.FERENCEIC/T3/39

U-

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

IT + 2y DECAY FOE A REAL PION

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

G. Furlan

F. Legovini

and

N. Paver

1973 MIRAM ARE-TRIESTE

IC/73/39

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

IHTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ir°'+ 2v DECAY FOE A REAL PION *

G. Furlan

International Centre for Theoretical Physics, Trieste, Italy,

and

Istituto di FiBica Teorica dell'Universita, Trieste, Italy,

F. Legovini and N. Paver

Istituto di Fisica Teorica dell'Universita, Trieste, Italy.

ABSTRACT

Using the method of Fiibini and Furlan, we discuss the extrapolation

of the soft-pion result for IT •* 2y decay, given by the Adler anomaly,

to the physical region, emphasising, in particular, the role played "by

vector dominance and chiral symmetry breaking. The extrapolation effects

turn out to be relatively small in our approach, in agreement with the PCAC

smoothness assumption. Some recent alternative treatments are briefly

commented.

MIRAMARE - TRIESTEApril 19T3

Submitted for publication.Supported in part by Istituto Nazionale di Fisica Nucleare, Sezionedi Trieste.

- 2 -

1. The decay of the neutral pion into two photons is an

outstanding process from the theoretical point of view. There

exist different descriptions of it, the most up-to-date ones

being based on vector dominance on one side and on PCAC on the

other.

Vector dominance, in its most refined version based on light-

cone mass dispersion relations , is claimed to give a satisfac—

tory result for the X° lifetime

On the other hand,the straightforward application of current

algebra and PCAC to this process leads, as is veil known, to the

disastrous result of a vanishing amplitude . Of course the result

holds for zero pion mass, so that a rapid variation of the amplitude

from C = 0 to ^ = ^ x can be invoked, which, hovever, would

spoil the simplicity of the whole PCAC approach. A very inter-

esting solution to this problem was proposed by Bell and Jackiw

and by Adler : the blame was not put on a possible r-apid variation

of the amplitude but on anomalous properties of familiar current

algebra when electromagnetic effects must be taken into account.

This can be phrased as anomalous commutators between current

densities or anomalous Ward identities or, finally, as an anomalous

form of the axial vector current divergence in the presence of

electromagnetism. All these (equivalent) formulations lead to the

final expression of the Tt° —> -2Q" amplitude ,, for a massless

pion:

*) "Actually the claim applies to the experimental value by Bellet-

9' Ttini _et al. 9'" T ^ tiEo-»2Jf-)« 41.^1.3. eV, while the recent Rosenfeldtables 3) quote the appreciably lower value T7 k^0""*^-?• 8 ±0 9eV

- 3 -

where S is the so-called Adler anomaly. £ 7C is related to the ;

observable width by JL (. TC ~^ - V~J — 1 "He I ^ / 6 ^ TX. and;using [

the Rosenfeld value, one finds 1 o I ~ vj.Ho while the result of )

Ref. 2 furnishes | S | 0.5& , both indications pointing [

towards the value 1 E> \ — /^ . ;

The theoretical evaluation of S involves a hard problem •'

of strong interaction dynamics. However,Adler has been able to ,

derive an exact formula for S in a quark-gluon model: the result |

is that, for such a renormalizable field theoretical model, S L

can be determined by simply evaluating the lowest order triangle |

graph with internal quark lines, at the point ^ = l»C = "p = O [

(k and p being the photons' four momenta, Q = k-V » ) . The great

virtue of Adler's result is that higher order radiative corrections

7)do not contribute in that limit . The formula is

S = 4C

.._.- >-J-3 / the charge of the underlying fermion of isospin

component l j . According to Eq. (l.2) the familiar quarks with

fractional charges, for which S = l/6, give a wrong prediction

(of a factor 3) for the 71,° amplitude. Thus,more complicated

models, like the"coloured" quarks recently introduced by Gell- ^

Mann or others similar^ predicting S = l/2, are being considered.

Eqs. (l.l,2) have successively received a firmer basis from more

general non-perturbative considerations based on the light-cone

9)behaviour of operator products , so that the anomaly S has become

an interesting "test body" for the nature of the underlying

constituents.

- 4 -

Of course, the possibility of a direct measurement of S depends

on the applicability of PCAC in taking T ^ O^ic J ^ V^J .

The relevance of the information for S has recently given origin

to some theoretical activity concerning the validity of such a

smoothness assumption for the TC°—^5.j amplitude. Various

formalisms and models have been used, and the conclusions show a

definite tendency to question the PCAC smoothness hypothesis or,

in other words, any approximate chiral invariance of the world.

In particular Drell has interpreted the well known

successes of PCAC,such as the Goldberger-Treiman relation and the

pion-nucleon scattering lengths, on a dynamical basis: pion pole

dominance holds only for the matrix elements of the divergence

of the axial vector current ow/"V between hadron states, but might

fail when <C2,}f*) ^ v A \ 0 / is considered. This amounts to intro-

ducing a large chiral symmetry breaking by emphasising

the contribution of the (3TC) continuum in the channel of the

axial divergence, so that the value S = l/6 can still be reconciled

with the observed lifetime, provided there are large corrections

from higher pion isobars.

Using a completely different approach based on light-cone

dominated mass dispersion relations, Preparata lias investiga-

ted the relation between the vector dominance description

and that based on the extrapolation of the PCAC anomaly to the

physioal point. Extrapolation effects, in his treatment, lead to

a completely different situation: the contribution of the anomaly

is drastically reduced, while the bulk is due to vector dominance.

One should remark however that the conclusion depends on a specific

choice of the parameters characterizing the symmetry breaking,

which implicitly assumes a large chiral violation.

In view of the importance of the TL "" d decay as possible

detector of fundamental properties, we believe that further analyses

of these points are of interest. In particular we focus on the

- 5 -

problem of the extrapolation of the "current algebra" result Eq.

(i.l) onto the pion mass-shell. To this end we adopt the method12)

proposed some time ago by Fubini and one of us , The idea is

to study an off-shell pion amplitude along a suitable integration

path connecting the soft pion point to the physical one. The method,

which has been successfully applied to several low-energy processes,

such as meson-baryon scattering at threshold, pion electroproduction

and so on, has its own virtues and faults;as we will recall later. i

We only mention the fact that the integration line we use spans

both the singularities in the channel of the axial vector divergence

(i.e. the pion plus any (3"K.) continuum) and in the channel of the

electromagnetic current (i.e. the vector mesons): thus a form of

vector dominance is naturally included in our formalism. The point <

the vector mesons' contribution, is weighted by a factor ^/n^" "° fbO) \*) V

which makes it small (a few percent of the experimental'.mplitude) , differently from the treatment of Ref. 11. It can ;

also be anticipated that the crucial role is played rather by the• i — > -» i

higher equal-time commutator \_ ^ f\ j V J f which depends I

linearly on the size of the chiral symmetry breaking (say the

"proton quark'1 mass); use of recent estimates indicates that [

also this contribution could be relatively small ( '""-' 20 % of the |

observed amplitude). In this spirit, extrapolation effects for {

7T_ —^ -2>X" should not appreciably affect the experimental determi1- I

nation of S (which requires \s \ ^ l / 2 ) and it seems possible

„__ .

* )In the case of — "- o decay the low-energy theorem is still2tdetermined by the value of S, -r^t^^-0) - ~ ~K

while the weight factor becomes rfT1'Tl/tvly ^ */large extrapolation effects are expected in this case.

- 6 -

to have a consistent scheme for controlling chiral violations.

This view is not shared by all theorists, and we mentioned above

a couple of recent contributions along a different line of thought.

The comparison will be made more precise in the text, but in any cad«

it is clear that also in our formalism large corrections can

.easily "be obtained by just increasing the symmetry breaking parameters.

For this reason the aim of the present work should be uriderstood

to be,rather than a definite statement on the size of those

corrections, the derivation of a relation among some interesting

quantities of particle physics, such as the observable ft ° lifetime,

the,Adler anomaly , the vector meson parameters, the "proton-quark"

mass (as a visualization of chiral symmetry breaking) and so on,

This connection has quite a general character, shared by other

representations for low—energy pion amplitudes, and we now proceedt

through the various steps for establishing it.

- 7 -

2. We start accepting the anomalous Ward identity which,

together with the electromagnetic current conservation, is at

the basis of the Adler and Bell and Jackiw theory of 1L° —*»

decay

«0 > (2.2)

where V . A are, respectively, the electromagnetic and the

neutral axial vector currents, x 5 C I f r is the

dual electromagnetic field tensor and T"* denotes the covariant

time-ordered product. As has just been remarked in the previous

section, the Adler anomaly S is outside the framework of conven-

tional current algebra, and therefore deeply tied to hadron dynamics

The subsequent procedure is quite standard. We introduce the

amplitudes, schematically represented in Fig. 1f

(2.3)

where Ib (^ ) S ^ v A (^ ) and \ V* Ct, £}*> is a real

photon s ta te of four-momentum r and polarization £* C^3 = 0

so that Eqs (2.1,2) give

- 8 -

s

= O . (2.6)

The most general structures, which ' autcanatieally take photon gauge

invariance into account, are

( 2: 7 )

^ I are free from kinematical

singularities and depend on the variables Q and K . After

removing the overall four-momentum conservation TC) O CV: + i> - *Vl,

Eqs (2.5,6) are translated into the constraints

(2.9)

We then take the soft pion limit Q —»> O (so that

- 9 -

• —S> O and so on for any other invariant) and get the relations

(2.12)

Eq. (2.11) is nothing but the low-energy theorem for X°—=>

decay.

The next step is to write down a dispersion relation for

the amplitude I ( t J, in order to turn Eq. (2.11) into a

sum rule. As is well known to experts^a possible choice of the

integration path is determined by requiring the soft-pion limit

to be reached first putting <Q ~ O and then taking the

limit Ho ~^ ° *The choice of the rest frame £f n O implies

so that, varying Q^o at fixed U) , ^ and *• vary along

the line •

(2.14)

If in particular we want the line Eq. (2.14) to connect the soft

pion point C = Ic5 ~ 0 to the physical point for 7t°—5» X )C~

decay, C^=Ona^ ; =: O , we have to choose Ui = ™ £ ( J6 »

works equally well), and our extrapolation path becomes

- 10 -

(2*15)

or . ,, (2.16)

Thus wo consider I as evaluated according to the particular

configuration (2.16),i.e.

and assuming an unsubtracted dispersion relation in the variable

V t which we will justify shortly, the low-energy theorem

Eq. (2.11) is easily converted into the sum rule

{ ^J y ^

where tCv) is determined after selection of the appropriate

covariant from

t^ =

OfYl

- 11 -

The above representation takesj into account singularities

both in the 0 Q -channel and in the 1 •<• -channel, thus intro- ;

ducing the pion and the vec':•<"> r meson uoTilr i.imtions all together,

in quite a natural way .

The drawback of this approach is that the lines of singular-

ities Q = M k = H are met twice by the integration path, so that |.

combinations of amplitudes taken at two different points appear, • \

rather than a single one. For instance, the line Q - T C (the \

pion pole) introduces both the contributions at It = O and at '

[c rr ZLO/Yi- , corresponding, respectively, to the direct and {i.

the Z graph} indeed , after explicit selection of the Single pion L

state in "t , one has ;

(2.20)

and Eq. (2.18) becomes

- ( ±L t(vl

where we have defined

(2.22)

so that T \Vc = O J is the physical TT —"^ - -?T* amplitude.

•'? t ,?•

- 12 -

The fact of having both the direct and the crossed mass

singularities in the game is the unavoidable feature of a method

which represents the most direct application of the saturation

approach to (in this case anomalous) equal-^time commutators.

A sensible simplification, as far as this point is concerned,

would be represented by the possibility of exploiting linear

paths. This is clearly related to the choice of a particular

reference frame. For instance the simplest " —^ jo° limit

leads to a fixed Q — O sum rule, which of course does not serve

our purposes. On general grounds one can see that, since the

parabola Eq.{ 2.15)represents the kinematical boundary of the

support region for our amplitudes, any straight path has to lie

externally; if it has to cross the Vc = O axis at = ry^tz it c*n

only be the tangent to the parabola at that particular point,

i.e. W*"s — CS-1"^ rc) a n d it is easy to check that it can be

obtained by considering the limit T' " ^ ^ with C^ * p •¥•

and ^ = 1 . O ^ •T]."*".= r^l^ ... In this case one has to talk of light-

cone behaviour rather than of equal time commutators: the simplicity

of the low-energy theorem in the form (2.1l), however, is lost.

For that reason this approach does not seem to be easily applicable

to the present problem and we shall prefer, therefore, the more

conventional equal-time formulation.

- 13 -

3. Let us come back to the amplitude » defined in

Eq. (2.4)• Its asymptotic behaviour can be established from the

application of the Bjorken-Johnson-Low theorem. In doing this

we assume, following the simplifying suggestion by Adler and13)Boulware , that there are no Q -number seagulls in the

—r- H. *j\definition of \ ' , so that we can write

which implies

(3.3)

Perhaps this point deserves some comment. The behaviour ofTTc\J'ifc*')at the soft-pion point HS'= ^ C , in particular thevalue of S as given by Eq. (1.2), are exactly determined bythe lowest order triangle diagram ( radiative corrections donot contribute at <^ - O } at least in some field theoreticalmodels). This can be phrased in a different way by saying thatthe equal time commutator \.A ) \} \ contains a non-canonicalpart, proportional to S, namely, in the spirit of the Bjorken-Johnson-Low theorem, that the amplitude T^ vhas an anomalousasymptotic behaviour. On the other hand, in order to have theasymptotic behaviour of the same amplitude \ LH > - ) weresort, this time, to a canonical argument, that is'"TCHo") ""^(^V5" <L^NLJ> . The reason forthis is that, as discussed in Ref. 13, no anomalies areexpected (or required) in the present case and there is noargument to appeal to a-specific diagram of perturbation theory.Use of the simple triangle graph would give "T" (Ho) "^ (fWT -^-(^ °A"»The same fact that a logarithmic factor appears is the typicalwarning of the fact that, in perturbation theory, the Bjorken-Johnson-Low limit and canonical equal time commutators quiteoften do not match.

These relations guarantee of course the unsubtractedness of the

dispersion relation for \ lV) and furnish the two further sum

rules

fclvfcCV) - O , (3.4)

(3.5)

The bonus of a typical superconvergence relation is not surprising

and is a consequence of the spin structure of the full amplitude

. We shall use Eq. (3.4) to disentangle T ^ {.\c^ O) from

^ ^ i m * ) (remark that T ^ t^-fc'™*) h a s i t s o w n right

to interest, since in principle it could be measured in a very

low—energy £<£."-*> R Q process). The disentanglement is auto-

matically achieved by combining Eqs. (2.l8) and (3*4) in the forms

(3.6)

TT ~ (3.7)

Finally there is the possibility of one more sum rule, i.e. Eq,

(3.5), provided one is ready to introduce an additional, model

dependent, element in the theory. Actually the appearance of higher

commutators, where chiral symmetry breaking enters explicitly

through the axial current divergence, is peculiar to all approaches

based on commutator-controlled mass extrapolation. Apparently

we do not need this extra piecej howf er, as we shall see shortly ,

it can be useful for an alternative description of some of the

off-shell corrections. Therefore we combine Eqs. (3.4) and i2-S)

- 15 -

in the form

(3.8)

In order to clarify the structure of the relations one can

derive, let us explicitly select, besides the pion, already intro-

duced in the previous seotion, the contribution of the vector

mesons in the k/ -channel, corresponding to \c — \\v

\J - Q (jj <4> . A glance at Eq. (2.19). shows that we need the matrix

elements

(3.10)

so that

C3.ll)

with

and,accordingly t

- 16 -

In order to take into account, in some way, the off-shell character^

of the (V^X^ vertex defined in Eq. (3.10), we adopt the (non~

*)unique) linear parametrization

The unknown quantity Q, will be eliminated, when possible,

in terms of the more meaningful higher commutator \JE> j V J -•

With these simplifying approximations the sum rules Eqs

(3.6 ,7) take the form

and similarly Eq. (3-8) becomes

V

*)Some support can be gained by the following argument.receives contributions from the continuum starting at the \3lC )threshold. A once subtracted dispersion relation in thevariable gives /i

<7**

If the 6" continuum is dominated by high masses, higher thanQ**'^ H\/ > then the integral is weakly dependent on ^ and

one can put

where ui is a n average 0" mass LL*

- 17 -

Remark that +ir and jr- (-2 'Vn ) differ only by terms of order1 as it can be expected from vector meson dominance.- )

In fact the difference of Eqs. (3.15, l6») reproduces the standard*) '

vector dominance result

•i C^AY\ ) - V 'N-' e y "•<*• T V ^> ^ ^ (3.18)

Eq. (3.17), moreover, looks like a somewhat sophisticated form

of vector dominance: the asymptotic behaviour is determined by

the equal-time commutator I.-**, V J t while the vector meson

contribution has the standard form plus an off-shell part, which

reminds Ua of the peculiar character of the approach.

Anyway, there is no point, for Eq. (3-17), in thinking of any

underlying chiral symmetry, all being 0{l) contributions (when„ a. • •

J> is considered to be <^ ^TC" , according to the "strong" version

of PCAC).

On the contrary Eqs. (3»l5>l6) exhibit the general features

of a low-energy theorem characterized by the /y denominators} in

Eqs .(.3.6,7), where the main contribution arises from the current

algebra input (even if anomalous in this case!), while the higher

state contributions, which represent the "corrections", are of

order T-/iJ^' with /-*- some characteristic mass.

*)Clearly our considerations can be generalized to the case ofan off-shell (timelike) photon kL>O ( 7L°^» jf" S + e " ") :as it is easily verified, our formalism would reproduce vectordominance in the form

which is the generalization of Iq. (3.18).

- 18 -

A more useful expression is obtained if

nated from Eqa; (3.15) and (3.17),-and we end up with the following

representation for the physical TC°—> ZL'ft* amplitude (,w© have put

(3.19)

This formula exhibits an explicit form of the finite pion mass

effects which is very similar, from the point of view of the

underlying spirit, to the estimates already performed for other

pion processes, like pion-nucleon scattering and pion'electro-

production. The corrections to the soft-pion result are Bummarized

"by the higher commutator, which embodies some general properties

of the symmetry breaking ID , and by some states, which contribute

to standard dispersion relations, damped however by the factor

Accordingly,no clash seems to arise between

tandard d*" I 1 *)

PCAC and vector dominance in the context of the present approach.

Vector mesons are always there, the only difference with a \<L

dispersion relation being that their contribution has the factor

/ front. As expected, the contribution of the

vector mesons is relatively small. Using the recent numerical

values for the relevant coupling constants (which, for convenience,

are collected in Table l) one finds that this piece

contributes less than 5 % of the experimental value of jr,

"i -X.ranging from 3.72 to 4.56 in units tTo tf fc) AO J according to

the available experimental values for the Tli*r*%£ rate. The bulk of

the corrections is thus represented by the higher (models-dependent)

commutator £_ , Eq. (3-3),

Of course the near validity of PCAC or, in other words, theapproximate chiral invariance of the physical world has beenimplicitly assumed in neglecting the (3TC.) continuum in the0" channel. We shall come back to this point later.

- 19 -

4. If one thought of X> and V as canonical fields, X

would be zero. However, since the franiework which leads to the

estimates O0 S is the quark model, we adopt the explicit expressions

which hold in that particular field theoretical model

(4.D

where p } the "proton quark" mass, is the parameter which

characterizes the chiral symmetry violation.

Making use of the canonical antlcommutation relations for

quark fields.one finds

o^VLCo>] =

where we have introduced the tensor currents

T (4.4)

and

_ Xaj^iT.IlT (45)

We assume,moreover, vector dominance,!.e.,

/ V J \ O > , ( 4 . 6 )

so that the final expression for /_ is

(4.7)

x «5-

- 20 -

where

(4.8)

Some rough estimates for 0? can be worked out just tracing

back to the l i terature concerndd with Z>V{b)yj and PCTC

By applying the treatment of Ref. 15 » the following expression

for 2-j can easily be derived; - ,

(4-9)

where is thetO-f* mixing angle, u ^ = — ( -M> y the vector anomalous

magnetic moment, »> N the nucleon mass; and A^2 d..

Thus apart from the other parameters involved in Eq. (4.9)*

the effect of X depends linearly on TH^ . This quantity has

recently become object of interest, in connection with the determina-

tion of the 'O* -term in pion-nucleon scattering and has received

several (sometimes contrasting) evaluations. Analyses of meson-16)

baryon scattering at threshold , and more recent considerations

based on light-cone physics and deep-inelastic scattering f

seem to suggest small values of r(T) p (typically, ^ 4 0 -r- 140 MeV),

corresponding to small values of the pion-nucleon<3*-term, and

indicating an approximate SUU)xSUtS)chiral symmetry of the hadrons.

Much larger values of 'WU ( Yy)-, 400 MeV), on the contrary,were18)

proposed by Brandt and Preparata , in their picture of'weak"

PCAC. This can makejT vary by a factor 10. On the other hand,

since the lower estimates ofrfl^are derived^ essentially, by apply—

- 21 -

ing the present extrapolation approach and making similar approxi**

mations, it should be more consistent, in our opinion, to adopt

(3*19))them here. This leads to a 2, contribution to jr

of the order of 2 5 % of the observed amplitude or less *)as

i l lustrated in Table I where the quantitative comparison of

the various contributions is also carried out.

An alternative determination of XT , which does not necessitate

the introduction °f the tensor currents, can be worked but

from the direct saturation of the commutator matrix element

J * ^V required in Eq. (.4*6). This approach, quite

that described in the previous, sections, wouldanalogous to

lead to

., 4 ft

* • } « • ]

A* I

where

( 4 - u )

* ) 19)A 9)In a recent paper a new version of PCTC has been pro-posed, according to which the 21 contribution might becomeappreciably larger.

•»•#)

In the limit of exact vector dominance, we have

^VTCX =" 2IV» v/* V V V / H ^ ! and> consequently,^,Indeed, as remarked at the beginning of this__ section.would be exactly zero in a model in which .2) and ywere associated, respectively, to the pion and the vectormeson fields.

- 22 -

It is easy to verify that Eq. (3.17), with the determination

(4.10) for^*" , takes on a form quite reminiscent of the Gell-

Mann-Sharp-Wagner model for ~K —*.3-jf" decay, based on vector

dominance. On the other hand, as far as the numerical evaluation

of the sum rule Eq. (3.19) is concerned, the use of Eq. (4.10),

combined with the available experimental information on Q ^

and 9 ,_ , would result in an overall effect ( 2- pl u s vector

meson contribution) ev«n more reduced compared vith the previous

determination.

Thus, in our conventional PCAC framework,the final corrections

turn out to be small, and in any case their size cannot modify

the arguments for a choice between different quark models, based

on an experimental determination of the Adler anomaly S.

Since the applicability of PCAC, in particular to the

%°-3>2,K" decay, is being considered with some suspicion, we devote

the next section to a few remarks on this point.

- 23 -

5. Drell has recently reconsidered the 'L ~*£\<decay-

in the framework of PCAC. With the aim of saving the fractional

quark model, he has shown that it is possible to invent some

dynamical mechanism which, while preserving the successful.'pre-

dictions of soft-pion theory, produces such large extrapolation

effects to give agreement with experiment by choosing S = l/6.

A simple schematization of the model consists in introducing a

pion isobar 7C* of high mass KX. , which simulates the (3 T )*)

continuum. This modifies the FCAC form of the vertex

(s.oIn order to reduce the number of unknowns, we choose a combination

9

i th ^of sum rules such that the contribution of the TC in the

channel is automatically eliminated, i.e. we consider

The pion Z graph can be further eliminated through vector meson

dominance, which has been shown to be consistent with our approach,

Eq. (3.18). The evaluation of the sum rule (5«2) is straightforward.

We include the vector meson contribution using the modified PCAC

f rro (5.1) for the vertices <C f" \ ID \ V^> • N o off-shell variation

- ;' the ^3 vA. coupling constant is at this stage allowed, since

the additional TL piece should account for that « The result is

*) It is perhaps worhtwhile to recall that Xj^/ has to be consideredas an " OC'Ti *') or an 0(l) according, respectively, to the"conventional" and to the "weak" views of PCAC.Here, however, weare not concerned with this point; rather, since the parametriczation Eq. (5.1) is applicable to both cases, we simply adoptit as the starting point for a quantitative discussion.

Eq. (5*1) amounts, in the spirit of the linear para-metrization, to taking•3^ « - ftn'/^xKavir'^/^n) r i.e. a slopemuch larger (of a factor >*•'*•/""»* ) than ) that assumed inEq. (3.14). This can be the source of large corrections as inDrell's case.

rtYV

+ — A +/,-*/ ^

" i-s/ O (^ _, j W \ (5.3)

In order to have an estimate of this new version of the corrections,

we begin by noticing that for values of pf in the range

21,5 ^ 2 (GeV)2the JE contribution is reduced by a factor v/Ai1Nl' /3

which means that even if a large chiral breaking is allowed,

nnf)? 400 MeV, its overall effect remains at most a 20 % -

30 % of "Wcexp. As far as the contribution a la Drell is concerned,

, can be eliminated through the experimental error in the

Goldberger-Treiman relation, i.e.

A I M'lNIl) (5-4)

If we assume, as a first reasonable indication, that Z. " '^f. Q "* 7,

the whole vector piece becomes of course larger than in the estimate

of the previous section, reaching 15 % ""~ 20 % of the experi-

mental amplitude.

However, if one is ready to accept higher values for ^ -jrlv t

everything is possible: not only the value S = l/6 can be made

to agree with experiment, the required additional contribution

being provided by the corrections, but, in principle, even the

Adler anomaly becomes unnecessary (a ratio of K ^-° ' coupling

constants of the order of 10 does the job of building the complete

amplitude ). These are the, more or less similar, conclusions

reached in Re£s, 10 and 11, starting from somewhat different

points of view. Although nothing prevents, in principle, such a

phenomenon, we are somewhat reluctant to accept it. Clearly a matter

of taste is involved here.

On the other hand,we have discussed in the previous sections

an, admittedly conservative, estimate of the corrections. I t

shows no clash with vector dominance and,although the total effect

is not completely negligible ( ^ 20 % of the experimental

amplitude), the argument for the choice among different models

of underlying constituents is not essentially modified. Anyway>

we believe that, apart from quantitative conclusions which,

at least at present, rely on uncertain parameters, the above

considerations should have pointed out which can be the important

parameters one has to include in a realistic determination of

the Adler anomaly from the experimental TC** lifetime, discussing

in particular the role of vector dominance*

ACKNOWLEDGMENTS

We are grateful to G. Preparata for interesting discussions at the

early stages of thiB work. Thanks are also due to Profs. AMus Salam and

P. Budini as veil as the International Atomic Energy Agency and UMESCO

for putting at our disposal the facilities of the International Centre for

Theoretical Physics, Trieste.

During the past years one of us (GF) has had many fruitful

discussions on current algebra problems vith his unforgettable friend

Bruno Renner. This vork 1B dedicated to his memory, -

REFERENCES

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G. Coame e t a l . / P h y s . Letters4_0B, 685 (1972) .

TABLE I

(Fjrj exp.

3-72 [Ref.3]

4-56 [Eef.2]

Current Algebra

3.85

(S = 1/2)

1.28

(S = 1/6)

2 contribution

o.soi^

0.10

0 . 2 7 ^ -

Vector Mesons

0.15

0.15

—

Drell

r

'

0.50

Refer to Eqs.

( 3 . 1 9 ) , ( 4 . 9 )

( 3 . 1 9 ) , ( 4 . 1 0 )

( 4 . 9 ) , ( 5 . 3 , 4 )

TABLE I : Summary of numerical est imates.

4-T£ <X <J-nTC} "10 un i t s are understood. The

following numerical values have been used 2 1 ' : 2-^ =. 0.&8 mTC '

!%= C.077;

F i g . 1 The a m p l i t u d e s E q s . ( 2 . 3 , 4 ) *

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