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Unitarity and vector meson contributions to K+ → π+γγ
.-__ EB
s
3 October 1996
PHYSICS
LETTERS 6
Physics Letters B 386 (1996) 403-412
ELSEYIER
Unitarity and vector meson contributions to K+ + 7;rfyy* G. D' Ambrosio, J. PortolCs lstituto Nazionale di Fisica Nucleare, Sezione di Napoli, Dipartamento di Scienze Fisiche. Universitci di Napoli, I-8012.5 Napoli, Italy
Received 5 June 1996 Editor: R. Gatto
Abstract We compute the one-loop unitarity corrections c3(p”) from K+ + T’T+Tto K’ + v~+yy and we find that they are relevant, increasing the leading order prediction for the width in a 30-40%. The contributions of local (3(p”) amplitudes, generated by vector meson exchange, are discussed in several models and we conclude that the vector resonance contribution should be negligible compared to the unitarity corrections.
1. Introduction
The phenomenology of radiative non-leptonic kaon decays provides crucial tests for the ability of Chiral Perturbation Theory ( xPT) [ 1 ] to explain weak low-energy processes and interesting possibilities to study CP violation in these channels. xPT is a natural framework that embodies together an effective theory (satisfying the basic chiral symmetry of QCD) and a perturbative Feynman-Dyson expansion. Its success in the study of radiative non-leptonic kaon decays has been remarkable (see [2,3] and references therein). Of course there are still open problems, but upcoming experiments should improve our phenomenological knowledge of these decays, in particular on K ---f~yy, K + d+t-, KL 4 l+tc- or KL + ye+e-. KL + r”yy is a very interesting channel by itself as a xPT test and also in order to establish the relative role of the CP conserving contribution to KL + r’e+eversus the CP violating contributions [ 2-81. KL + n-‘-yy has no O(p2) tree level contribution since the external particles are neutral. For this same reason there are no 0(p4) counterterms and the chiral meson loops are finite at this order [ 81. Consequently ,yPT gives an unambiguous prediction. Furthermore only an helicity suppressed amplitude for KL --+ r’e+eis generated, which is small compared to the CP violating ones. At higher orders in xPT a new invariant amplitude, generating an unsuppressed helicity amplitude to KL -+ n-‘e+e-, appears. As we will see in detail later there is a specific kinematical region in the diphoton invariant mass spectra of the K -+ ryy processes where only this last amplitude contributes. This is the region where both photons are nearly collinear, i.e. at small diphoton invariant mass. The experimental results showed [8] that while the S(p4) predicted spectrum *Work supported in part by HCM, EEC-Contract No. CHRX-CT920026 (EURODAQNE) 0370-2693/%/$12.00
Copyright 0 1996 Published by Elsevier Science B.V. All rights reserved.
PII SO370-2693(96)00970-7
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J. Portolb/Physics
Letters B 386 (1996) 403-412
looked in good agreement with the experiment, finding no contribution at small diphoton invariant mass, the predicted branching ratio was underestimated by a factor slightly bigger than two, two sigmas away from the experimental result. This fact prompted to several authors to consider higher chiral order corrections: i) Vector Meson Exchange contributions to the local 0(p6) amplitude [ 7,8], ii) Unitarity corrections from KL -+ z-O~+~T- [ 8,9] and iii) Complete unitarization of the 7rr intermediate states through a Khuri-Treiman treatment and inclusion of the experimental yy + 1~~1~~amplitude [IO]. It has to be emphasized that these higher order corrections do not represent a complete chiral order contribution. The size of the first correction is controversial [ 7,8], however a large contribution by itself is excluded by the experimental spectrum. The second contribution shows that unitarity corrections of KL + T’T+Tto KL ---f r”yy increase 20-30% the leading order amplitude and then have to be taken into account [ 8,9]. When both contributions are added it was possible to fit both the width and the spectrum [9]. The third correction increases the branching ratio by an additional 10 - 20%. The charged channel K+ -+ r+yy is experimentally less known (only an upper limit on the branching ratio exists [ II] ), but measurements with good precision are foreseen in the near future [ 3,121. The leading oneloop O(p4) result has been computed [5] and depends upon unknown weak local amplitudes which however could give important informations on vector meson weak interactions. Indeed while at c3(p4) it is well known that the finite part of the counterterms in the strong xPT lagrangian is saturated by the spectrum of lightest resonances (vector, axial-vector, scalar and pseudoscalar) [ 131, the situation in the 0(p4) weak lagrangian is less clear due to our ignorance on the weak couplings of vector mesons, however there are predictive models to test [ 141. We have evaluated the Kf + r+rr+nunitarity contribution to K+ + r+yy with the same procedure [S]. The dispersive contribution is used to compute the KL 4 T~~-+T~ unitarity corrections to KL --+ r”yy unambiguously computed up to a polynomial piece which is reabsorbed (together with the loop divergences) in the counterterms. Moreover various vector dominance exchange models are studied to saturate the bulk of the counterterms as in KL + ?r”yy . Then we can give a definite prediction for the kinematical region of nearly collinear photons where the unknown contribution of the 0(p”) local amplitude is negligible. The experimental in the DAQNE Q-factory [ 31 and other experiments [ 121 is going to improve the study of K+ + r+yy phenomenology of K + ryy and will allow us to analyze both channels in a correlated way.
2.
K -+ nyy amplitudes The general amplitude M(K(p)
+ n-(m)
for K ---f n-yy is given by y(ql,E~)
y(qz,c2))
and M p” has four invariant
where E 1,Q are the photon polarizations, MC””
=
A(z,Y)
~
41; - 41 .92f
(4;
) +
~B(z,Y) ,4
&
C(Z,Y> +
2
D(ZTY) ~p"P"qlpq2u + ~
mK
(1)
= ~1~~2Jf~~hw72)~
[e/wpv
C-P.91
amplitudes - 91 .q2pppy
P.42F
+P.ql
&P”
+P.92PpLql;)
K
(P .q2qlp
fp.9
42,)Po
+ (Pp&“apY
+P”&pnpy)
?wlp92,17
4 (2)
where v= P .
(41 -
m’K The physical
q2)
,
z=
(41 + q2J2
rn2,
region in the adimensional
variables y and z is given by
(3)
G. D’Ambrosio, J. Porto&/ 0 5
IYJ 5
Physics Letters B 386 (19961403-412
0 5 z L (1 -rd2,
$‘%J$Z),
405
(4)
with =a*+b*+c*-2(ab+ac+bc),
A(a,b,c)
rT=z.
(5)
Note that the invariant amplitudes A( z, y), B( Z, y) and C (z, y) have to be symmetric under the interchange of q1 and q2 as required by Bose symmetry, while D(z,y) is antisymmetric. In the limit where CP is conserved the amplitudes A and B contribute only to KL -+ ~‘yy , while C and D only contribute to KS + r”yy. In K+ + rr+yy all of them are involved. Using the definitions (2)) (3) the double differential rate for unpolarized photons is given by
(6) The processes K 3 ~yy have no tree level O(p*) contribution. At 0(p4) the amplitudes B( Z, y) and D( z, y) are still zero since there are not enough powers of momenta to generate the gauge structure, and therefore their leading contribution is 0(p6). As can be seen from (6) only the B and D terms contribute for small z (the invariant amplitudes are regular in the small y, z region). The antisymmetric character of the D( z, y) amplitude under the interchange of q1 and q2 means effectively that while its leading contribution is O(p6) this only can come from a finite loop calculation because the leading counterterms for the D amplitude are O($). However also this loop contribution is helicity suppressed compared to the B term. As shown in a similar situation in the electric direct emission of KL -+ T+T-~ [ 151, this antisymmetric 0(p6) loop contribution might be smaller than the local O(p*) contribution.
3.
KC --*dyy
at O(p4)
The leading AI = l/2 0(p4) in [5]. We review them here. The A amplitude reads A’4’(z)
=
G&&_, 2TZ
A(z,y)
and C(z,y)
amplitudes
(Z+l--Ij)F(f)+(z+r:,-l)F(Z)-Zz 7r
Here Gs is the effective weak coupling
constant
determined
for K+ -+ n-+yy have already been computed
I
function
(8)
is defined as
A 1 - z arcsin*( 2) F(x)
(7)
from K + TT decays at 0(p2)
]Gs] N 9.2 x 10-6GeV-2 and the F(x)
.
,
x14,
=
(9) In (7) the pion loop contribution F( z/r;) dominates by far over the kaon loop amplitude with F( z ) . The loop results are finite. However as we have already commented xPT allows an 0(p4) scale independent local contribution that in (7) is parameterized by
406
G. D’Ambrosio.
J. Portolk
/ Physics Letters B 386 (1996) 403-412
. c=~[3(L9+~,0)+h’,~-N,S_2N,s], that is a quantity 0( 1). The L9 and Lto are the local L?(p”) strong couplings and Nt4, Nt5 and Nts are 0(p4) weak couplings, still not fixed by the phenomenology, and that can be only computed in a model dependent way [ 141. The Weak Deformation Model (WDM) [ 81 predicts E = 0, while naive factorization in the Factorization Model (FM) [ 15,141 gives P = -2.3. In these models, due to the cancellation in the vector meson contribution in ( lo), the role of axial mesons could be relevant [ 141. The O(p4) contribution to the C( z, y) amplitude is
(11)
-+ yy). This amplitude is generated by the Wess-Zumino-Witten where r,, = m7/mK and Td = r(r” functional [ 161 (TO, q) --f yy through the sequence K+ --f T+(T’, v) ---f ~+yy. This contribution amounts roughly to less than 10% in the total width.
4. 0(p6)
local amplitudes
At 0(p6)
for K+ +
dyy
there are only four independent
F P FpA d”K+a*Y
.
Lorentz invariant
F P FpL”c?,,K+d%- ,
WI:,Fp,Fpv
local amplitudes K+v-
,
contributing
to K+ + rr+yy
PF P daFpYKf~-,
:
(12)
where the last one has no analogous in KL + ?r”yy. In general the couplings of these operators are not directly related with the ones in KL --f ~‘yy due to the fact that the electric charge matrix does not conmute with the generators of the electrically charged field. As we shall see, in specific models, relations between the two channels can be found. All of these operators contribute to A( z, y) but only the first one in ( 12) gives a B( z, y) amplitude. Loop divergences at O(p6) are absorbed in the counterterm coefficients, that thus renormalized are finite. Chiral dimensional analysis [ I] tells us that their contributions are suppressed compared to 0(p4) by a N 0.2. Nevertheless vector meson exchange was found to enhance this up to mi/rn$ w 0.4 factor mi/(4rFT)2 [ 131. Thus we try to estimate the contributions of the lightest resonances, i.e. vector mesons, and we assume that heavier resonances and non-resonant contributions give smaller corrections. This picture seems well verified in the strong coupling constants at 0(p4) [ 131, and it is likely to apply to the weak couplings. We are going to consider here the contribution of vector meson resonances to the weak 0(p6) counterterms for Kt + ~+yy (a more complete discussion and an extension to KL + z-‘yy is given in [ 161) and we find that only the terms in (12) with derivatives on the meson fields can be generated by vector meson exchange. in [8] we assume that the contribution to the local amplitudes for Following the study of KL --j ~‘yy is dominated by vector meson resonances and define their contribution with an adimensional Kf --) ~+yy parameter av generated by the first term in ( 12) as av = -
IT
2Ggmitxem
lim Bv(z).
: -0
where Bv(i) is the vector resonance contribution to the B amplitude. The A amplitude generated by exchange (Av) gets contributions from the first and second structure in (12). However as seen in we assume that these local amplitudes are generated through strong resonance exchange supplemented weak transition in the external legs, those two contributions are related and can be written in terms of parameter defined in ( 13) as
(13)
vector [8] if with a the av
G. D’Ambrosio,J. Portol&/ PhysicsLeners B 386 (19961403-412
Av
= G&pem av 7
(3 + r”, -
z>,
407
(14)
In [ 81 two different vector contributions to av in ( 13) have been proposed: the first one amounts for the weak counterterms generated by a strong vector resonance exchange with a weak transition in an external leg (ae;[‘), the second is generated by vector resonance exchange between a direct weak vector-pseudoscalar-photon (Vpy) vertex and a strong one (UT). While the first can be computed in a model independent way, the generation of direct Vpy weak vertices is still poorly known and therefore only models can be used. The external weak transition for K+ -+ n-+yy gives en = _ 128~~ h2V m2K 9m2,
=
%
-0.08,
(15)
where mv = mP and the strong VPy coupling
Jhv\ = (3.7 i 0.3) x 10v2 is defined by
(16) with UP = iutD,lJut U=exP
(il;
,
j’y $
= uFrut
Aj4.j)
+ u+Ff”u,
U=l.lll,
D,U = a,U
- ir,U + ilJlJ,, (17)
3
and u( 4) is an element of the coset space SU( 3) L @ SU( 3) R / SU( 3) v parameterized in terms of the Goldstone fields #i i = 1, . ..8. FL,: are the strength field tensors associated to the external rp and lp fields, VP is the nonet of vector meson resonances, F, 21 93 MeV is the decay constant of pion and, finally, (A) z Tr( A) In our case with two photon external fields !, = rfi = eQA, ’ . The model dependent contribution to av from direct weak vertices in the WDM and FM quoted above is atrwDM = -av ext ,
4
FM = -2kF a”v”’,
where in the FM kF is the unknown fudge factor that is not fixed by the model and satisfies 0 < kF 2 1. Naive factorization predicts kF = 1 . By adding (15) and (18) we see that av=ae;[‘+&? is flV,WDM
=O,
av,FM=(1-2k~)ae,x’.
(19)
We note that for the allowed range of values of kF in the FM both models agree in predicting a very small vector and therefore the important conclusion of our exercise is that meson contribution to O(p6) in K+ -+ dyy the vector contribution to the local B( Z, y) amplitude can be neglected. If, as it is reasonable to assume, other resonance interchange corrections are even smaller, we can conclude that the small z region of Kt + n+-yy is completely predictable at c3(p6) through chiral loops and it is likely to be dominated by the unitarity corrections to K+ -+ z-+yy . Indeed in that region the 0(p6> is dominant and our ignorance on the of K+ ---t n-+&s-? amplitude is irrelevant.
’ For a full discussion of the definitions and notations see
[ 13,8]
G. D’Ambrosio, J. Portolks/Physics
408
5. Unitarity
corrections
The amplitude variables
K+ + m+n+n- to K+ + n+yy
for the process K(p)
$2 - Sl -, m:
x=
of
y=-
Letters B 386 f1996) 403-412
+ ?r(pr )?r(p2)r(p3)
can be expanded
in powers of the Dalitz plot
s3 - so rni
(20)
’
where Si = (p - pi)2 for i = 1,2,3, so = (sr + sz + ss) /3 and the subscript 3 indicates the odd pion. For the decay K+(p) --f z-+(PI)~+(P~)~-(P~) the isospin decomposition, neglecting the small phase shifts and up to quadratic terms, is written as [ 17,171 A(K+
-+ T+TT+T-
I= 2cr1 - (~3 + (PI - $3
- 2(51 + 53)(Y2 + ix”,
- (51 +53
-
+ ~6~3) Y
5;>v2
-
ix’,
where the subscripts 1 and 3 refer to Al = l/2, 3/2 transitions been fitted to the data [ 171. The 0(p2) amplitude Ac2’(Ki.
+ T+T+T-)
= Gsm;
(f - r;Y)
>
respectively,
(21)
and the coefficients
,
in (21) have
(22)
with the value of Gs in (8) underestimates by 20-30s the experimental linear slopes for K+ + V+TT+T-. At the next chiral order, the experimental linear and quadratic slopes in (21) are recovered (with predictive power too) [ 171. Since the 0(p4) Joop contribution (7) to K+ + n-‘yy , is generated by (22)) it seems natural to try to include all the contributions to K+ + n-+yy generated by the experimental slopes in (21)) i.e. we induced by the 0(p4) corrections to K+ + TT~T+T~. This evaluate the 0(p6) contributions to K+ + r+yy can be done similarly to the case of KL + n-‘yy in [ 8,9] : (21) is considered as an effective K+ -+ rTT+v+rTTchiral vertex, the kinematical variables are replaced by the appropriate covariant derivatives, the QED scalar vertices are added through minimal coupling and then the usual Feynman diagrams approach can be used. There are 13 Feynman diagrams that have two different topologies. There is a subset of 9 diagrams where one or both photons are radiated by the external legs and therefore do not give and absorptive contribution. The sum of these bremsstrahlung-like diagrams is not gauge invariant and cancels with an analogous term from the 4 remaining diagrams (two of which have also an absorptive part) and generate the A( z, y) and B( z, y) amplitudes. This loop result for the A (z, y) and B (z, y) amplitudes is divergent and needs to be regularized and renormalized. We have used dimensional regularization and divergences are absorbed by the four counterterms discussed earlier. Alternatively one could use a subtracted dispersion relation. While the absorptive contribution is uniquely determined, the dispersive one, due to the presence of subtraction constants, can be computed only up to a polynomial. This is precisely the same ambiguity that happens in the effective vertex method that we have used where the finite part of the counterterms is a priori unknown. We assume that the dominant contribution should come from the non-polynomial amplitude generated by the cut of the two-pion intermediate state. Otherwise contributions generated by a vanishing on-shell K -+ 3~ amplitude can be reabsorbed in the unknown polynomial amplitude. in the MS subtraction scheme are The final result for the A and B amplitudes for K’ -+ dyy A(z,y)
= 2
.
2(2crl -
a3)
+
I+$-+
(PI
-$3+hY3)
{[ -
32jl 77
(In($) -51) [r-f,
-1)
+8(1;6(rj-zI+9(r:-z~2jiF(i)]
I?(;>
G. D’Ambrosio, J. Portolk/Physics
+c1
-r2,+z) 1
-’
(
1, 24
z 72r* ,+$(1+$)
r2 --.z+ 24~
( 36
- C+2r$ql
[
= F
+f
($
where the F(x)
{
$(411+&)
- 10) &)] function
(tR(g+3F(;)))
i+l-$)R(;)+$(l-$)F($)] ( r;
+Gsm2, (z + ri - l)iF(z)
B(z,y)
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Letters B 386 (1996) 403-412
[-A
,
+2q2
(23)
(1+21n($))++
+Gm:r)i}
9
(24)
has been defined in (9) and
x14, R(x)
=
(25) x>4,
’
with /3(x) defined in (9) (F(z) N -AZ and R(z) Y &z for z ---) 0). we have spoken before. In (23), (24) the qi, i = 1,2,3 stand for the unknown polynomial contribution Though, in principle, four different subtraction constants could appear, these are the only ones that are necessary in order to absorb the loop divergences generated by this unitarity correction. We emphasize that these local amplitudes are not generated by the vector mesons (already taken into account by a~) and therefore are expected to be suppressed by &/A: with A, N 4rF,, over the previous chiral order. If as a naive chiral dimensional analysis we choose as coefficients ifor definiteness) we get
N 2.06,
77*= ; %i N 0.24, X
of the suppression
3 m* 73
=
-5g
3
N
factor the factor accompanying
-0.26.
ln(mz/p2)
(26)
X
We note the big numerical value for ~1, but looking into the A amplitude (23) we see that r)i is suppressed by mz/m$. The order of magnitude we get indeed is the one expected by chiral counting. In (23), (24) we only have included the AZ = 312 coefficients of the O(p*> contribution of K+ + yrfn+rTbecause the bigger errors in the determination of these coefficients in the 0(p4) amplitudes. We notice that at this order while the B amplitude is y-independent the A amplitude gets its first y dependence. As we have seen the vector meson contribution to the B amplitude is likely to be very small. If we use the same definition (13) for our B amplitude (24) we get
410
G. D’Ambrosio, J. Portol&/Physics
Fig. 1. Br(K+
- r+yy ) as a function of the 2 amplitude. The dashed line corresponds to 0(p4) ,@T amplitude. The full line corresponds to the amplitude including the O(ph) unitarity corrections
Letters B 386 (1996) 403-412
Fig. 2. Comparison of the normalized z-distribution for K+ -+ ~+yy at 0(p4) (dashed line) with our 0(p6) correction (full line) for C = 0.
a[ =
(27)
The first term amounts to cv 0.52 if p = mp is taken. Then from (26) we can conclude that due to the expected vanishing of the vector meson contributions, the polynomial non-resonant amplitude, though small compared with the unitarity corrections, could be in principle tested in this channel at low 2. We note that this situation is different to the KL ---in-Oyy case where the vector meson contribution seems to be more relevant. We have not included the A1 = 3/2 couplings in the 0(p6) contribution because of the big errors in the experimental fit. But we should stress that inputting those amplitudes could modify the low-z behaviour of the B amplitude even a 30%. The argument about the relevance of the correction to KL 4 ?r”yy coming from the inclusion of the experimental yy + 7r07ro amplitude [ IO] is weakened here due to the relative suppression of K+ --f n-i~Oflo compared to Ki --+ TT~TT~TF.
6. K+ -+ rr+yy
: branching
ratio and spectra
We can now proceed to study the numerical results for K+ 3 v+yy taking into account the 0(p4) and our 0(p6) loop evaluation of the unitarity corrections. As we have shown the 0(p6) vector resonance dominated local contributions are negligible. In our numerical discussion we will input q = 0, i = 1,2,3 and ,u = mp. At present there is only an upper bound for Br( K+ h n-+yy) which depends on the shape of the spectrum
[I81 Br( K+ i
dyy
> jexp 5 1.5 x 10e4
Br( K+ d n-+yy ) jexp < 1.O x 10e6
(xa
amplitude)
(constant
,
amplitude)
.
(28)
The uncertainty in the theoretical prediction is dominated by the unknown 0(p4) counterterm generated amplitude C in (7). In Fig. 1 we show Br( K+ + rr+yy) as a function of E with and without the 0(p6) corrections that we have computed. We remind that WDM predicts C!= 0 while naive FM gives E = -2.3 with the following results: Br(K+
+ ~+yy ) /WDM = 7.24 x lo-‘,
Br(K+ --f n-+yy ) lnFM = 6.20 x 10-7.
(29)
G. D’Ambrosio, J. Portok/
/ .__
0 0
05
1
_*_
15
I
_~ _..~ 8
m,
411
Physics Letters B 386 (1996) 403-412
/ ,’
25
3
35 (GN)
-+
Fig. 3. Normalized di-photon mass spectrum of K+ -t ~+yy for ? = -2.3 (naive FM, dashed line) and ? = 0 (WDM, full line).
IYI
Fig. 4.
d Br( K+ + P+YY ) r?v
spectrum for ? = 0 as a function of
Ivl.
When comparing with the O(p4) predictions [5] we find that the unitarity corrections increase around 30-40% the branching ratio. In Fig. 2 we show the z-distribution at S(p4> and with our 0(p6> unitarity correction for ? = 0. As can be seen the correction is noticeable. The uncertainty on the 2 amplitude also translates into the spectra. In Fig. 3 we show the spectrum of the invariant mass of the two photons for the two values of i; predicted by the WDM and naive FM. We notice that the main dependence on E arises in the z-region where there is the bulk of the absorptive contribution. Finally in Fig. 4 we show the spectrum in the “asymmetric” _vvariable for f = 0. At this order this spectrum has much less structure than the one in the di-photon invariant mass.
7. Conclusion and the expected experimental measurement of The situation of the chiral prediction for KL ---f r”yy l( K’ -+ r+yy) at DAQNE and BNL make interesting to enlarge our theoretical knowledge about this last process, We have reviewed and studied the vector resonance dominated S(p6) local amplitudes which main role is to determine the low-z region of the invariant di-photon mass spectrum. The conclusion is that this contribution is likely to be negligible. The O(p6) unitarity corrections from K+ --f QT+TT+TTto K’ + n-+yy have been computed and found to be relevant increasing the branching ratio (for a fixed value of the E amplitude) around 30-40%. The shape of the z-distribution is shown to be sensitive to the evaluated corrections (Fig. 2 and Fig. 3). as a function of 2. The included corrections In Fig. 1 we have shown the branching ratio for Kf + byy will allow to get a more accurate determination of 2 once the branching ratio of Kf + n+yy is measured and therefore will provide a new independent relation between 0(p4) weak coupling constants that will improve our predictive power in this sector.
Acknowledgements The authors thank G. Ecker for interesting correspondence about the subject and the Particle Theory Group at MIT where part of this work was done. J.P also thanks the hospitality of the Particle Theory Group at Rutherford-Appleton Laboratory where this work was started. J.P. is supported by an INFN Postdoctoral fellowship and also partially supported by DGICYT under grant PB94-0080.
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Letters B 386 (1996) 403-412
References ] I ] S. Weinberg, Physica, A 96 ( 1979) 327: J. Gasser and H. Leutwyler, Ann. of Phys. 1.58 (1984) 142; A.V. Manohar and H. Georgi, Nucl. Phys. B 234 ( 1984) 189: J. Gasser and H. Leutwyler, Nucl. Phys. B 2.50 (1985) 465. 121 J.L. Ritchie and SC. Wojcicki, Rev. Mod. Phys., 65 ( 1993) 1149; L. Littenberg and G. Valencia, Ann. Rev. Nucl. Part. Sci. 43 (1993) 729; A.J. Buras and M.K. Harlander, in: Heavy Flavours, eds. A.J. Buras and M. Lindner, Advanced Physics (World Scientific, Singapore 1992). Vol. IO. p. 58; R. Battiston, D. Cocolicchio, G.L. Fogli and N. Paver, Phys. Rep. 214 ( 1992) 293; B. Winstein and L. Wolfenstein, Rev. Mod. Phys., 65 ( 1993) 1113. [3
I G. D’Ambrosio,
G. Ecker, G. Isidori and H. Neufeld, Radiative non-leptonic kaon decays, ed. L. Maiani, G. Pancheri and N. Paver, LNF (1995). p. 265. [4] J.F. Donoghue, B.R. Holstein and G. Valencia, Phys. Rev.. D 35 ( 1987) 2769. [S] G. Ecker, A. Pith and E. de Rafael, Nucl. Phys., B 303 (1988) 665. 161 J. Flynn and L. Randall, Nucl. Phys., B 326 ( 1989) 31: C. Dib. I. Dunietz and EJ. Gilman, Phys. Len. B 218 ( 1989) 487; T. Morozumi, H. Iwasaki, Progr. Theor. Phys. 82 ( 1989) 371. 17 1 L.M. Sehgal, Phys. Rev. D 38 (1988) 808; P. Heiliger and L.M. Sehgal, Phys. Rev. D 47 ( 1993) 4920. [ 8 ] G. Ecker, A. Pith and E. de Rafael, Phys. Lett. B 237 ( 1990) 48 I. [9] G. Ecker, A. Pith and E. de Rafael, Phys. Ixtt. B 189 (1987) 363; L. Cappiello and G. D’Ambrosio, Nuovo Cimento A 99 ( 1988) 15.5. ] IO ] E731 Collab., V. Papadimitriou et al., Phys. Rev. D 44 ( 1991) 573; NA31 Collab., G. Barr et al., Phys. Lett. B 284 (1992)
Series on Directions
in: Second DAQNE
in High Energy
Physics
Handbook,
440.
( I I ] L. Cappiello. G. D’Ambrosio and M. Miragliuolo, Phys. Len. B 298 ( 1993) 423. [ 121A.G. Cohen, G. Ecker, A. Pith, Phys. Lett. B 304 (1993) 347. [ 13 1J. Kambor, B.R. Holstein, Phys. Rev. D 49 ( 1994) 2346. 1141 Review of Particle Properties, L. Montanet et al., Phys. Rev. D SO ( 1994) 1173. [ 15) F?Franzini, Preprint LNF-95/054, to be Published in the Proceedings of XVII International Symposium on Lepton-Photon Beijing (China), IO-15 June, (1995). [ 16 ] G. Ecker, J. Gasser, A. Pith and E. de Rafael, Nucl. Phys. B 321 ( 1989) 311. 1171 G. Ecker, J. Kambor and D. Wyler, Nucl. Phys. B 394 (1993) 101; See also G. Isidori and A. Pugliese. Nucl. Phys. B 385 (1992) 437; C. Bruno and J. Prades, 2. Phys. C 57 (1993) S85. [ I8 ] G. D’Ambrosio and G. Isidori, 2. Phys. C 65 ( 1995) 649. 1191 A. Pith and E. de Rafael, Nucl. Phys. B 358 (1991) 31 I. ] 201 J. Wess and B. Zumino, Phys. Len. B 37 ( 1971) 95; E. Witten, Nucl. Phys. B 223 (1983) 422. I21 ] G. D’Ambrosio and J. Portoles, Preprint INFNNA-lV-96/21 ( 1996). 1221 C. Zemach, Phys. Rev. 133 (1964) B1201; T.J. Devlin and J.O. Dickey. Rev. Mod. Phys. 51 ( 1979) 237. I23 1 J. Kambor, J. Missimer and D. Wyler. Phys. Lett. B 261 ( 1991) 496; L. Maiani and N. Paver, CP conserving nonleptonic K + 37r decays, in: Second DAQNE Pancheri and N. Paver. LNF (1995). p. 239. 1241 MS. Atiya et al.. Phys. Rev. Len. 65 (1990) 1188.
Physics Handbook,
Interactions,
eds. L. Maiani,
G.
s
3 October 1996
PHYSICS
LETTERS 6
Physics Letters B 386 (1996) 403-412
ELSEYIER
Unitarity and vector meson contributions to K+ + 7;rfyy* G. D' Ambrosio, J. PortolCs lstituto Nazionale di Fisica Nucleare, Sezione di Napoli, Dipartamento di Scienze Fisiche. Universitci di Napoli, I-8012.5 Napoli, Italy
Received 5 June 1996 Editor: R. Gatto
Abstract We compute the one-loop unitarity corrections c3(p”) from K+ + T’T+Tto K’ + v~+yy and we find that they are relevant, increasing the leading order prediction for the width in a 30-40%. The contributions of local (3(p”) amplitudes, generated by vector meson exchange, are discussed in several models and we conclude that the vector resonance contribution should be negligible compared to the unitarity corrections.
1. Introduction
The phenomenology of radiative non-leptonic kaon decays provides crucial tests for the ability of Chiral Perturbation Theory ( xPT) [ 1 ] to explain weak low-energy processes and interesting possibilities to study CP violation in these channels. xPT is a natural framework that embodies together an effective theory (satisfying the basic chiral symmetry of QCD) and a perturbative Feynman-Dyson expansion. Its success in the study of radiative non-leptonic kaon decays has been remarkable (see [2,3] and references therein). Of course there are still open problems, but upcoming experiments should improve our phenomenological knowledge of these decays, in particular on K ---f~yy, K + d+t-, KL 4 l+tc- or KL + ye+e-. KL + r”yy is a very interesting channel by itself as a xPT test and also in order to establish the relative role of the CP conserving contribution to KL + r’e+eversus the CP violating contributions [ 2-81. KL + n-‘-yy has no O(p2) tree level contribution since the external particles are neutral. For this same reason there are no 0(p4) counterterms and the chiral meson loops are finite at this order [ 81. Consequently ,yPT gives an unambiguous prediction. Furthermore only an helicity suppressed amplitude for KL --+ r’e+eis generated, which is small compared to the CP violating ones. At higher orders in xPT a new invariant amplitude, generating an unsuppressed helicity amplitude to KL -+ n-‘e+e-, appears. As we will see in detail later there is a specific kinematical region in the diphoton invariant mass spectra of the K -+ ryy processes where only this last amplitude contributes. This is the region where both photons are nearly collinear, i.e. at small diphoton invariant mass. The experimental results showed [8] that while the S(p4) predicted spectrum *Work supported in part by HCM, EEC-Contract No. CHRX-CT920026 (EURODAQNE) 0370-2693/%/$12.00
Copyright 0 1996 Published by Elsevier Science B.V. All rights reserved.
PII SO370-2693(96)00970-7
G. D’Ambrosio,
404
J. Portolb/Physics
Letters B 386 (1996) 403-412
looked in good agreement with the experiment, finding no contribution at small diphoton invariant mass, the predicted branching ratio was underestimated by a factor slightly bigger than two, two sigmas away from the experimental result. This fact prompted to several authors to consider higher chiral order corrections: i) Vector Meson Exchange contributions to the local 0(p6) amplitude [ 7,8], ii) Unitarity corrections from KL -+ z-O~+~T- [ 8,9] and iii) Complete unitarization of the 7rr intermediate states through a Khuri-Treiman treatment and inclusion of the experimental yy + 1~~1~~amplitude [IO]. It has to be emphasized that these higher order corrections do not represent a complete chiral order contribution. The size of the first correction is controversial [ 7,8], however a large contribution by itself is excluded by the experimental spectrum. The second contribution shows that unitarity corrections of KL + T’T+Tto KL ---f r”yy increase 20-30% the leading order amplitude and then have to be taken into account [ 8,9]. When both contributions are added it was possible to fit both the width and the spectrum [9]. The third correction increases the branching ratio by an additional 10 - 20%. The charged channel K+ -+ r+yy is experimentally less known (only an upper limit on the branching ratio exists [ II] ), but measurements with good precision are foreseen in the near future [ 3,121. The leading oneloop O(p4) result has been computed [5] and depends upon unknown weak local amplitudes which however could give important informations on vector meson weak interactions. Indeed while at c3(p4) it is well known that the finite part of the counterterms in the strong xPT lagrangian is saturated by the spectrum of lightest resonances (vector, axial-vector, scalar and pseudoscalar) [ 131, the situation in the 0(p4) weak lagrangian is less clear due to our ignorance on the weak couplings of vector mesons, however there are predictive models to test [ 141. We have evaluated the Kf + r+rr+nunitarity contribution to K+ + r+yy with the same procedure [S]. The dispersive contribution is used to compute the KL 4 T~~-+T~ unitarity corrections to KL --+ r”yy unambiguously computed up to a polynomial piece which is reabsorbed (together with the loop divergences) in the counterterms. Moreover various vector dominance exchange models are studied to saturate the bulk of the counterterms as in KL + ?r”yy . Then we can give a definite prediction for the kinematical region of nearly collinear photons where the unknown contribution of the 0(p”) local amplitude is negligible. The experimental in the DAQNE Q-factory [ 31 and other experiments [ 121 is going to improve the study of K+ + r+yy phenomenology of K + ryy and will allow us to analyze both channels in a correlated way.
2.
K -+ nyy amplitudes The general amplitude M(K(p)
+ n-(m)
for K ---f n-yy is given by y(ql,E~)
y(qz,c2))
and M p” has four invariant
where E 1,Q are the photon polarizations, MC””
=
A(z,Y)
~
41; - 41 .92f
(4;
) +
~B(z,Y) ,4
&
C(Z,Y> +
2
D(ZTY) ~p"P"qlpq2u + ~
mK
(1)
= ~1~~2Jf~~hw72)~
[e/wpv
C-P.91
amplitudes - 91 .q2pppy
P.42F
+P.ql
&P”
+P.92PpLql;)
K
(P .q2qlp
fp.9
42,)Po
+ (Pp&“apY
+P”&pnpy)
?wlp92,17
4 (2)
where v= P .
(41 -
m’K The physical
q2)
,
z=
(41 + q2J2
rn2,
region in the adimensional
variables y and z is given by
(3)
G. D’Ambrosio, J. Porto&/ 0 5
IYJ 5
Physics Letters B 386 (19961403-412
0 5 z L (1 -rd2,
$‘%J$Z),
405
(4)
with =a*+b*+c*-2(ab+ac+bc),
A(a,b,c)
rT=z.
(5)
Note that the invariant amplitudes A( z, y), B( Z, y) and C (z, y) have to be symmetric under the interchange of q1 and q2 as required by Bose symmetry, while D(z,y) is antisymmetric. In the limit where CP is conserved the amplitudes A and B contribute only to KL -+ ~‘yy , while C and D only contribute to KS + r”yy. In K+ + rr+yy all of them are involved. Using the definitions (2)) (3) the double differential rate for unpolarized photons is given by
(6) The processes K 3 ~yy have no tree level O(p*) contribution. At 0(p4) the amplitudes B( Z, y) and D( z, y) are still zero since there are not enough powers of momenta to generate the gauge structure, and therefore their leading contribution is 0(p6). As can be seen from (6) only the B and D terms contribute for small z (the invariant amplitudes are regular in the small y, z region). The antisymmetric character of the D( z, y) amplitude under the interchange of q1 and q2 means effectively that while its leading contribution is O(p6) this only can come from a finite loop calculation because the leading counterterms for the D amplitude are O($). However also this loop contribution is helicity suppressed compared to the B term. As shown in a similar situation in the electric direct emission of KL -+ T+T-~ [ 151, this antisymmetric 0(p6) loop contribution might be smaller than the local O(p*) contribution.
3.
KC --*dyy
at O(p4)
The leading AI = l/2 0(p4) in [5]. We review them here. The A amplitude reads A’4’(z)
=
G&&_, 2TZ
A(z,y)
and C(z,y)
amplitudes
(Z+l--Ij)F(f)+(z+r:,-l)F(Z)-Zz 7r
Here Gs is the effective weak coupling
constant
determined
for K+ -+ n-+yy have already been computed
I
function
(8)
is defined as
A 1 - z arcsin*( 2) F(x)
(7)
from K + TT decays at 0(p2)
]Gs] N 9.2 x 10-6GeV-2 and the F(x)
.
,
x14,
=
(9) In (7) the pion loop contribution F( z/r;) dominates by far over the kaon loop amplitude with F( z ) . The loop results are finite. However as we have already commented xPT allows an 0(p4) scale independent local contribution that in (7) is parameterized by
406
G. D’Ambrosio.
J. Portolk
/ Physics Letters B 386 (1996) 403-412
. c=~[3(L9+~,0)+h’,~-N,S_2N,s], that is a quantity 0( 1). The L9 and Lto are the local L?(p”) strong couplings and Nt4, Nt5 and Nts are 0(p4) weak couplings, still not fixed by the phenomenology, and that can be only computed in a model dependent way [ 141. The Weak Deformation Model (WDM) [ 81 predicts E = 0, while naive factorization in the Factorization Model (FM) [ 15,141 gives P = -2.3. In these models, due to the cancellation in the vector meson contribution in ( lo), the role of axial mesons could be relevant [ 141. The O(p4) contribution to the C( z, y) amplitude is
(11)
-+ yy). This amplitude is generated by the Wess-Zumino-Witten where r,, = m7/mK and Td = r(r” functional [ 161 (TO, q) --f yy through the sequence K+ --f T+(T’, v) ---f ~+yy. This contribution amounts roughly to less than 10% in the total width.
4. 0(p6)
local amplitudes
At 0(p6)
for K+ +
dyy
there are only four independent
F P FpA d”K+a*Y
.
Lorentz invariant
F P FpL”c?,,K+d%- ,
WI:,Fp,Fpv
local amplitudes K+v-
,
contributing
to K+ + rr+yy
PF P daFpYKf~-,
:
(12)
where the last one has no analogous in KL + ?r”yy. In general the couplings of these operators are not directly related with the ones in KL --f ~‘yy due to the fact that the electric charge matrix does not conmute with the generators of the electrically charged field. As we shall see, in specific models, relations between the two channels can be found. All of these operators contribute to A( z, y) but only the first one in ( 12) gives a B( z, y) amplitude. Loop divergences at O(p6) are absorbed in the counterterm coefficients, that thus renormalized are finite. Chiral dimensional analysis [ I] tells us that their contributions are suppressed compared to 0(p4) by a N 0.2. Nevertheless vector meson exchange was found to enhance this up to mi/rn$ w 0.4 factor mi/(4rFT)2 [ 131. Thus we try to estimate the contributions of the lightest resonances, i.e. vector mesons, and we assume that heavier resonances and non-resonant contributions give smaller corrections. This picture seems well verified in the strong coupling constants at 0(p4) [ 131, and it is likely to apply to the weak couplings. We are going to consider here the contribution of vector meson resonances to the weak 0(p6) counterterms for Kt + ~+yy (a more complete discussion and an extension to KL + z-‘yy is given in [ 161) and we find that only the terms in (12) with derivatives on the meson fields can be generated by vector meson exchange. in [8] we assume that the contribution to the local amplitudes for Following the study of KL --j ~‘yy is dominated by vector meson resonances and define their contribution with an adimensional Kf --) ~+yy parameter av generated by the first term in ( 12) as av = -
IT
2Ggmitxem
lim Bv(z).
: -0
where Bv(i) is the vector resonance contribution to the B amplitude. The A amplitude generated by exchange (Av) gets contributions from the first and second structure in (12). However as seen in we assume that these local amplitudes are generated through strong resonance exchange supplemented weak transition in the external legs, those two contributions are related and can be written in terms of parameter defined in ( 13) as
(13)
vector [8] if with a the av
G. D’Ambrosio,J. Portol&/ PhysicsLeners B 386 (19961403-412
Av
= G&pem av 7
(3 + r”, -
z>,
407
(14)
In [ 81 two different vector contributions to av in ( 13) have been proposed: the first one amounts for the weak counterterms generated by a strong vector resonance exchange with a weak transition in an external leg (ae;[‘), the second is generated by vector resonance exchange between a direct weak vector-pseudoscalar-photon (Vpy) vertex and a strong one (UT). While the first can be computed in a model independent way, the generation of direct Vpy weak vertices is still poorly known and therefore only models can be used. The external weak transition for K+ -+ n-+yy gives en = _ 128~~ h2V m2K 9m2,
=
%
-0.08,
(15)
where mv = mP and the strong VPy coupling
Jhv\ = (3.7 i 0.3) x 10v2 is defined by
(16) with UP = iutD,lJut U=exP
(il;
,
j’y $
= uFrut
Aj4.j)
+ u+Ff”u,
U=l.lll,
D,U = a,U
- ir,U + ilJlJ,, (17)
3
and u( 4) is an element of the coset space SU( 3) L @ SU( 3) R / SU( 3) v parameterized in terms of the Goldstone fields #i i = 1, . ..8. FL,: are the strength field tensors associated to the external rp and lp fields, VP is the nonet of vector meson resonances, F, 21 93 MeV is the decay constant of pion and, finally, (A) z Tr( A) In our case with two photon external fields !, = rfi = eQA, ’ . The model dependent contribution to av from direct weak vertices in the WDM and FM quoted above is atrwDM = -av ext ,
4
FM = -2kF a”v”’,
where in the FM kF is the unknown fudge factor that is not fixed by the model and satisfies 0 < kF 2 1. Naive factorization predicts kF = 1 . By adding (15) and (18) we see that av=ae;[‘+&? is flV,WDM
=O,
av,FM=(1-2k~)ae,x’.
(19)
We note that for the allowed range of values of kF in the FM both models agree in predicting a very small vector and therefore the important conclusion of our exercise is that meson contribution to O(p6) in K+ -+ dyy the vector contribution to the local B( Z, y) amplitude can be neglected. If, as it is reasonable to assume, other resonance interchange corrections are even smaller, we can conclude that the small z region of Kt + n+-yy is completely predictable at c3(p6) through chiral loops and it is likely to be dominated by the unitarity corrections to K+ -+ z-+yy . Indeed in that region the 0(p6> is dominant and our ignorance on the of K+ ---t n-+&s-? amplitude is irrelevant.
’ For a full discussion of the definitions and notations see
[ 13,8]
G. D’Ambrosio, J. Portolks/Physics
408
5. Unitarity
corrections
The amplitude variables
K+ + m+n+n- to K+ + n+yy
for the process K(p)
$2 - Sl -, m:
x=
of
y=-
Letters B 386 f1996) 403-412
+ ?r(pr )?r(p2)r(p3)
can be expanded
in powers of the Dalitz plot
s3 - so rni
(20)
’
where Si = (p - pi)2 for i = 1,2,3, so = (sr + sz + ss) /3 and the subscript 3 indicates the odd pion. For the decay K+(p) --f z-+(PI)~+(P~)~-(P~) the isospin decomposition, neglecting the small phase shifts and up to quadratic terms, is written as [ 17,171 A(K+
-+ T+TT+T-
I= 2cr1 - (~3 + (PI - $3
- 2(51 + 53)(Y2 + ix”,
- (51 +53
-
+ ~6~3) Y
5;>v2
-
ix’,
where the subscripts 1 and 3 refer to Al = l/2, 3/2 transitions been fitted to the data [ 171. The 0(p2) amplitude Ac2’(Ki.
+ T+T+T-)
= Gsm;
(f - r;Y)
>
respectively,
(21)
and the coefficients
,
in (21) have
(22)
with the value of Gs in (8) underestimates by 20-30s the experimental linear slopes for K+ + V+TT+T-. At the next chiral order, the experimental linear and quadratic slopes in (21) are recovered (with predictive power too) [ 171. Since the 0(p4) Joop contribution (7) to K+ + n-‘yy , is generated by (22)) it seems natural to try to include all the contributions to K+ + n-+yy generated by the experimental slopes in (21)) i.e. we induced by the 0(p4) corrections to K+ + TT~T+T~. This evaluate the 0(p6) contributions to K+ + r+yy can be done similarly to the case of KL + n-‘yy in [ 8,9] : (21) is considered as an effective K+ -+ rTT+v+rTTchiral vertex, the kinematical variables are replaced by the appropriate covariant derivatives, the QED scalar vertices are added through minimal coupling and then the usual Feynman diagrams approach can be used. There are 13 Feynman diagrams that have two different topologies. There is a subset of 9 diagrams where one or both photons are radiated by the external legs and therefore do not give and absorptive contribution. The sum of these bremsstrahlung-like diagrams is not gauge invariant and cancels with an analogous term from the 4 remaining diagrams (two of which have also an absorptive part) and generate the A( z, y) and B( z, y) amplitudes. This loop result for the A (z, y) and B (z, y) amplitudes is divergent and needs to be regularized and renormalized. We have used dimensional regularization and divergences are absorbed by the four counterterms discussed earlier. Alternatively one could use a subtracted dispersion relation. While the absorptive contribution is uniquely determined, the dispersive one, due to the presence of subtraction constants, can be computed only up to a polynomial. This is precisely the same ambiguity that happens in the effective vertex method that we have used where the finite part of the counterterms is a priori unknown. We assume that the dominant contribution should come from the non-polynomial amplitude generated by the cut of the two-pion intermediate state. Otherwise contributions generated by a vanishing on-shell K -+ 3~ amplitude can be reabsorbed in the unknown polynomial amplitude. in the MS subtraction scheme are The final result for the A and B amplitudes for K’ -+ dyy A(z,y)
= 2
.
2(2crl -
a3)
+
I+$-+
(PI
-$3+hY3)
{[ -
32jl 77
(In($) -51) [r-f,
-1)
+8(1;6(rj-zI+9(r:-z~2jiF(i)]
I?(;>
G. D’Ambrosio, J. Portolk/Physics
+c1
-r2,+z) 1
-’
(
1, 24
z 72r* ,+$(1+$)
r2 --.z+ 24~
( 36
- C+2r$ql
[
= F
+f
($
where the F(x)
{
$(411+&)
- 10) &)] function
(tR(g+3F(;)))
i+l-$)R(;)+$(l-$)F($)] ( r;
+Gsm2, (z + ri - l)iF(z)
B(z,y)
409
Letters B 386 (1996) 403-412
[-A
,
+2q2
(23)
(1+21n($))++
+Gm:r)i}
9
(24)
has been defined in (9) and
x14, R(x)
=
(25) x>4,
’
with /3(x) defined in (9) (F(z) N -AZ and R(z) Y &z for z ---) 0). we have spoken before. In (23), (24) the qi, i = 1,2,3 stand for the unknown polynomial contribution Though, in principle, four different subtraction constants could appear, these are the only ones that are necessary in order to absorb the loop divergences generated by this unitarity correction. We emphasize that these local amplitudes are not generated by the vector mesons (already taken into account by a~) and therefore are expected to be suppressed by &/A: with A, N 4rF,, over the previous chiral order. If as a naive chiral dimensional analysis we choose as coefficients ifor definiteness) we get
N 2.06,
77*= ; %i N 0.24, X
of the suppression
3 m* 73
=
-5g
3
N
factor the factor accompanying
-0.26.
ln(mz/p2)
(26)
X
We note the big numerical value for ~1, but looking into the A amplitude (23) we see that r)i is suppressed by mz/m$. The order of magnitude we get indeed is the one expected by chiral counting. In (23), (24) we only have included the AZ = 312 coefficients of the O(p*> contribution of K+ + yrfn+rTbecause the bigger errors in the determination of these coefficients in the 0(p4) amplitudes. We notice that at this order while the B amplitude is y-independent the A amplitude gets its first y dependence. As we have seen the vector meson contribution to the B amplitude is likely to be very small. If we use the same definition (13) for our B amplitude (24) we get
410
G. D’Ambrosio, J. Portol&/Physics
Fig. 1. Br(K+
- r+yy ) as a function of the 2 amplitude. The dashed line corresponds to 0(p4) ,@T amplitude. The full line corresponds to the amplitude including the O(ph) unitarity corrections
Letters B 386 (1996) 403-412
Fig. 2. Comparison of the normalized z-distribution for K+ -+ ~+yy at 0(p4) (dashed line) with our 0(p6) correction (full line) for C = 0.
a[ =
(27)
The first term amounts to cv 0.52 if p = mp is taken. Then from (26) we can conclude that due to the expected vanishing of the vector meson contributions, the polynomial non-resonant amplitude, though small compared with the unitarity corrections, could be in principle tested in this channel at low 2. We note that this situation is different to the KL ---in-Oyy case where the vector meson contribution seems to be more relevant. We have not included the A1 = 3/2 couplings in the 0(p6) contribution because of the big errors in the experimental fit. But we should stress that inputting those amplitudes could modify the low-z behaviour of the B amplitude even a 30%. The argument about the relevance of the correction to KL 4 ?r”yy coming from the inclusion of the experimental yy + 7r07ro amplitude [ IO] is weakened here due to the relative suppression of K+ --f n-i~Oflo compared to Ki --+ TT~TT~TF.
6. K+ -+ rr+yy
: branching
ratio and spectra
We can now proceed to study the numerical results for K+ 3 v+yy taking into account the 0(p4) and our 0(p6) loop evaluation of the unitarity corrections. As we have shown the 0(p6) vector resonance dominated local contributions are negligible. In our numerical discussion we will input q = 0, i = 1,2,3 and ,u = mp. At present there is only an upper bound for Br( K+ h n-+yy) which depends on the shape of the spectrum
[I81 Br( K+ i
dyy
> jexp 5 1.5 x 10e4
Br( K+ d n-+yy ) jexp < 1.O x 10e6
(xa
amplitude)
(constant
,
amplitude)
.
(28)
The uncertainty in the theoretical prediction is dominated by the unknown 0(p4) counterterm generated amplitude C in (7). In Fig. 1 we show Br( K+ + rr+yy) as a function of E with and without the 0(p6) corrections that we have computed. We remind that WDM predicts C!= 0 while naive FM gives E = -2.3 with the following results: Br(K+
+ ~+yy ) /WDM = 7.24 x lo-‘,
Br(K+ --f n-+yy ) lnFM = 6.20 x 10-7.
(29)
G. D’Ambrosio, J. Portok/
/ .__
0 0
05
1
_*_
15
I
_~ _..~ 8
m,
411
Physics Letters B 386 (1996) 403-412
/ ,’
25
3
35 (GN)
-+
Fig. 3. Normalized di-photon mass spectrum of K+ -t ~+yy for ? = -2.3 (naive FM, dashed line) and ? = 0 (WDM, full line).
IYI
Fig. 4.
d Br( K+ + P+YY ) r?v
spectrum for ? = 0 as a function of
Ivl.
When comparing with the O(p4) predictions [5] we find that the unitarity corrections increase around 30-40% the branching ratio. In Fig. 2 we show the z-distribution at S(p4> and with our 0(p6> unitarity correction for ? = 0. As can be seen the correction is noticeable. The uncertainty on the 2 amplitude also translates into the spectra. In Fig. 3 we show the spectrum of the invariant mass of the two photons for the two values of i; predicted by the WDM and naive FM. We notice that the main dependence on E arises in the z-region where there is the bulk of the absorptive contribution. Finally in Fig. 4 we show the spectrum in the “asymmetric” _vvariable for f = 0. At this order this spectrum has much less structure than the one in the di-photon invariant mass.
7. Conclusion and the expected experimental measurement of The situation of the chiral prediction for KL ---f r”yy l( K’ -+ r+yy) at DAQNE and BNL make interesting to enlarge our theoretical knowledge about this last process, We have reviewed and studied the vector resonance dominated S(p6) local amplitudes which main role is to determine the low-z region of the invariant di-photon mass spectrum. The conclusion is that this contribution is likely to be negligible. The O(p6) unitarity corrections from K+ --f QT+TT+TTto K’ + n-+yy have been computed and found to be relevant increasing the branching ratio (for a fixed value of the E amplitude) around 30-40%. The shape of the z-distribution is shown to be sensitive to the evaluated corrections (Fig. 2 and Fig. 3). as a function of 2. The included corrections In Fig. 1 we have shown the branching ratio for Kf + byy will allow to get a more accurate determination of 2 once the branching ratio of Kf + n+yy is measured and therefore will provide a new independent relation between 0(p4) weak coupling constants that will improve our predictive power in this sector.
Acknowledgements The authors thank G. Ecker for interesting correspondence about the subject and the Particle Theory Group at MIT where part of this work was done. J.P also thanks the hospitality of the Particle Theory Group at Rutherford-Appleton Laboratory where this work was started. J.P. is supported by an INFN Postdoctoral fellowship and also partially supported by DGICYT under grant PB94-0080.
G. D’Ambrosio, J. Portol&/Physics
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Letters B 386 (1996) 403-412
References ] I ] S. Weinberg, Physica, A 96 ( 1979) 327: J. Gasser and H. Leutwyler, Ann. of Phys. 1.58 (1984) 142; A.V. Manohar and H. Georgi, Nucl. Phys. B 234 ( 1984) 189: J. Gasser and H. Leutwyler, Nucl. Phys. B 2.50 (1985) 465. 121 J.L. Ritchie and SC. Wojcicki, Rev. Mod. Phys., 65 ( 1993) 1149; L. Littenberg and G. Valencia, Ann. Rev. Nucl. Part. Sci. 43 (1993) 729; A.J. Buras and M.K. Harlander, in: Heavy Flavours, eds. A.J. Buras and M. Lindner, Advanced Physics (World Scientific, Singapore 1992). Vol. IO. p. 58; R. Battiston, D. Cocolicchio, G.L. Fogli and N. Paver, Phys. Rep. 214 ( 1992) 293; B. Winstein and L. Wolfenstein, Rev. Mod. Phys., 65 ( 1993) 1113. [3
I G. D’Ambrosio,
G. Ecker, G. Isidori and H. Neufeld, Radiative non-leptonic kaon decays, ed. L. Maiani, G. Pancheri and N. Paver, LNF (1995). p. 265. [4] J.F. Donoghue, B.R. Holstein and G. Valencia, Phys. Rev.. D 35 ( 1987) 2769. [S] G. Ecker, A. Pith and E. de Rafael, Nucl. Phys., B 303 (1988) 665. 161 J. Flynn and L. Randall, Nucl. Phys., B 326 ( 1989) 31: C. Dib. I. Dunietz and EJ. Gilman, Phys. Len. B 218 ( 1989) 487; T. Morozumi, H. Iwasaki, Progr. Theor. Phys. 82 ( 1989) 371. 17 1 L.M. Sehgal, Phys. Rev. D 38 (1988) 808; P. Heiliger and L.M. Sehgal, Phys. Rev. D 47 ( 1993) 4920. [ 8 ] G. Ecker, A. Pith and E. de Rafael, Phys. Lett. B 237 ( 1990) 48 I. [9] G. Ecker, A. Pith and E. de Rafael, Phys. Ixtt. B 189 (1987) 363; L. Cappiello and G. D’Ambrosio, Nuovo Cimento A 99 ( 1988) 15.5. ] IO ] E731 Collab., V. Papadimitriou et al., Phys. Rev. D 44 ( 1991) 573; NA31 Collab., G. Barr et al., Phys. Lett. B 284 (1992)
Series on Directions
in: Second DAQNE
in High Energy
Physics
Handbook,
440.
( I I ] L. Cappiello. G. D’Ambrosio and M. Miragliuolo, Phys. Len. B 298 ( 1993) 423. [ 121A.G. Cohen, G. Ecker, A. Pith, Phys. Lett. B 304 (1993) 347. [ 13 1J. Kambor, B.R. Holstein, Phys. Rev. D 49 ( 1994) 2346. 1141 Review of Particle Properties, L. Montanet et al., Phys. Rev. D SO ( 1994) 1173. [ 15) F?Franzini, Preprint LNF-95/054, to be Published in the Proceedings of XVII International Symposium on Lepton-Photon Beijing (China), IO-15 June, (1995). [ 16 ] G. Ecker, J. Gasser, A. Pith and E. de Rafael, Nucl. Phys. B 321 ( 1989) 311. 1171 G. Ecker, J. Kambor and D. Wyler, Nucl. Phys. B 394 (1993) 101; See also G. Isidori and A. Pugliese. Nucl. Phys. B 385 (1992) 437; C. Bruno and J. Prades, 2. Phys. C 57 (1993) S85. [ I8 ] G. D’Ambrosio and G. Isidori, 2. Phys. C 65 ( 1995) 649. 1191 A. Pith and E. de Rafael, Nucl. Phys. B 358 (1991) 31 I. ] 201 J. Wess and B. Zumino, Phys. Len. B 37 ( 1971) 95; E. Witten, Nucl. Phys. B 223 (1983) 422. I21 ] G. D’Ambrosio and J. Portoles, Preprint INFNNA-lV-96/21 ( 1996). 1221 C. Zemach, Phys. Rev. 133 (1964) B1201; T.J. Devlin and J.O. Dickey. Rev. Mod. Phys. 51 ( 1979) 237. I23 1 J. Kambor, J. Missimer and D. Wyler. Phys. Lett. B 261 ( 1991) 496; L. Maiani and N. Paver, CP conserving nonleptonic K + 37r decays, in: Second DAQNE Pancheri and N. Paver. LNF (1995). p. 239. 1241 MS. Atiya et al.. Phys. Rev. Len. 65 (1990) 1188.
Physics Handbook,
Interactions,
eds. L. Maiani,
G.