- Email: [email protected]
Nc expansion
5 October 2000
Physics Letters B 490 Ž2000. 213–222 www.elsevier.nlrlocaternpe
K 0 –K 0 mixing in the 1rNc expansion Santiago Peris a , Eduardo de Rafael b a
Grup de Fısica Teorica and IFAE, UniÕersitat Autonoma de Barcelona, 08193 Barcelona, Spain ´ ` ` Centre de Physique Theorique, CNRS-Luminy, Case 907, F-13288 Marseille Cedex 9, France ´
b
Received 14 June 2000; accepted 15 August 2000 Editor: R. Gatto
Abstract We present the result for the invariant BˆK factor of K 0 –K 0 mixing in the chiral limit and to next-to-leading order in the 1rNc expansion. We explicitly demonstrate the cancellation of the renormalization scale and scheme dependences between short- and long-distance contributions in the final expression. Numerical estimates are then given, by taking into account increasingly refined short- and long-distance constraints of the underlying QCD Green’s function which governs the BˆK factor. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The Standard Model predicts strangeness changing transitions with D S s 2 via two virtual W-exchanges between quark lines, the so-called box diagrams. The low-energy physics of these transitions is governed by an effective Hamiltonian which is proportional to the local four-quark operator, Žsummation over colour indices 1y g within brackets is understood and q L ' 2 5 q,. QDSs2 Ž x . ' Ž sLŽ x . g md LŽ x . . Ž sLŽ x . gm d LŽ x . . ,
Ž 1.1 . l q s Vqd) Vqs
modulated by a quadratic form of the flavour mixing matrix elements , q s u,c,t , with coefficient functions F1,2,3 of the heavy masses of the fields t, Z 0 , W ", b, and c which have been integrated out 1 : D Ss2 Heff s
GF2 MW2 4p 2
l2c F1 q l2t F2 q 2 l c l t F3 CD Ss2 Ž m . QD Ss2 Ž x . .
Ž 1.2 .
The operator QDSs2 is multiplicatively renormalizable and has an anomalous dimension g Ž a s . which in perturbative QCD ŽpQCD. is defined by the equation
m2
1
d dm
² QDSs2 : s y 12 g Ž a s . ² QD Ss2 : , 2
g Ž as . s
as p
g1 q
as
2
žp/g
For a detailed discussion, see e.g. Refs. w1,2x and references therein.
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 9 7 7 - 1
2 q PPP
.
Ž 1.3 .
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
214
with
1 6
Ž b1 s Žy11 Nc q 2 n f . . g1 s
3 2
ž
1y
1 Nc
/
g2 s
and
1 32
ž
1y
1 Nc
/
y 193 Nc q 43 n f y 21 q 57
1 Nc
q 4b 1 k ,
Ž 1.4 .
where k s 0 in the naıÕe ¨ dimensional renormalization scheme ŽNDR. and k s y4 in the ’t Hooft–Veltman renormalization scheme ŽHV.. The renormalization m-scale dependence of the Wilson coefficient CDSs2 Ž m . in 1 5 2 2 Eq. Ž1.2. is then Ž b 2 s y 17 12 Nc q 4 Ž Nc y 1 . r2 NC n f q 12 Nc n f . CDSs2 Ž m . s 1 q
™
1q
as Ž m . 1
as Ž m . p
p
Ž
1433 1936
b1
ž
g2 q
q 18 k .
ž
g1 yb 1
b2
/ž
g 1 r Žy b 1 .
1
as Ž m .
/
Ž 1.5 .
Ž9r11 .Ž 1rN c .
1
as Ž m .
/
,
Ž 1.6 .
where the second line gives the result to next-to-leading order in the 1rNc expansion, which is the approximation at which we shall be working here. The matrix element ² K 0 < QDSs2 Ž 0 . < K 0 : ' 43 f K2 MK2 BK Ž m . , defines the so-called BK -parameter parameters in the phenomenological QCD, the four quark operator QDSs2 to lowest order in chiral perturbation
Ž sL g m d L . s
d LQC D Ž x . d lm Ž x . 32
Ž 1.7 .
of K 0 y K 0 mixing at short distances, which is one of the crucial studies of CP-violation in the Standard Model. In the large-Nc limit of factorizes into a product of two current operators. Each of these currents, theory, has a simple bosonic realization:
´ 2i F tr l 2 0
32
Ž D mU † . U s
1
'2
F0 E m K 0 q PPP ,
Ž 1.8 .
where F0 denotes the coupling constant of the pion in the chiral limit, l32 is given by the matrix l32 s d i3 d 2 j in flavour space and U is the 3 = 3 unitary matrix in flavour space, U s 1 q PPP , which collects the Goldstone fields and which under chiral rotations transforms like U VR UVL†. The large-Nc approximation, with inclusion of the chiral corrections in the factorized contribution, leads to the result:
™
BK < Nc ™ ` s 34
and
CD Ss2 Ž m . < Nc ™ ` s 1 .
Ž 1.9 .
In full generality, the bosonization of the four-quark operator QDSs2 Ž x . to lowest order in the chiral expansion is described by an effective operator which is of O Ž p 2 . 2 : QDSs2 Ž x .
2
´ y F4 g
See e.g. Refs. w3,4x.
4 0
D Ss2
Ž m . tr l32 Ž D m U † . Ul32 Ž Dm U † . U ,
Ž 1.10 .
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
215
where g DSs2 Ž m . is a dimensionless coupling constant which depends on the underlying dynamics of spontaneous chiral symmetry breaking ŽS x SB. in QCD. The relation between the coupling g DSs2 Ž m . and the phenomenological BK factor defined in Eq. Ž1.7. is simply BK Ž m ,m u , d , s
™ 0. s
3 4
g D Ss2 Ž m . .
Ž 1.11 .
Notice that the coupling g DSs2 Ž m ., and hence BK , is perfectly well defined in the chiral limit, while the matrix element in Eq. Ž1.7. vanishes in the chiral limit as a chiral power. To lowest order in the chiral expansion, the relation to the so called inÕariant BˆK -factor is as follows: BˆK s 34 CD Ss2 Ž m . = g D Ss2 Ž m . ,
Ž 1.12 .
which means that the coupling g DSs2 Ž m . must have a m dependence Žand a scheme dependence. which must cancel with the m dependence Žand scheme dependence. of the Wilson coefficient CDSs2 Ž m .. The purpose of this note is to show how the mechanism of cancellations works in practice within the framework of the 1rNc expansion. This will allow us to give a numerical result for BˆK which is valid to lowest order in the chiral expansion and to next-to-leading order in the 1rNc expansion. We wish to emphasize that the novel feature of this work is that, to our knowledge, it is the first calculation of BˆK which explicitly shows the cancellations of scale and scheme dependences.
2. Bosonization of four-quark operators The QCD Lagrangian in the presence of external chiral sources lm , rm of left- and right-currents, but with neglect of scalar and pseudoscalar sources which is justified in the chiral limit approximation we shall be working here, has the form Ž a. Ž a. mn LQC D s y 14 Gmn G q iq Dx q ,
Ž 2.1 .
with Dx the Dirac operator Dx s g m Ž Em q iGm . y ig m lm
1 y g5 2
q rm
1 q g5 2
.
Ž 2.2 .
The bosonization of the four-quark operator QDSs2 Ž x . is formally defined by the functional integral w3x ² QDSs2 Ž x . : s Tr Dxy1 i
4
y d y
d Dx d lm Ž x . 32 d4q
H H Ž 2p .
4
Tr Dxy1 i
d Dx m
d l Ž x . 32
ž
eyi qPŽ xyy. Tr Dxy1 i
d Dx d lm Ž x . 32
Dxy1 i
d Dx m
d l Ž y . 32
/
,
Ž 2.3 .
where the trace Tr here also includes the functional integration over gluons in large Nc . The first term corresponds to the factorized pattern and gives the contributions of O Ž Nc2 .; the second term corresponds to the unfactorized pattern and it involves an integral, Žwhich is regularization dependent,. over all virtual momenta q. This is the term which gives the next-to-leading O Ž Nc . contribution we are interested in.
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
216
To proceed further, it is convenient to use the Schwinger’s operator formalism. With Pu the conjugate momentum operator, in the absence of external chiral fields, the full quark propagator is
Ž x<
1 Dx
< y. s Ž x < Pu q g
a
la
i 1 y g5 2
q ra
1 q g5
< y.
Ž x
with
E i
Em
y Gm d Ž x y y . .
2
Ž 2.4 . The chiral expansion is then defined as an expansion in the la and ra external sources. The precise relation between the formal bosonization in Eq. Ž2.3. and the explicit chiral realization in Eq. Ž1.10. can best be seen from the fact that: D mU †U s E mU †U q iU † r mU y il m. This shows that there are a priori six different ways to compute the constant g DSs2 Ž m ., although they are all related by chiral gauge invariance. One possible choice is the term3 D Ss2 L27 Ž x. sy
F04 4
g D Ss2 Ž m . PPP ytr l32 U † r m Ul32 U † rm U q PPP .
½
5
Ž 2.5 .
This is a convenient choice because the underlying QCD Green’s function is the four-point function, w L smd Ž x . ' Ý a s a Ž x .g m 1 y2 g 5 d a Ž x . and R dms Ž x . ' Ý a d a Ž x .g m 1 q2 g 5 s a Ž x .x, 3 W Lmanb d 4 x d 4 y d 4 z e i qP x e i lP y eyi lP z ²0 < T L smd Ž x . R ad s Ž y . Lns d Ž 0 . R dbs Ž z . 4 <0: , R L R Ž q,l . s lim i
™0 H
l
Ž 2.6 .
Ž . In fact, what we need, as seen in Eq. Ž2.3., is the integral of the unfactorized four-point function W Lmanb R L R q,l over the four-vector q with the Lorentz indices of the two left-currents contracted. This is a quantity which is a good order parameter of S x SB. The integral over the solid angle has the form Ž Q 2 ' yq 2 .
H
< d V q gmn W LmRanb L R Ž q,l . unfactorized s
ž
l al b l2
/
2 y g a b W LŽ1. R LR ŽQ . ,
Ž 2.7 .
where the transversality in the four-vector l follows from current algebra Ward identities. We are still left with Ž 2 . over the full euclidean range: 0 F Q 2 F ` which has to be done an integral of the invariant function W LŽ1. R LR Q in the same renormalization scheme as the calculation of the short-distance Wilson coefficient CDSs2 Ž x . in Eq. Ž1.2. has been done, i.e. in the MS-scheme. The coupling constant g DSs2 Ž m . defined with dimensional regularization in d s 4 y e dimensions is then given by the following integral, g DSs2 Ž m , e . s 1 y
m2had . 32p 2 F02
Ž 4pm2rm2had . .
er2
G Ž 2 y er2 .
`
y e r2
H0 dz z
ž
W w z x 'z
m2had . F02
/
W LŽ 1R.L R Ž z m2had . . ,
Ž 2.8 .
where, for convenience, we have normalized Q 2 to a characteristic hadronic scale m2had. , Ž z ' Q 2rm2had. ,. which in practice we shall choose within the range: 1 GeV F m had.F 2 GeV.
3 Other choices are indeed possible but, in general, the underlying QCD Green’s functions have pieces which are not order parameters and spurious contributions which depend on the regularization. We have shown the equivalence among the various possible choices, but we postpone the detailed discussion which is rather technical to a longer publication.
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
217
3. Bˆ K to next-to-leading order in the 1 r Nc expansion In full generality, Green’s functions in QCD at large Nc w5x are given by sums over an infinite number of hadronic poles w6x. In the particular case of the function W w z x, wsee Eqs. Ž2.8., Ž2.7. and Ž2.6.x, it can be shown that the most general ansatz in large-Nc QCD is an infinite sum of poles and double poles of the form `
W w z x s6y
`
1
Ý ai r
is1
1
`
i
is1
Ý bi r 2 q Ý
y i
is1
½
ai
1
Ž z q ri .
q bi
1
Ž z q ri .
2
5
,
Ž 3.1 .
where r i s Mi2rm 2had. and Mi is the mass of the i-th narrow hadronic state. The first term on the r.h.s. is the contribution from the Goldstone poles to the W w z x-function. This constant term is known from previous calculations, Žsee Refs. w7–9x,. with which we agree. Constant terms, however, do not contribute to the integral in Eq. Ž2.8. defined in dimensional regularization. In the absence of a solution of large-Nc QCD, the actual values of the mass-ratios r i , and residues a i and bi , remain unknown. As discussed below, there are, however, short-distance and long-distance constraints that the function W w z x has to obey. With W w z x given by Eq. Ž3.1., the integral in Eq. Ž2.8. can be done analytically, with the result g DSs2 Ž m , e . s 1 y
m2had . 32p
2
½ž
F02
q log4p y g E q log m2rm2had . q 1
e
`
y
`
2
`
1
Ý a i log ri q Ý bi r
is1
is1
i
5
/
Ý ai is1
.
Ž 3.2 .
We shall next explore the short-distance properties and long-distance properties which are presently known about the function W w z x, and hence about the parameters a i , bi and r i . v
Ž 2 . is governed by the OPE. We find the result The large-Q 2 behaviour of the function W LŽ1. R LR Q 2 2 lim W LŽ1. R L R Ž Q . s 24p
™`
Q2
as
1q
p
e 1 2 6
Ž5qk . qO
F04
2
as
žp/
Q4
q O Ž 1rQ 8 . ,
Ž 3.3 .
with k s 0 in the NDR scheme and k s y4 in the HV scheme The fact that the residue of the in the OPE is known, entails the constraint `
1 Q4
power term
e
Ý ai s R q 2 S ,
Ž 3.4 .
is1
with R s 24p 2
as p
qO
Nc a s2
F02
p2
m2had .
ž /
S s 4p 2 Ž 5 q k .
and
as p
qO
Nc a s2
F02
p2
m2had .
ž /
.
Ž 3.5 .
. Ž m . in the MS-scheme: This allows us to define the renormalized coupling constant g DŽ rSs2 . g DŽ rSs2 Ž m. s1y
´ž a Žm
had .
32p
2
F02
`
½
R log m2rm2had . q R q S q
.
/
Ý is1
Ž9r11 .Ž 1rNc .
as Ž m .
s
m2had .
½
1y
1
m2had .
2 16p
2
F02
ya i log r i q bi
1
ri
`
ž
RqSq
Ý is1
ya i log r i q bi
1
ri
5
Ž 3.6 .
/5
,
Ž 3.7 .
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
218
where in the second line we have written the renormalization group improved result. We insist on the fact that this result, contrary to the various large-Nc inspired calculations which have been published so far w3,7–12x is the full result to next-to-leading order in the 1rNc expansion. The renormalization m scale dependence as well as the scheme dependence in the factor S cancel exactly, at the next-to-leading log approximation, with the short distance m and scheme dependences in the Wilson coefficient CDSs2 Ž m . in Eq. Ž1.6.. Therefore, the full expression of the invariant BˆK -factor to lowest order in the chiral expansion and to next-to-leading order in the 1rNc expansion, which takes into account the hadronic contribution from light quarks below a mass scale m had. is then BˆK s
ž
3r11
1
a s Ž m had . .
y
m2had . 32p
2
3
/ ½ 4
1y
a s Ž m had . . 1229 p
`
ya i log r i q bi
Ý
F02 is1
1936 1
ri
5
qO
ž
Nc a s2 Ž m had . .
.
p2
/ Ž 3.8 .
The numerical choice of the hadronic scale m had. is, a priori, arbitrary. In practice, however, m had. has to be sufficiently large so as to make meaningful the truncated pQCD series in the first line. Our final error in the numerical evaluation of BˆK will include the small effect of fixing m had. within the range 1 GeV F m had.F 2 GeV. Ž 2 . implies the sum rule The fact that there is no 1rQ 2 term in the OPE expansion of the function W LŽ1. RLR Q v
` is1
`
1
Ý ai r
q i
1
Ý bi r 2 s 6 .
is1
Ž 3.9 .
i
This sum rule plays an equivalent role ˆ to the 1st Weinberg sum rule in the case of the P L R Ž Q 2 . two-point 4 function . Like in this case, it guarantees the cancellation of quadratic divergences in the integral in Eq. Ž2.8. and it relates the contribution from Goldstone particles to a specific sum of resonance state contributions. What this means in practice is that, although the Goldstone pole does not contribute to the integral in Eq. Ž2.8. defined in dimensional regularization, the normalization of the function W w z x at z s 0 is, precisely, fixed by the residue of the Goldstone pole contribution; i.e., W w0x s 6. Ž 2. We have found that there is no 1rQ 6 power correction in the OPE expansion of the function W LŽ1. RLR Q , which implies the additional short-distance sum rule v
`
Ý
`
a i ri s
is1
Ý bi .
Ž 3.10 .
is1
The slope at the origin of the function W w z x is fixed by a linear combination of coupling constants of the O Ž p 4 . Gasser–Leutwyler Lagrangian w14x, with the result 5 v
`
1
`
i
is1
1
m2had .
i
F02
Ý a i r 2 q 2 Ý bi r 3 s 24
is1
w 2 L1 q 5L2 q L3 q L9 x .
Ž 3.11 .
The numerical evaluation of BˆK in Eq. Ž3.8. requires the input of the mass-ratios r i and couplings a i and bi of the narrow states which fill the details of the hadronic spectrum of light flavours. Our goal is to find the minimal hadronic ansatz of pole and double pole terms of the form shown in Eq. Ž3.1. which satisfies the
4 5
See e.g. Ref. w13x. The r.h.s. in Eq. Ž3.11. agrees with the expression reported in Ref. w9x.
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
219
short-distance and long-distance constraints discussed above, and use then this minimal hadronic ansatz to evaluate the integral in Eq. Ž2.8.. The fact that the function W w z x is an order parameter of S x SB ensures that it has a smooth behaviour in the ultraviolet and it is then reasonable to approximate it by a few states. ŽThere is no need here for an infinite number of narrow states to reproduce the asymptotic behaviour of pQCD, as it would be the case with a Green’s function which is not an order parameter of S x SB.. This procedure has been successfully tested in other examples, like the calculation of the electroweak pqy p 0 mass difference w13x and the decay of pseudoscalars into lepton pairs w15x, and it has been applied w16x to the evaluation of matrix elements of electroweak penguin operators as well 6 .
4. Numerical evaluation and conclusions We now proceed to the numerical evaluation of the analytic result for BˆK in Eq. Ž3.8. by successive improvements in the hadronic input. v
The crudest approximation is the one where the only terms of the next-to-leading order in the 1rNc expansion which are retained are those from the first line in Eq. Ž3.8.. This implicitly assumes that all the way down to the hadronic m had. scale there is only pQCD running and that the hadronic contribution below the m had. scale due to light quarks is negligible. One can see that from the integral in Eq. Ž2.8., if we write it in an equivalent cut-off form: 2
1
H0 dz W w z x q H1L rm
2 had .
dz WOPE w z x ,
Ž 4.1 .
with L2 s m2 exp 1 q S , and where in the second integral we use the asymptotic OPE expression in Eq. R Ž3.3.. The approximation we are discussing is the one where the first low-energy integral is simply neglected. The result of this next-to-leading log Žnll. approximation is to bring up the large-Nc prediction in Eq. Ž1.9. to a value which, including an estimate of higher order corrections, Žwe use L M SŽ3. s Ž372 " 40. MeV,. but ignoring the systematic errors involved in the neglect of the hadronic contribution, would be
ž
BˆK < nll ; 0.96 " 0.04
v
/
for
m had .s 1.4 GeV .
Ž 4.2 .
The problem, however, with this simple estimate is that the underlying assumption of neglecting entirely the low-energy hadronic integral does not satisfy any of the long-distance matching constraints discussed in the previous section. We can improve on the previous estimate by taking into account the contribution from the lowest hadronic state, the r vector meson, alone. In this approximation one can fix the large-Nc hadronic ansatz in Eq. Ž3.1. to satisfy the two matching constraints in Eqs. Ž3.9. and Ž3.11.. In fact, it has been shown that the phenomenological values of the L i constants which appear on the r.h.s. of Eq. Ž3.11. agree well with those obtained from the integration of the lowest Õector state w23–25x alone7. This fixes de r mass to Mr s 760 MeV, in the chiral limit where F0 s 85.3 MeV. Using this approximation w25x, the sum rule in Eq. Ž3.11. becomes simply a 1 r 1 q 2 b 1 s 21 r 12 . We can then solve for a 1 and b 1 in terms of r 1 s M V2rm 2had. with the result: a 1 s y9r 1 and b 1 s 15 r 12 . The resulting W w z x function, normalized to its value at the origin
6 7
See also Refs. w17–19x and Refs. w20–22x for other recent evaluations of electroweak penguin operators. The constant L3 gets also an extra contribution from the lowest scalar state, but it is rather small.
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
220
W w0x s 6, is plotted in Fig. 1a below as a function of z s Q 2rm2had. . Also shown in the same plot are the chiral perturbation behaviour of the W w z x function from the knowledge of its value at the origin and the slope at the origin, Žthe dotted line,. and the corresponding behaviour from the OPE expression in Eq. Ž3.3., Žthe dashed line.. We can now make an improved estimate of BˆK by calculating the integral
H0ˆdz W w z x qHzˆL rm z
2
2 had .
dz WOPE w z x ,
Ž 4.3 .
with W w z x in the first integral approximated by the lowest hadronic state ansatz just discussed. The choice of zˆ which minimizes the dependence of BˆK on zˆ is the one at which the two curves W w z x and WOPE w z x intersect. As seen in Fig. 1a, Žwhich corresponds to m had., 1.4 GeV, F0 s 85.3 MeV and Mr s 760 MeV,. this happens at a value zˆ , 0.15 i.e. Q , 540 MeV; a little bit too low but not unreasonable. This improved approximation brings down the simple next-to-leading log prediction in Eq. Ž4.2. to BˆK < VM D , 0.48 " 0.04 ,
v
Ž 4.4 .
where, again, the error corresponds to an estimate of higher order perturbative corrections only. Incomplete as it may be, this estimate shows a crucial qualitative issue, which is the fact that the low-energy hadronic contribution below a mass scale m had.; 1.4 GeV brings down, rather dramatically, the value of BˆK < nll evaluated at that scale. We can already conclude that any calculation which ‘ignores’ the details of the low-energy hadronic contribution will give an overestimate of BˆK . In order to improve on the previous estimate and give a plausible range of systematic errors we have examined the effect of including other hadronic states beyond the r . Inserting the contribution from the lowest axial-vector meson A1 , as well as from the r , improves the quality of the short-distance matching which now happens around zˆ , 0.2 with the result BˆK , 0.44 " 0.04
Ž for
MA21 s 1.5Mr2 .
and
BˆK , 0.46 " 0.04
Ž for
MA21 s 2.2 Mr2 . .
Ž 4.5 .
The minimal hadronic ansatz which satisfies all the matching conditions discussed previously i.e., the short-distance constraints in Eqs. Ž3.3., Ž3.9. and Ž3.10., as well as the long-distance constraint W w0x s 6 and the constraint in Eq. Ž3.11. which fixes the slope of the W w z x-function at the origin, requires the presence of at least three hadronic states. Using the phenomenological fact w25x that the new states do not modify significantly the slope at the origin of the function W w z x results in the further sum rules: a i r i q 2 bi s 0 for
Fig. 1. Plot of the hadronic function W w z x in Eq. Ž3.1. versus z s Q 2 rm2had. for m had.s 1.4 GeV. The solid curves are the normalized function W w z x corresponding to the Õector meson dominance ansatz in Ža., and to the minimal hadronic ansatz in Žb., Žsee the text.. The dotted line represents the low-energy chiral behaviour and the dashed curve the prediction from the OPE.
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
221
i s 2, 3. The corresponding couplings a i and bi for i s 1, 2, 3 can then be solved analytically as functions of the ratios of masses r i only. The new improved hadronic function W w z x intersects the OPE curve at higher zˆ values, with a substantial improvement in the quality of the matching. As an example, we show in Fig. 1b the shape of the W w z x function corresponding to the minimal hadronic ansatz, normalized to its value at the origin, for the particular choice of input values: m had.s 1.4 GeV Mr s 760 MeV, MA 1 s 1.1 GeV and MrX s 1.3 GeV. Also shown in the same plot are the chiral perturbation and short-distance OPE curves, Žthe same curves as in Fig. 1a.. Varying Mr s 760 MeV in "10 MeV and m had. and the mass ratios r i within a reasonable domain: 1 GeV F m had.F 2 GeV; 1.5 r 1 F r 2 F 2.2 r 1 and 2.2 r 1 F r 3 F 3.2 r 1 , gives us the possibility of exploring the error range in the determination of the BˆK factor, which is now given by the following expression: BˆK s
ž
3r11
1
a s Ž m had . .
y
/ ½
m2had . 32p
2
3 4
Ý
F02 is1,2,3
1y
a i log
a s Ž m had . . p zˆ q r i
ri
y bi
1229 1936
ž
y 34 log zˆ q O 1
1 y
zˆ q r i
ri
/5
.
ž
Nc a s2 Ž m had . .
p2
/ Ž 4.6 .
The net effect of the minimal hadronic ansatz which satisfies all the matching conditions discussed previously is to further lower the value of the Õector meson dominance prediction in Eq. Ž4.4. with the result BˆK s 0.41 " 0.09 ,
Ž 4.7 .
where the error is the combined limit of errors from short- and long-distances contributions to BˆK for a choice of m had. in the range 1 GeV F m had.F 2 GeV. This result, of course, does not include the error due to next-to-next-to-leading terms in the 1rNc expansion, nor the error due to chiral corrections in the unfactorized contribution, but we consider it to be a rather robust prediction of BˆK in the chiral limit and at the next-to-leading order in the 1rNc expansion.
™
Our final result in Eq. Ž4.7. is compatible with the current algebra prediction w26x which, to lowest order in chiral perturbation theory, relates the BK -factor to the Kq pq p 0 decay rate. As discussed in Ref. w27x the bosonization of the four-quark operator QDSs2 and the bosonization of the operator Q 2 y Q1 which generates D I s 1r2 transitions are related to each other in the combined chiral limit and next-to-leading order 1rNc expansion. Lowering the value of BˆK from the large-Nc prediction in Eq. Ž1.9. to the result in Eq. Ž4.7. is correlated with an increase of the coupling constant in the lowest order effective chiral Lagrangian which generates D I s 1r2 transitions, and provides a first step towards a quantitative understanding of the dynamical origin of the D I s 1r2 rule. We plan to make a detailed comparison between our result here and other previous estimates in a longer publication. Finally, we wish to point out that the techniques applied here can be extended as well to include higher order terms in the chiral expansion. It remains to be seen if chiral corrections to the unfactorized pattern in Eq. Ž2.3. could be so large, Žof order 100%?. as to increase our result in Eq. Ž4.7. to the values favoured by the lattice QCD determinations8 as well as by recent phenomenological arguments9.
8 9
See e.g. the latest review in Ref. w28x and references therein. See e.g. Refs. w29,30x.
222
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
Acknowledgements We wish to thank Marc Knecht for very helpful discussions on many topics related to the work reported here and to Hans Bijnens, Maarten Golterman, Michel Perrottet, Toni Pich and Ximo Prades for discussions at various stages of this work. One of the authors ŽS.P.. is also grateful to C. Bernard and Y. Kuramashi for conversations and to the Physics Department of Washington University in Saint Louis for the hospitality extended to him while this work was being finished. This work has been supported in part by TMR, EC-Contract No. ERBFMRX-CT980169 ŽEURODA f NE.. The work of S. Peris has also been partially supported by the research project CICYT-AEN99-0766.
References w1x A.J. Buras, M. Jamin, P. Weisz, Nucl. Phys. B 347 Ž1990. 491; S. Herrlich, U. Nierste, Nucl. Phys. B 419 Ž1994. 292; B 476 Ž1996. 27. w2x G. Buchalla, A.J. Buras, M.E. Lautenbacher, Rev. Mod. Phys. 68 Ž1996. 1125; A.J. Buras, in: R. Gupta, A. Morel, E. de Rafael, F. David ŽEds.., Les Houches Lectures, Session LXVIII, Probing the Standard Model of Particle Interactions, North-Holland, Amsterdam, 1999. w3x A. Pich, E. de Rafael, Nucl. Phys. B 358 Ž1991. 311. w4x E. de Rafael, Chiral Lagrangians and Kaon CP-Violation, in: J.F. Donoghue ŽEd.., CP Violation and the Limits of the Standard Model, Proc. TASI’94, World Scientific, Singapore, 1995. w5x G. ’t Hooft, Nucl. Phys. B 72 Ž1974. 461; B 73 Ž1974. 461. w6x E. Witten, Nucl. Phys. B 79 Ž1979. 57. w7x J.P. Fatelo, J.-M. Gerard, Phys. Lett. B 347 Ž1995. 136. ` w8x T. Hambye, G.O. Kohler, P.H. Soldan, Eur. Phys. J. C 10 Ž1999. 271. ¨ w9x J. Bijnens, J. Prades, JHEP 01 Ž1999. 023. w10x W.A. Bardeen, A.J. Buras, J.-M. Gerard, Phys. Lett. B 211 Ž1988. 343. ` w11x J.-M. Gerard, Acta Phys. Polon. B 21 Ž1990. 257. ` w12x S. Bertolini, J.O. Egg, M. Fabbrichesi, E.I. Lashin, Nucl. Phys. B 514 Ž1998. 63. w13x M. Knecht, S. Peris, E. de Rafael, Phys. Lett. B 443 Ž1998. 255. w14x J. Gasser, H. Leutwyler, Nucl. Phys. B 250 Ž1985. 465. w15x M. Knecht, S. Peris, M. Perrottet, E. de Rafael, Phys. Rev. Letters 83 Ž1999. 5230. w16x M. Knecht, S. Peris, E. de Rafael, Phys. Lett. B 457 Ž1999. 227. w17x L. Conti et al., Phys. Lett. B 421 Ž1998. 273. w18x C.R. Allton et al., hep-latr9806016. w19x L. Lellouch, C.-J. David Lin, hep-latr9809142. w20x J.F. Donoghue, E. Golowich, hep-phr9911309. w21x S. Narison, hep-phr0004247. w22x J. Bijnens, J. Prades, hep-phr0005189. w23x G. Ecker, J. Gasser, A. Pich, E. de Rafael, Nucl. Phys. B 321 Ž1989. 311. w24x G. Ecker, J. Gasser, H. Leutwyler, A. Pich, E. de Rafael, Phys. Lett. B 223 Ž1989. 425. w25x S. Peris, M. Perrottet, E. de Rafael, JHEP 05 Ž1998. 011; M. Golterman, S. Peris, Phys. Rev. D 61 Ž2000. 034018. w26x J.F. Donoghue, E. Golowich, B.R. Holstein, Phys. Lett. B 119 Ž1982. 412. w27x A. Pich, E. de Rafael, Phys. Lett. B 374 Ž1996. 186. w28x Y. Kuramashi, Nucl. Phys. B ŽProc. Suppl.. 83–84 Ž2000. 24. w29x M. Cuchini et al hep-phr9910236. w30x F. Caravaglios, F. Parodi, P. Roudeau, A. Stocchi, hep-phr0002171.
Physics Letters B 490 Ž2000. 213–222 www.elsevier.nlrlocaternpe
K 0 –K 0 mixing in the 1rNc expansion Santiago Peris a , Eduardo de Rafael b a
Grup de Fısica Teorica and IFAE, UniÕersitat Autonoma de Barcelona, 08193 Barcelona, Spain ´ ` ` Centre de Physique Theorique, CNRS-Luminy, Case 907, F-13288 Marseille Cedex 9, France ´
b
Received 14 June 2000; accepted 15 August 2000 Editor: R. Gatto
Abstract We present the result for the invariant BˆK factor of K 0 –K 0 mixing in the chiral limit and to next-to-leading order in the 1rNc expansion. We explicitly demonstrate the cancellation of the renormalization scale and scheme dependences between short- and long-distance contributions in the final expression. Numerical estimates are then given, by taking into account increasingly refined short- and long-distance constraints of the underlying QCD Green’s function which governs the BˆK factor. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The Standard Model predicts strangeness changing transitions with D S s 2 via two virtual W-exchanges between quark lines, the so-called box diagrams. The low-energy physics of these transitions is governed by an effective Hamiltonian which is proportional to the local four-quark operator, Žsummation over colour indices 1y g within brackets is understood and q L ' 2 5 q,. QDSs2 Ž x . ' Ž sLŽ x . g md LŽ x . . Ž sLŽ x . gm d LŽ x . . ,
Ž 1.1 . l q s Vqd) Vqs
modulated by a quadratic form of the flavour mixing matrix elements , q s u,c,t , with coefficient functions F1,2,3 of the heavy masses of the fields t, Z 0 , W ", b, and c which have been integrated out 1 : D Ss2 Heff s
GF2 MW2 4p 2
l2c F1 q l2t F2 q 2 l c l t F3 CD Ss2 Ž m . QD Ss2 Ž x . .
Ž 1.2 .
The operator QDSs2 is multiplicatively renormalizable and has an anomalous dimension g Ž a s . which in perturbative QCD ŽpQCD. is defined by the equation
m2
1
d dm
² QDSs2 : s y 12 g Ž a s . ² QD Ss2 : , 2
g Ž as . s
as p
g1 q
as
2
žp/g
For a detailed discussion, see e.g. Refs. w1,2x and references therein.
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 9 7 7 - 1
2 q PPP
.
Ž 1.3 .
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
214
with
1 6
Ž b1 s Žy11 Nc q 2 n f . . g1 s
3 2
ž
1y
1 Nc
/
g2 s
and
1 32
ž
1y
1 Nc
/
y 193 Nc q 43 n f y 21 q 57
1 Nc
q 4b 1 k ,
Ž 1.4 .
where k s 0 in the naıÕe ¨ dimensional renormalization scheme ŽNDR. and k s y4 in the ’t Hooft–Veltman renormalization scheme ŽHV.. The renormalization m-scale dependence of the Wilson coefficient CDSs2 Ž m . in 1 5 2 2 Eq. Ž1.2. is then Ž b 2 s y 17 12 Nc q 4 Ž Nc y 1 . r2 NC n f q 12 Nc n f . CDSs2 Ž m . s 1 q
™
1q
as Ž m . 1
as Ž m . p
p
Ž
1433 1936
b1
ž
g2 q
q 18 k .
ž
g1 yb 1
b2
/ž
g 1 r Žy b 1 .
1
as Ž m .
/
Ž 1.5 .
Ž9r11 .Ž 1rN c .
1
as Ž m .
/
,
Ž 1.6 .
where the second line gives the result to next-to-leading order in the 1rNc expansion, which is the approximation at which we shall be working here. The matrix element ² K 0 < QDSs2 Ž 0 . < K 0 : ' 43 f K2 MK2 BK Ž m . , defines the so-called BK -parameter parameters in the phenomenological QCD, the four quark operator QDSs2 to lowest order in chiral perturbation
Ž sL g m d L . s
d LQC D Ž x . d lm Ž x . 32
Ž 1.7 .
of K 0 y K 0 mixing at short distances, which is one of the crucial studies of CP-violation in the Standard Model. In the large-Nc limit of factorizes into a product of two current operators. Each of these currents, theory, has a simple bosonic realization:
´ 2i F tr l 2 0
32
Ž D mU † . U s
1
'2
F0 E m K 0 q PPP ,
Ž 1.8 .
where F0 denotes the coupling constant of the pion in the chiral limit, l32 is given by the matrix l32 s d i3 d 2 j in flavour space and U is the 3 = 3 unitary matrix in flavour space, U s 1 q PPP , which collects the Goldstone fields and which under chiral rotations transforms like U VR UVL†. The large-Nc approximation, with inclusion of the chiral corrections in the factorized contribution, leads to the result:
™
BK < Nc ™ ` s 34
and
CD Ss2 Ž m . < Nc ™ ` s 1 .
Ž 1.9 .
In full generality, the bosonization of the four-quark operator QDSs2 Ž x . to lowest order in the chiral expansion is described by an effective operator which is of O Ž p 2 . 2 : QDSs2 Ž x .
2
´ y F4 g
See e.g. Refs. w3,4x.
4 0
D Ss2
Ž m . tr l32 Ž D m U † . Ul32 Ž Dm U † . U ,
Ž 1.10 .
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
215
where g DSs2 Ž m . is a dimensionless coupling constant which depends on the underlying dynamics of spontaneous chiral symmetry breaking ŽS x SB. in QCD. The relation between the coupling g DSs2 Ž m . and the phenomenological BK factor defined in Eq. Ž1.7. is simply BK Ž m ,m u , d , s
™ 0. s
3 4
g D Ss2 Ž m . .
Ž 1.11 .
Notice that the coupling g DSs2 Ž m ., and hence BK , is perfectly well defined in the chiral limit, while the matrix element in Eq. Ž1.7. vanishes in the chiral limit as a chiral power. To lowest order in the chiral expansion, the relation to the so called inÕariant BˆK -factor is as follows: BˆK s 34 CD Ss2 Ž m . = g D Ss2 Ž m . ,
Ž 1.12 .
which means that the coupling g DSs2 Ž m . must have a m dependence Žand a scheme dependence. which must cancel with the m dependence Žand scheme dependence. of the Wilson coefficient CDSs2 Ž m .. The purpose of this note is to show how the mechanism of cancellations works in practice within the framework of the 1rNc expansion. This will allow us to give a numerical result for BˆK which is valid to lowest order in the chiral expansion and to next-to-leading order in the 1rNc expansion. We wish to emphasize that the novel feature of this work is that, to our knowledge, it is the first calculation of BˆK which explicitly shows the cancellations of scale and scheme dependences.
2. Bosonization of four-quark operators The QCD Lagrangian in the presence of external chiral sources lm , rm of left- and right-currents, but with neglect of scalar and pseudoscalar sources which is justified in the chiral limit approximation we shall be working here, has the form Ž a. Ž a. mn LQC D s y 14 Gmn G q iq Dx q ,
Ž 2.1 .
with Dx the Dirac operator Dx s g m Ž Em q iGm . y ig m lm
1 y g5 2
q rm
1 q g5 2
.
Ž 2.2 .
The bosonization of the four-quark operator QDSs2 Ž x . is formally defined by the functional integral w3x ² QDSs2 Ž x . : s Tr Dxy1 i
4
y d y
d Dx d lm Ž x . 32 d4q
H H Ž 2p .
4
Tr Dxy1 i
d Dx m
d l Ž x . 32
ž
eyi qPŽ xyy. Tr Dxy1 i
d Dx d lm Ž x . 32
Dxy1 i
d Dx m
d l Ž y . 32
/
,
Ž 2.3 .
where the trace Tr here also includes the functional integration over gluons in large Nc . The first term corresponds to the factorized pattern and gives the contributions of O Ž Nc2 .; the second term corresponds to the unfactorized pattern and it involves an integral, Žwhich is regularization dependent,. over all virtual momenta q. This is the term which gives the next-to-leading O Ž Nc . contribution we are interested in.
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
216
To proceed further, it is convenient to use the Schwinger’s operator formalism. With Pu the conjugate momentum operator, in the absence of external chiral fields, the full quark propagator is
Ž x<
1 Dx
< y. s Ž x < Pu q g
a
la
i 1 y g5 2
q ra
1 q g5
< y.
Ž x
with
E i
Em
y Gm d Ž x y y . .
2
Ž 2.4 . The chiral expansion is then defined as an expansion in the la and ra external sources. The precise relation between the formal bosonization in Eq. Ž2.3. and the explicit chiral realization in Eq. Ž1.10. can best be seen from the fact that: D mU †U s E mU †U q iU † r mU y il m. This shows that there are a priori six different ways to compute the constant g DSs2 Ž m ., although they are all related by chiral gauge invariance. One possible choice is the term3 D Ss2 L27 Ž x. sy
F04 4
g D Ss2 Ž m . PPP ytr l32 U † r m Ul32 U † rm U q PPP .
½
5
Ž 2.5 .
This is a convenient choice because the underlying QCD Green’s function is the four-point function, w L smd Ž x . ' Ý a s a Ž x .g m 1 y2 g 5 d a Ž x . and R dms Ž x . ' Ý a d a Ž x .g m 1 q2 g 5 s a Ž x .x, 3 W Lmanb d 4 x d 4 y d 4 z e i qP x e i lP y eyi lP z ²0 < T L smd Ž x . R ad s Ž y . Lns d Ž 0 . R dbs Ž z . 4 <0: , R L R Ž q,l . s lim i
™0 H
l
Ž 2.6 .
Ž . In fact, what we need, as seen in Eq. Ž2.3., is the integral of the unfactorized four-point function W Lmanb R L R q,l over the four-vector q with the Lorentz indices of the two left-currents contracted. This is a quantity which is a good order parameter of S x SB. The integral over the solid angle has the form Ž Q 2 ' yq 2 .
H
< d V q gmn W LmRanb L R Ž q,l . unfactorized s
ž
l al b l2
/
2 y g a b W LŽ1. R LR ŽQ . ,
Ž 2.7 .
where the transversality in the four-vector l follows from current algebra Ward identities. We are still left with Ž 2 . over the full euclidean range: 0 F Q 2 F ` which has to be done an integral of the invariant function W LŽ1. R LR Q in the same renormalization scheme as the calculation of the short-distance Wilson coefficient CDSs2 Ž x . in Eq. Ž1.2. has been done, i.e. in the MS-scheme. The coupling constant g DSs2 Ž m . defined with dimensional regularization in d s 4 y e dimensions is then given by the following integral, g DSs2 Ž m , e . s 1 y
m2had . 32p 2 F02
Ž 4pm2rm2had . .
er2
G Ž 2 y er2 .
`
y e r2
H0 dz z
ž
W w z x 'z
m2had . F02
/
W LŽ 1R.L R Ž z m2had . . ,
Ž 2.8 .
where, for convenience, we have normalized Q 2 to a characteristic hadronic scale m2had. , Ž z ' Q 2rm2had. ,. which in practice we shall choose within the range: 1 GeV F m had.F 2 GeV.
3 Other choices are indeed possible but, in general, the underlying QCD Green’s functions have pieces which are not order parameters and spurious contributions which depend on the regularization. We have shown the equivalence among the various possible choices, but we postpone the detailed discussion which is rather technical to a longer publication.
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
217
3. Bˆ K to next-to-leading order in the 1 r Nc expansion In full generality, Green’s functions in QCD at large Nc w5x are given by sums over an infinite number of hadronic poles w6x. In the particular case of the function W w z x, wsee Eqs. Ž2.8., Ž2.7. and Ž2.6.x, it can be shown that the most general ansatz in large-Nc QCD is an infinite sum of poles and double poles of the form `
W w z x s6y
`
1
Ý ai r
is1
1
`
i
is1
Ý bi r 2 q Ý
y i
is1
½
ai
1
Ž z q ri .
q bi
1
Ž z q ri .
2
5
,
Ž 3.1 .
where r i s Mi2rm 2had. and Mi is the mass of the i-th narrow hadronic state. The first term on the r.h.s. is the contribution from the Goldstone poles to the W w z x-function. This constant term is known from previous calculations, Žsee Refs. w7–9x,. with which we agree. Constant terms, however, do not contribute to the integral in Eq. Ž2.8. defined in dimensional regularization. In the absence of a solution of large-Nc QCD, the actual values of the mass-ratios r i , and residues a i and bi , remain unknown. As discussed below, there are, however, short-distance and long-distance constraints that the function W w z x has to obey. With W w z x given by Eq. Ž3.1., the integral in Eq. Ž2.8. can be done analytically, with the result g DSs2 Ž m , e . s 1 y
m2had . 32p
2
½ž
F02
q log4p y g E q log m2rm2had . q 1
e
`
y
`
2
`
1
Ý a i log ri q Ý bi r
is1
is1
i
5
/
Ý ai is1
.
Ž 3.2 .
We shall next explore the short-distance properties and long-distance properties which are presently known about the function W w z x, and hence about the parameters a i , bi and r i . v
Ž 2 . is governed by the OPE. We find the result The large-Q 2 behaviour of the function W LŽ1. R LR Q 2 2 lim W LŽ1. R L R Ž Q . s 24p
™`
Q2
as
1q
p
e 1 2 6
Ž5qk . qO
F04
2
as
žp/
Q4
q O Ž 1rQ 8 . ,
Ž 3.3 .
with k s 0 in the NDR scheme and k s y4 in the HV scheme The fact that the residue of the in the OPE is known, entails the constraint `
1 Q4
power term
e
Ý ai s R q 2 S ,
Ž 3.4 .
is1
with R s 24p 2
as p
qO
Nc a s2
F02
p2
m2had .
ž /
S s 4p 2 Ž 5 q k .
and
as p
qO
Nc a s2
F02
p2
m2had .
ž /
.
Ž 3.5 .
. Ž m . in the MS-scheme: This allows us to define the renormalized coupling constant g DŽ rSs2 . g DŽ rSs2 Ž m. s1y
´ž a Žm
had .
32p
2
F02
`
½
R log m2rm2had . q R q S q
.
/
Ý is1
Ž9r11 .Ž 1rNc .
as Ž m .
s
m2had .
½
1y
1
m2had .
2 16p
2
F02
ya i log r i q bi
1
ri
`
ž
RqSq
Ý is1
ya i log r i q bi
1
ri
5
Ž 3.6 .
/5
,
Ž 3.7 .
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
218
where in the second line we have written the renormalization group improved result. We insist on the fact that this result, contrary to the various large-Nc inspired calculations which have been published so far w3,7–12x is the full result to next-to-leading order in the 1rNc expansion. The renormalization m scale dependence as well as the scheme dependence in the factor S cancel exactly, at the next-to-leading log approximation, with the short distance m and scheme dependences in the Wilson coefficient CDSs2 Ž m . in Eq. Ž1.6.. Therefore, the full expression of the invariant BˆK -factor to lowest order in the chiral expansion and to next-to-leading order in the 1rNc expansion, which takes into account the hadronic contribution from light quarks below a mass scale m had. is then BˆK s
ž
3r11
1
a s Ž m had . .
y
m2had . 32p
2
3
/ ½ 4
1y
a s Ž m had . . 1229 p
`
ya i log r i q bi
Ý
F02 is1
1936 1
ri
5
qO
ž
Nc a s2 Ž m had . .
.
p2
/ Ž 3.8 .
The numerical choice of the hadronic scale m had. is, a priori, arbitrary. In practice, however, m had. has to be sufficiently large so as to make meaningful the truncated pQCD series in the first line. Our final error in the numerical evaluation of BˆK will include the small effect of fixing m had. within the range 1 GeV F m had.F 2 GeV. Ž 2 . implies the sum rule The fact that there is no 1rQ 2 term in the OPE expansion of the function W LŽ1. RLR Q v
` is1
`
1
Ý ai r
q i
1
Ý bi r 2 s 6 .
is1
Ž 3.9 .
i
This sum rule plays an equivalent role ˆ to the 1st Weinberg sum rule in the case of the P L R Ž Q 2 . two-point 4 function . Like in this case, it guarantees the cancellation of quadratic divergences in the integral in Eq. Ž2.8. and it relates the contribution from Goldstone particles to a specific sum of resonance state contributions. What this means in practice is that, although the Goldstone pole does not contribute to the integral in Eq. Ž2.8. defined in dimensional regularization, the normalization of the function W w z x at z s 0 is, precisely, fixed by the residue of the Goldstone pole contribution; i.e., W w0x s 6. Ž 2. We have found that there is no 1rQ 6 power correction in the OPE expansion of the function W LŽ1. RLR Q , which implies the additional short-distance sum rule v
`
Ý
`
a i ri s
is1
Ý bi .
Ž 3.10 .
is1
The slope at the origin of the function W w z x is fixed by a linear combination of coupling constants of the O Ž p 4 . Gasser–Leutwyler Lagrangian w14x, with the result 5 v
`
1
`
i
is1
1
m2had .
i
F02
Ý a i r 2 q 2 Ý bi r 3 s 24
is1
w 2 L1 q 5L2 q L3 q L9 x .
Ž 3.11 .
The numerical evaluation of BˆK in Eq. Ž3.8. requires the input of the mass-ratios r i and couplings a i and bi of the narrow states which fill the details of the hadronic spectrum of light flavours. Our goal is to find the minimal hadronic ansatz of pole and double pole terms of the form shown in Eq. Ž3.1. which satisfies the
4 5
See e.g. Ref. w13x. The r.h.s. in Eq. Ž3.11. agrees with the expression reported in Ref. w9x.
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
219
short-distance and long-distance constraints discussed above, and use then this minimal hadronic ansatz to evaluate the integral in Eq. Ž2.8.. The fact that the function W w z x is an order parameter of S x SB ensures that it has a smooth behaviour in the ultraviolet and it is then reasonable to approximate it by a few states. ŽThere is no need here for an infinite number of narrow states to reproduce the asymptotic behaviour of pQCD, as it would be the case with a Green’s function which is not an order parameter of S x SB.. This procedure has been successfully tested in other examples, like the calculation of the electroweak pqy p 0 mass difference w13x and the decay of pseudoscalars into lepton pairs w15x, and it has been applied w16x to the evaluation of matrix elements of electroweak penguin operators as well 6 .
4. Numerical evaluation and conclusions We now proceed to the numerical evaluation of the analytic result for BˆK in Eq. Ž3.8. by successive improvements in the hadronic input. v
The crudest approximation is the one where the only terms of the next-to-leading order in the 1rNc expansion which are retained are those from the first line in Eq. Ž3.8.. This implicitly assumes that all the way down to the hadronic m had. scale there is only pQCD running and that the hadronic contribution below the m had. scale due to light quarks is negligible. One can see that from the integral in Eq. Ž2.8., if we write it in an equivalent cut-off form: 2
1
H0 dz W w z x q H1L rm
2 had .
dz WOPE w z x ,
Ž 4.1 .
with L2 s m2 exp 1 q S , and where in the second integral we use the asymptotic OPE expression in Eq. R Ž3.3.. The approximation we are discussing is the one where the first low-energy integral is simply neglected. The result of this next-to-leading log Žnll. approximation is to bring up the large-Nc prediction in Eq. Ž1.9. to a value which, including an estimate of higher order corrections, Žwe use L M SŽ3. s Ž372 " 40. MeV,. but ignoring the systematic errors involved in the neglect of the hadronic contribution, would be
ž
BˆK < nll ; 0.96 " 0.04
v
/
for
m had .s 1.4 GeV .
Ž 4.2 .
The problem, however, with this simple estimate is that the underlying assumption of neglecting entirely the low-energy hadronic integral does not satisfy any of the long-distance matching constraints discussed in the previous section. We can improve on the previous estimate by taking into account the contribution from the lowest hadronic state, the r vector meson, alone. In this approximation one can fix the large-Nc hadronic ansatz in Eq. Ž3.1. to satisfy the two matching constraints in Eqs. Ž3.9. and Ž3.11.. In fact, it has been shown that the phenomenological values of the L i constants which appear on the r.h.s. of Eq. Ž3.11. agree well with those obtained from the integration of the lowest Õector state w23–25x alone7. This fixes de r mass to Mr s 760 MeV, in the chiral limit where F0 s 85.3 MeV. Using this approximation w25x, the sum rule in Eq. Ž3.11. becomes simply a 1 r 1 q 2 b 1 s 21 r 12 . We can then solve for a 1 and b 1 in terms of r 1 s M V2rm 2had. with the result: a 1 s y9r 1 and b 1 s 15 r 12 . The resulting W w z x function, normalized to its value at the origin
6 7
See also Refs. w17–19x and Refs. w20–22x for other recent evaluations of electroweak penguin operators. The constant L3 gets also an extra contribution from the lowest scalar state, but it is rather small.
S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
220
W w0x s 6, is plotted in Fig. 1a below as a function of z s Q 2rm2had. . Also shown in the same plot are the chiral perturbation behaviour of the W w z x function from the knowledge of its value at the origin and the slope at the origin, Žthe dotted line,. and the corresponding behaviour from the OPE expression in Eq. Ž3.3., Žthe dashed line.. We can now make an improved estimate of BˆK by calculating the integral
H0ˆdz W w z x qHzˆL rm z
2
2 had .
dz WOPE w z x ,
Ž 4.3 .
with W w z x in the first integral approximated by the lowest hadronic state ansatz just discussed. The choice of zˆ which minimizes the dependence of BˆK on zˆ is the one at which the two curves W w z x and WOPE w z x intersect. As seen in Fig. 1a, Žwhich corresponds to m had., 1.4 GeV, F0 s 85.3 MeV and Mr s 760 MeV,. this happens at a value zˆ , 0.15 i.e. Q , 540 MeV; a little bit too low but not unreasonable. This improved approximation brings down the simple next-to-leading log prediction in Eq. Ž4.2. to BˆK < VM D , 0.48 " 0.04 ,
v
Ž 4.4 .
where, again, the error corresponds to an estimate of higher order perturbative corrections only. Incomplete as it may be, this estimate shows a crucial qualitative issue, which is the fact that the low-energy hadronic contribution below a mass scale m had.; 1.4 GeV brings down, rather dramatically, the value of BˆK < nll evaluated at that scale. We can already conclude that any calculation which ‘ignores’ the details of the low-energy hadronic contribution will give an overestimate of BˆK . In order to improve on the previous estimate and give a plausible range of systematic errors we have examined the effect of including other hadronic states beyond the r . Inserting the contribution from the lowest axial-vector meson A1 , as well as from the r , improves the quality of the short-distance matching which now happens around zˆ , 0.2 with the result BˆK , 0.44 " 0.04
Ž for
MA21 s 1.5Mr2 .
and
BˆK , 0.46 " 0.04
Ž for
MA21 s 2.2 Mr2 . .
Ž 4.5 .
The minimal hadronic ansatz which satisfies all the matching conditions discussed previously i.e., the short-distance constraints in Eqs. Ž3.3., Ž3.9. and Ž3.10., as well as the long-distance constraint W w0x s 6 and the constraint in Eq. Ž3.11. which fixes the slope of the W w z x-function at the origin, requires the presence of at least three hadronic states. Using the phenomenological fact w25x that the new states do not modify significantly the slope at the origin of the function W w z x results in the further sum rules: a i r i q 2 bi s 0 for
Fig. 1. Plot of the hadronic function W w z x in Eq. Ž3.1. versus z s Q 2 rm2had. for m had.s 1.4 GeV. The solid curves are the normalized function W w z x corresponding to the Õector meson dominance ansatz in Ža., and to the minimal hadronic ansatz in Žb., Žsee the text.. The dotted line represents the low-energy chiral behaviour and the dashed curve the prediction from the OPE.
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i s 2, 3. The corresponding couplings a i and bi for i s 1, 2, 3 can then be solved analytically as functions of the ratios of masses r i only. The new improved hadronic function W w z x intersects the OPE curve at higher zˆ values, with a substantial improvement in the quality of the matching. As an example, we show in Fig. 1b the shape of the W w z x function corresponding to the minimal hadronic ansatz, normalized to its value at the origin, for the particular choice of input values: m had.s 1.4 GeV Mr s 760 MeV, MA 1 s 1.1 GeV and MrX s 1.3 GeV. Also shown in the same plot are the chiral perturbation and short-distance OPE curves, Žthe same curves as in Fig. 1a.. Varying Mr s 760 MeV in "10 MeV and m had. and the mass ratios r i within a reasonable domain: 1 GeV F m had.F 2 GeV; 1.5 r 1 F r 2 F 2.2 r 1 and 2.2 r 1 F r 3 F 3.2 r 1 , gives us the possibility of exploring the error range in the determination of the BˆK factor, which is now given by the following expression: BˆK s
ž
3r11
1
a s Ž m had . .
y
/ ½
m2had . 32p
2
3 4
Ý
F02 is1,2,3
1y
a i log
a s Ž m had . . p zˆ q r i
ri
y bi
1229 1936
ž
y 34 log zˆ q O 1
1 y
zˆ q r i
ri
/5
.
ž
Nc a s2 Ž m had . .
p2
/ Ž 4.6 .
The net effect of the minimal hadronic ansatz which satisfies all the matching conditions discussed previously is to further lower the value of the Õector meson dominance prediction in Eq. Ž4.4. with the result BˆK s 0.41 " 0.09 ,
Ž 4.7 .
where the error is the combined limit of errors from short- and long-distances contributions to BˆK for a choice of m had. in the range 1 GeV F m had.F 2 GeV. This result, of course, does not include the error due to next-to-next-to-leading terms in the 1rNc expansion, nor the error due to chiral corrections in the unfactorized contribution, but we consider it to be a rather robust prediction of BˆK in the chiral limit and at the next-to-leading order in the 1rNc expansion.
™
Our final result in Eq. Ž4.7. is compatible with the current algebra prediction w26x which, to lowest order in chiral perturbation theory, relates the BK -factor to the Kq pq p 0 decay rate. As discussed in Ref. w27x the bosonization of the four-quark operator QDSs2 and the bosonization of the operator Q 2 y Q1 which generates D I s 1r2 transitions are related to each other in the combined chiral limit and next-to-leading order 1rNc expansion. Lowering the value of BˆK from the large-Nc prediction in Eq. Ž1.9. to the result in Eq. Ž4.7. is correlated with an increase of the coupling constant in the lowest order effective chiral Lagrangian which generates D I s 1r2 transitions, and provides a first step towards a quantitative understanding of the dynamical origin of the D I s 1r2 rule. We plan to make a detailed comparison between our result here and other previous estimates in a longer publication. Finally, we wish to point out that the techniques applied here can be extended as well to include higher order terms in the chiral expansion. It remains to be seen if chiral corrections to the unfactorized pattern in Eq. Ž2.3. could be so large, Žof order 100%?. as to increase our result in Eq. Ž4.7. to the values favoured by the lattice QCD determinations8 as well as by recent phenomenological arguments9.
8 9
See e.g. the latest review in Ref. w28x and references therein. See e.g. Refs. w29,30x.
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S. Peris, E. de Rafaelr Physics Letters B 490 (2000) 213–222
Acknowledgements We wish to thank Marc Knecht for very helpful discussions on many topics related to the work reported here and to Hans Bijnens, Maarten Golterman, Michel Perrottet, Toni Pich and Ximo Prades for discussions at various stages of this work. One of the authors ŽS.P.. is also grateful to C. Bernard and Y. Kuramashi for conversations and to the Physics Department of Washington University in Saint Louis for the hospitality extended to him while this work was being finished. This work has been supported in part by TMR, EC-Contract No. ERBFMRX-CT980169 ŽEURODA f NE.. The work of S. Peris has also been partially supported by the research project CICYT-AEN99-0766.
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