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An estimate of the large distance contribution to the K0−K0 mixing matrix element
Volume 236, number 2
PHYSICS LETTERS B
15 February 1990
AN ESTIMATE OF THE LARGE DISTANCE CONTRIBUTION TO THE K°-K ° MIXING MATRIX ELEMENT A.A. P I V O V A R O V l Theory Division. CERN. CH- 1211 Geneva 23, Switzerland Received 24 October 1989
The matrix element of the K°-K° mixing is calculated with the help of an asymptotic wave function of the K meson. The comparison with the small distance contribution only is made.
1. Introduction L
The i m p o r t a n t role the K ° - K ° system plays for investigating m a n y delicate questions o f the structure of the theory o f strong and electroweak interactions is often stressed in the literature (see e.g. ref. [ 1 ] and references quoted t h e r e i n ) . The s t a n d a r d SU ( 3 ) × SUL(2)×U(1) model with three generations o f quarks and leptons allows mixing between pure hadronic K ° and K ° states due to an exchange with the intermediate vector boson. This results in the appearance o f the weak states KL and Ks which have p a r a m e t e r s measured with good accuracy at present. The theoretical analyses o f the K L - K s system - the d e t e r m i n a t i o n o f the mass split, for example - encounters difficulties due to the necessity o f performing some calculations in the large distance area for strong interactions, i.e. in the area o f strong coupling o f q u a n t u m c h r o m o d y n a m i c s . In ref. [2] only the small distance contribution from the full matrix element out(K~ I K ° ) i . was extracted and the p r o b l e m ( o f the d e t e r m i n a t i o n o f the mass split, for e x a m p l e ) was reduced to the calculation o f an hadronic matrix element o f the form ( ~ o I Herr I K ° ) , where tterr is the effective local h a m i l t o n i a n o b t a i n e d from the famous box diagram. This h a m i l t o n i a n can be presented as a product o f a coefficient function C ( M w , tnt, m ..... ) d e p e n d i n g on the W-boson mass, on the Permanent address: Institute for Nuclear Research of the USSR Academy of Sciences, 117 312 Moscow, USSR. 214
heavy quark masses M,, Mc, on the p a r a m e t e r s o f the quark mixing matrix and some others, and on a local o p e r a t o r O = (&Y, dL)2: Herr= C ( M w , mt, m ..... ) O . The coefficient function C(...) can be reliably calculated perturbatively; this was done in the leading logarithmic a p p r o x i m a t i o n in ref. [ 3 ]. Even within this simplified formulation o f the problem, in order to get the full answer for the mixing matrix element one has to calculate the object ( K ° I 0 1 K ° ) , i.e. the hadronic matrix element at low energies. Many papers were devoted to this p r o b l e m [ 4 - 1 4 ]. F o r example, the p a r a m e t e r B defined by the equation 3 B= 2f~m~
(K°[OI K°)
was o b t a i n e d within the Q C D sum rule approach for the three-point function in ref. [ 11 ] and turns out to be B = (0.9_+0.1)as( 1.2 GeV z)
--2/9,
where as(/z 2) is the strong coupling constant. The value B = I corresponds to the v a c u u m d o m i n a n c e a p p r o x i m a t i o n suggested in the pioneering p a p e r ref. [2]. Thus the mass difference between KL and Ks can be p a r a m e t r i z e d as Am = m L --/rt s -----Ares +
AIH L ,
Volume 236, number 2
PHYSICS LETTERS B
where Ams=cB with c being a known constant and AmL being connected with the large distance contribution [ 15,16 ] and being completely undetermined within this approach. The parameter AmL was calculated in refs. [ 17,18 ] in a model dependent way by saturating the matrix element with low lying states. At present there is no reliable estimate of AmL based on first principles of QCD only and therefore there is no such estimate for Am either. At this point I should like to repeat once more that this estimate is of paramount importance for the precision comparison of the standard model predictions with the existing experimental data. In the present paper the matrix element of the K °~0 transition is calculated with the help of an asymptotic kaon wave function. The large distance contribution happens to be small enough.
15 February 1990
Tl,~(px, X 2, p2)
=i
f (0l Tg(O)y~ySd(x)d(y)y~ySs(y)10)
X e x p ( - i p y ) dy
( 1)
has a dispersion representation of the form f ~PP(px, Y 2, S) ds
TU~(px, x2, p2) =
s_p2
iO
'
where the possible subtraction terms have been omitted. Saturating the spectral density a~'"(px, x 2, s) with the contribution of the K meson only and using the idea of duality between quarks and hadrons we get the following representation for the wave function: So
f 2 p~OU(u, v)= ~ at'~(u, v,s) ds. 0
2. The asymptotic wave function of the K meson In this section the asymptotic wave function of the K meson is calculated using the finite energy sum rule technique of quantum chromodynamics [ 19,20 ]. We restrict ourselves to the leading approximation in the strong coupling constant as and the nonperturbative vacuum condensates. So we do not take into account any vacuum condensates at all. The kaon wave function is defined by the equation (01Y(0)P(0, x)~,,Tsd(x)lK°(p) ) = ifK(~p(p)(, X 2 ) ,
The spectral density aU~(u, v, s) in the leading approximation is determined by the only diagram and equals to 3 (
aU~(px, x 2, s) = ~ 2
P(O,x)=Pexp(igfA~'(y)dy.) and an integration contour is a straight line between the points 0 and x. Here the function OU(u, v) is gauge invariant due to the presence of the P exponent. We, however, will calculate this function in the gauge fixed by the condition xuAU(x ) =0. The use of the finite energy sum rule technique for determination of the wave function is straightforward and generalizes slightly the ordinary approach [20]. Namely, the correlator
f'
dR(s) dR(s) f " ) exp( - ½ipx) + d x ~ dx,
(2)
[in eq. (2) only the relevant tensor structures are maintained]. Here R(s)=½x/(px)2-sx z, f= s i n ( R ) / R , f ' =df/dR. For the scalar components A and B of the wave function 0 u we have the following result after performing the integration:
O'( u, v) =p'A ( u, v) + xUB( u, v) , where
dZR(s)
~pUp,f+ ~
3
So
f~
(So) × \(sinR (-~o)R
cos R(So)
)
exp( - ½ipx)
and
f~B(px, x 2) 3Sopxr(cosR(So) - arc2 x 2 [ 8 (sinR(So) + -Sox - ~ \ n-~o)
sin R(So)~
R(S~
1
J R2(So)
sinR(O)~lexp(_½ipx)" n(o) ]A
The parameter So is free and can be determined 215
Volume 236, number 2
PHYSICS LETTERS B
from the normalization condition A(0, 0 ) = 1. Then So= (2~/'K) 2. We have put the light quark masses equal to zero everywhere in the calculation. Thus at this stage we have the wave function of the K meson originated from Q C D directly without using any model or any additional assumptions except the duality hypotheses. Having this function in hands one can calculate some matrix element with the kaons on mass shell. In the next section one of these matrix elements determining the K ° - K ° mixing will be calculated.
3. The K°-K-6 mixing matrix element The effective four-fermion lagrangian obtained after taking the limit Mw~o~ has the form
jwl,j~t
L= ~
where Gv is Fermi's constant, J~'~=~ATu Vq~ is the weak charged hadronic current, qAm(u, C, t) T, qK= (d, s, b ) r and Vis the quark mixing matrix. The K°-K~ transition is determined by an equation of the form om( K ° ( k ' ) 1 K ° ( k ) )in =
dx.
(3)
If we expand the correlator (3) in the series in m~-~ and m c ~ then we get the effective local lagrangian which is responsible for the small distance contribution to the transition. There is, however, another contribution to the expression (3) which cannot be expanded in such a series. In the leading approximation this contribution is described by the diagram with massless u-quarks in the intermediate state. The effective cutofffor this diagram is connected not with the quark mass but with the properties of the kaon wave function. Strictly speaking an expansion in m ~ ~ also requires some care since the expansion parammer is actually the ratio of the kaon wave runelion scale, which is x/-~ in our case, and the charmed quark mass I H c . For the sake of simplicity we consider a simplified model with two generations only. This simple model allows to demonstrate all basic features of the calcu216
lation. The generalization to the realistic model with three generations is straightforward and will be done elsewhere. The leading term of the light cone expansion for the correlator (3) has the following simple form:
TL(x)L(O)=(4Gvsin Occ°sOc)2
42 × 4&(x)2de(O)&(O)2dc(x) x [ 4:,(x, O) - ¢,(x, mc)
]2
where 2=xUTu, ~(x, m)=]x-2tr[2S(x, r e ) l , and S(x, m) is the propagator of the fermion with the mass m. The next term of the expansion contains some operators of the form, for example, SL{X)y'UGlw()')dL(O)sL(O)7"dL{X)and their role will be briefly discussed later. A matrix element of the (nonlocal) operator O(x, O)=SL(X)2dL(O)~L(O)S'dL(X) between the K ° and K ° states can be calculated with the help of the kaon wave function. In the simplest approximation of the vacuum dominance for this matrix element the answer is ( K ° l O ( x , O ) l K °) 1 "2 X 2 ) exp(~ipx) ] 2 =~JK([PI'O,,(px, + ~ [ -- (px)2--/-- ½1H~x2] cos(px) ) .
i(2zr)4fi(k - k ' )M,
i; (K~(p)ITL(x)L(O)IK°(p))
M= ~
15 February 1990
Bearing in mind the further integration of this quantity over x one can average it over directions of the vector x. Then keeping the leading terms in m.~ only we get ( K ° 10(x, 0 ) l K ° ) "v __ I / ' 2 t ~ 2 --g~/KtrtK
w2f [A(0 "" t
' ~..2 . )+X
2
B ' ( 0 , X "' ) ] e + j }I
= W(x 2, & ) . The superscript " a v " marks the quantity averaged over angles and
B' (0, x z ) = (px)-lB(px,
x 2 ) Ipx=O.
Finally the representation for M takes the form
{ 4Gv sin_0c cos 0c ,]2
\
,f
×~
,/2
:
W(x2, So)[q)(x,O)-clo(x, mc)]2dx.
(4)
Volume 236, number 2
PHYSICS LETTERS B
This expression describes the full K ° - K ° transition matrix element and its division on the separate contributions o f small and large distances is an artificial one. This division has sense only as an approximate method of the calculation o f the integral (4). Namely, the quantity (4) can be rewritten as a sum, M = M s + ML, where Ms and ML are proportional to f ~/]'(x 2, So) [ - 2 ~ ( . \ , O ) ~ ( x , m~)+ ~ ( x ,
IHc)
2 ]
dx
and
f
W ( x , So) qb(x, 0) 2 d x
respectively. After such a decomposition the first term can be calculated a p p r o x i m a t e l y as a series in So/ m~ and it is this part that is ordinary called the small distance contribution. On the contrary, the quantity ML is fully d e t e r m i n e d by the p a r a m e t e r So only and is known under the nickname "the large distance contribution". Note that both integrals ML and Ms do not exist separately due to ultraviolet divergencies and require some regularization procedure for their correct determination. The sum M is finite and, o f course, does not depend on any regularization chosen for ML and Ms. The calculation o f these integrals was performed in the euclidean space using dimensional regularization. In this particular case, the integral M s ( 0 ) = Ms l xo=o, i.e. in the zeroth order approximation in the ratio So/m~, is finite due to some accidental causes and namely this quantity ordinary used in the literature. The result has the form Ms(0)=-
4GvsinOccosO~
m~
x/2
, ,,
,
16~'~/kink
(5)
and M :/Is ( 0 )
3So \~ / So
1+ ~ l l n -
t}l c
-
9t._~
157.1 in 4
"~
420
--
--
7
)
15 February 1990
(i) due to the higher terms in the light cone expansion at x 2 ~ 0 , i.e. due to the o p e r a t o r o f the form gGdYd, e.g. This forces us to take into account the c o m p o n e n t s o f the wave function of the K meson more complicated than the two-quark only. If some p a r a m e t e r g 2 is defined by the relation (Olg.~)'"G~,,d[ K ° ( p ) ) = ifK62p~, then the corrections will be o f the order offi2/tn 2" the numerical value o f d~ was d e t e r m i n e d [21 ] to be 62= 0.16 + 0.02 GeV 2; (ii) due to a more accurate calculation o f the kaon wave function, which means including the perturbarive and n o n p e r t u r b a t i v e corrections to the theoretical expression for the correlator ( 1 ) and taking into account the contribution of higher states in the physical spectral density, for example the one of the a, meson; (iii) due to the d e t e r m i n a t i o n o f the matrix element of the nonlocal o p e r a t o r O(x, 0) in a more accurate way than the vacuum d o m i n a n c e approximation. Though these corrections can hardly be controlled with good accuracy, the result o b t a i n e d tells us that the contribution o f large distances is small enough in the model with two generations of quarks and leptons.
4. Conclusion To conclude, we note that the smallness o f the large distance contribution justifies the application o f the result (5) for the estimation o f the parameters of the standard model.
Acknowledgement The author thanks the CERN TH Division, where this work was completed, for its kind hospitality. I am also indebted to F. Karsch for teaching me how to use the CERN c o m p u t e r facilities.
•
Here C=0.5772... is Euler's constant. Numerically for M~= 1.25 GeV and j~, = 1.22f~,f,~= 133 MeV, we have
M / Ms( O )= 1 - 0 . 0 8 7 . Which are the possible uncertainties o f the result obtained? They are
References [ l ] J.F. Donoghue, E. Golowich and B. Holstein, Phys. Rep. 131 (1986) 319. [2] M.K. Galliard and B.W. Lee, Phys. Rev. D 10 (1974) 897. [3] F.J. Gilman and M.B. Wise, Phys. Len. B 93 (1980) 129; Phys. Rev. D 27 (1982) 1128.
217
Volume 236, number 2
PHYSICS LETTERS B
[4] R.E. Shrock and S.B. Treiman, Phys. Rev. D 19 (1979) 2148. [5] B. Mcwilliams and O. Shancar, Phys. Rev. D 22 (1980) 2853. [6] J.F. Donoghue, E. Golowich and B. Holstein, Phys. Lett. B 119 (1982) 412. [7] B. Guberina, B. Machet and E. de Rafael, Phys. Lett. B 128 (1983) 269. [ 8 ] N. Cabibbo, G. Martinelli and R. Petronzio, Nucl. Phys. B 244 (1984) 381. [9] P. Cea, G. Nardulli and G. Preparata, Phys. Lett. B 148 (1984) 477. [ 10] A. Pich and E. de Rafael, Phys. Lett. B 158 (1985) 477. [ 11 ] K.G. Chetyrkin, A.L. Kataev, A.B. Krasulin and A.A. Pivovarov, INR preprint P-390 (Moscow, 1985 ); Phys. Lett. B 174 (1986) 104.
218
15 February 1990
[ 12] R. Decker, Nucl. Phys. B 277 (1986) 661. [ 13 ] L.J. Reinders and S. Yazaki, Nucl. Phys. B 288 (1987) 789. [ 14] M.M. Gavela et al., Nucl. Phys. B (Proc. Suppl.) 4 (1988) 466. [15] L. Wolfenstein, Nucl. Phys. B 160 (1979) 501. [ 16] C. Hill, Phys. Lett. B 97 (1980) 275. [ 17 ] D. Hochberg and R.G. Sachs, Phys. Rev. D 27 (1983) 606. [ 18 ] G.F. Donoghue, E. Golowich and B. Holstein, Phys. Lett. B 135 (1984) 481. [ 19] K.G. Chetyrkin, N.V. Krasnikov and A.N. Tavkhelidze, Phys. Lett. B 76 (1978) 83. [20] N.V. Krasnikov, A.A. Pivovarov and N.N. Tavkhelidze, Z. Phys. C 19 (1983) 301. [21 ] A.A. Ovchinnikov and A.A. Pivovarov, INR preprint P-574 (Moscow, 1987).
PHYSICS LETTERS B
15 February 1990
AN ESTIMATE OF THE LARGE DISTANCE CONTRIBUTION TO THE K°-K ° MIXING MATRIX ELEMENT A.A. P I V O V A R O V l Theory Division. CERN. CH- 1211 Geneva 23, Switzerland Received 24 October 1989
The matrix element of the K°-K° mixing is calculated with the help of an asymptotic wave function of the K meson. The comparison with the small distance contribution only is made.
1. Introduction L
The i m p o r t a n t role the K ° - K ° system plays for investigating m a n y delicate questions o f the structure of the theory o f strong and electroweak interactions is often stressed in the literature (see e.g. ref. [ 1 ] and references quoted t h e r e i n ) . The s t a n d a r d SU ( 3 ) × SUL(2)×U(1) model with three generations o f quarks and leptons allows mixing between pure hadronic K ° and K ° states due to an exchange with the intermediate vector boson. This results in the appearance o f the weak states KL and Ks which have p a r a m e t e r s measured with good accuracy at present. The theoretical analyses o f the K L - K s system - the d e t e r m i n a t i o n o f the mass split, for example - encounters difficulties due to the necessity o f performing some calculations in the large distance area for strong interactions, i.e. in the area o f strong coupling o f q u a n t u m c h r o m o d y n a m i c s . In ref. [2] only the small distance contribution from the full matrix element out(K~ I K ° ) i . was extracted and the p r o b l e m ( o f the d e t e r m i n a t i o n o f the mass split, for e x a m p l e ) was reduced to the calculation o f an hadronic matrix element o f the form ( ~ o I Herr I K ° ) , where tterr is the effective local h a m i l t o n i a n o b t a i n e d from the famous box diagram. This h a m i l t o n i a n can be presented as a product o f a coefficient function C ( M w , tnt, m ..... ) d e p e n d i n g on the W-boson mass, on the Permanent address: Institute for Nuclear Research of the USSR Academy of Sciences, 117 312 Moscow, USSR. 214
heavy quark masses M,, Mc, on the p a r a m e t e r s o f the quark mixing matrix and some others, and on a local o p e r a t o r O = (&Y, dL)2: Herr= C ( M w , mt, m ..... ) O . The coefficient function C(...) can be reliably calculated perturbatively; this was done in the leading logarithmic a p p r o x i m a t i o n in ref. [ 3 ]. Even within this simplified formulation o f the problem, in order to get the full answer for the mixing matrix element one has to calculate the object ( K ° I 0 1 K ° ) , i.e. the hadronic matrix element at low energies. Many papers were devoted to this p r o b l e m [ 4 - 1 4 ]. F o r example, the p a r a m e t e r B defined by the equation 3 B= 2f~m~
(K°[OI K°)
was o b t a i n e d within the Q C D sum rule approach for the three-point function in ref. [ 11 ] and turns out to be B = (0.9_+0.1)as( 1.2 GeV z)
--2/9,
where as(/z 2) is the strong coupling constant. The value B = I corresponds to the v a c u u m d o m i n a n c e a p p r o x i m a t i o n suggested in the pioneering p a p e r ref. [2]. Thus the mass difference between KL and Ks can be p a r a m e t r i z e d as Am = m L --/rt s -----Ares +
AIH L ,
Volume 236, number 2
PHYSICS LETTERS B
where Ams=cB with c being a known constant and AmL being connected with the large distance contribution [ 15,16 ] and being completely undetermined within this approach. The parameter AmL was calculated in refs. [ 17,18 ] in a model dependent way by saturating the matrix element with low lying states. At present there is no reliable estimate of AmL based on first principles of QCD only and therefore there is no such estimate for Am either. At this point I should like to repeat once more that this estimate is of paramount importance for the precision comparison of the standard model predictions with the existing experimental data. In the present paper the matrix element of the K °~0 transition is calculated with the help of an asymptotic kaon wave function. The large distance contribution happens to be small enough.
15 February 1990
Tl,~(px, X 2, p2)
=i
f (0l Tg(O)y~ySd(x)d(y)y~ySs(y)10)
X e x p ( - i p y ) dy
( 1)
has a dispersion representation of the form f ~PP(px, Y 2, S) ds
TU~(px, x2, p2) =
s_p2
iO
'
where the possible subtraction terms have been omitted. Saturating the spectral density a~'"(px, x 2, s) with the contribution of the K meson only and using the idea of duality between quarks and hadrons we get the following representation for the wave function: So
f 2 p~OU(u, v)= ~ at'~(u, v,s) ds. 0
2. The asymptotic wave function of the K meson In this section the asymptotic wave function of the K meson is calculated using the finite energy sum rule technique of quantum chromodynamics [ 19,20 ]. We restrict ourselves to the leading approximation in the strong coupling constant as and the nonperturbative vacuum condensates. So we do not take into account any vacuum condensates at all. The kaon wave function is defined by the equation (01Y(0)P(0, x)~,,Tsd(x)lK°(p) ) = ifK(~p(p)(, X 2 ) ,
The spectral density aU~(u, v, s) in the leading approximation is determined by the only diagram and equals to 3 (
aU~(px, x 2, s) = ~ 2
P(O,x)=Pexp(igfA~'(y)dy.) and an integration contour is a straight line between the points 0 and x. Here the function OU(u, v) is gauge invariant due to the presence of the P exponent. We, however, will calculate this function in the gauge fixed by the condition xuAU(x ) =0. The use of the finite energy sum rule technique for determination of the wave function is straightforward and generalizes slightly the ordinary approach [20]. Namely, the correlator
f'
dR(s) dR(s) f " ) exp( - ½ipx) + d x ~ dx,
(2)
[in eq. (2) only the relevant tensor structures are maintained]. Here R(s)=½x/(px)2-sx z, f= s i n ( R ) / R , f ' =df/dR. For the scalar components A and B of the wave function 0 u we have the following result after performing the integration:
O'( u, v) =p'A ( u, v) + xUB( u, v) , where
dZR(s)
~pUp,f+ ~
3
So
f~
(So) × \(sinR (-~o)R
cos R(So)
)
exp( - ½ipx)
and
f~B(px, x 2) 3Sopxr(cosR(So) - arc2 x 2 [ 8 (sinR(So) + -Sox - ~ \ n-~o)
sin R(So)~
R(S~
1
J R2(So)
sinR(O)~lexp(_½ipx)" n(o) ]A
The parameter So is free and can be determined 215
Volume 236, number 2
PHYSICS LETTERS B
from the normalization condition A(0, 0 ) = 1. Then So= (2~/'K) 2. We have put the light quark masses equal to zero everywhere in the calculation. Thus at this stage we have the wave function of the K meson originated from Q C D directly without using any model or any additional assumptions except the duality hypotheses. Having this function in hands one can calculate some matrix element with the kaons on mass shell. In the next section one of these matrix elements determining the K ° - K ° mixing will be calculated.
3. The K°-K-6 mixing matrix element The effective four-fermion lagrangian obtained after taking the limit Mw~o~ has the form
jwl,j~t
L= ~
where Gv is Fermi's constant, J~'~=~ATu Vq~ is the weak charged hadronic current, qAm(u, C, t) T, qK= (d, s, b ) r and Vis the quark mixing matrix. The K°-K~ transition is determined by an equation of the form om( K ° ( k ' ) 1 K ° ( k ) )in =
dx.
(3)
If we expand the correlator (3) in the series in m~-~ and m c ~ then we get the effective local lagrangian which is responsible for the small distance contribution to the transition. There is, however, another contribution to the expression (3) which cannot be expanded in such a series. In the leading approximation this contribution is described by the diagram with massless u-quarks in the intermediate state. The effective cutofffor this diagram is connected not with the quark mass but with the properties of the kaon wave function. Strictly speaking an expansion in m ~ ~ also requires some care since the expansion parammer is actually the ratio of the kaon wave runelion scale, which is x/-~ in our case, and the charmed quark mass I H c . For the sake of simplicity we consider a simplified model with two generations only. This simple model allows to demonstrate all basic features of the calcu216
lation. The generalization to the realistic model with three generations is straightforward and will be done elsewhere. The leading term of the light cone expansion for the correlator (3) has the following simple form:
TL(x)L(O)=(4Gvsin Occ°sOc)2
42 × 4&(x)2de(O)&(O)2dc(x) x [ 4:,(x, O) - ¢,(x, mc)
]2
where 2=xUTu, ~(x, m)=]x-2tr[2S(x, r e ) l , and S(x, m) is the propagator of the fermion with the mass m. The next term of the expansion contains some operators of the form, for example, SL{X)y'UGlw()')dL(O)sL(O)7"dL{X)and their role will be briefly discussed later. A matrix element of the (nonlocal) operator O(x, O)=SL(X)2dL(O)~L(O)S'dL(X) between the K ° and K ° states can be calculated with the help of the kaon wave function. In the simplest approximation of the vacuum dominance for this matrix element the answer is ( K ° l O ( x , O ) l K °) 1 "2 X 2 ) exp(~ipx) ] 2 =~JK([PI'O,,(px, + ~ [ -- (px)2--/-- ½1H~x2] cos(px) ) .
i(2zr)4fi(k - k ' )M,
i; (K~(p)ITL(x)L(O)IK°(p))
M= ~
15 February 1990
Bearing in mind the further integration of this quantity over x one can average it over directions of the vector x. Then keeping the leading terms in m.~ only we get ( K ° 10(x, 0 ) l K ° ) "v __ I / ' 2 t ~ 2 --g~/KtrtK
w2f [A(0 "" t
' ~..2 . )+X
2
B ' ( 0 , X "' ) ] e + j }I
= W(x 2, & ) . The superscript " a v " marks the quantity averaged over angles and
B' (0, x z ) = (px)-lB(px,
x 2 ) Ipx=O.
Finally the representation for M takes the form
{ 4Gv sin_0c cos 0c ,]2
\
,f
×~
,/2
:
W(x2, So)[q)(x,O)-clo(x, mc)]2dx.
(4)
Volume 236, number 2
PHYSICS LETTERS B
This expression describes the full K ° - K ° transition matrix element and its division on the separate contributions o f small and large distances is an artificial one. This division has sense only as an approximate method of the calculation o f the integral (4). Namely, the quantity (4) can be rewritten as a sum, M = M s + ML, where Ms and ML are proportional to f ~/]'(x 2, So) [ - 2 ~ ( . \ , O ) ~ ( x , m~)+ ~ ( x ,
IHc)
2 ]
dx
and
f
W ( x , So) qb(x, 0) 2 d x
respectively. After such a decomposition the first term can be calculated a p p r o x i m a t e l y as a series in So/ m~ and it is this part that is ordinary called the small distance contribution. On the contrary, the quantity ML is fully d e t e r m i n e d by the p a r a m e t e r So only and is known under the nickname "the large distance contribution". Note that both integrals ML and Ms do not exist separately due to ultraviolet divergencies and require some regularization procedure for their correct determination. The sum M is finite and, o f course, does not depend on any regularization chosen for ML and Ms. The calculation o f these integrals was performed in the euclidean space using dimensional regularization. In this particular case, the integral M s ( 0 ) = Ms l xo=o, i.e. in the zeroth order approximation in the ratio So/m~, is finite due to some accidental causes and namely this quantity ordinary used in the literature. The result has the form Ms(0)=-
4GvsinOccosO~
m~
x/2
, ,,
,
16~'~/kink
(5)
and M :/Is ( 0 )
3So \~ / So
1+ ~ l l n -
t}l c
-
9t._~
157.1 in 4
"~
420
--
--
7
)
15 February 1990
(i) due to the higher terms in the light cone expansion at x 2 ~ 0 , i.e. due to the o p e r a t o r o f the form gGdYd, e.g. This forces us to take into account the c o m p o n e n t s o f the wave function of the K meson more complicated than the two-quark only. If some p a r a m e t e r g 2 is defined by the relation (Olg.~)'"G~,,d[ K ° ( p ) ) = ifK62p~, then the corrections will be o f the order offi2/tn 2" the numerical value o f d~ was d e t e r m i n e d [21 ] to be 62= 0.16 + 0.02 GeV 2; (ii) due to a more accurate calculation o f the kaon wave function, which means including the perturbarive and n o n p e r t u r b a t i v e corrections to the theoretical expression for the correlator ( 1 ) and taking into account the contribution of higher states in the physical spectral density, for example the one of the a, meson; (iii) due to the d e t e r m i n a t i o n o f the matrix element of the nonlocal o p e r a t o r O(x, 0) in a more accurate way than the vacuum d o m i n a n c e approximation. Though these corrections can hardly be controlled with good accuracy, the result o b t a i n e d tells us that the contribution o f large distances is small enough in the model with two generations of quarks and leptons.
4. Conclusion To conclude, we note that the smallness o f the large distance contribution justifies the application o f the result (5) for the estimation o f the parameters of the standard model.
Acknowledgement The author thanks the CERN TH Division, where this work was completed, for its kind hospitality. I am also indebted to F. Karsch for teaching me how to use the CERN c o m p u t e r facilities.
•
Here C=0.5772... is Euler's constant. Numerically for M~= 1.25 GeV and j~, = 1.22f~,f,~= 133 MeV, we have
M / Ms( O )= 1 - 0 . 0 8 7 . Which are the possible uncertainties o f the result obtained? They are
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