lowest order, this arises from the linear coupling g2 p, L= — 'trB p XB"X + m, =&2@ . ' Many theoretical attempts have been made to calculate properties of the pseudoscalar rnesons which arise at order E' ', such as the direct radiative pion decay, the pion charge radius, and polarizability which were discussed in the previous section. DifFerent procedures, each of which claims to follow from chiral symmetry, are found to yield quite different predictions, and it is a confusing matter to understand how results of these various "chiral" calculational schemes, e.g. , (i) chiral quark model, (ii) chiral field theory, (iii) current-algebra sum rules, (iv) effective Lagrangian, (v) o model, (vi) vector dominance, (vii) leading nonanalytic corrections (chiral logs), etc. , are interrelated. Nevertheless, that is our goal in this work. As emphasized above, all chiral-symmetric models, regardless of their origin must agree at O(L' '), i.e., order m, since the form of L' ' is required in order to reproduce the free pseudoscalar Lagrangian. However, alternative models can have very different results at O(L ' ') and these differences can be exploited in order to choose between such models. A simple example is the familiar linear o. model, which can be written in the form 2383 (64) h~=hv= 4m &2F (iii) Linear o model. In this approach, as outlined above, one considers the pion to be grouped together with a heavier scalar meson, the o. , in a chirally invariant fashion. Here we consider the meson sector alone. Often the linear o. model is used with fermion fields in addition, but we do not study this case. Calculations then yield 2384 F. DONOGHUE AND BARRY R. HOLSTEIN JOHN (r')= 2 16m ln F tions of order m lnm over those of order m in the expansion in the energy and/or mass. This leads to 11 6 m —1 CX 24m m F (65) 12~2v'2F —vector dominance: In this picture one uses current-algebra —PCAC techniques to yield exact sum rules, which are then approximated using a vector —axial-vector dominance assumption. Calculations have found, in this approach, 6 +E a ~M m mz h„=&2F, hi, = m„4n &2F„ (v) Leading nonanalytic corrections. Here one keeps or s 1ns, only the nonanalytic terms, such as m lnm which are found when calculating meson loops. The motivation for this is the formal dominance of correc- DqDqexp exp[i&(4)] = i . d 4xq exp i f Dq Dqexp 1 i A, —1 (r'„) = ln a~ = — pM =0, h~ (iv) Current-algebra m 40 m 2 p2 2 mz 1 +— ln 2 p2 (67) =0 . There are additional models but these are sufficient to make our point;. Analysis of the first two models is easiest to perform within the framework of effective-Lagrangian techniques which have been widely used recently in order to attempt to understand the restrictions which QCD places on effective low-energy field theories. We shall outline here the approach due to Balog, ' but other workers have obtained identical results. We consider the state obtained from the vacuum by means of a local chiral transformation on the quark fields and interpret the energy difference between this state and the vacuum as the lowenergy effective action. In this way we find @y5 %exp i 1 4y5 X q (68a) L f d x q7q where 7 =& —— .1 —. 1 — i g(1+— y 5 ) i 2 g(1 y 5 ) 2 (68b) . and L„,R„=V„+A„ (68c) represent external fiavor gauge fields. Performing ing up to the a2 coefficient) one finds L' '=L, „, „„+ 384m Tr[ 4(D„UD" U the functional integration where X, is the number of colors, L», ~»~ is the anomalous piece of the Lagrangian as given by Witten, and the remaining O(E' ') terms can be put into the form given in Eq. (11) with the identification ' 2L3 =L9 = 2L ]o N, 48m U ) ]+ 16m 2 for à (70) Examination of Eq. (67) suggests that integrating out the quark fields corresponds directly to the simple chiralquark-model calculations performed in earlier times. (69) This identification is further secured by noting that hq =32vr (L9+L,o) =1, V (r'll 1 (and keep- — ) +2D„U D, UD„U D, U+4(D„D„U )(D,D, U)+4U L„UR„, + 8(R „,D„U D„U+ L„,D„UD 8L i =4L2 = using a heat kernal expansion ' 12L9 F2 3 (71) 4' F as given by direct calculation in such models. We observe that the predicted charge radius, (r )'"' =0. 33 fm, is somewhat too small and the ratio h~/hz somewhat too large in this chiral quark model. Some derivations of Eq. (69) claim that gluonic effects have also been taken into account in these coeKcients. PION TRANSITIONS AND MODELS OF CHIRAL SYMMETRY We would like to devote a rather technical aside on why this is not the case. This paragraph is intended mainly for theorists who have experience with the heat-kernel expansion. Consider first the functional integral over the fermion variables, prior to the gluonic integration. This result has been calculated in terms of a set of operators of increasing dimensions. These operators can contain gluon and/or chiral fields. Schematically one might give examples of terms in the expansion in the following manner: 8', s(F„,U)=a, +azF„'g'"'+a3(TrD„UD" Ut)2 +a4I'„'g'""[Tr(D UD" Ut)] + Wept (a) unity and an SU(2) charge matrix The a3 and a2 terms are of a dimension (i.e., dim=4) that they could be found in the heat-kernel expansion in the a2 coefficient. The a3 term is of the form given in Eq. (69). In contrast the a4 term has dimension=8 and could only appear in the a4 coefficient, which is not included because it is beyond present calculational capabilities. The a4 operator is given pictorially in Fig. 1(c), involving two gluons plus the chiral field, while the az operator is shown in Fig. 1(b). Yet higher-order operators with more gluon fields are also pictured. Next consider the functional integral over the gluonic degrees of freedom. This removes the gluonic operators and we are left with an expansion in the chiral field + (73) Now the coefficient p2 will contain the effect of a3 and also that of a4. Pictorially these give the gluonic corrections pictured in Fig. 2. The result in Eq. (69) is obtained from only the free quark loop Fig. 2(a), as the gluonic portions were contained in the part of the expression which was not calculated. For the anomaly we have the Adler-Bardeen theorem to guarantee that gluonic corrections do not modify the result. However, no such theorem applies to the remaining portion of the effective Lagrangian. In QCD we really need to include the full set of gluonic diagrams. This explanation makes clear why the sophisticated heat-kernel evaluation of Balog and others yields the same coefficients as the simple chiral quark model: They both represent simply a free quark loop calculation. An identical procedure to that given in Eq. (69) can be used to yield the results of chiral baryon field-theory calculations. In this case, however, N, must be set equal to (b) (c) (c) (b) FIG. 2. Schematic picture of some contributions to the effective action obtained after integrating over the gluon fieMs. (a) is a free quark loop which yields the results obtained from the a2 coefficient in the heat-kernal expansion. (b), (c), (d) are gluonic corrections to this. 0 0 0 1 (72) W, s( U) =P, +P~[Tr(B„UBi'Ut)] 2385 (d) FIG. 1. Schematic picture of some of the contributions to the full fermion determinant obtained by integrating out the quark fields coupled to gluons and to a background chiral field. The X indicates a factor of a chiral field in the resulting action, while a wavy line indicates a factor of a gluon field. (74) Also we include the axial-vector coupling is employed. g„. We find then 2Lio= 8Li =4L2 = — 2L3 =L9 = — constant 2 48m. (75) Likewise the coefficient preceding L», m»~ is smaller than in the chiral quark model by a factor of g~/3. It is perhaps surprising then that the ~ ~yy decay amplitude is essentially the same in the chiral field theory and chiral quark models. This results obtains, however, because the value of N, Trl, 3Q (76) to which the amplitude is proportional, is equal to unity in both models. The same "miracle" does not occur in evaluation of the polarizability since these results involve and axial radiative decay N, Tr4&[Q, 4] or N, Tr[Q, C&][Q, @] (77) which differ by a factor of X, . Thus, we predict, in such a model, 327T' 4L9 gg which agrees with the results of the model calculations. It has often been asserted that the agreement between chiral quark model and chiral field theory calculations cited above is "accidental. This seems clearly to be the case since the chiral coefficients L, L2, L, 3, which determine m-m scattering, for example, differ by a factor 3. We observe that the predicted pion charge radius is much too small. The ratio h„/hv is in good agreement with the level measured experimentally. However, this must be regarded as accidental. The basic structure of the o. model has been quoted above. At tree level only the o.-pole diagram contributes, while loop effects have been evaluated by Gasser and We observe that the structure in this case is Leutwyler. completely ruled out by experiment. Indeed the charge radius is too small. Also the axial-vector to vector structure constant h & and the pion polarizability aE are found " 2386 JOHN F. DONOGHUE AND BARRY R. HOLSTEIN to have the wrong sign. In the model which we label as vector dominance, one integrates out the effects of the vector and axial-vector mesons, in a similar fashion to that performed for the scalar meson in Eq. (62) producing an effective chiral Lagrangian of the form Eq. (11) but now with~6'33 L] = 2L2 = 6L3 =6+ F =F +F (81) The charge radius then becomes while h ~ /h v is given by I vF 2 mv hz /hv=32n 12 (79) FvGv ', 2M2v F~ L 10 4Mv' find Fv = 150 MeV F~ lim p~0 (80) is given by the Weinberg sum rule (r = ) 6 F2 F2 4M 4Mv (83) =0 39fm Mv 7T x e'~ (Ol 'T[ 2V&( x) V„( 0) (84) in good agreement with its experimentally measured value. Also, by use of current-algebra —PCAC we can relate h„ to the difference of vector and axial-vector twopoint functions: &2F g„— +(&2F ,'(r ) —h—~)(q„q„g„q M„(q,p)=&2F„q„q P v Jd (r )+ These expressions may not appear familiar in this form. However, vector dominance demands FvGv=F so that the charge radius assumes its expected form, Fv Here Gv=60 MeV is the pm. m coupling while Fv, F~ are the constants of proportionality between vector, axialvector currents, and the corresponding meson fields. We while (82) F Mv —&„+(x)& (0)ll0) )— . (85) I Use of vector —axial-vector dominance for the vacuum expectation value gives lim M„(q, p) = —&2 Using the approximate h„=&2F Fvz g„ result Fv/F =2 we mg find (90) or qpqv &pv hq mg FA'pqv q 2 —m„~ ~ (86) We find then from the coefficients of (i) g„,: Fv F„=F„, q„q: h~ =&2F '(r )— —, hv F2 2 mv 2 m~ Fv 2 mv F 1 mv 1 2 mg 2 mg =0.43 in good agreement with experiment. Finally we include an approximation F2 which corresponds to the result given in Eq. (83) and is a sum rule first given by Das, Mathur, and Okubo. Using the vector-dominance approximation and the Weinberg sum rule we find then 2 which, for an axial meson with the reasonable mass value m ~ —1260 MeV yields (87) (88) h„=&2F (91) theo — which is the Weinberg sum rule, and of (ii) F„ m~~ 1 2 =8m (89) (92) based on the loop structure of chiral perturbation theory. In an expansion in the energy, a term of order m lnm /p or q lnm„/p is technically larger than one of order m or q, if p is chosen at some fixed hadronic scale, i.e. , p-m or 1 GeV. Such terms arise when meson loop effects are calculated. An approximation which is widely used is to keep only these nonanalytic contributions and to disregard all of the polynomical terms. This is effectively the same as setting L; =0. How well does this work in practice? A typical example is the pion charge radius, where chiral logs would predict PION TRANSITIONS AND MODELS OF CHIRAL SYMMETRY 2387 TABLE I. Tabulated are various O(L' ') properties of the pion as predicted in various chiral models and as measured experimentally. Note that only the vector-dominance picture is uniformly successful. (r ) Chiral Vector Quark loop field theory dominance 0.33 0.026 0. 17 0.033 0.41 1.0 (fm~) ' h~(m ) h„/hv 1 &~ /'h v aE+pM (10 aE (10 fm fm ) ) 1.9 0 6. 1 Linear cr' Chiral logs Expt 0.05 0.026' —0.33' 0.3 0 0. 12 0.026 0.44+0.01 0 029+0.019 0.46+0.02 2.3+0.6 1.4+3.1 6.8+1.4 0.39 0.026 0.43 2.3 0 2.6 0 2.5 —2.0 0 0.7 0 0 m =0.7 GeV and have used Eq. (44) for h& even though the anomaly is strictly not present in the mesonic sector of the o. model. 'In the linear o. model, we have used —1 16772/'2 ln m 2 p2 2 m& 1 +— ln 2 p2 0. 12 fm2, p=1 GeV, 0. 06 frn, p= —,' GeV, (93) which is very much smaller than the experimental result. From our previous discussion it is clear that this must fail for r~/hi, also. In the case of h„/hi, and aE, the leading-log approximation predicts Ag Oy AE 0 (94) again in conAict with the data. It should also be mentioned that the leading nonanalytic corrections to m~ and are also in disagreescattering have been studied ment with the data, being too small and not effective in the right channels. Overall we conclude that the leading nonanalytic term approximation has very little in cornmon with the real world. Rather the reverse seems true; i.e., the tree-level coefFicients L,. are dominant and the effects of chiral logs are small. Our results for chiral calculations of pionic properties are summarized in Table I. It is clear that only the vector —axial-vector dominance picture gives overall a satisfactory representation of experimental results (with the exception of the pion polarizability which urgently needs to be remeasured, as noted above). This point is made even more vividly by comparing the calculated chiral expansion parameters with values determined empirically by Gasser and Leutwyler. Strictly speaking, this comparison is somewhat ambiguous since the empirically determined values are not fundamental constants but rather are determined at a given mass scale p, because of the renorrnalization-point dependence arising from pion loop corrections. Nevertheless, as noted above, the scale dependence and residual effects from loops are not large and one can use the overall agreement as a gauge of how well a given theoretical picture is able to represent experimental reality. A summary is given in Table II. We observe that both the chiral quark model and vector dominance give an approximate picture of experimental reality. However, the vector-dominance calculation is in much better agreement with the results of pion radiative decay and the pion charge radius, as already noted. Chiral field theory predictions are consistently a factor of 3 too small, associated with the lack of color degrees of freedom, while the linear 0 model or chiral logs simply do not provide an adequate representation of nature. V. CONCLUSION We have noted that higher-order effects in models possessing chiral symmetry can be expressed in terms of a number of phenomenological small parameters L &, . . . , L &0. In particular we examined a range of pion interactions, involving the charge radius, radiative decay, and polarizability, and found that all could be understood in terms of only L9 and L,o, requiring a relationship between some of these experimental quantities in chiral theories. At the present time a discrepancy exists between the measured and predicted pion polarizability, and remeasurement of aE is strongly suggested. Such chiral parameters also offer a concise and useful way to characterize theoretical calculations of high-order chiral effects. In particular, we noted that previous quark loop calculations can be characterized by the use of a recently calculated effective chiral Lagrangian. A similar Lagrangian is found to represent previous calculations in so-called chiral field theory. The former is found to offer but hardly precise description of exa semiquantitative TABLE II. Listed are values of the chiral expansion parameters L; as calculated in various chiral models and as measured experimentally. All should be multiplied by a factor of 10 '. Note that only the vector-domance picture is uniformly successful. L, + L2 L9 Llo L3 Vector dominance Quark loop Chiral field theory —0.8 —0.4 —2. 1 1.6 6.3 —3.2 0.8 3.3 +2. 1 —1.7 7.3 —5.8 Linear 0. Empirical —0.5 —2.3+0.5 1.5 2.0+0.5 6.9+0.2 —5.2+0.3 0.9 —2.0 JOHN 2388 F. DONOGHUE AND BARRY R. HOLSTEIN 40 tions of the low-energy constants work so well. After all, vector dominance has long been a successful phenomenological tool, even if its fundamental origin is not completely clear. What is interesting about recent works on the origins of effective Lagrangians is that they extend vector-dominance ideas to the rigorous techniques of chiral symmetry. Further tests of this picture are also suggested. Indeed for any of the higher-order properties of the pion discussed above, there exist corresponding parameters describing the structure of the charged kaon, and only some of these have been measured. The polarizability of the neutral pion and kaon are also predicted unambiguously in chiral models in terms of (finite) meson loop contributions. Also worthy of study is a more direct connecunion tion of the vector-dominance —chiral-Lagrangian perimental results, while the latter is too small by a factor of 3. Satisfactory agreement is found, however, in a picture based upon vector dominance. Very often the linear o. model or the leading nonanalytic approximation are used to study some consequences of chiral symmetry. This is tempting because they offer simple and easily manageable theories. However, it is clear from the comparison with data that they bear little relation to reality. They are not a good representation of low-energy QCD, and consequences derived from them are suspect. The pattern found here agrees well with the results of an analysis of m~ scattering. There also, the vectormeson picture provides an excellent description of the low-energy constants involved, in that case L, and L2. The scale dependence and effects of loops are greater in that analysis, but the comparison was made to a tree-level fit to the data. The chiral quark model predicts both L& and L2 too small, with the ratio L /Lz differing by a factor of 2 from the fit. 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