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K → πππ decays in large Nc chiral perturbation theory
Volume 238, number 2,3,4
PHYSICS LETTERS B
5 April 1990
K - . I t n n DECAYS I N L A R G E Arc C H I R A L P E R T U R B A T I O N T H E O R Y Hai-Yang C H E N G Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA and lnstitute of Physics, Academia Sinica, Taipei, Taiwan 11529, Rep. China Received 30 November 1989
Corrections to the current-algebra analysis of A/= ~ and 3 K~nnn decay amplitudes including Ks--,n+n-n ° are calculated using the large Nc higher-order effective chiral lagrangians for strong and weak interactions. Neglecting gluonic corrections to the coupling constants of the higher-derivative chiral lagrangians, we find an agreement between theory and experiment within 3% for the constant and linear terms of A/= ½ transitions. Quadratic coefficients of various AJ= 3 amplitudes in the Dalitz plot are predicted. The branching ratio and the slope parameter ofKs~n+n-n ° are found to be 3.9 × 10 - 7 and 4.4 X 10-2, respectively.
1. Introduction
A higher-order effective chiral lagrangian for A S = 1 nonleptonic weak interactions was recently derived within the framework o f the 1~No a p p r o a c h a n d has been applied to the A / = ½ K + - ~ x + r t + x - decay [ 1 ]. The purpose o f the present p a p e r is to complete the study for other K--, 3x decay modes and also for A / = 3 transitions. The K (k) -~ x (Pl) x (P2) rt (P3) a m p l i t u d e s in the Dalitz plot are conventionally p a r a m e t r i z e d as A(K~3rt)=a+bY+c(Y2+]X
2) + d ( Y Z - ~ X 2) ,
(1)
where Y= ( s 3 - s o ) / m 2, X = ( & - & ) / m ~ , si= ( k - p i ) 2, So= (s, +SR+S3)/3 (the subscript " 3 " is assigned to the " o d d " p i o n ) . It has been known for more than two decades that the constant and linear parameters a and b respectively can be related to K ~ x x a m p l i t u d e s via current algebra or the lowest order chiral lagrangian [2]. However, theoretical predictions for those two coefficients in both A / = ½ and A / = 3 a m p l i t u d e s though reasonable are too small by ( 2 0 - 3 5 ) % when c o m p a r e d to experiment (see tables 1 and 2). Moreover, more accurate experiments in the seventies have uncovered the presence o f quadratic m o m e n t u m terms [ 3 ] (i.e. the p a r a m e ters c and d). This means that, within the framework o f chiral p e r t u r b a t i o n theory ( C h P T ) , higher order lagrangian terms with higher derivatives must be taken into consideration in order to account for non-vanishing quadratic coefficients a n d the discrepancy between theory a n d experiment for the constant and linear terms. However, it is known that no first-principle predictions can be m a d e in the standard scenario o f the chirallagrangian a p p r o a c h except for two correlations between the p a r a m e t e r s a and c as well as b and d (see below). Basically, there exist two p r e d i c a m e n t s with C h P T in regard to the study of K--, 3x decays: First, coupling constants o f higher order strong and weak chiral lagrangians are a priori unknown; second, calculations o f chiralloop contributions to K ~ 37t, which must be included in order to ensure the scale independence o f the physical amplitude, are extremely h o r r e n d o u s % In principle, the running couplings of the p4 strong chiral lagrangian 5e~4) can be empirically d e t e r m i n e d from various low-energy hadronic processes s u p p l e m e n t e d with the Zweig rule argument, as done in the classic work o f Gasser a n d Leutwyler [5]. However, there is only one process, n a m e l y K--, xrc~, relevant for the d e t e r m i n a t i o n o f the effective nonleptonic weak lagrangian 5P~ ) . Furthermore, in the limit o f CP and chiral s y m m e t r y there are seven i n d e p e n d e n t terms in L f ~ ) responsible for A / = ½ ~ ChiraMoop contributions to K~3n were calculated recently by Kambor, Missimer and Wyler [4]. 399
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PHYSICS LETTERS B
5 April 1990
Table 1 Large N~ chiral lagrangian predictions for the parameters a, b, c and d in the A/= ~ decay amplitudes of K ~ n n n . Effects of gluonic corrections to the coupling constants are not considered. The parameters are in units of 10-6. The experimental numbers are taken from ref. [3].
( 1 ) K+~n+n+n -
(2) K+---,n+n°n °
a
b
c
d
Lg~ ), Lg~2) c~4) 5°~ ) total expt.
+ 1.500 - 0.276 +0.553 + 1.777 1.829 + 0.005
-0.167 - 0.049 -0.049 - 0.265 - 0.258 _+0.004
+ 0.0044 -0.0089 - 0.0045 - 0.0074 + 0.0022
+ 0.0044 +0.0044 + 0.0088 0.0125 + 0.0012
5o~), ~ z )
50~ ) total expt.
+0.750 -- 0.138 + 0.276 + 0.888 0.915_+0.0024
+0.167 + 0.049 + 0.049 + 0.265 0.258_+0.004
+ 0.0022 - 0.0044 - 0.0022 -0.0037_+0.001 l
- 0.0044 - 0.0044 - 0.0088 -0.0125_+0.0012
~), ~z) LP~4) LP~) total expt.
--0.750 + 0.138 -0.276 --0.888 -- 0.915 -+0.0024
--0.167 -- 0.049 -0.049 -0.265 -- 0.258 -+0.004
-- 0.0022 +0.0044 +0.0022 0.0037 _+0.0011
+ 0.0044 +0.0044 +0.0088 0.0125 _+0.0012
~),
--2.249 +0.414 -0.829 - 2.664 -2.744_+0.007
o ~ 4)
(3) KL-+n+n-n °
(4) Kn-*n°n°n °
o~7~4)
LP~ ) total expt.
LP~2)
-0.0066 +0.0133 + 0.0067 0.0111 _+0.0033
A S = 1 w e a k t r a n s i t i o n s , w h i c h c e r t a i n l y c a n n o t b e e m p i r i c a l l y e x t r a c t e d f r o m a single p h y s i c a l process. T h e a b o v e - m e n t i o n e d flaws a r e d r a m a t i c a l l y c i r c u m v e n t e d i n t h e l i m i t o f large Arc: C o u p l i n g c o n s t a n t s o f ~ 4 ) a n d 5 a ~ ) are t h e o r e t i c a l l y c a l c u l a b l e a n d c h i r a l - l o o p effects are n e g l i g i b l e i n t h e l e a d i n g 1 ~No e x p a n s i o n . I n o t h e r w o r d s , a b s o l u t e p r e d i c t i o n s i n C l a P T c a n b e a c h i e v e d i n t h e large Arc limit. C a l c u l a t i o n o f K ~ 3 n D a l i t z a m p l i t u d e s in t h e large Nc a p p r o a c h o f C h P T h a s b e e n d o n e in t h e l i t e r a t u r e [ 6 , 7 ] . T h e effects o f 5 ¢ ~ ) ( 1 / N ~ ) w e r e c o m p u t e d i n ref. [ 6 ] b y m e a n s o f t h e e f f e c t i v e w e a k h a m i l t o n i a n in c o n j u n c t i o n w i t h t h e v a c u u m - i n s e r t i o n m e t h o d ~2. T h e h a d r o n i c m a t r i x e l e m e n t s o f q u a r k c u r r e n t s are d e t e r m i n e d f r o m ~ s 2) + 5°~4)( 1 / N ~ ) c o u p l e d to e x t e r n a l W fields. T h e r e are t w o w e a k p o i n t s w i t h t h i s c a l c u l a t i o n : First, m a n y p h y s i c i s t s felt v e r y u n e a s y w i t h t h e v a c u u m - s a t u r a t i o n m e t h o d s i n c e it fails to e x p l a i n t h e n o n l e p t o n i c w e a k d e c a y s o f c h a r m e d a n d s t r a n g e m e s o n s , t h o u g h t h e u s e o f f a c t o r i z a t i o n in ref. [ 6 ] is e n t i r e l y legitim a t e ~3. S e c o n d , t h e e f f e c t i v e w e a k h a m i l t o n i a n e m p l o y e d in ref. [ 6 ] is n o t t r u l y e v a l u a t e d in t h e large N~ limit. I n t h e p r e s e n t p a p e r , we will w r i t e d o w n e x p l i c i t l y t h e large N~ c h i r a l l a g r a n g i a n r e s p o n s i b l e for n o n l e p t o n i c A S = 1 w e a k t r a n s i t i o n s a n d o b t a i n K ~ n n n a m p l i t u d e s u n a m b i g u o u s l y in t h e l e a d i n g 1/Nc e x p a n s i o n .
~2 Fajfer and G6rard [ 7 ] have calculated the contribution of ~ ) ( 1/N~) directly. [Though the weak chiral lagrangian LP~4) ( 1/N¢) was not explicitly written down by them, it is nevertheless uniquely determined from LP~2) + 50~4~( 1~No ). ] They concluded that the "QCDinspired" lagrangian approach in which L,°~4)( 1~No) is obtained by the integration of spurious chiral anomalies, as we adopt in the present paper, predicts a too small coefficient b and yields a wrong sign for the coefficient d for the A/= ½ K ~ 3 n amplitudes. As pointed out in ref. [ 1 ], this is attributed to the fact that the contribution due to the L 9 term ofCJ~ 4) [see eq. ( 5 ) ] was missed by them. ~3 This is because the factorization method is used there only for the purpose of relating K-> 3n to K ~ 2n, not for the direct evaluation of kaon decay amplitudes. 400
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PHYSICS LETTERS B
Table 2 Same of table 1 but for A/= 3 decay amplitudes. The parameters are in units of 10-8 a ( 1) K+ ~ + ~ + ~
~,
j,~2)
~4) ~W~) total expt.
2~), ,,~2)
(2) K+~n+rt°n°
cj~4) .~) total expt. (3) KL~X+X-~°
5°~ ), LP~2)
~(~4) 5G~) total expt. ~), ~2) cj~4) ~) total expt.
(4) KL--~x°~°x°
b
c
d
+4.777 --0.880 + 1.759 + 5.656 7.14 _+0.36
+4.264 --0.155 + 1.237 + 5.346 5.59 _+0.66
+0.0141 --0.0282 -- 0.0141
+0.0141 --0.1128 -- 0.0987
+2.388 - 0.440 +0.880 + 2.828 3.57_+0.18
+2.738 + 0.183 +0.502 + 3.423 3.11 _+0.66
+ 0.0070 -0.0141 - 0.0071
- 0.0166 -0.0458 - 0.0624
+4.777 --0.880 + 1.759 + 5.656 7.14_+0.36
-- 1.525 +0.337 -0.735 - 1.923 -2.48_+0.48
+0.0141 -0.0282 -0.0141
--0.0307 +0.0670 +0.0363
+ 14.330 -2.640 +5.279 + 16.969 21.42 _+1.08
+0.0423 -0.0846 -0.0423
2. Large N~ higher order chiral lagrangians The lowest order chiral lagrangian including explicit chiral-symmetry breaking for low-energy Q C D is given by ~s~2) = _ _~f2 Tr
L,LU+ ~-sf~T r ( M U * + UM*),
(2)
where U=exp(2iq)/f~), 0 = ( 1 / x / 2 )0a2a, Tr(Aa2 b) = 2d ab, f ~ = 132 MeV, L , - (OuU) U* is an SU (3)R singlet, and M is a meson mass matrix with the n o n - v a n i s h i n g matrix elements M11 = M 2 2 ~-- m 2, M33 = 2m ~ - m ~. The lowest order nonleptonic AS-- 1 weak lagrangian reads
o ~ ) = _g~l/2) Tr(26LuLU ) _g~/2)(Lul3LUzi +4Lu23L~ l + 5Lu23L~2) - g 2 (73 / 2 ) (Lul3L~'l +Lu23LS{l -Lu23L~2)
,
(3)
where the subscript of the weak coupling constant g denotes the SU (3) representation u n d e r the left-handed chiral rotation, and the superscript refers to the isospin representation. Those u n k n o w n couplings are determ i n e d from the measured K ~ 2~ rates to be [ 8 ]
g(I/2).a_~(l/2) 8 1~27
~
_ 0 . 2 6 × 1 0 - S m 2,
g~3/2)~ _ 0 . 8 6 × 1 0 - 1 O m 2K
(4)
where the sign is fixed by the v a c u u m - i n s e r t i o n method. To study the higher derivative lagrangian, consider Q C D coupled to external vector (V,) and axial-vector (A u) gauge fields. The variation of the fermionic d e t e r m i n a n t u n d e r a local chiral transformation is governed by chiral anomalies. The effective action is then the integration of chiral anomalies. There are two categories of global chiral anomalies: proper (Bardeen) anomalies which contain the totally antisymmetric tensor eu,-p and spurious anomalies which do not. It is well known that the integration of topological (Bardeen) anomalies gives 401
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PHYSICS LETTERS B
5 April 1990
rise to the Wess-Zumino-Witten effective action [9 ]. Likewise, the integration of non-topological (spurious) chiral anomalies yields an action for p4 non-anomalous strong chiral lagrangians [ 10]
~(4)(1/N~) =L~ [Tr(D~'U*DuU) ]2+L2[Tr(DuU*D.U ) ] 2 + L 3 Tr(DuUeDu U) 2 +L9 Tr( FR.D~'UtD"U+ F~.DUUD"U *) + Lto Tr( UtF].UF I'~L) ,
(5)
where 8Ll =4L2 = - 2 L 3 = L 9 = - 2 L l o =Nc/48/r 2 ,
(6)
D,,U=OuU+A~U_UA], Fue;R=O~,Al.,~ r~I_,R,.., 4L,R1j, andA~t.,R = V u+_A~,.Inprinciple,thep4cou, -Ù,A L,R u +~-~ pling constants Lj can be affected by the gluon effects arising from all planar diagrams without internal quark loops. It was shown recently in ref. [ 11 ] that to the order of a 2, the couplings L3 and L~o do receive gluonic corrections. One of the great advantages of the 1/N~ expansion approach is that the structure of the weak chiral lagrangian 50~) (1/N~) is uniquely determined by ~ 2 ) + 5~4)(1/N~) based on the following three ingredients: an extremely simple structure of the weak hamiltonian at the quark level in the large Arc limit, bosonization of quark currents extracted from 5° (s2) + 5 ~ 4) ( 1/N~), and factorization valid in the large Nc approximation. The resulting p4 AS= 1 nonleptonic weak chiral lagrangians have the form [ 1 ] ~ ) ( 1/Nc) = O(8~/2~ + Ot 3/2) with
0(8'/2) = f g8 ~ {hi Tr(26Lt, LUL.L") +h2 Tr(Z6L~,L.LUL") + h 3 Tr(26LI, L.L"LU ) +h4 Tr(26LuL~) Tr(L~'L ") +h5 Tr(26 YY) +h6 Tr( [Z6, Y]LuLU) +h7 Tr( [26, Y~,~]LUL~)}, (7)
02(3/2) =g27(L,,131L~, +L~z23£{'l-Lu23£~2 +/~al3L~I +£az3L~,-/~g, z3L~2) ,
and N~
3hi = - h 2 - - - h 4 = ~
(l-q),
h6=-h7 =
£ , , = ~ {N~ ( 12 - r / ) ( L . L ~ L . + L , , L " L . ) + [ Y ,
Nc
24n2,
h3=hs=0
L u l + [ L ~,17 1},
(8)
where Y~,,,=(OI,0.U)U* , Yu~=Yg~- Y~,*,,, Y=g~'~'~,~ (ref. [ 12] ), ~/=64n2fiL3 (ilL3 being the gluonic modification to L3), and Az =2nf==830 MeV, which is the scale of chiral symmetry breaking, is fixed in this large Nc approach [ 13]. Since in the SU (3) limit g~/2) =g~3/2)/5 and g~l/2) >> g~37/e)' we thus neglected the AI= ½27plet weak operator in eq. (7) and denoted g8 =-g~l/2), g:7 = g~3/2).
3. K--, nnn
Dalitz amplitudes
The CP-conserving K - , nnn Dalitz amplitudes have the following isospin structure:
A ++-=i(n+n+n - LHw IK + ) =2(a~ +a3) - (b~ +b3 - b g ) Y+ 2 (c, +c3) ( Y~+ ~x ~) - (d, +d3 - d g ) ( y 2 _ ~X ~) ,
A +°°=i(n+n°n°iHw IK + ) = (al +a3) + (b, +b3 +b~) Y+ (c, +c3) ( y 2 + ~X 2 ) + (d, +d3 +d'3) ( y 2
A ~ - O = i ( n + n - n O l H w l K s ) = -- 2~ b .3 X + 34d 3. X Y , 402
~X 2 ) ,
(9)
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A~ °=i(n+n-n°lHwlKc) = - (al - 2 a 3 ) - (bl - 2 6 3 ) Y - (cl - 2 c 3 ) ( y 2 + i X 2 ) _ (dl - 2 d 3 ) ( y 2
A°°°=i(~°~°~°[Hw lKc ) = - 3(al -2a3)- 3(G --2c3) ( y2 + ½X2)
~X 2 ) , (9 cont'd)
,
where the subscript 1 (3) refers to A I = ½ (3) transitions. Our parametrization of K - , 3re amplitudes is the same as that of Delvin and Dickey [ 3 ] and that of Zemach [ 14 ] except that we have also included quadratic terms for A / = 3 amplitudes and corrected a sign error in ref. [ 3 ] for the quadratic coefficient of ( y 2 _ ~X 2) in A + + The K-~ 3~ decay receives two contributions: direct weak transitions and meson pole diagrams. Contributions due to the lowest order effective lagrangians ~ 9 ~ 2 ) and 5°~ ) are (Ks--, 3~ will be treated separately)
4m (l+Ym-~KY
A(K-.3n)=-gs~a where
g=gs (g27) for 2 d =
,
½ (3) transitions, and the parameters a, 7 are channel dependent [6]
A + + - : ~1 =2, yl = - 3 ; or3 = 2 , y3 = 12, A~--°"• al = -
(10)
A +°°: al = 1, ?q = 3, a 3 - 1, T3 = ~ - ~ Z z ,
1, yl = 3 ; a3 = 2 , ~3 = - ~ -15 7 ~ , 27_
AOOO:
a~ -7- - 3 , ~ = 0 ; a3 = 6 , y3 = 0 ,
(11)
where z = m,J(rnK 2 2 --m,O. 2 Eqs. (10) and ( 11 ) agree with the current-algebra analysis [ 15] except that contributions proportional to z in the slope parameters o f the A / = ~ amplitudes do not show up in the latter framework, as pointed out in ref. [6]. The numerical results shown in table 1 show clearly that the lowest order predictions for the constant and linear terms of LXI=½amplitudes are 18% and 35% respectively below the experimental values. The p4 strong chiral lagrangian 5 ~ 4) ( 1/N~) contributes to K ~ 3 ~ via the pole diagram. The resulting corrections read
4m2
4(mK~4(1--3 m~2~'] 2(mK~ 2 4 2 2 H=mJ(mKAz),
where al = a 3 = 3 H , A + + - " • 1~1 = 53H ; / ] 3
•
53H ,
A +oo: fll
A~_O.fll=_3H;fl3=_3H(1 •
9.
= -
"1- g-: ) ,
3 ~m) f2l Y +
(y2+ff~_) +fl(y2
2~)],
(12)
and 3 H ; ]~3 =
--
31t( 1 + ~z)
A ° ° ° : fll ~ - 0 ; ~]3 = 0 .
(13)
(Because of the theoretical uncertainty in the estimate of ilL> we will not consider in the present paper the effects ofgluonic corrections to K--,~nrt). It is extremely useful to apply eq. (9) to check the consistency of the results of eqs. (10) - ( 13 ). Eq. ( 12 ) implies interesting correlations between the parameters a and c as well as b and d. These correlations indicate that the higher-order chiral corrections to the constant (linear) term also fix the quadratic coefficient c(d), as first pointed out in ref. [ 16]. From tables 1 and 2 it is evident that the 5Ps(4)( l / N c ) corrections to the K ~ 3n amplitudes have wrong signs for the A I = ½ coefficients a, c, and the A / = ~3 parameters a and b (except for b of A +00). This shows that ~ 4 ) alone does not suffice to reproduce the main features of the experimental data and the inclusion of the fourderivative weak chiral lagrangian 5 : ~ ) is called for. To evaluate the effects of 5el~ ) ( 1/Arc), we first notice that only the h~, h2 and h4 operators ofO8~/2) contribute to the A I = ½ transitions. (The h6 contribution to K~3rc is compensated by that of h7.) Second, contrary to ~ s (4) , S ~ ~ contributes only via the direct-emission diagram. The computation is lengthy but otherwise straightforward. The K - , 3~ Dalitz amplitudes induced by Y ~ ) ( 1 ~Arc)are the same as eq. ( 12 ) except that a and fl are 403
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replaced by a ' a n d fl', respectively, with ~t4 c(~=-3H,
fl'~=3H,
fl,3(A+OO)=_~H,
o~=-3H,
fl'3(A++-)=-12H,
fl,3(A~-O)=~H,
fl~(AOOO)=0.
(14)
For A l = ½amplitudes, numerical results displayed in table 1 present a remarkable agreement between theory and experiment within 3% for the constant a n d linear terms ~s. This means that very little room is left for chiralloop corrections. The predicted coefficient c is just marginally in accord with data within the experimental errors, while the other coefficient d is off by three standard deviations. Clearly more accurate K 3 n data are urgent to clarify this discrepancy. In the A / = ~ sector, we see from table 2 that the linear term is generally in agreement with data, whereas the constant term is four standard deviations off from experiment. Obviously, more high-statistics experiments are required to improve the d e t e r m i n a t i o n of A/=-32 coefficients a and b, and to extract the quadratic terms c and d in order to test the chiral-lagrangian approach. We now turn to the decay K s - - , n + n - n°. In absence of CP violation, this mode is a A / = 3 transition into the I = 2 state of three pions. The CP-violating part of K s ~ n + n - n o can proceed through the A / = l part of the weak hamiltonian. We find the CP-even K s ~ n + n - n ° amplitude to be _ A+-°=2gz7 _fm2 4
9+3...~ 2+-25 mK - m ~ Az
1+3
)] I X+6g 7 5+ml~_rn,~j_ ~.~5
5-t-
m K2 - --- m ~ ]2
XY
• (15)
Again, the reader may use eq. (9) to check the consistency ofA ~--o with other K ~ 3n amplitudes. It is of interest to note that the energy dependence of the A~- - ° is of the form **6 X ( I + a Y ) with the slope parameter a = 4.4 × 10 -2. We see that the higher order chiral corrections enhance the amplitude by 24%. After the phasespace integration, the branching ratio is calculated to be
B r ( K s - , n + n - n °) = 3 . 9 × 10 -7 ,
(16)
which is to be compared with the current-algebra prediction 2.4 X 10- 7 and the present experimental upper limit 4.9 × 10- 5 [ 17 ]. The CP-violating amplitude of this decay mode approximated by eA (KL ~ n + n - n °) leads to a branching ratio of order 1 × l 0 -9, much smaller than the CP-conserving part. To summarize, the gross features of K 3 n data are satisfactorily described by the p 4 large N~ effective chiral lagrangians for weak and strong interactions. Quadratic parameters of A / = 3 K--, 3n Dalitz amplitudes are predicted. The branching ratio and the slope parameter ofKs-~n+n-n ° are found to be 3.9X 10 -7 and 4.4X 10 -2, respectively.
~:4 A different result fl'l = - 3H was obtained in ref. [ 7 ] since the L9 term of ~ 4 ) ( 1~No) was not taken into consideration in the same reference. Results ofeq. (14) may be also obtained from eqs. (4.18)-(4.21) of the first article in ref. [6] evaluated in the vacuuminsertion method by puttingfK=f~ and f= 0. Note that the parameter fin ref. [6 ] should be interpreted as the fraction of contributions to the A/= ½K-~2n amplitude due to higher-orderpenguin operators. ,5 Mass terms of ~ 4 ) ( 1~No) cannot be determined by the integration of chiral anomalies since external scalar and pseudoscalar fields do not contribute to anomalies. It should be stressed that mass operators only affect the coefficientsa and b because they do not have enough derivatives to induce quadratic terms. The effect of L5 Tr[OUUtOuU(MtU+UtM)], which is also responsible for SU (3) breaking in the ratio offK/f~, on K~ 3n has been estimated in ref. [7 ] ; it improves the discrepancyfor the AJ= ~ coefficienta. ,t6 SinceXis asymmetric in Sl and s2, it is obvious that CP-conservingKs-~n°n°n° is prohibited by Bose symmetry. Just as KL---~Kn, UP nonvaniance in Ks~n°n°n° receivestwo contributions: direct CP violation in K 1-*x°n°K° and CP violation in the K°-I(° mass matrix. 404
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Acknowledgement T h e a u t h o r wishes to t h a n k C.Y. C h e u n g a n d W.B. Y e u n g for the c o l l a b o r a t i o n d o n e in ref. [ 6] w h i c h m a d e the p r e s e n t task possible. H e is also grateful to the Physics D e p a r t m e n t o f the B r o o k h a v e n N a t i o n a l L a b o r a t o r y for their hospitality. T h i s w o r k was s u p p o r t e d in part by the U S D e p a r t m e n t o f Energy ( c o n t r a c t n u m b e r D E A C 0 2 - 7 6 C H 0 0 0 1 6 ) a n d the N a t i o n a l Science C o u n c i l o f the R e p u b l i c o f China.
References [ 1] H.Y. Cheng, preprint BNL-43295 ( 1989 ). [2] J.A. Cronin, Phys, Rev. 101 (1967) 1483. [3] T.J. Devlin and J.O. Dickey, Rev. Mod. Phys. 51 (1979) 237. [4] J. Kambor, J. Missimer and D. Wyler, preprint PSI-PR-89-18 (1989). [5] J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158 (1984) 142; Nucl. Phys. B 250 (1985) 465, 517. [6] H.Y. Cheng, C.Y. Cheung and W.B. Yeung, Z. Phys. C 43 (1989) 391; Mod. Phys. Lett. A 4 (1989) 869. [ 7 ] S. Fajfer and J.-M. G6rard, Z. Phys. C 42 (1989) 425, 431. [8] H.Y. Cheng, J. Mod. Phys. A 4 (1989) 495. [ 9 ] See, e.g.N.K. Pak and P. Rossi, Nucl. Phys. B 250 ( 1985 ) 279. [ 10] J. Balog, Phys. Lett. B 149 (1984) 197; Nucl. Phys. B 258 (1985) 361; K. Seo, Gifu preprint GWJC-1 (1984), unpublished; A.A. Andrianov, Phys. Lett. B 157 (1985) 425. [ 11 ] D. Espriu, R. de Rafael and J. Taron, CERN preprint CERN-TH-5599 (1989). [ 12 ] J.F. Donoghue, E. Golowich and B.R. Holstein, Phys. Rev. D 30 (1984) 587. [ 13] H.Y. Cheng, Phys, Rev. D 36 (1987) 2056. [ 14] See C. Zemach, Phys. Rev. 133 (1964) B 1201 (especially table 4). [ 15 ] See e.g.L.F. Li and L. Wolfenstein, Phys. Rev. D 21 ( 198 ) 178. [ 16 ] E. Golowich, Phys. Rev. D 35 (1987) 2764; D 36 ( 1987 ) 3516. [ 17 ] V.V. Barmin et al., Nuovo Cimento 85 A ( 1985 ) 67.
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K - . I t n n DECAYS I N L A R G E Arc C H I R A L P E R T U R B A T I O N T H E O R Y Hai-Yang C H E N G Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA and lnstitute of Physics, Academia Sinica, Taipei, Taiwan 11529, Rep. China Received 30 November 1989
Corrections to the current-algebra analysis of A/= ~ and 3 K~nnn decay amplitudes including Ks--,n+n-n ° are calculated using the large Nc higher-order effective chiral lagrangians for strong and weak interactions. Neglecting gluonic corrections to the coupling constants of the higher-derivative chiral lagrangians, we find an agreement between theory and experiment within 3% for the constant and linear terms of A/= ½ transitions. Quadratic coefficients of various AJ= 3 amplitudes in the Dalitz plot are predicted. The branching ratio and the slope parameter ofKs~n+n-n ° are found to be 3.9 × 10 - 7 and 4.4 X 10-2, respectively.
1. Introduction
A higher-order effective chiral lagrangian for A S = 1 nonleptonic weak interactions was recently derived within the framework o f the 1~No a p p r o a c h a n d has been applied to the A / = ½ K + - ~ x + r t + x - decay [ 1 ]. The purpose o f the present p a p e r is to complete the study for other K--, 3x decay modes and also for A / = 3 transitions. The K (k) -~ x (Pl) x (P2) rt (P3) a m p l i t u d e s in the Dalitz plot are conventionally p a r a m e t r i z e d as A(K~3rt)=a+bY+c(Y2+]X
2) + d ( Y Z - ~ X 2) ,
(1)
where Y= ( s 3 - s o ) / m 2, X = ( & - & ) / m ~ , si= ( k - p i ) 2, So= (s, +SR+S3)/3 (the subscript " 3 " is assigned to the " o d d " p i o n ) . It has been known for more than two decades that the constant and linear parameters a and b respectively can be related to K ~ x x a m p l i t u d e s via current algebra or the lowest order chiral lagrangian [2]. However, theoretical predictions for those two coefficients in both A / = ½ and A / = 3 a m p l i t u d e s though reasonable are too small by ( 2 0 - 3 5 ) % when c o m p a r e d to experiment (see tables 1 and 2). Moreover, more accurate experiments in the seventies have uncovered the presence o f quadratic m o m e n t u m terms [ 3 ] (i.e. the p a r a m e ters c and d). This means that, within the framework o f chiral p e r t u r b a t i o n theory ( C h P T ) , higher order lagrangian terms with higher derivatives must be taken into consideration in order to account for non-vanishing quadratic coefficients a n d the discrepancy between theory a n d experiment for the constant and linear terms. However, it is known that no first-principle predictions can be m a d e in the standard scenario o f the chirallagrangian a p p r o a c h except for two correlations between the p a r a m e t e r s a and c as well as b and d (see below). Basically, there exist two p r e d i c a m e n t s with C h P T in regard to the study of K--, 3x decays: First, coupling constants o f higher order strong and weak chiral lagrangians are a priori unknown; second, calculations o f chiralloop contributions to K ~ 37t, which must be included in order to ensure the scale independence o f the physical amplitude, are extremely h o r r e n d o u s % In principle, the running couplings of the p4 strong chiral lagrangian 5e~4) can be empirically d e t e r m i n e d from various low-energy hadronic processes s u p p l e m e n t e d with the Zweig rule argument, as done in the classic work o f Gasser a n d Leutwyler [5]. However, there is only one process, n a m e l y K--, xrc~, relevant for the d e t e r m i n a t i o n o f the effective nonleptonic weak lagrangian 5P~ ) . Furthermore, in the limit o f CP and chiral s y m m e t r y there are seven i n d e p e n d e n t terms in L f ~ ) responsible for A / = ½ ~ ChiraMoop contributions to K~3n were calculated recently by Kambor, Missimer and Wyler [4]. 399
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Table 1 Large N~ chiral lagrangian predictions for the parameters a, b, c and d in the A/= ~ decay amplitudes of K ~ n n n . Effects of gluonic corrections to the coupling constants are not considered. The parameters are in units of 10-6. The experimental numbers are taken from ref. [3].
( 1 ) K+~n+n+n -
(2) K+---,n+n°n °
a
b
c
d
Lg~ ), Lg~2) c~4) 5°~ ) total expt.
+ 1.500 - 0.276 +0.553 + 1.777 1.829 + 0.005
-0.167 - 0.049 -0.049 - 0.265 - 0.258 _+0.004
+ 0.0044 -0.0089 - 0.0045 - 0.0074 + 0.0022
+ 0.0044 +0.0044 + 0.0088 0.0125 + 0.0012
5o~), ~ z )
50~ ) total expt.
+0.750 -- 0.138 + 0.276 + 0.888 0.915_+0.0024
+0.167 + 0.049 + 0.049 + 0.265 0.258_+0.004
+ 0.0022 - 0.0044 - 0.0022 -0.0037_+0.001 l
- 0.0044 - 0.0044 - 0.0088 -0.0125_+0.0012
~), ~z) LP~4) LP~) total expt.
--0.750 + 0.138 -0.276 --0.888 -- 0.915 -+0.0024
--0.167 -- 0.049 -0.049 -0.265 -- 0.258 -+0.004
-- 0.0022 +0.0044 +0.0022 0.0037 _+0.0011
+ 0.0044 +0.0044 +0.0088 0.0125 _+0.0012
~),
--2.249 +0.414 -0.829 - 2.664 -2.744_+0.007
o ~ 4)
(3) KL-+n+n-n °
(4) Kn-*n°n°n °
o~7~4)
LP~ ) total expt.
LP~2)
-0.0066 +0.0133 + 0.0067 0.0111 _+0.0033
A S = 1 w e a k t r a n s i t i o n s , w h i c h c e r t a i n l y c a n n o t b e e m p i r i c a l l y e x t r a c t e d f r o m a single p h y s i c a l process. T h e a b o v e - m e n t i o n e d flaws a r e d r a m a t i c a l l y c i r c u m v e n t e d i n t h e l i m i t o f large Arc: C o u p l i n g c o n s t a n t s o f ~ 4 ) a n d 5 a ~ ) are t h e o r e t i c a l l y c a l c u l a b l e a n d c h i r a l - l o o p effects are n e g l i g i b l e i n t h e l e a d i n g 1 ~No e x p a n s i o n . I n o t h e r w o r d s , a b s o l u t e p r e d i c t i o n s i n C l a P T c a n b e a c h i e v e d i n t h e large Arc limit. C a l c u l a t i o n o f K ~ 3 n D a l i t z a m p l i t u d e s in t h e large Nc a p p r o a c h o f C h P T h a s b e e n d o n e in t h e l i t e r a t u r e [ 6 , 7 ] . T h e effects o f 5 ¢ ~ ) ( 1 / N ~ ) w e r e c o m p u t e d i n ref. [ 6 ] b y m e a n s o f t h e e f f e c t i v e w e a k h a m i l t o n i a n in c o n j u n c t i o n w i t h t h e v a c u u m - i n s e r t i o n m e t h o d ~2. T h e h a d r o n i c m a t r i x e l e m e n t s o f q u a r k c u r r e n t s are d e t e r m i n e d f r o m ~ s 2) + 5°~4)( 1 / N ~ ) c o u p l e d to e x t e r n a l W fields. T h e r e are t w o w e a k p o i n t s w i t h t h i s c a l c u l a t i o n : First, m a n y p h y s i c i s t s felt v e r y u n e a s y w i t h t h e v a c u u m - s a t u r a t i o n m e t h o d s i n c e it fails to e x p l a i n t h e n o n l e p t o n i c w e a k d e c a y s o f c h a r m e d a n d s t r a n g e m e s o n s , t h o u g h t h e u s e o f f a c t o r i z a t i o n in ref. [ 6 ] is e n t i r e l y legitim a t e ~3. S e c o n d , t h e e f f e c t i v e w e a k h a m i l t o n i a n e m p l o y e d in ref. [ 6 ] is n o t t r u l y e v a l u a t e d in t h e large N~ limit. I n t h e p r e s e n t p a p e r , we will w r i t e d o w n e x p l i c i t l y t h e large N~ c h i r a l l a g r a n g i a n r e s p o n s i b l e for n o n l e p t o n i c A S = 1 w e a k t r a n s i t i o n s a n d o b t a i n K ~ n n n a m p l i t u d e s u n a m b i g u o u s l y in t h e l e a d i n g 1/Nc e x p a n s i o n .
~2 Fajfer and G6rard [ 7 ] have calculated the contribution of ~ ) ( 1/N~) directly. [Though the weak chiral lagrangian LP~4) ( 1/N¢) was not explicitly written down by them, it is nevertheless uniquely determined from LP~2) + 50~4~( 1~No ). ] They concluded that the "QCDinspired" lagrangian approach in which L,°~4)( 1~No) is obtained by the integration of spurious chiral anomalies, as we adopt in the present paper, predicts a too small coefficient b and yields a wrong sign for the coefficient d for the A/= ½ K ~ 3 n amplitudes. As pointed out in ref. [ 1 ], this is attributed to the fact that the contribution due to the L 9 term ofCJ~ 4) [see eq. ( 5 ) ] was missed by them. ~3 This is because the factorization method is used there only for the purpose of relating K-> 3n to K ~ 2n, not for the direct evaluation of kaon decay amplitudes. 400
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Table 2 Same of table 1 but for A/= 3 decay amplitudes. The parameters are in units of 10-8 a ( 1) K+ ~ + ~ + ~
~,
j,~2)
~4) ~W~) total expt.
2~), ,,~2)
(2) K+~n+rt°n°
cj~4) .~) total expt. (3) KL~X+X-~°
5°~ ), LP~2)
~(~4) 5G~) total expt. ~), ~2) cj~4) ~) total expt.
(4) KL--~x°~°x°
b
c
d
+4.777 --0.880 + 1.759 + 5.656 7.14 _+0.36
+4.264 --0.155 + 1.237 + 5.346 5.59 _+0.66
+0.0141 --0.0282 -- 0.0141
+0.0141 --0.1128 -- 0.0987
+2.388 - 0.440 +0.880 + 2.828 3.57_+0.18
+2.738 + 0.183 +0.502 + 3.423 3.11 _+0.66
+ 0.0070 -0.0141 - 0.0071
- 0.0166 -0.0458 - 0.0624
+4.777 --0.880 + 1.759 + 5.656 7.14_+0.36
-- 1.525 +0.337 -0.735 - 1.923 -2.48_+0.48
+0.0141 -0.0282 -0.0141
--0.0307 +0.0670 +0.0363
+ 14.330 -2.640 +5.279 + 16.969 21.42 _+1.08
+0.0423 -0.0846 -0.0423
2. Large N~ higher order chiral lagrangians The lowest order chiral lagrangian including explicit chiral-symmetry breaking for low-energy Q C D is given by ~s~2) = _ _~f2 Tr
L,LU+ ~-sf~T r ( M U * + UM*),
(2)
where U=exp(2iq)/f~), 0 = ( 1 / x / 2 )0a2a, Tr(Aa2 b) = 2d ab, f ~ = 132 MeV, L , - (OuU) U* is an SU (3)R singlet, and M is a meson mass matrix with the n o n - v a n i s h i n g matrix elements M11 = M 2 2 ~-- m 2, M33 = 2m ~ - m ~. The lowest order nonleptonic AS-- 1 weak lagrangian reads
o ~ ) = _g~l/2) Tr(26LuLU ) _g~/2)(Lul3LUzi +4Lu23L~ l + 5Lu23L~2) - g 2 (73 / 2 ) (Lul3L~'l +Lu23LS{l -Lu23L~2)
,
(3)
where the subscript of the weak coupling constant g denotes the SU (3) representation u n d e r the left-handed chiral rotation, and the superscript refers to the isospin representation. Those u n k n o w n couplings are determ i n e d from the measured K ~ 2~ rates to be [ 8 ]
g(I/2).a_~(l/2) 8 1~27
~
_ 0 . 2 6 × 1 0 - S m 2,
g~3/2)~ _ 0 . 8 6 × 1 0 - 1 O m 2K
(4)
where the sign is fixed by the v a c u u m - i n s e r t i o n method. To study the higher derivative lagrangian, consider Q C D coupled to external vector (V,) and axial-vector (A u) gauge fields. The variation of the fermionic d e t e r m i n a n t u n d e r a local chiral transformation is governed by chiral anomalies. The effective action is then the integration of chiral anomalies. There are two categories of global chiral anomalies: proper (Bardeen) anomalies which contain the totally antisymmetric tensor eu,-p and spurious anomalies which do not. It is well known that the integration of topological (Bardeen) anomalies gives 401
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rise to the Wess-Zumino-Witten effective action [9 ]. Likewise, the integration of non-topological (spurious) chiral anomalies yields an action for p4 non-anomalous strong chiral lagrangians [ 10]
~(4)(1/N~) =L~ [Tr(D~'U*DuU) ]2+L2[Tr(DuU*D.U ) ] 2 + L 3 Tr(DuUeDu U) 2 +L9 Tr( FR.D~'UtD"U+ F~.DUUD"U *) + Lto Tr( UtF].UF I'~L) ,
(5)
where 8Ll =4L2 = - 2 L 3 = L 9 = - 2 L l o =Nc/48/r 2 ,
(6)
D,,U=OuU+A~U_UA], Fue;R=O~,Al.,~ r~I_,R,.., 4L,R1j, andA~t.,R = V u+_A~,.Inprinciple,thep4cou, -Ù,A L,R u +~-~ pling constants Lj can be affected by the gluon effects arising from all planar diagrams without internal quark loops. It was shown recently in ref. [ 11 ] that to the order of a 2, the couplings L3 and L~o do receive gluonic corrections. One of the great advantages of the 1/N~ expansion approach is that the structure of the weak chiral lagrangian 50~) (1/N~) is uniquely determined by ~ 2 ) + 5~4)(1/N~) based on the following three ingredients: an extremely simple structure of the weak hamiltonian at the quark level in the large Arc limit, bosonization of quark currents extracted from 5° (s2) + 5 ~ 4) ( 1/N~), and factorization valid in the large Nc approximation. The resulting p4 AS= 1 nonleptonic weak chiral lagrangians have the form [ 1 ] ~ ) ( 1/Nc) = O(8~/2~ + Ot 3/2) with
0(8'/2) = f g8 ~ {hi Tr(26Lt, LUL.L") +h2 Tr(Z6L~,L.LUL") + h 3 Tr(26LI, L.L"LU ) +h4 Tr(26LuL~) Tr(L~'L ") +h5 Tr(26 YY) +h6 Tr( [Z6, Y]LuLU) +h7 Tr( [26, Y~,~]LUL~)}, (7)
02(3/2) =g27(L,,131L~, +L~z23£{'l-Lu23£~2 +/~al3L~I +£az3L~,-/~g, z3L~2) ,
and N~
3hi = - h 2 - - - h 4 = ~
(l-q),
h6=-h7 =
£ , , = ~ {N~ ( 12 - r / ) ( L . L ~ L . + L , , L " L . ) + [ Y ,
Nc
24n2,
h3=hs=0
L u l + [ L ~,17 1},
(8)
where Y~,,,=(OI,0.U)U* , Yu~=Yg~- Y~,*,,, Y=g~'~'~,~ (ref. [ 12] ), ~/=64n2fiL3 (ilL3 being the gluonic modification to L3), and Az =2nf==830 MeV, which is the scale of chiral symmetry breaking, is fixed in this large Nc approach [ 13]. Since in the SU (3) limit g~/2) =g~3/2)/5 and g~l/2) >> g~37/e)' we thus neglected the AI= ½27plet weak operator in eq. (7) and denoted g8 =-g~l/2), g:7 = g~3/2).
3. K--, nnn
Dalitz amplitudes
The CP-conserving K - , nnn Dalitz amplitudes have the following isospin structure:
A ++-=i(n+n+n - LHw IK + ) =2(a~ +a3) - (b~ +b3 - b g ) Y+ 2 (c, +c3) ( Y~+ ~x ~) - (d, +d3 - d g ) ( y 2 _ ~X ~) ,
A +°°=i(n+n°n°iHw IK + ) = (al +a3) + (b, +b3 +b~) Y+ (c, +c3) ( y 2 + ~X 2 ) + (d, +d3 +d'3) ( y 2
A ~ - O = i ( n + n - n O l H w l K s ) = -- 2~ b .3 X + 34d 3. X Y , 402
~X 2 ) ,
(9)
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A~ °=i(n+n-n°lHwlKc) = - (al - 2 a 3 ) - (bl - 2 6 3 ) Y - (cl - 2 c 3 ) ( y 2 + i X 2 ) _ (dl - 2 d 3 ) ( y 2
A°°°=i(~°~°~°[Hw lKc ) = - 3(al -2a3)- 3(G --2c3) ( y2 + ½X2)
~X 2 ) , (9 cont'd)
,
where the subscript 1 (3) refers to A I = ½ (3) transitions. Our parametrization of K - , 3re amplitudes is the same as that of Delvin and Dickey [ 3 ] and that of Zemach [ 14 ] except that we have also included quadratic terms for A / = 3 amplitudes and corrected a sign error in ref. [ 3 ] for the quadratic coefficient of ( y 2 _ ~X 2) in A + + The K-~ 3~ decay receives two contributions: direct weak transitions and meson pole diagrams. Contributions due to the lowest order effective lagrangians ~ 9 ~ 2 ) and 5°~ ) are (Ks--, 3~ will be treated separately)
4m (l+Ym-~KY
A(K-.3n)=-gs~a where
g=gs (g27) for 2 d =
,
½ (3) transitions, and the parameters a, 7 are channel dependent [6]
A + + - : ~1 =2, yl = - 3 ; or3 = 2 , y3 = 12, A~--°"• al = -
(10)
A +°°: al = 1, ?q = 3, a 3 - 1, T3 = ~ - ~ Z z ,
1, yl = 3 ; a3 = 2 , ~3 = - ~ -15 7 ~ , 27_
AOOO:
a~ -7- - 3 , ~ = 0 ; a3 = 6 , y3 = 0 ,
(11)
where z = m,J(rnK 2 2 --m,O. 2 Eqs. (10) and ( 11 ) agree with the current-algebra analysis [ 15] except that contributions proportional to z in the slope parameters o f the A / = ~ amplitudes do not show up in the latter framework, as pointed out in ref. [6]. The numerical results shown in table 1 show clearly that the lowest order predictions for the constant and linear terms of LXI=½amplitudes are 18% and 35% respectively below the experimental values. The p4 strong chiral lagrangian 5 ~ 4) ( 1/N~) contributes to K ~ 3 ~ via the pole diagram. The resulting corrections read
4m2
4(mK~4(1--3 m~2~'] 2(mK~ 2 4 2 2 H=mJ(mKAz),
where al = a 3 = 3 H , A + + - " • 1~1 = 53H ; / ] 3
•
53H ,
A +oo: fll
A~_O.fll=_3H;fl3=_3H(1 •
9.
= -
"1- g-: ) ,
3 ~m) f2l Y +
(y2+ff~_) +fl(y2
2~)],
(12)
and 3 H ; ]~3 =
--
31t( 1 + ~z)
A ° ° ° : fll ~ - 0 ; ~]3 = 0 .
(13)
(Because of the theoretical uncertainty in the estimate of ilL> we will not consider in the present paper the effects ofgluonic corrections to K--,~nrt). It is extremely useful to apply eq. (9) to check the consistency of the results of eqs. (10) - ( 13 ). Eq. ( 12 ) implies interesting correlations between the parameters a and c as well as b and d. These correlations indicate that the higher-order chiral corrections to the constant (linear) term also fix the quadratic coefficient c(d), as first pointed out in ref. [ 16]. From tables 1 and 2 it is evident that the 5Ps(4)( l / N c ) corrections to the K ~ 3n amplitudes have wrong signs for the A I = ½ coefficients a, c, and the A / = ~3 parameters a and b (except for b of A +00). This shows that ~ 4 ) alone does not suffice to reproduce the main features of the experimental data and the inclusion of the fourderivative weak chiral lagrangian 5 : ~ ) is called for. To evaluate the effects of 5el~ ) ( 1/Arc), we first notice that only the h~, h2 and h4 operators ofO8~/2) contribute to the A I = ½ transitions. (The h6 contribution to K~3rc is compensated by that of h7.) Second, contrary to ~ s (4) , S ~ ~ contributes only via the direct-emission diagram. The computation is lengthy but otherwise straightforward. The K - , 3~ Dalitz amplitudes induced by Y ~ ) ( 1 ~Arc)are the same as eq. ( 12 ) except that a and fl are 403
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replaced by a ' a n d fl', respectively, with ~t4 c(~=-3H,
fl'~=3H,
fl,3(A+OO)=_~H,
o~=-3H,
fl'3(A++-)=-12H,
fl,3(A~-O)=~H,
fl~(AOOO)=0.
(14)
For A l = ½amplitudes, numerical results displayed in table 1 present a remarkable agreement between theory and experiment within 3% for the constant a n d linear terms ~s. This means that very little room is left for chiralloop corrections. The predicted coefficient c is just marginally in accord with data within the experimental errors, while the other coefficient d is off by three standard deviations. Clearly more accurate K 3 n data are urgent to clarify this discrepancy. In the A / = ~ sector, we see from table 2 that the linear term is generally in agreement with data, whereas the constant term is four standard deviations off from experiment. Obviously, more high-statistics experiments are required to improve the d e t e r m i n a t i o n of A/=-32 coefficients a and b, and to extract the quadratic terms c and d in order to test the chiral-lagrangian approach. We now turn to the decay K s - - , n + n - n°. In absence of CP violation, this mode is a A / = 3 transition into the I = 2 state of three pions. The CP-violating part of K s ~ n + n - n o can proceed through the A / = l part of the weak hamiltonian. We find the CP-even K s ~ n + n - n ° amplitude to be _ A+-°=2gz7 _fm2 4
9+3...~ 2+-25 mK - m ~ Az
1+3
)] I X+6g 7 5+ml~_rn,~j_ ~.~5
5-t-
m K2 - --- m ~ ]2
XY
• (15)
Again, the reader may use eq. (9) to check the consistency ofA ~--o with other K ~ 3n amplitudes. It is of interest to note that the energy dependence of the A~- - ° is of the form **6 X ( I + a Y ) with the slope parameter a = 4.4 × 10 -2. We see that the higher order chiral corrections enhance the amplitude by 24%. After the phasespace integration, the branching ratio is calculated to be
B r ( K s - , n + n - n °) = 3 . 9 × 10 -7 ,
(16)
which is to be compared with the current-algebra prediction 2.4 X 10- 7 and the present experimental upper limit 4.9 × 10- 5 [ 17 ]. The CP-violating amplitude of this decay mode approximated by eA (KL ~ n + n - n °) leads to a branching ratio of order 1 × l 0 -9, much smaller than the CP-conserving part. To summarize, the gross features of K 3 n data are satisfactorily described by the p 4 large N~ effective chiral lagrangians for weak and strong interactions. Quadratic parameters of A / = 3 K--, 3n Dalitz amplitudes are predicted. The branching ratio and the slope parameter ofKs-~n+n-n ° are found to be 3.9X 10 -7 and 4.4X 10 -2, respectively.
~:4 A different result fl'l = - 3H was obtained in ref. [ 7 ] since the L9 term of ~ 4 ) ( 1~No) was not taken into consideration in the same reference. Results ofeq. (14) may be also obtained from eqs. (4.18)-(4.21) of the first article in ref. [6] evaluated in the vacuuminsertion method by puttingfK=f~ and f= 0. Note that the parameter fin ref. [6 ] should be interpreted as the fraction of contributions to the A/= ½K-~2n amplitude due to higher-orderpenguin operators. ,5 Mass terms of ~ 4 ) ( 1~No) cannot be determined by the integration of chiral anomalies since external scalar and pseudoscalar fields do not contribute to anomalies. It should be stressed that mass operators only affect the coefficientsa and b because they do not have enough derivatives to induce quadratic terms. The effect of L5 Tr[OUUtOuU(MtU+UtM)], which is also responsible for SU (3) breaking in the ratio offK/f~, on K~ 3n has been estimated in ref. [7 ] ; it improves the discrepancyfor the AJ= ~ coefficienta. ,t6 SinceXis asymmetric in Sl and s2, it is obvious that CP-conservingKs-~n°n°n° is prohibited by Bose symmetry. Just as KL---~Kn, UP nonvaniance in Ks~n°n°n° receivestwo contributions: direct CP violation in K 1-*x°n°K° and CP violation in the K°-I(° mass matrix. 404
Volume 238, number 2,3,4
PHYSICS LETTERS B
5 April 1990
Acknowledgement T h e a u t h o r wishes to t h a n k C.Y. C h e u n g a n d W.B. Y e u n g for the c o l l a b o r a t i o n d o n e in ref. [ 6] w h i c h m a d e the p r e s e n t task possible. H e is also grateful to the Physics D e p a r t m e n t o f the B r o o k h a v e n N a t i o n a l L a b o r a t o r y for their hospitality. T h i s w o r k was s u p p o r t e d in part by the U S D e p a r t m e n t o f Energy ( c o n t r a c t n u m b e r D E A C 0 2 - 7 6 C H 0 0 0 1 6 ) a n d the N a t i o n a l Science C o u n c i l o f the R e p u b l i c o f China.
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