Dispersive analysis of the decay η → 3π

16 May 19% PHYSICS

ELSEVIER

LElTERS

6

Physics Letters B 375 (1996) 335-342

Dispersive analysis of the decay 33+ 37r* A.V. Anisovich a, H. Leutwyler b a Academy of Sciences, Petersburg Nuclear Physics Institute, Gatchina. St. Petersburg, 1883.50 Russia h lnstitut fir theoretische Physik der Universitiit Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland and CERN, CH-121 I Geneva, Switzerland Received 11 January 1996 Editor: R. Gatto

Abstract We demonstrate that the decay 7 masses. The transition amplitude is is calculated by means of dispersion theoretical uncertainties in the result this ratio of quark masses.

1. Chid

perturbation

--+ 37r represents a sensitive probe for the breaking of chiral symmetry by the quark proportional to the mass ratio (MS - mi)/(mf - ri2’). The factor of proportionality relations, using chiral perturbation theory to determine the subtraction constants. The are shown to be remarkably small, so that v-decay may be used to accurately measure

theory

* = -3

Since Bose statistics does not allow three pions to form a configuration where the total angular momentum and the total isospin both vanish, the decay 7 + 37~ violates isospin symmetry. The transition amplitude contains terms proportional to the quark mass difference md - m, as Well contributions CCe*, generated by the electromagnetic interaction. The latter are strongly suppressed by chiral symmetry - the transition is due almost exclusively to the isospin breaking part i(m, - md) (iiu - c&i) of the QCD Hamiltonian [ 11. The relevant matrix element was worked out in one loop approximation of chiral perturbation theory in Ref. [ 21. Extracting a suitable ratio of quark masses, the amplitude of the decay r] + &r-r0 is given by

*Work

supported

in part by Schweizerischer

Nationalfonds.

0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved PII SO370-2693(96)00192-X

1

_=

Q* -

1 M:,M2,44; 3J?;F; ~2,

M(s, t, u) 1

rni - rni

(1)

rn: - ih* ’

where the dimensionless factor M( s, t, u) exclusively involves measured quantities. It is of the form M(s,t,u) + Mz(t)

=Mo(s)+(s-u)Ml(t)+(s-t)Ml(u) +

M2(u)

-

$M2W

,

(2)

with s = (p,+ + pT- >*, t = (p,+ pro )‘, U = (p+~ +p,t)*. The functions Mo(s),Ml(s),M2(~) represent contributions with isospin I = 0, 1,2, respectively. They contain branch point singularities generated by the final state interaction in the S- and P-waves. The imaginary parts due to partial waves with angular momentum e 2 2 only start showing up if the chiral perturbation series is evaluated to three or more loops. For this reason, the amplitude retains the structure (2) also at next-to-next-to-leading order. We stick to this

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A.V. Anisouich. H. Leunvyler/Physics

1

2

4

6

a

I. Real part of the amplitude along the line s = U. The dashed line is the current algebra prediction. The dash-dotted curve corresponds to the one loop approximationof chiral perturbationtheory. The full line represents the result of the dispersive calculation described in the text. Fig.

approximation throughout the following, i.e. represent the amplitude in terms of the S- and P-waves and neglect the discontinuities from partial waves with 0 > 2. ’ The one loop result shows that, up to and including first order corrections, the decay rate is inversely proportional to Q4, with a known factor of proportionality. A measurement of the rate thus amounts to a measurement of the quark mass ratio Q. Apart from experimental uncertainties, the accuracy of the result for Q is determined by the accuracy of the theoretical prediction for the factor M( s, t, u). This motivates the present work. In Fig. I, the real part of the one loop amplitude is plotted along the line s = u (dash-dotted). For small values of s, the curve very closely follows the current algebra formula (dashed) - there, the correction is small. The cusp at s = 4 Mi arises from the final state interaction, which is dominated by the isospin zero Swave. The singularity requires the amplitude to grow more strongly with s than the current algebra result. At the maximal energy accessible in v-decay, ,,6 = M, M,, the correction amounts to 3 of the leading term. This indicates that (a) the current algebra formula ’ The same approximation is commonly used also in n-z= scattering. where the Roy equations [3] provide a rigorous starting point for the dispersive analysis. The validity of the representation analogous to (2) at two loop order of chiral perturbation theory was pointed out in Ref. [ 4 1 and an analysis of the ~VT data on this basis is given in Ref. 151.The explicit two-loop result 161 not only confirms this structure, but shows that reasonable estitnates for the relevant effective coupling constants lead to very sharp predictions for the scattering lengths and effective ranges.

Letfers B 375 (1996) 335-342

underestimates the rate by a substantial factor and (b) the first two terms of the chiral perturbation series represent a good approximation only at small values of s. The reason why, in part of the physical region, the corrections are unusually large is understood. Chiral symmetry implies that the pions are subject to an interaction with a strength determined by F,. In the chiral limit the corresponding I = 0 S-wave phase shift is given by I$ = s/lfk~~F~ and thus grows with the square of the center of mass energy. Accordingly, the corrections rapidly grow with the energy of the charged pion pair.

2. Dispersion

relations and sum rules

There is a well-known method which allows one to calculate the final state interaction effects, also if they are large: dispersion relations. Indeed, this method was applied to three-body decays long ago [ 71 and many papers concerning the decay 7 + 37r have appeared in the meantime - they may be traced with Refs. [ 8111. Crudely speaking, analyticity and unitarity determine the decay amplitude up to the subtraction constants. Chiral perturbation theory is needed only for these. The additional information contained in the one loop result merely reflects the fact that the chiral perturbation series obeys unitarity, order by order. In the following, we determine the absorptive part with unitarity and use dispersion relations to express the full amplitude in terms of the absorptive part and a set of subtraction constants. The latter are calculated by matching the dispersive and one loop representations at low energies. In principle, the same steps must be taken whenever unitarity is used to extend the range of validity of the effective theory - our analysis outlines the general procedure in the context of a nontrivial example. For earlier work concerning the marriage of chiral perturbation theory and dispersion relations, we refer to [ 12-141 and the references therein. We first set up the dispersion relatizs obe2l_ed by fhe one loop amplitude. The functions MO( s ) , MI ( s) , M2 (s) are analytic except for a cut along the real axis. For s ---f 00, M,,(s)and Mu grow like s2 logs, while M,(s) is of order slogs. Accordingly, three subtractions are needed for fio( s), E?(s) and two for El(s),

A.V. Anisouich, H. Leutwyler/

/i&(s) = a0

+

bos +

s

s3 O” ds’ Im&(

cos2+ ;

slj,I

337

Physics Letters B 375 (19%) 335-342

4L3 - 1/64T2 d = b1 + c2 = - F; (M; _ M2,)

s’)

- s-k

4M;

s2 M,(s)=UI+bls+;

+I

ds’ IrnMl (s’) cc S’2S,_s_iiE!

s

4Mq

i&(s) = a2

+

b2s + c2s2

O” ds’ Imfi2( FsS’-s-iiE’

s’)

IT s

(3)

4M?,

Inserting this representation in Eq. (2)) the one loop amplitude takes the form M(s, t,u) = P(s, I, u) + D( s, t,u), where the polynomial P(s, t, u) contains the contributions from the subtraction constants QT..., c2 while D( s, t, u) collects the dispersion integrals. On kinematic grounds, the polynomial only involves four independent terms, which may be written intheformP(s,t,u) =a+bs+cs2-d(s2+2tu): Only four combinations of the eight subtraction constants ~10,. . . , c2 are of physical significance. The dispersive representation of the one loop amplitude thus appears to involve four subtraction constants. In fact, however, the constant c is determined by the imaginary parts through a_sum rule, for the following reason. The amplitudes MO(S) and Mu individually grow like s210gs for s -+ co, but the combination Go(s) + i M2( s) only diverges in proportion to s logs. For the above representation to reproduce this behaviour at large values of s, the term 0; s2 arising from the dispersion integrals must be compensated by the one from the subtraction polynomials, i.e.

c=co+~c:! 1 ?l =-J’

O3 ds’ F{Im&(

s’) + 4 Imi2(s’)}.

r.

;;;i{s’ImMl(s’)

+ImM:!(s’)}.

(5)

This shows that the one loop amplitude is fully determined by its imaginary parts and by the three constants a, b and L3. The dispersive representation explicitly expresses the fact that an analytic function is uniquely characterized by its singularities: While the functions Im&( s) describe those associated with the branch cut along the real axis, the subtraction constants account for the singularities at infinity.

3. Unitarity In the one loop approximation, the amplitude obeys unitarity only modulo contributions of order p6. The corresponding expressions for the discontinuities across the branch cut only hold to leading order of the chiral expansion. The unitarity condition may be solved more accurately, using an approximation which holds up to and including two loops of chiral perturbation theory. The approximation accounts for two-body collisions, but disregards multiparticle interactions. In the case of elastic scattering, the approximation is well-known and is referred to as elastic unitarity. The extension of this notion to three-body decays is not trivial, however: Some of the early papers contain erroneous prescriptions. A coherent framework is obtained with analytic continuation in the mass of the 11 and is discussed in detail in the literature quoted above. Denoting the discontinuity of the function MI(S) by disc MI(S) z {MI( s + k) - MI( s - ie)}/2i, the approximate form of unitarity used in the present work reads 2

(4)

4M$

discMl(s)

= sinSI

e -i61(s){M,(s)

+ A,(s)}. (6)

The terms_of order2 logs also cancel in the combination s Ml (s) + M2( s), but this function does not obey a twice subtracted dispersion relation, because the term LXL3 gives rise to a contribution which grows like s2. Hence d is related to the effective coupling constant L3,

* In the standard notation, where the phase shift of the partial wave with isospin and angular momentum quantum numbers I, 1 is denoted by S!(S), the unitarity condition for the amplitudes Mn(s),Ml(s) and MS(S) involves 6n(s) E #(s).Sl(.s) E 6:(s) and 82(s) 3 6;(s), respectively.

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The first term in the curly brackets accounts for repeated collisions in the s-channel, while the second arises from two-particle final state interactions in the t- and u-channels and involves angular projections of the functions Ma(s) , Ml (s) , A42(s). Explicit expressions for the relevant angular integrals are given in Ref. [ IO], together with a detailed discussion of the steps required to analytically continue these in M,. We briefly comment on the three phase shifts 60 (s) ,SI (s) , 82(s) , which represent an important input of our calculation. Chiral perturbation theory yields remarkably sharp predictions near threshold. Together with the experimental information available at higher energies, the Roy equations then determine their behaviour up to KK threshold, to within small uncertainties [ 151. The function ??a(s) passes through kr at fi = 850-900 MeV and continues rising beyond this point. In the context of our calculation, the behaviour at higher energies does not play a crucial role. We use a parametrization where So(s) tends to rr for s + 00, so that the discontinuity, which is proportional to sin Sa( s), tends to zero. A similar representation of the high energy behaviour is also used for the P-wave, where the phase shift rapidly passes through $r at s = A$ The exotic partial wave with I = 2, on the other hand, describes a repulsive interaction and does not exhibit resonance behaviour. We use a parametrization where the function S,(s) asymptotically tends to zero.

4. Ambiguities relations

in the solution of the dispersion

If the phase shifts are known, the unitarity conditions determine the discontinuity across the branch cut in terms of the amplitude itself. Analyticity then implies that the functions Ma(s), Ml (s) and M2( s) obey dispersion relations analogous to Eq. (3)) Ml(s)

= 4(s) oc,

+L ds’ r. I’

sin&(s’)e-i61(.s’) s,

_

s_

ie

pm’)

+&(W.

4.k;

(7) For simplicity, we assume that the phase shifts rapidly reach their asymptotic values, so that the dispersion

Letters L3375 (1996) 335-342

integrals converge without subtractions, but this is not essential - we might just as well write the integrals in subtracted form, absorbing the difference in the polynomials Pt (s) . It is crucial that the dispersion relations used uniquely determine the amplitude. In fact for the above relations this is not the case: The corresponding homogeneous equations - obtained by dropping the polynomials S(s) - possess nontrivial solutions. In its simplest form, the problem shows up if the contributions to the discontinuity from the crossed channels are dropped. Unitarity then reduces to three independent constraints of the form discA = sin St epiSiMt, or, equivalently, Mr( s + ie) = e2i6iMt( s - ie). This condition is well-known from the dispersive analysis of form factors and can be solved explicitly: The Omnbs function, defined by

obeys 0, (s + ie) = e2isi s1t (s - ie) , so that the ratio mI (s) = MI(S) /a~(s) is continuous across the cut. Since 01 (s) does not have any zeros, ml(s) is an entire function. In view of the asymptotic behaviour of the phase shifts specified above, s20 (s) , 0 I (s) tend to zero in inverse proportion to s, while (12(s) approaches a constant. Assuming that Ml(s) does not grow faster than a power of s, the same then also holds for rnr (s) . Being entire, mo (s) , ml (s)and m2 (s)thus represent polynomials: The genera1 solution of the simplified unitarity conditions is of the form Mt (s) = rnr (s)RI (s) , where rnr (s) is a polynomial. The ambiguity mentioned above arises because the Omnbs factors belonging to the partial waves with I = 0, I tend to zero for s -+ co. The dispersion relation for Ma ( s) , e.g. implies Ma(s) + Pa(s) . If PO( s) is a polynomial of degree n, mo (s) is of degree n + 1. Hence the above dispersion relation for Mo( s) admits a one parameter family of solutions. The same is true of MI (s), while the solution of the dispersion relation for A42( s) is unique. The same problem also occurs if the functions tir( s) are not discarded. The preceding discussion points the way towards a solution of the problem: It suffices to replace the above integral equations with the dispersion relations obeyed by the functions

A.V. Anisovich, H. Leuiwyler/

ml(s)

-_ M,(s)/&(s),

which obey

rnl(S + 2) - rn[(S - 2) =

M,( s + ie)e- 6 _

.

MI ( s _

i.c)

.@I

PII

,.

The integral equations

then take the form

w

+IT

s

sin &( s’) &( s’)

dS’/R~(s’)~(~‘-.s-i~)

(8)

4MX

In the simplified situation considered above, these equations indeed unambiguously fix the solution in terms of the polynomials pt( s). Our numerical results indicate that the same is true also for the full set of coupled integral equations, but we do not have an analytic proof of this statement. In the present context, the asymptotic behaviour of the phase shifts is an academic issue. The fact that the solution of Eq. (7) is unique if all of the phase shifts tend to zero, but becomes ambiguous if some of them tend to r, shows that this form of the dispersive framework is deficient. As an illustration, we return to the simplified problem and discuss the change in the function Mu(s) which results if the high energy behaviour of the phase shift 60(s) is modified. Suppose that So(s) is taken to rapidly drop to zero at some large value SO, such that the corresponding Omnes function differs from the original one by the factor 1 -S/SO. If the dispersion relations are written in the form (8)) the solution merely picks up this factor and thus only receives a small correction at low energies. With Eq. (7), however, the modification generates a qualitative change, as it selects a unique member from a one parameter family of solutions - the result cannot be trusted, because it is sensitive to physically irrelevant modifications of the input.

5. Subtractions The chiral perturbation series yields an oversubtracted representation: The number of subtractions

Physics Letters B 375 (1996) 335-342

339

grows with the order at which the series is evaluated. The corresponding singularity at infinity is an artefact, which occurs outside the range where this representation is meaningful. The high energy behaviour of the physical amplitude is the same as in the case of elastic scattering and is dominated by Pomeron exchange (since the amplitude ZV,J + mn- violates isospin symmetry, the coupling to the Pomeron is suppressed by a factor of md - m,, but this holds for the entire amplitude). Comparison with rr-scattering indicates that the dispersive representation of the functions Mo( S) , A41(s) , A42(s) should require only two subtractions. If only two subtractions are made, the contributions from the region above KEthreshold are quite important and need to be analyzed in detail (compare e.g. Ref. [ 131, where the scalar form factors are studied by using unitarity also for the quasi-elastic transition 7rr -+ K@. The uncertainties associated with the high energy part of the dispersion integrals do not limit the accuracy of the measurement of Q, however. One may use more than the minimal number of subtractions and determine the extra subtraction constants with the observed Dalitz plot distribution. With sufficiently many subtractions, the contributions from the high energy region become negligibly small. In principle, a measurement of Q requires theoretical input only for the overall normalization, while the remaining subtraction constants, which specify the dependence of the amplitude on s, t, U, may be determined experimentally. We fix the number of subtractions by requiring that the function M( s, t, U) grows at most linearly in all directionss,t,u --+co:Mo(s) =O(s),Ml(s) =0(l), Mz (s) = O(s). In view of the Omnb factors, this implies that the subtraction polynomials are of the form Fe(s) =ao+pos+~&~,(S) =&I +plS,4(S) = cr2+PZS and thus involve altogether seven subtraction constants. As is the case with the one loop amplitude, the decomposition into the three isospin components Mo( s) , Ml (s) , M3( s) is not unique. Since the transformationMl(s) + Ml(s)+kl,M~(s)+ Mu+ k:! + k3s preserves the asymptotic behaviour and can be absorbed in Mo( s),only four of the seven subtraction constants are of physical significance. Without loss of generality we exploit this freedom in the isospin decomposition and set 1~1= LY:!= p2 = 0. The dispersion relations then take the form

340

A.V. Anisovich. H. Leutwyler/Physics

+-

s2

m

ds’

7T J

sin&(s’)

Ao(s’)

S’[fla(S’)I(S’-F-_iE)

4M;

J

+s O”-ds' lr

JIM;

sinSt(s’)

s’ Ifl,(s’)l

Ai

(s’-

O” J ds’

s - ie) > ’

sin&(s’)

Az(s’)

s’2~.n,(.s’)~(G--_iE)

4A4;

(9) The four subtraction constants (~0, PO, yu, pt play the same role as the terms a, b, c, d occurring in the one loop result. The constants CYO, PO already appear in the current algebra approximation, which corresponds to aa + ,& s = (3s - 4Mi) /( Mt - Mz). At leading order, the amplitude thus vanishes at s = :M$ a feature which is beautifully confirmed by the observed Dalitz plot distribution. The phenomenon is a consequence of SU(2),(xSU( 2),,-symmetry and does not rely on the expansion in powers of nzY, which is generating the main uncertainties in the chiral representation: In the limit m, = rnd = 0, where the pions are massless, the decay amplitude contains two Adler zeros, one at ~~1 = O(s = u = 0), the other at pp- = O(s = t = 0). The perturbations generated by m,,md produce a small imaginary part, but the line ReM(s, t, u) = 0 still passes through the vicinity of the above two points. The figure shows that along the line s = u, the one loop corrections shift the Adler zero from 1.33 Mi to sA = 1.4 1 M$ The position of the zero is related to the value of the constant LYO,while pa determines the slope of the curve there. The figure also shows that the corrections of order m,y barely modify the latter: The slope of the one loop amplitude differs from the current algebra result, /3a = 3/(Mi - Mi), by less than a percent. We therefore assume chiral perturbation theory to be reliable in the vicinity of the Adler zero and determine the subtraction constants cyc and /?a by matching the dispersive representation of the amplitude with the one loop representation there.

Letters B 375 (1996) 335-342

Concerning the value of (~0, this procedure is safe, because LYOis protected by the low energy theorem mentioned above. The slope PO, on the other hand, is not controlled by SU( 2) Rx SU( 2) L. The one loop result does account for the corrections of order m,s, but disregards higher order contributions. According to a general rule of thumb, first order SU( 3) breaking effects are typically of order 25% while second order contributions are of the order of the square of this. It so happens that in the case of the slope, the various first order corrections nearly cancel one another. The second order effects are discussed in some detail in Ref. [ 161, on the basis of the large NC expansion. In particular, it is shown there that the one loop result fully accounts for ~7’ mixing, except for a factor of cos Oso~ pv 0.93. This represents a second order correction of typical size, consistent with the rule of thumb. In the convention adopted above, the term /3i determines the slope of Ml (s). More generally, j31 is related to the convention independent quantity M’, (0) + ;M;(O)

(10) In the one loop approximation, the lhs coincides with 61 +c2. The low energy theorem (5) thus relates pi to the coupling constant Ls, whose value is known rather accurately from Ke, decay, L3 = (-3.5 & 1.1) . 10-j [171. Both for the dispersive representation and for the one loop approximation, the quantity M’, (0) + i M;(O) receives two contributions: one from an integral over the discontinuities of the amplitude, which accounts for the effects generated by the low lying states, and one from a subtraction term {pt c-) -(4Ls - 1/647r2)/F:(MG - M$)}, which incorporates all other singularities, including those at infinity. The main difference between the two representations is that the integral in I$. (10) exclusively accounts for the TV discontinuities, while the one in&q. (5) includes the singularities generated by KK, TV and 77 intermediate states, albeit only to one

A.V. Anisouich, H. Leutwyler/Physics

loop. One readily checks that the integrals over the elastic region 4Mz < s < (M, + M,)2 only differ by terms of order p2, which are beyond the accuracy of the one loop prediction. This is an immediate consequence of the fact that chiral perturbation theory is consistent with unitarity. In the elastic region, the dispersive calculation yields a more accurate representation of the discontinuities than the one loop approximation. Accordingly, we use this approximation only to estimate the remainder. The one loop result shows that the contributions from rv intermediate states are proportional to Mi and therefore tiny - inelastic channels generate significant effects only for 4@. < s < co. The entire contribution from this interval to the integral in Eq. (5) amounts to a shift in PI of +I .7 GeV4, too small to stick out from the uncertainties in the term from L.7. We conclude that /?I is dominated by this term,

PI = -

4Lj - l/645-2 FZ(Mi-M2,)

= 6.5 * 1.8 GeV-”

and take the above shift as an estimate for the uncertainties associated with the inelastic channels. The constant ‘yo is related to the convention independent combination M{ (0) + $ M$‘( 0) of Taylor coefficients and may be evaluated in the same manner. In particular, one may check that the elastic contributions to the relevant integrals coincide within the accuracy of the one loop representation also in this case. The only difference is that the low energy theorem (4) does not contain a term analogous to L3, so that the arguments outlined in the preceding paragraph now imply yo N 0. The uncertainties to be attached to this result may again be estimated with the contribution from the interval 4M’, < s < co to the integral over the imaginary part of the one loop amplitude, which amounts to a shift in the value of yo by +8.6 GeVe4. The net effect of the uncertainties in the subtraction constants PI, ‘yo is small. At the center of the Dalitz plot, the above shifts due to inelastic discontinuities increase the amplitude by about 5%. The order of magnitude is comparable with the shift in /SO due to 77’ mixing, which acts in the opposite direction.

Letters B 375 (1996) 335-342

341

6. Results The integral Eqs. (9) may be solved iteratively, starting e.g. with MI(S) = 0. The results are illustrated in Fig. 1: The solid line shows the real part of the amplitude along the line s = u, which connects one of the Adler zeros with the center of the Dalitz plot. As can be seen from this figure, the corrections to the one loop result are rather modest. The calculation confirms the crude estimates given in Ref. [2]. The limiting factor of the dispersive evaluation is the accuracy within which the low energy theorems for the subtraction constants can be trusted. Including the uncertainties associated with inelastic channels and with the phase shifts, we estimate the error in the result for the rate at 10%. In principle, the method thus allows a measurement of Q to an accuracy of 2.5%. Using the overall fit of the particle data group 3 , I’s_-nt T- ,+ = 283 f 28 eV, and adding errors quadratically, we obtain Q = 22.7 + 0.8. A more detailed account of our numerical results is in preparation [ 191.

Acknowledgements We are indebted to V.V. Anisovich, Hans Bebie, Jiirg Gasser and Peter Minkowski for useful comments and generous help with computer ethonography and to Joachim Kambor, Christian Wiesendanger and Daniel Wyler for keeping us informed about closely related work [ 111.

[ I ] D.G. Sutherland, Phys. Lett. 23 ( 1966) 384; J.S. Bell and D.G. Sutherland, Nucl. Phys. B 4 ( 1968) 3 1.5; For a recent analysis of the electromagnetic contributions, see, R. Baur, J. Kambor and D. Wyler, Electromagnetic corrections to the decays 7) + 3~r, preprint hep-ph/9510396. 121 J. Gasser and H. Leutwyler,

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and N.H. Fuchs,

Phys.

Rev. D 47

.7Note that the experimental situation is not satisfactory: is based on mutually inconsistent data sets [ I8 1,

the fit

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1.51 M. Knecht, B. Moussalam,

.I. Stern and N.H. Fuchs, The low energy rn- amplitude to one and two loops, preprint hep-ph/9507319. [ 61 J. Bijnens. G. Colangelo, G. Ecker, .I. Gasser and M.E. Sainio, Elastic ?r7r scattering to two loops, preprint hepph/9511397. 7 1 V.N. Gribov, Nucl. Phys. 5 ( 1958) 653; N. Khuri and S. Treiman, Phys. Rev. 119 (1960) 1115; V.V. Anisovich and A.A. Anselm, Sov. Phys. Usp. 9 (1966) 287 [ Usp. Fyz. Nauk 88 ( 1966) 2871. 8 1 A. Neveu and J. Scherk, Ann. Phys. 57 (1970) 39. 9 I C. Roiesnel and T.N. Truong, Nucl. Phys. B 187 ( 1981) 293; T.N. Truong, Nucl. Phys. B (Proc. Suppl.) A 24 (1991) 93. I 101 A.V. Anisovich, Physics of Atomic Nuclei 58 ( 1995) 1383 I Yad. Fiz. 58 ( 1995) 14671. I 11 [ J. Kambor, C. Wiesendanger and D. Wyler. Final state interactions and Khuri-Treiman equations in 7 -+ 37r decays, preprint hep-ph/9509374.

Leners B 375 (1996) 335-342 [12J L.S. Brown and R.L. Goble, Phys. Rev. Len 20 (1968) 346; Phys. Rev. D 4 ( 1971) 723. [ 131 J.F. Donoghue, J. Gasser and H. Leutwyler, Nucl. Phys. B 343 (1990) 341. [ 141 J.F. Donoghue, On the marriage of ,$T and dispersion relations, preprint hep_ph/9506205. [ 151 B. Ananthanarayan and P. Btittiker, The chiral coupling constants et and ea from rr phase shifts, in preparation. [ 161 H. Leutwyler, Implications of ~7’ mixing for the decay ~7+ 31r, hep-ph/9601236. [ 171 J. Bijnens, G. Colangelo and J. Gasser, Nucl. Phys. B 427 (1994) 427. [ 181 For a review of the experimental situation, see the note by N.A. Roe in Review of Particle Properties, Phys. Rev. D 50 (1994) 1173. [ 191 A.V. Anisovich and H. Leutwyler, Measuring the quark mass ratio (tni - tn~)/(rn~ - r%*) by means of 7 + 3~. in preparation.