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How large can direct CP violation in K→3π be from chiral perturbation theory?
Physics Letters B 273 ( 1991 ) 497-500 North-Holland
How large can direct CP violation in from chiral perturbation theory?
PHYSICS LETTERS B
3rt be
G. D'Ambrosio lstituto Nazionale di Fisica Nucleare, Sezione di Napoli, 1-80125 Naples, Italy
G. Isidori Dipartimento di Fisica, University of Rome "La Sapienza", 1-00185 Rome, Italy
and N. P a v e r Dipartimento di Fisica Teorica, University of Trieste, 1-34100 Trieste, Italy and lstituto Nazionale di Fisica Nucleare, Sezione di Trieste, 1-34127 Trieste, Italy
Received 8 October 1991
Using a general parametrization, we discuss the size of direct CP violation in charged K~3n decays, as it should be expected from chiral perturbation theory including chiral corrections to order p4. These terms are required in order to remove the AI= 3 suppression. We argue that the magnitude of these effects for the slope asymmetry, although enhanced by the AI= ½rule, should be at most of the order of 10-5.
In the standard model of weak interactions direct C P violation is predicted to be different from zero. As an alternative to K - . 2 x decays, where so far the experimental results are contradictory [ 1,2 ], it is interesting to study the charge asymmetries in the decays K +--,3rt [3,4], which are n o n v a n i s h i n g only if there is direct C P violation. These studies seem very suitable at a ~-factory, where with 10 m_ 10 ~ qb's one could expect a statistical error on the asymmetry of about 10- 3_ 10- 4 [ 5 ]. In K - , 2 r t there are only two i n d e p e n d e n t isospin amplitudes, namely At=o and At= 2. Therefore, direct C P violation is suppressed in this mode by a factor 2~ since it can occur only through the interference between these two amplitudes. In K + --. 3rt there are three i n d e p e n d e n t amplitudes, i.e. A1= ~s, AI= ~U and A~=2. Thus one might hope to overcome the ~ suppression by the interference between the two A / = ½amplitudes. In what follows we discuss this aspect of direct C P violation in the general framework of chiral perturbation theory ( C H P T ) . We find results different from those obtained by previous authors [6 ], and we discuss the possible origin of this difference. Making reference to the c o n v e n t i o n a l parametrization of the Dalitz plot distribution [ 7 ] of K ~ 3x
]A(K~3x)]2oc 1 + g Y + j X
,
( 1)
one can define two CP-odd observables, i.e. the partial rate asymmetry, F(K+--,3~) -F(K --, 3x) AF= F(K+~3r~) +F(K___,3x),
(2)
and the slope asymmetry, 0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
497
Volume 273, n u m b e r 4
Ag= g(K+ ~3n)-g(K-~3n) g(K+~3n) +g(K-~3n)
PHYSICS LETTERS B
26 D e c e m b e r 1991
(3) "
In eq. ( 1 ) we have defined the Dalitz variables Y= (s3--So)/m~ and X = ( S l - s 2 ) / m 2, where s,= ( p - p i ) 2 with p and pi the four-momenta of the kaon and o f the pion i ( i = 3 indicates the "odd charge" pion), and SO = 1 (S 1 ..~_$2..]_$3 ).
We will concentrate on the decay channel K -+--,n-+n-+n~v, which can be directly related to K -+~rc-+n°n ° by using the isospin decomposition [8], and in particular on the slope asymmetry Ag where the CP-odd effect should be larger. The isospin decomposition of K -+--,n-+n + nT decays up to linear terms, in the notation of refs. [7,9,10 ], is written as follows: A (K+-~Tt+g+rt - ) = ( 2 a l - a 3 ) exp (iOl) + (ill
-- lf13)
exp(i6M) Y+X/3 73 exp (i62) Y,
A ( K - ~rt-rc-rt + ) = (2aT - a S ) exp (i61) + (fit - ½fl$) exp (i6M) Y + x / ~ 75 exp (i62) Y,
(4)
where a,, fli and 7, correspond to the three different final states: I = 1 symmetric, I = 1 with mixed symmetry and I = 2. The subscript i = 1,3 refers to the A I = ½ and A I = 3 transitions respectively. We have included the phases due to final state strong interactions, which are necessary in order to induce nonvanishing CP violating asymmetries. These phases are expected to be very small, due to the smallness of the available phase space: indeed they have been estimated both in the nonrelativistic limit [ 1 1 ] and in C H P T [ 4 ]. The two calculations coincide at the center of the Dalitz plot, and the result is 61 --6M = ½(6, --62) ~ 0.07.
(5)
From eq. (4) we obtain Ag=
I m [ ( 2 a l - a3)*(fl~ - ~f13) ]sin(fi, --6M) + I m [ (2a~ -- a3)*x/3 r3sin(6, --32) ] Re[ (2al - a3)*(fl~ - ½f13) ]cos(6. --6M) + R e [ (2a~ -- a3)*x/3 Y3cos(6~ - 6 2 ) ]
(6)
The constants a, fl and y must be estimated in a theoretical model. Since K - , 3 n is a low energy process involving would-he Goldstone bosons of the strong interaction symmetry SU (3)L × SU (3)R, the natural framework to estimate the relevant hadronic matrix elements ( 3 r ~ l H w l K ) , where Hw is the effective electroweak nonleptonic hamiltonian, is represented by C H P T [ 12 ]. In this approach transition amplitudes are expanded in powers ofpseudoscalar meson masses and momenta, by means of phenomenological chiral lagrangians [ 13,10 ]. Such lagrangians are particularly convenient computational tools to evaluate hadronic matrix elements in agreement with the low energy theorems of current algebra and PCAC, and allow the extrapolation of these theorems to higher orders in m o m e n t a consistent with the chiral symmetry of strong interactions. At the leading order p2, neglecting electromagnetic corrections, there are only two A S = 1 meson operators: one octet for A I = 1 transitions and a 27-plet for A I = ½ and A I = 3 transitions. These operators relate the K--,3n amplitudes to the K~2rc amplitudes both for the CP conserving and the CP violating parts. The fact that there are only two operators implies that there is only one relative CP violating (electroweak) phase, so that we cannot have, at this order p2, any interference between the two unsuppressed A I = ½ amplitudes. Consistent with the usual parametrization of the C K M matrix, we attribute this phase to the octet operator, and accordingly we have ImAo Imaz Imfl~ ReAo - Rea~ - Refl~ "
(7)
Im Ao/Re Ao is related to the CP violating parameter ¢' of K ~ 2~: ( 6 ~ - 6 o ) ] Im ( A ~ ) _~ e ' = i e x p [ i x/~
498
exp(~in) x/~ ImAo c oKeAo ~ (l--g2,),
(8)
Volume 273, number 4
PHYSICS LETTERS B
26 December 1991
where o)= Re A2/Re Ao-~ ~ , and ~ t takes into account isospin breaking and electromagnetic effects, which we will neglect ~'. Taking Re (E'/E) -~ 2.3 × 10- 3 from ref. [ 1 ], and using it as an upper bound, we obtain Im Ao < ~l.6×l
0_ 4
.
(9)
Then, using lowest order results o f C H P T for K + -~ 3n and eq. (5), expression (6) for Ag reduces to o) I m A o ( 6 3 9 m2 --IAgl--~(~--~M)~ R e A o k 4 + 4 m K -2 m J~
~<0.7X 10-'
(10)
2
We emphasize that the large coefficient ~ is just due to a large A / = 3 C l e b s c h - G o r d a n coefficient. Since in the CP conserving a m p l i t u d e s there is a ( 2 0 - 3 0 ) % discrepancy between the lowest order theoretical predictions o f C H P T and experimental results [ 7 ], it is i m p o r t a n t to consider higher order corrections both in the real and in the imaginary parts o f the amplitudes. By power counting we see that the n u m e r a t o r o f e q . ( 6 ) is at least o f order p 6 , resulting from an order p2 for each a m p l i t u d e and an extra p2 for the final state interaction phases 3. The next order, which will include loops and counterterms, is o f order p 8. Since higher order operators can have different electroweak phases one might hope to beat the ~2 suppression factor by chiral corrections. F o r this reason we will concentrate our analysis only on the A I = ½ a m p l i t u d e s (so that in what follows we drop the subscript 1 in a and fl). We write any o f the K--, 3n a m p l i t u d e s c~ or fl in the following form: A--A_,,tree(2)+<4) + A (4o)p+~6) ,
(11)
where superscripts ( 2 ) , ( 4 ) and ( 6 ) denote the order in the chiral expansion and o f course the tree a m p l i t u d e s include also the order p 4 counterterms. To separate out the effect o f the final state strong interactions, we further d e c o m p o s e A~oovinto absorptive plus dispersive parts: A (4o)+ (6)
_; j (4)+ -- ~a abs
(6)
.a_A (4)+ --za disp
(6)
(12)
,
so that O/ a(4)'(6) = ~1 gv (2)'(4) bs ~ tree
,
R~4)'(6) = ~M R (2)'(4) /-' a b s P" t r e e
(13)
•
As a consequence o f the fact that there is only one A I = ½o p e r a t o r at order p 2, we have the following identities: Im ~,~(2) Im t_~4) Im ~ru(4) tree ~disp abs Re ,~(2) - Re u"~. d(4) - Re ~"~- a(4) tree isp bs
|m /~(2) "'"/-'tree --
Re/~(2) /~ tree
--
i m fl~4)° Pta /~(4) *~"/Jd isp
--
i* '~" a~4) Pabs 1~ /~(4) x'~.~, b ' a b s
(14) "
Using these relations one can easily verify that at order p8, extracting the final state interaction O (p2), only the interference between A~e~ and A~,4~ remains in the n u m e r a t o r of Ag in eq. (6). We observe that A(d2p does not contribute, because it has the same electroweak phase as A ~)e, and A di,p ~6) contributes only at O (p ~o). Finally, we obtain I m A o ( R e f l (4) A g = ( ~ l - - d M ) R~-e~° Reefl(2)
R e a C4) I m a ~ 4) (Reo¢ 2~ + ima(2~
Imfl(4)'~ Imfl(2)J '
(15)
where for simplicity we have d r o p p e d the subscript tree in c~ and ft. We know from experiments that the chiral expansion works pretty well for the CP conserving amplitudes, implying that ]Refl(4)/Refl(2)], ]Rec~(4)/Re o~(2) ] ~< 1.
These corrections may be relevant only if the interference between 2d= 3 and AI= 1 amplitudes [ 14]. Since we are interested only in the order of magnitude of Ag we can disregard them. 499
Volume 273, number 4
PHYSICS LETTERS B
26 December 1991
Now, if we make the plausible assumption that the chiral expansion also works for the CP violating amplitudes, we see that [Agl~<4.5×10 - s .
(16)
This assumption, though not yet verified by experiments, is justified theoretically if the electroweak phase arises mainly from the gluon penguin operator [ 15 ], which separately satisfies current algebra relations together with the current-current quark operators [ 9 ]. Our estimate for Ag is consistent with the conclusions of refs. [ 16,17 ], but not with ref. [6 ] which gives a result about 30 times larger. The reason of this discrepancy could be that in ref. [ 6 ] the weak mesonic lagrangian is derived by applying a hadronization procedure to the quark effective nonleptonic lagrangian. It appears from eqs. ( 13 ) and (10) of ref. [ 6 ] that the resulting matrix element of the gluon penguin operator (~rt I Os I K ) does not vanish in the flavour-SU (3) limit, in contradiction with the Cabibbo-Gell-Mann theorem [ 18 ], and this incorrect chiral behaviour is also present in the matrix element (rtnTt 1051K). In this way the current-current quark operators O~, 02, 03 have different chiral behaviour from 05, so that there are two different meson operators at order p2 which could lead to a large interference between two unsuppressed A/= ½amplitudes for K ± ~ 3It. Indeed, in the subsequent ref. [ 19] by the same authors, devoted to K ° ~ 3n decays, the correct behaviour of 05 is implemented, while the amplitudes of O~, 02 and 03 remain the same. One can see that in this case the contribution from 05 becomes different by a factor of about ~o, and that a result for IAgl quite compatible with our eq. (16) is obtained. In conclusion, although probably beyond the reach of present experimental capabilities, we believe it still interesting to improve the existing, rather poor limits on Ag (and AF) at a D-factory and/or at intense kaon beams, in order to test the prediction in eq. (16). Any experimental result for the slope asymmetry larger than 10- 5 (in order of magnitude) is unexpected, and would represent a great surprise for chiral perturbation theory. We thank Professor R. Barbieri, Professor L. Maiani and Professor A. Pugliese for useful discussions.
References [ 1 ] G.D. Barr (NA31 Collab.), talk at LP-HEP 91 Geneva Conf. [ 2 ] B. Winstein (E731 Collab. ), talk at LP-HEP 91 Geneva Conf. [ 3 ] C. Avilez, Phys. Rev. D 23 ( 1981 ) 1124. [4] B. Grinstein, S.J. Rey and M. Wise, Phys. Rev. D 33 (1986) 1495. [ 5 ] R. Barbieri, L. Maiani, G. Martinelli, L. Paoluzi, N. Paver, R. Petronzio and E. Remiddi, report from the qb-factory working group LNF-91/020(R) ( 1991 ). [6] A.A. Bel'kov, G. Bohm, D. Ebert and A.Y. Lanjov, Phys. Len. B 232 (1989) 118. [ 7 ] T.J. Devlin and J.O. Dickey, Rev. Mod. Phys. 51 ( 1979 ) 237. [8] C. Zemach, Phys. Rev. 133 (1964) 1201. [9] L.F. Fi and L. Wolfenstein, Phys. Rev. D 31 (1980) 178. [ 10] J. Kambor, J. Missimer and D. Wyler, Phys. Lett. B 261 ( 1991 ) 496. [ 11 ] B.Ya Zeldovich, Sov. J. Nucl. Phys. 6 (1968) 611. [ 12 ] J. Gasser and H. Leutwyler, Nucl. Phys. B 250 ( 1985 ) 465; H. Georgi, in: Weak interactions and modern particle theory (Benjamin/Cummings, Menlo Park, 1984). [13] J.A. Cronin, Phys. Rev. 161 (1967) 1483. [ 14 ] G. Buchalla, A.J. Buras and M.K. Harlander, Nucl. Phys. B 337 (1990) 313; M. Lusignoli, L. Maiani, G. Martinelli and L. Reina, University of Rome preprint n.792 6/3/91; G. lsidori, L. Maiani and A. Pugliese, in preparation. [ 15 ] M.A. Shifrnan, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 120 ( 1977 ) 316. [ 16] J.F. Donoghue, B. Holstein and G. Valencia, Phys. Rev. D 36 (1987) 798. [17] H.-Y. Cheng, Phys. Rev. D 43 (1991) 1579. [ 181 N. Cabibbo, Phys. Rev. Lett. 12 (1964) 62; M. Gell-Mann, Phys. Rev. Lett. 12 (1964) 155. [ 19 ] A.A. Bel'kov, G. Bohm, D. Ebert and A.Y. Lanjov, Nucl. Phys. B (Proc. Suppl. ) 10 (1990) 367; Nucl. Phys. B 359 ( 1991 ) 322.
500
How large can direct CP violation in from chiral perturbation theory?
PHYSICS LETTERS B
3rt be
G. D'Ambrosio lstituto Nazionale di Fisica Nucleare, Sezione di Napoli, 1-80125 Naples, Italy
G. Isidori Dipartimento di Fisica, University of Rome "La Sapienza", 1-00185 Rome, Italy
and N. P a v e r Dipartimento di Fisica Teorica, University of Trieste, 1-34100 Trieste, Italy and lstituto Nazionale di Fisica Nucleare, Sezione di Trieste, 1-34127 Trieste, Italy
Received 8 October 1991
Using a general parametrization, we discuss the size of direct CP violation in charged K~3n decays, as it should be expected from chiral perturbation theory including chiral corrections to order p4. These terms are required in order to remove the AI= 3 suppression. We argue that the magnitude of these effects for the slope asymmetry, although enhanced by the AI= ½rule, should be at most of the order of 10-5.
In the standard model of weak interactions direct C P violation is predicted to be different from zero. As an alternative to K - . 2 x decays, where so far the experimental results are contradictory [ 1,2 ], it is interesting to study the charge asymmetries in the decays K +--,3rt [3,4], which are n o n v a n i s h i n g only if there is direct C P violation. These studies seem very suitable at a ~-factory, where with 10 m_ 10 ~ qb's one could expect a statistical error on the asymmetry of about 10- 3_ 10- 4 [ 5 ]. In K - , 2 r t there are only two i n d e p e n d e n t isospin amplitudes, namely At=o and At= 2. Therefore, direct C P violation is suppressed in this mode by a factor 2~ since it can occur only through the interference between these two amplitudes. In K + --. 3rt there are three i n d e p e n d e n t amplitudes, i.e. A1= ~s, AI= ~U and A~=2. Thus one might hope to overcome the ~ suppression by the interference between the two A / = ½amplitudes. In what follows we discuss this aspect of direct C P violation in the general framework of chiral perturbation theory ( C H P T ) . We find results different from those obtained by previous authors [6 ], and we discuss the possible origin of this difference. Making reference to the c o n v e n t i o n a l parametrization of the Dalitz plot distribution [ 7 ] of K ~ 3x
]A(K~3x)]2oc 1 + g Y + j X
,
( 1)
one can define two CP-odd observables, i.e. the partial rate asymmetry, F(K+--,3~) -F(K --, 3x) AF= F(K+~3r~) +F(K___,3x),
(2)
and the slope asymmetry, 0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
497
Volume 273, n u m b e r 4
Ag= g(K+ ~3n)-g(K-~3n) g(K+~3n) +g(K-~3n)
PHYSICS LETTERS B
26 D e c e m b e r 1991
(3) "
In eq. ( 1 ) we have defined the Dalitz variables Y= (s3--So)/m~ and X = ( S l - s 2 ) / m 2, where s,= ( p - p i ) 2 with p and pi the four-momenta of the kaon and o f the pion i ( i = 3 indicates the "odd charge" pion), and SO = 1 (S 1 ..~_$2..]_$3 ).
We will concentrate on the decay channel K -+--,n-+n-+n~v, which can be directly related to K -+~rc-+n°n ° by using the isospin decomposition [8], and in particular on the slope asymmetry Ag where the CP-odd effect should be larger. The isospin decomposition of K -+--,n-+n + nT decays up to linear terms, in the notation of refs. [7,9,10 ], is written as follows: A (K+-~Tt+g+rt - ) = ( 2 a l - a 3 ) exp (iOl) + (ill
-- lf13)
exp(i6M) Y+X/3 73 exp (i62) Y,
A ( K - ~rt-rc-rt + ) = (2aT - a S ) exp (i61) + (fit - ½fl$) exp (i6M) Y + x / ~ 75 exp (i62) Y,
(4)
where a,, fli and 7, correspond to the three different final states: I = 1 symmetric, I = 1 with mixed symmetry and I = 2. The subscript i = 1,3 refers to the A I = ½ and A I = 3 transitions respectively. We have included the phases due to final state strong interactions, which are necessary in order to induce nonvanishing CP violating asymmetries. These phases are expected to be very small, due to the smallness of the available phase space: indeed they have been estimated both in the nonrelativistic limit [ 1 1 ] and in C H P T [ 4 ]. The two calculations coincide at the center of the Dalitz plot, and the result is 61 --6M = ½(6, --62) ~ 0.07.
(5)
From eq. (4) we obtain Ag=
I m [ ( 2 a l - a3)*(fl~ - ~f13) ]sin(fi, --6M) + I m [ (2a~ -- a3)*x/3 r3sin(6, --32) ] Re[ (2al - a3)*(fl~ - ½f13) ]cos(6. --6M) + R e [ (2a~ -- a3)*x/3 Y3cos(6~ - 6 2 ) ]
(6)
The constants a, fl and y must be estimated in a theoretical model. Since K - , 3 n is a low energy process involving would-he Goldstone bosons of the strong interaction symmetry SU (3)L × SU (3)R, the natural framework to estimate the relevant hadronic matrix elements ( 3 r ~ l H w l K ) , where Hw is the effective electroweak nonleptonic hamiltonian, is represented by C H P T [ 12 ]. In this approach transition amplitudes are expanded in powers ofpseudoscalar meson masses and momenta, by means of phenomenological chiral lagrangians [ 13,10 ]. Such lagrangians are particularly convenient computational tools to evaluate hadronic matrix elements in agreement with the low energy theorems of current algebra and PCAC, and allow the extrapolation of these theorems to higher orders in m o m e n t a consistent with the chiral symmetry of strong interactions. At the leading order p2, neglecting electromagnetic corrections, there are only two A S = 1 meson operators: one octet for A I = 1 transitions and a 27-plet for A I = ½ and A I = 3 transitions. These operators relate the K--,3n amplitudes to the K~2rc amplitudes both for the CP conserving and the CP violating parts. The fact that there are only two operators implies that there is only one relative CP violating (electroweak) phase, so that we cannot have, at this order p2, any interference between the two unsuppressed A I = ½ amplitudes. Consistent with the usual parametrization of the C K M matrix, we attribute this phase to the octet operator, and accordingly we have ImAo Imaz Imfl~ ReAo - Rea~ - Refl~ "
(7)
Im Ao/Re Ao is related to the CP violating parameter ¢' of K ~ 2~: ( 6 ~ - 6 o ) ] Im ( A ~ ) _~ e ' = i e x p [ i x/~
498
exp(~in) x/~ ImAo c oKeAo ~ (l--g2,),
(8)
Volume 273, number 4
PHYSICS LETTERS B
26 December 1991
where o)= Re A2/Re Ao-~ ~ , and ~ t takes into account isospin breaking and electromagnetic effects, which we will neglect ~'. Taking Re (E'/E) -~ 2.3 × 10- 3 from ref. [ 1 ], and using it as an upper bound, we obtain Im Ao < ~l.6×l
0_ 4
.
(9)
Then, using lowest order results o f C H P T for K + -~ 3n and eq. (5), expression (6) for Ag reduces to o) I m A o ( 6 3 9 m2 --IAgl--~(~--~M)~ R e A o k 4 + 4 m K -2 m J~
~<0.7X 10-'
(10)
2
We emphasize that the large coefficient ~ is just due to a large A / = 3 C l e b s c h - G o r d a n coefficient. Since in the CP conserving a m p l i t u d e s there is a ( 2 0 - 3 0 ) % discrepancy between the lowest order theoretical predictions o f C H P T and experimental results [ 7 ], it is i m p o r t a n t to consider higher order corrections both in the real and in the imaginary parts o f the amplitudes. By power counting we see that the n u m e r a t o r o f e q . ( 6 ) is at least o f order p 6 , resulting from an order p2 for each a m p l i t u d e and an extra p2 for the final state interaction phases 3. The next order, which will include loops and counterterms, is o f order p 8. Since higher order operators can have different electroweak phases one might hope to beat the ~2 suppression factor by chiral corrections. F o r this reason we will concentrate our analysis only on the A I = ½ a m p l i t u d e s (so that in what follows we drop the subscript 1 in a and fl). We write any o f the K--, 3n a m p l i t u d e s c~ or fl in the following form: A--A_,,tree(2)+<4) + A (4o)p+~6) ,
(11)
where superscripts ( 2 ) , ( 4 ) and ( 6 ) denote the order in the chiral expansion and o f course the tree a m p l i t u d e s include also the order p 4 counterterms. To separate out the effect o f the final state strong interactions, we further d e c o m p o s e A~oovinto absorptive plus dispersive parts: A (4o)+ (6)
_; j (4)+ -- ~a abs
(6)
.a_A (4)+ --za disp
(6)
(12)
,
so that O/ a(4)'(6) = ~1 gv (2)'(4) bs ~ tree
,
R~4)'(6) = ~M R (2)'(4) /-' a b s P" t r e e
(13)
•
As a consequence o f the fact that there is only one A I = ½o p e r a t o r at order p 2, we have the following identities: Im ~,~(2) Im t_~4) Im ~ru(4) tree ~disp abs Re ,~(2) - Re u"~. d(4) - Re ~"~- a(4) tree isp bs
|m /~(2) "'"/-'tree --
Re/~(2) /~ tree
--
i m fl~4)° Pta /~(4) *~"/Jd isp
--
i* '~" a~4) Pabs 1~ /~(4) x'~.~, b ' a b s
(14) "
Using these relations one can easily verify that at order p8, extracting the final state interaction O (p2), only the interference between A~e~ and A~,4~ remains in the n u m e r a t o r of Ag in eq. (6). We observe that A(d2p does not contribute, because it has the same electroweak phase as A ~)e, and A di,p ~6) contributes only at O (p ~o). Finally, we obtain I m A o ( R e f l (4) A g = ( ~ l - - d M ) R~-e~° Reefl(2)
R e a C4) I m a ~ 4) (Reo¢ 2~ + ima(2~
Imfl(4)'~ Imfl(2)J '
(15)
where for simplicity we have d r o p p e d the subscript tree in c~ and ft. We know from experiments that the chiral expansion works pretty well for the CP conserving amplitudes, implying that ]Refl(4)/Refl(2)], ]Rec~(4)/Re o~(2) ] ~< 1.
These corrections may be relevant only if the interference between 2d= 3 and AI= 1 amplitudes [ 14]. Since we are interested only in the order of magnitude of Ag we can disregard them. 499
Volume 273, number 4
PHYSICS LETTERS B
26 December 1991
Now, if we make the plausible assumption that the chiral expansion also works for the CP violating amplitudes, we see that [Agl~<4.5×10 - s .
(16)
This assumption, though not yet verified by experiments, is justified theoretically if the electroweak phase arises mainly from the gluon penguin operator [ 15 ], which separately satisfies current algebra relations together with the current-current quark operators [ 9 ]. Our estimate for Ag is consistent with the conclusions of refs. [ 16,17 ], but not with ref. [6 ] which gives a result about 30 times larger. The reason of this discrepancy could be that in ref. [ 6 ] the weak mesonic lagrangian is derived by applying a hadronization procedure to the quark effective nonleptonic lagrangian. It appears from eqs. ( 13 ) and (10) of ref. [ 6 ] that the resulting matrix element of the gluon penguin operator (~rt I Os I K ) does not vanish in the flavour-SU (3) limit, in contradiction with the Cabibbo-Gell-Mann theorem [ 18 ], and this incorrect chiral behaviour is also present in the matrix element (rtnTt 1051K). In this way the current-current quark operators O~, 02, 03 have different chiral behaviour from 05, so that there are two different meson operators at order p2 which could lead to a large interference between two unsuppressed A/= ½amplitudes for K ± ~ 3It. Indeed, in the subsequent ref. [ 19] by the same authors, devoted to K ° ~ 3n decays, the correct behaviour of 05 is implemented, while the amplitudes of O~, 02 and 03 remain the same. One can see that in this case the contribution from 05 becomes different by a factor of about ~o, and that a result for IAgl quite compatible with our eq. (16) is obtained. In conclusion, although probably beyond the reach of present experimental capabilities, we believe it still interesting to improve the existing, rather poor limits on Ag (and AF) at a D-factory and/or at intense kaon beams, in order to test the prediction in eq. (16). Any experimental result for the slope asymmetry larger than 10- 5 (in order of magnitude) is unexpected, and would represent a great surprise for chiral perturbation theory. We thank Professor R. Barbieri, Professor L. Maiani and Professor A. Pugliese for useful discussions.
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