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2) in the chiral limit
6 April 2000
Physics Letters B 478 Ž2000. 172–184
Dispersive calculation of B7Ž3r2. and B8Ž3r2. in the chiral limit John F. Donoghue a
a,b
, Eugene Golowich
a
Department of Physics and Astronomy, UniÕersity of Massachusetts, Amherst, MA 01003, USA b TH DiÕision, CERN, GeneÕa, Switzerland Received 2 December 1999; accepted 16 February 2000 Editor: R. Gatto
Abstract We show how the isospin vector and axialvector current spectral functions r V,3 and rA,3 can be used to determine in leading chiral order the low energy constants B7Ž3r2. and B8Ž3r2.. This is accomplished by matching the Operator Product Expansion to the dispersive analysis of vacuum polarization functions. The data for the evaluation of these dispersive integrals has been recently enhanced by the ALEPH measurement of spectral functions in tau decay, and we update our previous phenomenological determination. Our calculation yields in the NDR renormalization scheme and at renormalization scale m s 2 GeV the values B7Ž3r2. s 0.55 " 0.07 " 0.10 and B8Ž3r2. s 1.11 " 0.16 " 0.23 for the quark mass values ms q m ˆ s 0.1 GeV. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The recent KTeV and NA48 findings that e Xre , 20 P 10y4 raise the important question whether a value so large can be consistent with Standard Model expectations w1x. One of the key quantities upon which the Standard Model prediction is based is the B-factor B8Ž3r2.. In this paper, we work in the chiral limit to obtain an analytic expression for B8Ž3r2. Žand also for B7Ž3r2. .. Our results take the form of sum rules involving the difference r V y rA of isospin-one spectral functions. The constants B7Ž3r2. Ž m . and B8Ž3r2. Ž m . are defined in terms of the matrix elements ²2p < QiŽ3r2. < K :m ' BiŽ3r2. Ž m . ²2p < QiŽ3r2. < K :mvac
Ž i s 7,8 . , Ž 1 .
where m is the renormalization scale, Q7Ž3r2. and Q8Ž3r2. are the D I s 3r2 electroweak penguin operators Q7Ž3r2. ' s a G Lm d a u b GmR u b y d b GmR d b
ž
/
q s a G Lm u a u b GmR d b , Q8Ž3r2. ' s a G Lm d b u b GmR u a y d b GmR d a
ž
q s a G Lm u b u b GmR d a ,
/ Ž 2.
a,b are color labels and G Lm ' g m Ž1 q g 5 ., G Rm ' g m Ž1 y g 5 .. Analogous B-factors are defined for the other weak operators. The most important contributions to e Xre are the matrix elements of the penguin operator Q6 and the electroweak penguin operator
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 2 3 9 - 2
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
Q83r2 . As expressed in terms of the B-factors this is seen in the approximate numerical relation,
eX e
s 10 = 10y4 = Ž 2.5B6 y 1.3 B8Ž 3r2 . .
ž
100 MeV ms q md
/
,
Ž 3.
when evaluated at m s 2 GeV in the MS-NDR renormalization scheme. Alternatively, in terms of the operator matrix elements themselves one has
eX e
s 10 = 10y4 y3.1 GeVy3 P ² Q6 : 0 y0.51 GeVy3 P ² Q8Ž 3r2 . : 2 ,
Ž 4.
where O1 , O8 are the local four-quark operators t3 t3 t3 t3 O1 ' qgm q qg m q y qgm g 5 q qg mg 5 q , 2 2 2 2 t t 3 3 O8 ' qgm l A q qg ml A q 2 2 t t3 3 y qgmg 5 l A q qg mg 5 l A q . Ž 7. 2 2 In the above, q s u,d, s, t 3 is a Pauli Žflavor. matrix, l A 4 are the Gell Mann color matrices and the subscripts on O1 , O8 refer to the color carried by their currents. 1 A chiral evaluation of B7Ž3r2. and B8Ž3r2. is thus seen Žcf. Eq. Ž6.. to depend upon ² O1 :m and ² O8 :m . The K 0 ™ pp matrix elements are likewise recoverable in the chiral limit from the vacuum matrix elements,
where ² Q6 : 0 ' ² Ž pp . Is0 < Q6 < K
0:
lim ² Ž pp . Is2 < Q7Ž3r2. < K 0 :m s y
and
² Q8Ž3r2. : 2 ' ² Ž pp . Is2 < Q8Ž3r2. < K 0 : ,
ps0
Ž 5.
again defined in the NDR scheme at m s 2 GeV. A kaon-to-pion weak matrix element can be analyzed by chiral methods and expressed as an expansion in momenta and quark masses w2x. Here we shall calculate the leading term in such an expansion, valid in the limit of exact chiral symmetry where the Goldstone bosons become massless. In fact, our analysis yields values for the relevant K-to-p matrix elements themselves. However, we shall transcribe this information into the equivalent form of B-factors in order to express our results in the more conventional form and thus allow comparison with other techniques. When considered in the chiral limit, the content of Eq. Ž1. reduces to lim B7Ž3r2. Ž m .
ps0
3 sy
Fp4
P
mu q md mp2
P
mu q ms m2K
² O1 :m ,
lim B8Ž3r2. Ž m .
ps0
1 sy =
Fp4 1 3
P
mu q md mp2
P
173
4 Fp3
² O1 :m ,
lim ² Ž pp . Is2 < Q8Ž3r2. < K 0 :m
ps0
4 sy
Fp3
1 3
² O1 :m q 12 ² O8 :m .
Ž 8.
The rest of this paper describes a calculational procedure to obtain analytic expressions for the vacuum matrix elements. In Section 2, we show how to extract ² O1 :m and ² O8 :m from the isospin-one vector and axialvector correlators by deriving sum rules for ² O1 :m and ² O8 :m in a momentum cutoff scheme. This renormalization is especially well suited for comparing theory to experiment. In Section 3 we introduce a four-quark operator ODSs1 , distinct from the familiar nonleptonic hamiltonian HDSs1 , whose kaon-to-pion matrix element is related to ² O1 :m and ² O8 :m in the chiral limit. We demonstrate consistency of this information with the sum rules of Section 2. Section 4 describes a procedure for obtaining ² O1 :m and ² O8 :m in MS renormalization, which is commonly used in lattice-theoretic simulations. Our final numerical results and concluding statements appear, respectively, in Section 5 and Section 6.
mu q ms m2K
² O1 :m q 12 ² O8 :m ,
1
Ž 6.
Throughout we denote vacuum expectation values as ²0 < O <0:m ' ² O :m .
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
174
2. Dispersive analysis of vacuum polarization functions In seeking values for ² O1 :m and ² O8 :m , it is natural to study the vacuum polarization functions as these are also defined in terms of vacuum matrix elements of four-quark operators Žbut generally not all at the same spacetime point.. Thus we consider the combination P V ,3 y P A,3 Žthe subscript ‘3’ denotes the isospin flavor., i d 4 x e i qP x ²0 < T Ž V3m Ž x . V3n Ž 0 . y A 3m Ž x . An3 Ž 0 . . <0:
H
Ž r V ,3 y rA ,3 . Ž s . ;
1
Ž 9.
; ln
Q2
ž / 2
H0L ds s
q
w P V ,3 y P A ,3 x Ž Q . s
Q4
2
2
w r V ,3 y rA ,3 x Ž s . ,
2
² a s2 O1 :m y ² a s2 O8 :m
w r V ,3 y rA ,3 x Ž s . q O Ž Qy2 . .
Ž 10 . 2.1. Correlators in d-dimensions
2
where Q ' yq . In writing this spectral relation, we have made use of the first and second Weinberg sum rules w3x, which are both valid in the chiral limit Due to the complexity of QCD, there exist no analytic expressions for the correlators and spectral functions that are valid over the entire energy domain. However, some crucial information is available. At low energies, r V ,3 and rA,3 are determined from t-lepton decays and from eqey scattering. As one proceeds from the resonance region of nonperturbative physics to larger energies, the effect of individual channels becomes indistinguishable and perturbative QCD ŽpQCD. becomes operative. The boundary between nonperturbative and perturbative regions defines a scale L ; 2 ™ 3 GeV. In the pQCD domain, the leading-log behavior of Ž P V ,3 y P A,3 .Ž Q 2 . is given by w4x Q 6 Ž P V ,3 y P A ,3 . Ž Q 2 .
Q
m2
q... .
4 d i qP x mddy .r . i d x e
H
=²0 < T Ž V3m Ž x . V3n Ž 0 . y A 3m Ž x . An3 Ž 0 . . <0: s Ž q mq n y q 2 g mn . Ž P V ,3 y P A ,3 . Ž q 2 . 2 y q mq nP AŽ0. ,3 Ž q . .
²0 < T Ž V3m Ž x . Vm ,3 Ž 0 . y A 3m Ž x . Am ,3 Ž 0 . . <0:
Ž d y 1 . md4y.r .d
s
dr2
`
H0 dQ
8 3
Ž 14 .
The energy scale md.r. Ž‘d.r.’ denotes dimensional regularization. has been introduced to maintain the proper dimensions away from d s 4. It is straightforward to invert Eq. Ž14. and we find
=
2
ž /
Consider the definition of P V ,3 y P A,3 as expressed in d-dimensions,
Ž 4p .
; 2p ² a s O8 :m q ln
Ž 12.
Ž 13 .
s2
H0 ds s q Q
8 3
L2
2
`
² a s2 O1 :m y ² a s2 O8 :m q . . . .
Together, the spectral relations of Eqs. Ž10. and Ž12. imply
Associated with this correlator is the difference of spectral functions r V ,3 y rA,3 ,
1
8 3
s3
Q 6 Ž P V ,3 y P A ,3 . Ž Q 2 .
s Ž q mq n y q 2 g mn . Ž P V ,3 y P A ,3 . Ž q 2 . 2 y q mq nP AŽ0. ,3 Ž q . .
This asymptotic relation will be of special value to our determination of B8Ž3r2. since it contains information on the vacuum matrix element ² O8 :m Žcf. Eq. Ž8... The large-s behavior of the spectral functions can be inferred from the logarithmic term in Eq. Ž11. via continuation to the real q 2-axis,
2
G Ž dr2 . eyi qP x Q d Ž P V ,3 y P A ,3 . Ž Q 2 . .
Ž 15 .
² a s2 O1 :m y ² a s2 O8 :m
Ž 11.
Up to this point the procedure is well defined, as all quantities are finite-valued.
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
To obtain a relation for ² O1 :m , we need to evaluate Eq. Ž15. in the limit of x ™ 0. However, the asymptotic condition of Eq. Ž11. implies that unless the integral on the right-hand-side of Eq. Ž15. is regularized, it will diverge as x ™ 0. There are a number of ways to perform the regularization, and we shall consider two particularly useful approaches – first a momentum space cutoff directly below and then MS renormalization in Section 4. We shall distinguish vacuum matrix elements in the two schemes by means of the superscripts ‘Žc.o..’ for momentum-cutoff and ‘ŽMS.’ for modified minimal subtraction. 2.2. Two sum rules in momentum-space cutoff renormalization Let us remove the divergence which occurs for d s 4 in Eq. Ž15. by cutting off the Q 2-integral at Q 2 s m2 , where m is the renormalization scale and for convenience we set md.r.s m. It is valid to take d s 4 in this case since the integral is finite. We find ² O1 :mŽc .o .. s
3 16p
2
2
H0m dQ
2
Q 4 Ž P V ,3 y P A ,3 . Ž Q 2 . .
175
must be true for the scale m. Then by combining Eq. Ž18. with Eq. Ž10., we obtain the sum rule 2p ² a s O8 :mŽc .o .. `
s I8 '
H0 ds s
2
m2 s q m2
w r V ,3 y rA ,3 x Ž s . .
Ž 19 .
Despite their apparent similarity, it is important to understand that there is a basic difference between the sum rules for ² O1 :mŽc.o.. and ² a s O8 :mŽc.o... The former is obtained rather directly by taking the x ™ 0 limit of Eq. Ž15. and using a cutoff in momentum to regularize the procedure. However, the latter rests upon assuming the dominance of the leading Qy6 term in the OPE of Eq. Ž11.. This assumption becomes increasingly questionable as m is lowered to energies just above the resonance region. It leads to an uncertainty in the value of ² a s O8 :mŽc.o.. which is not present in ² O1 :mŽc.o... We postpone discussion to Section 5 regarding numerical evaluation of the integrals I1 , I8 appearing in Eqs. Ž17. and Ž19..
3. Kaon-to-pion matrix elements of a left-right operator
Ž 16 . Using Eq. Ž10. to express this relation in terms of spectral functions, we arrive immediately at the following sum rule, 16p 2 3
² O1 :mŽc .o .. `
s I1 '
H0
ds s 2 ln
ž
s q m2 s
/
w r V ,3 y rA ,3 x Ž s . . Ž 17 .
It is equally straightforward to derive a sum rule for ² a s O8 :mŽc.o... We first set Q 2 s m2 in Eq. Ž11. to obtain ² a s O8 :mŽc .o .. s
m6 2p
A distinct but equivalent path to learn about ² O1 :m and ² O8 :m is to perform a chiral analysis of the kaon-to-pion matrix elements themselves. However, the usual ŽV y A. = ŽV y A. weak hamiltonian HDSs1 would be of no help in the chiral limit since its K-to-pi matrix elements vanish there. Instead we introduce a ŽV y A. = ŽV q A. nonleptonic operator ODSs1 defined as ODSs1 '
g 22 8
Hd
4
x Dmn Ž x , MW2 . J mn Ž x . ,
J mn Ž x . ' 12 T d Ž x . g m Ž 1 q g 5 . u Ž x . = u Ž 0. g n Ž 1 y g 5 . s Ž 0. m m s 12 T Ž V1y i2 Ž x . q A1yi2 Ž x . .
Ž P V ,3 y P A ,3 . Ž m2 . .
Ž 18 .
Because the variable Q 2 is constrained in Eq. Ž11. to lie in the range where pQCD makes sense, the same
n n = Ž V4q i5 Ž 0 . y A 4qi5 Ž 0 . . ,
Ž 20 .
where Dmn is the W-boson propagator and Vam, A am Ž a s 1, . . . ,8. are the flavor-octet vector, axialvector
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
176
currents. Operators similar to ODSs1 have received some previous attention in the literature w5,6x. The LR chiral structure of ODSs1 ensures the survival of the K-to-pi matrix element M Ž p . s ²pyŽ p .< ODSs1 < KyŽ p .: in the p ™ 0 limit, where we obtain
where a s ' g 32r4p is the QCD fine structure constant and m - MW . Using standard techniques w2x, we use the renormalization group ŽRG. to provide a summation of the leading-log dependence over the range from MW down to m ,
M ' lim M Ž p .
M,
ps0
s
g 22 16 Fp2
Hd
= ²0 < T Ž
4
Ž x . Vm ,3 Ž 0 .
y A 3m
Ž x . Am ,3 Ž 0 . . <0: . Ž 21 .
In the following, we perform a leading-log calculation of QCD corrections to the chiral matrix element M . This leads naturally to renormalization group equations ŽRGE. for the quantities ² O1 :m and ² a s O8 :m . Since the W-boson propagator in Eq. Ž21. acts as a cutoff for contributions with < x < G My1 W , we consider the leading term of the following operator product expansion, V3m Ž x . V3n Ž 0 . y A 3m Ž x . An3 Ž 0 .
Ž 0.
V3n
Ž 0.
y A 3m
Ž 0.
An3
c1 Ž m . s
c8 Ž m . s
3.1. Leading-log analysis of QCD corrections
s V3m
c1 Ž m . ² O1 :m q c8 Ž m . ² O8 :m ,
2'2 Fp2
Ž 0. q O Ž x . .
1
ž ž
9 1 6
8r9
asŽ m.
/ ž / ž q8
a s Ž MW .
8r9
asŽ m.
y
a s Ž MW .
y1r9
asŽ m. a s Ž MW . asŽ m.
a s Ž MW .
/ y1r9
/
asŽ m. s 1 q 9
asŽ m. 4p
ln
MW2
ž / m2
a s Ž MW . .
ž /
0 16r3
3r2 7
O1 , O8
Ž 25 .
Ž 28 .
An expansion of Eq. Ž26. through second order in a s Ž m . gives GF
3
2'2 Fp2 3 q 32p
² O1 :m q
ln2 2
8p
ln
MW2
8
2
3
ž /ž m
MW2
ž / m2
² a s O8 :m
² a s2 O1 :m y ² a s2 O8 :m
/
.
Ž 29 .
H
In order to express this vacuum matrix element at some lower energy m , we must take QCD radiative corrections into account. The effect of these will be to mix O1 with O8 . The result of mixing at one-loop order is
,
with
Ž 22 .
so that the matrix element specified at energy scale MW becomes GF ² O1 :M W . M, Ž 24 . 2'2 Fp2
,
Ž 27 .
M,
Evaluation of the spacetime integral in Eq. Ž21. is straightforward, gmn d 4 x Dmn Ž x , MW2 . s 2 , Ž 23 . MW
as MW2 O1 O1 ™ q ln O8 O8 4p m2
Ž 26 .
where
x D Ž x , MW2 .
V3m
GF
Let us gain some feeling for the numbers involved. The minimum value of renormalization scale considered in this paper is m 0 s 2 GeV. Adopting this scale and taking a s Ž MW . s 0.119 and a s Ž m 0 . s 0.334 w7x, we find for the RG coefficients in Eq. Ž26., M,
GF 2'2 Fp2
1.071² O1 :m 0 q 0.268² O8 :m 0 ,
Ž 30 .
whereas the coefficients in the perturbative expression of Eq. Ž29. become M,
GF 2'2 Fp2
² O1 :m 0 q 0.88² a s O8 :m 0
y0.52 Ž ² a s2 O8 :m 0 y 83 ² a s2 O1 :m 0 . .
Ž 31 .
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
Finally, the condition of scale independence for the matrix element M , E m2 2 M s 0 , Ž 32 . Em leads directly to the renormalization group equations E 3 ² a s O8 :m , m2 2 ² O1 :m s Ž 33 . 8p Em E 1 16 ² a s2 O1 :m y 2² a s2 O8 :m . m2 2 ² a s O8 :m s 4p 3 Em Ž 34 . To summarize – the above operator-product analysis involves computing radiative corrections perturbatively to one-loop order in QCD Žcf. Eq. Ž25.. and retaining only the dependence on leading logarithms in the evolution from scale MW down to scale m. The value of m cannot be taken too small, otherwise the perturbative framework breaks down. 3.2. Verification of the operator product expansion Despite the explicit difference between the procedures of Section 2 and that carried out directly above, the two are equivalent. In particular, we can show that the ² O1 :m sum rule of Eq. Ž17. and the ² O8 :m sum rule of Eq. Ž19. reproduce the OPE to the leading log level. This verifies both the derivations that we provided and gives a direct insight into the workings of the OPE. Consider a partition of M characterized by the scale m , M s M- Ž m . q M) Ž m . , Ž 35 . where M- Ž m . and M) Ž m . are dependent respectively on contributions with Q - m and Q ) m. Also, in addition to maintaining the requirement that m lie in the pQCD domain, we further constrain it to obey m < MW . We then obtain M- Ž m . s
3GF MW2
m2
32'2 p 2 Fp2
H0
2
dQ 2
Q4 Q 2 q MW2
= P V ,3 Ž Q . y P A ,3 Ž Q 3GF m2 s dQ 2 Q 4 2 2 32'2 p Fp 0
2
and M) Ž m . s
M) Ž m . s
3
'2 Fp2
8p
3 32p
Q4
2
2
Q 2 q MW2
ln2 2
MW2
ž / ž /
ln
m2
Ž 37 .
² a s O8 :m
MW2
m2
= Ž ² a s2 O8 :m y 83 ² a s2 O1 :m . .
Ž 38 .
Comparison of Eq. Ž29. with Eq. Ž38. yields the relation, M- Ž m . s
GF 2'2 Fp2
² O1 :m .
Ž 39 .
We see that the operators that we originally defined independently of the weak interaction are in fact the ones that appear in the Operator Product Expansion. Of course, this is to be expected, but it proves an explicit pedagogical demonstration of the nature of the OPE. 3.3. Sum rules and RG relations To complete the chain of logic, we demonstrate consistency of the spectral function sum rules for ² O1 :mŽc.o.. and ² O8 :mŽc.o.. with the corresponding renormalization group relations obtained previously from the operator product expansion Žcf. Eqs. Ž33. and Ž34... The RG equation for ² O1 :mŽc.o.. is immediately recovered upon differentiating the sum rule of Eq. Ž17. and making use of Eq. Ž19.,
Em2
² O1 :mŽc .o .. 3
s 8p 3
Ž 36 .
GF
y
= P V ,3 Ž Q 2 . y P A ,3 Ž Q 2 . q O Ž m2rM W2 .
32'2 p 2 Fp2
`
Hm dQ
Upon inserting the large-Q form of Eq. Ž11. into Eq. Ž37., we obtain
E
H
3GF MW2
= P V ,3 Ž Q 2 . y P A ,3 Ž Q 2 . .
m2
.
177
s 8p
m2 P
`
s2
H ds s q m 2p 0
² a s O8 :mŽc .o .. .
2
w r V ,3 y rA ,3 x Ž s . Ž 40 .
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
178
To derive the RG relation for ² O8 :mŽc.o.. we start with the sum rule of Eq. Ž19.,
E m
2
Em
below m2 , we can let d ™ 4 and recover exactly the cutoff integrand of Eq. Ž16., ² O1 :mŽd .r .. s ² O1 :m
² O8 :mŽc .o .. 2
Ž d y 1 . m4y d
q
m2
E
s 2p Em
m2
` 2
ds
H0
sqm
2
Ž 4p .
s 2 w r V ,3 y rA ,3 x Ž s . .
`
=
Hm dQ
Ž 41 . The integral in the above is seen to be m6 Ž P V ,3 Ž m2 . y P A ,3 Ž m2 . . . We replace it using the asymptotic expression of Eq. Ž13. to obtain
E m2
Em2
² O8 :mŽc .o ..
m2 s
2p Em2
=
8 3
ln
ž / L2
2
2
2
G Ž dr2 .
Q d Ž P V ,3 y P A ,3 . Ž Q 2 . .
Ž 43 . The asymptotic tail can be analysed in d dimensions. This introduces scheme dependence depending on which method is used to define Dirac algebra away from four dimensions, e.g. the naive dimensional regularization ŽNDR. and ’t Hooft-Veltman ŽHV. schemes in which g 5 is respectively anticommuting and commuting w8x. We find 1 3
² a s2 O1 :mŽ c .o . . y ² a s2 O8 :mŽ c .o . .
H0L ds s
q
m2
E
2
dr2
Ž d y 1 . Q d P Ž P V ,3 y P A ,3 . Ž Q 2 . s 2pa s² O8 :m Q dy 6 1 q Ž d s q 14 . e q O Ž a s2 . . Ž 44 .
w r V ,3 y rA ,3 x Ž s . q O Ž my2 . , Ž 42 .
where e ' 4 y d and d s ' d scheme is associated with the loop integration and scheme-dependence. The values of d s in the NDR and HV schemes are
from which the RG relation of Eq. Ž34. follows directly.
ds s
4. MS renormalization
The Q 2 ) m2 integral can then be performed with the result
½
y5r6 1r6
Ž NDR. Ž HV. .
Ž 45 .
² O1 :mŽd .r .. The work of the preceding sections was based on a momentum-space cutoff renormalization scheme, which is useful in yielding sum rules directly related to experimental data. At the same time, however, it is distinct from the more standard MS prescription. In this section, we demonstrate how to relate the two approaches. 4.1. Short distance analysis Let us reconsider Eq. Ž15.. We can show how the cutoff renormalization is related to dimensional regularization by keeping the high-Q 2 part of the integral in Eq. Ž15. and analyzing it in terms of an e-expansion. We divide the integral into integration ranges below and above m2 . For the part of integral with Q 2
s ² O1 :m 3
2
q 8p 4 y d ² = a s O8 :m .
y g q ln4p q 32 q 2 d s
Ž 46 .
The MS prescription is a subcase of dimensional regularization in which the terms 2rŽ4 y d . y g q ln4p in Eq. Ž46. are removed in the renormalization procedure. This gives our desired relation in a given scheme, 3as 3 Žc .o .. ² O1 :MS q Ž q 2 d s . ² O8 :m . Ž 47. m s ² O1 :m 8p 2 To derive an analogous relation between ² O8 :m MS and ² O8 :mŽc.o.. we employ the leading behavior of correlators and spectral functions in the MS renor-
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
179
Fig. 1. Some QCD corrections: full theory Ža.-Žb., effective theory Žc.-Žd..
malization prescription, which has been calculated using the NDR scheme w4x, Q 6 Ž P V ,3 y P A ,3 . Ž Q 2 . ;2p ² a s O8 :mŽ MS. 8
q 2q
3
119
ln
Q2
ž / ž / m2
Q2
y ln
q 12
² a s2 O1 :mŽ MS.
m2
² a s2 O8 :mŽ MS.
Ž 48 .
and Ž r V ,3 y rA ,3 . Ž s . 1 ; 3 83 ² a s2 O1 :mŽMS . y ² a s2 O8 :mŽMS . s q... . Ž 49. Then by setting Q s m in Eq. Ž48. and combining the result with Eq. Ž49. we find
m
2
H0 ds s q m s 2pa s q
as p
ž
GF
2
1q
119a s 24p
/
² qq < O1 < qq :tree
2'2
w r V ,3 y rA ,3 x Ž s .
3as 8p
ž
ln
MW2 m2
/
y 1 ² qq < O8 < qq :tree
O Ž a s2 . . qO
² O8 :mŽMS .
² O1 :mŽMS . .
Ž 50 .
Comparison with Eq. Ž47. then implies that the NDR matrix element is given to first order in a s by 119a s as ² O8 :mŽ MS. s 1 y ² O8 :mŽc .o .. y ² O1 :m . 24p p Ž 51 .
ž
Ms
q
s2
`
Determination of the coefficients c1 and c8 proceeds in two steps: first calculate QCD radiative corrections in both the full and effective theories, and then ‘match’ the two calculations. We shall carry out this procedure at one-loop order of the QCD radiative corrections. Since c1 s 1 q O Ž a s2 . and c8 s O Ž a s ., this will yield a determination of c8 . For definiteness, we consider the free scattering of zero momentum quarks and adopt a common quark mass m to serve as the infrared cutoff. In the full theory, evaluations of the one-loop radiative corrections like those displayed in Fig. 1Ža., Fig. 1Žb. are finite and yield
/
4.2. MS matching at one loop In this section we perform the matching at one loop and verify the scheme independence of the . result. The effective operator ODŽeff Ss1 , is expressed in O O terms of local operators 1 and 8 , GF . ODŽeff c1 Ž m . O1 q c8 Ž m . O8 . Ž 52 . Ss1 s 2'2
Ž 53 .
The analogous calculation in the effective theory is divergent and must be regularized. We employ dimensional regularization which introduces the scheme dependence mentioned above. Our calculation of amplitudes like those in Fig. 1Žc., Fig. 1Žd. gives Ms
GF
² qq < O1 < qq :tree
2'2 q
=
3as 8p 2
e
Ž1 q ds e q d l e .
y g q ln4p y ln
qc8 ² qq < O8 < qq :tree ,
m2
m2
² qq < O8 < qq :tree
Ž 54 .
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
180
After the removal of the divergent term in MS renormalization, we compare the full theory and the effective theory to identify the coefficient function as c8 Ž m . s
3as
ln
8p
MW2
m2
y 32 y 2 d s .
Ž 55 .
. When ODŽeff Ss1 is applied to our problem of vacuum matrix elements, we have the amplitude
Ms
GF 2'2 Fp2
MS c1 Ž m . ² O1 :MS . m q c 8 Ž m . ² O8 :m
Given our previous identification of the scheme dependent operator O 1 in Eq. Ž47., it can be seen that the scheme dependence cancels between that of the matrix element and of the coefficient in the operator product expansion.
5. Numerical analysis and uncertainties We base our numerical determination of ² O8 :mŽc.o.. and ² O1 :mŽc.o.. respectively on the sum rules in Eq. Ž19. and Eq. Ž17.. This involves calculation of the integrals I8 and I1 , which contain the combination of spectral functions r V ,3 y rA,3 , `
2
H0 ds K Ž s, m . w r i
V ,3 y r A ,3
x Ž s . Ž i s 1,8 . , Ž 57 .
with K8 s s2
m2 s q m2
,
K 1 ' s 2 ln
ž
s q m2 s
/
.
Ž 58 .
As such, I8 and I1 belong to the family of spectral integrals which include the DMO sum rule w9x, the two Weinberg sum rules w3x and the sum rule for the pion electromagnetic mass splitting w10x. The kernels occurring in these ‘classical’ sum rules are K DM O s 1rs , K W2 s s ,
K W1 s 1 ,
K em s sln s .
I8 s y Ž 0.30 " 0.04 . P 10y2 , I1 s y Ž 0.42 " 0.06 . P 10y2 ,
Ž 60 .
At the higher renormalization scale, m s 4 GeV, we obtain I8 s y Ž 0.43 " 0.06 . P 10y2 ,
Ž 56 .
Ii s
set of constraints for any evaluation of I8 and I1. Using an updated form of our earlier study w11x of chiral sum rules, we find for renormalization scale m s 2 GeV the values
Ž 59 .
This happenstance is most fortunate as the integrals defined by the kernels in Eq. Ž59. form a powerful
I1 s y Ž 0.97 " 0.12 . P 10y2 .
Ž 61 .
5.1. Uncertainties from data analysis The error bars quoted above correspond to our estimate of the uncertainty in the sum rules due to imprecision in our knowledge of the spectral functions. Before proceeding let us describe how these were arrived at, and assess other sources of uncertainty. The data at lower values of s are extremely well known, and they introduce very little uncertainty compared to other sources which we are concerned with. The high energy tail of r V y rA is small above s s 5 GeV 2 . In the m s 2 GeV integrals, there remains essentially no sensitivity to this high energy tail once the constraints are imposed 2 . It is in the matching of these two regions that one encounters the greatest uncertainties. Fortunately, the four integral constraints described above are very stringent and allow us to limit the uncertainties in this region. We have used several methods to construct spectral functions which match the data within error bars and yet satisfy our sum rule constraints. These give variations in our integrals of under 6%. In addition, we have considered the situation where
2 At higher values of m there occurs more sensitivity to the asymptotic tail, and it is the tail that describes the logarithmic running of the O 8 matrix element. While we have a good handle on the size of the 1r s 3 component of the tail, we know little about the 1r s 4 component. However as long as the 1r s 4 portion is not much larger than the 1r s 3 behavior for s) 5 GeV 2 , its effect is within our quoted error bars.
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
the asymptotic form of the spectral function is reached only on the average, with a damped oscillating term that provides deviations from the average. Since our sum rules are equivalent to transforming back to euclidean Q 2 , these oscillations give exponentially suppressed effects at large m2 once integrated. Again the constraints are very powerful in further limiting this effect, and our studies lead us to increase the uncertainty in the fit to 10% to account for this form of variation. We also must account for the fact that the data and the input into the constraints are measured in a world where mp2 is not zero, yet we are interested in the result in the chiral limit. This introduces corrections of order mp2 rmr2 which is of order 3%. In fact, since we know some of the physics involved in passing to the chiral limit, we could attempt to make a realistic correction for the extrapolation to the chiral limit. However, since this would appear to introduce some model dependence into our procedure, we prefer to simply include the uncertainty as an error bar. In practice, the effect which has the most sensitivity for our results is the constraint of the pion electromagnetic mass difference, since the kernel K em bears the greatest resemblance to K 1 and K 8 . Work on the pion and kaon electromagnetic mass differences indicates that the quark mass corrections are somewhat larger than average. Therefore, to be conservative we triple the canonical error estimate, leading us to quote a 9% uncertainty for the extrapolation to the chiral limit. We have added this in quadrature to the statistical uncertainty to arrive at the error bars cited above. 5.2. Uncertainty from the operator product expansion Finally, we need to address the fact that it has become common to cite matrix elements at a scale m s 2 GeV, which is a rather low scale for perturbative QCD to be fully in the asymptotic region. In fact, our method can be used for any m , and we can check if the asymptotic QCD behavior is obtained. For example, the renormalization group equations relate the m-dependence to the magnitudes of the operator matrix elements. Although one of the relations ŽEqs. Ž33. and Ž40.. is automatically satisfied,
181
we explicitly showed above that the second holds only if m is large enough, i.e. that it is well into the region where the asymptotic tail of the spectral functions becomes applicable. It is easy to see from the data alone that this is not the case at m s 2 GeV. Another way to state the same result is that there remain power corrections in the sum rule, although the renormalization group states that the running with m should be only logarithmic. We believe Žbecause of the generality of our framework. that this issue must also be present in the lattice results, and we urge the evaluation of lattice matrix elements at larger values of m. We do see such non-asymptotic behavior in our results. Actually, for O1 our method of cutting off the high frequency modes of the current-current product is in accord with Wilson’s original idea of the definition of a scale-dependent matrix element. Therefore, our sum rule for ² O1 :mŽc.o.. can be treated as a definition of this amplitude at any scale m , even if that scale is not yet asymptotic. For O8 , however, there is some uncertainty as to an ideal definition in the non-asymptotic region. For example, the RG relation of Eqs. Ž33. and Ž40. requires that we use exactly our definition, yet this only is foolproof if the RGEs are fully valid. Equivalently, if this matrix element is defined via the coefficient of Qy6 in the vacuum polarization, there can be order Qy8 corrections remaining if one works in the non-asymptotic region. We see evidence of such power corrections. Moreover, attempting to discard the Qy8 effects leads to a larger matrix element. At m s 4 GeV, the corrections are rather modest, in line with other uncertainties that we have described. However, at m s 2 GeV, these non-asymptotic effects represent a significant intrinsic uncertainty. We haÕe taken these into account by combining two eÕaluations, one obtained by eÕaluating the sum rule at m s 4 GeV and using the RGE to transform down to m s 2 GeV, and the other by direct eÕaluation of the sum rule at the lower scale. We aÕerage these two and assign the difference as an independent error bar. The error bar is chosen such that a one-sigma variation reproduces the full range between the two methods of evaluation. We do this for both matrix elements. The quoted error bar at m s 4 GeV is scaled down from the m s 2 GeV values by a factor of four, as appropriate for quadratic power corrections.
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
182
5.3. ConÕersion to MS renormalization
In the NDR scheme with m s 2 GeV this translates into the following B-factor determinations,
We now transform to the MS matrix elements. The results of our direct evaluation at m s 2 GeV leads to the matrix elements
B7Ž3r2.
w NDR, m s 2 GeVx
ž
2GeV
Ž 62 .
B8Ž3r2.
w NDR, m s 2 GeVx
ž
s 1.11 " 0.16 " 0.23 . where we have taken a s Ž2 GeV. , 0.334. When we evaluate the integrals at m s 4 GeV and use the RGE to rescale back to m s 2 GeV, we instead obtain ² O8 : ŽMS. s y Ž 1.29 " 0.15 . P 10y3 GeV 6 , 2GeV
² O1 : ŽMS. s y Ž 1.02 " 0.10 . P 10y4 GeV 6 , 2GeV
Ž 63 .
which is a measure of the potential non-asymptotic corrections found at low values of m. As explained above, this leads us to quote our result as ² O8 : ŽMS. s y Ž 0.98 " 0.13 " 0.23 . P 10y3 GeV 6 , 2GeV
² O1 : ŽMS. s y Ž 0.86 " 0.10 " 0.16 . P 10y4 GeV 6 , 2GeV
Ž 64 . The first error bar corresponds to the uncertainty in the evaluation of the sum rule whereas the second is the potential non-asymptotic uncertainty defined above. Note that the two matrix elements differ by an order of magnitude. The related K 0 ™ pp matrix elements are then ² Ž pp . Is2 < Q7Ž3r2. < K 0 : 2 GeV s Ž 0.43 " 0.05 " 0.08 . GeV 3 ,
/
0.1 GeV ms q m ˆ
2
/ Ž 66 .
Note that the combination of B-factor and quark masses is ‘physical’, appearing in the formula for e Xre . The comparison of these results with some lattice evaluations is hampered by the fact that our evaluation is of the full matrix elements, while most lattice calculations are of the B-factors directly w12,13x. If large values of quark masses are used, our results are larger that other estimates, yet for the currently favored smaller quark masses the results are not inconsistent. There is one recent lattice evaluation which provides absolute matrix elements which we can compare to. The Rome group w14x quotes the matrix element in in the the quenched approximation using K ™ p matrix elements plus the chiral relation between K ™ p and K ™ pp . When the meson masses are taken as the kaon mass they find, in the NDR scheme at m s 2 GeV, ² Ž pp . Is2 < Q7Ž3r2. < K 0 : 2 GeV s Ž 0.22 " 0.04 . GeV 3 , ² Ž pp . Is2 < Q8Ž3r2. < K 0 : 2 GeV s Ž 1.02 " 0.10 . GeV 3 . Ž 67 . The quoted error does not include estimates of the effect of quenching nor the extrapolation to the continuum limit. Their results seem to be systematically smaller than ours. Finally at m s 4 GeV we have the vacuum matrix elements, y3 ² O8 :ŽMS. GeV 6 , 4G eV s y Ž 1.63 " 0.20 " 0.06 . P 10
² Ž pp . Is2 < Q8Ž3r2. < K 0 : 2 GeV s Ž 2.58 " 0.37 " 0.47 . GeV 3 .
ms q m ˆ
2
s 0.55 " 0.07 " 0.10 ,
² O8 : ŽMS. s y Ž 0.67 " 0.09 . P 10y3 GeV 6 , ² O1 : ŽMS. s y Ž 0.70 " 0.10 . P 10y4 GeV 6 , 2GeV
0.1 GeV
Ž 65 .
y4 ² O1 :ŽMS. GeV 6 . 4G eV s y Ž 1.71 " 0.20 " 0.04 . P 10 Ž 68 .
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
The corresponding K ™ pp matrix elements are ² Ž pp . Is2 < Q7Ž3r2. < K 0 :4 GeV s Ž 0.85 " 0.10 " 0.02 . GeV 3 , ² Ž pp . Is2 < Q8Ž3r2. < K 0 :4 GeV s Ž 4.34 " 0.56 " 0.15 . GeV 3
Ž 69 .
and for the B-factors we obtain, B7Ž3r2. w NDR, m s 4 GeV x
ž
0.1 GeV ms q m ˆ
2
/
s 1.10 " 0.13 " 0.03 , B8Ž3r2.
w NDR, m s 4 GeVx
ž
s 1.87 " 0.25 " 0.07 .
0.1 GeV ms q m ˆ
2
/
O 8 matrix element. This raises the concern that when one is working at such a low value of m , there may be significant corrections even in lattice evaluations. Certainly, use of m - 1 GeV, as occurs in many model dependent evaluations, appears extremely dubious. The values displayed above are based on working in the chiral limit of massless quarks. One must, however, add to these the chiral corrections. Work has begun on this important problem w15x. Nevertheless, it is interesting to look at the phenomenological consequences of the results reported in this paper.. While we cannot give a full evaluation of e Xre because we have not evaluated the contribution of B6 , we can give the contribution arising from the electroweak penguin,
eX
Ž 70 .
That B7Ž3r2. has a large variation with m is expected from the RGE of Eq. Ž34., given our previous result that the vacuum matrix element of O8 is much larger than that of O1.
183
že/
B8
s Ž y12 " 3 . P 10y4 .
Ž 71 .
The effect of B6 is expected to be positive, and needs to be almost three times larger than that of B8 if the Standard Model is to account for the experimental value.
6. Concluding comments Acknowledgements The method that we have described has the virtue of being a fully rigorous framework. Moreover, the input data is largely taken from experiment, and hence represents an evaluation that is model independent. Besides the direct comparison with the results with lattice calculations, there may also be other lessons in this calculation. Since in our method the matrix elements are evaluated by constructing the Euclidean vacuum polarization function, lattice calculations may also be able to directly follow many of the steps in our procedure, and thereby test their methods in more detail. By explicitly studying the product of currents at non-zero values of the spatial separation, the matrix elements can be evaluated without some of the operator mixing problems that occur on the lattice when using local operators. Moreover, by studying vacuum matrix elements as well as hadronic matrix elements, the chiral relations can be checked on the lattice. Finally, we recall the lesson, discussed above, that power-law corrections still appear to exist at m s 2 GeV, especially in the
This work was supported in part by the National Science Foundation. We thank Guido Martinelli for useful comments.
References w1x KTeV Collaboration, A. Alari-Harati et al., Phys. Rev. Lett. 83 Ž1999. 22; G. Barr, Plenary talk at XIX Intl. Symp. on Lepton-Photon Interactions at High Energies, Stanford University, 8r7r99-8r14r99. w2x E.g. see J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model, Cambridge University Press, Cambridge, England 1992. w3x S. Weinberg, Phys. Rev. Lett. 18 Ž1967. 507. w4x L.V. Lanin, V. P Spirodonov, K.G. Chetyrkin, Yad. Fiz. 44 Ž1986. 1372. For earlier work, see also M. Shifman, A. Vainshtein, V. Zakarov, Nucl. Phys. B 147 Ž1979. 385. w5x J.F. Donoghue, E. Golowich, Phys. Lett. B 315 Ž1993. 406. w6x M. Davier, A. Hocker, L. Girlanda, J. Stern, Phys. Rev. D 58 ¨ Ž1998. 096014. w7x ALEPH Collaboration, Eur. Phys. Jnl. C 4 Ž1998. 409.
184
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
w8x See e.g. A.J. Buras, P.H. Weisz, Nucl. Phys. B 333 Ž1990. 66; M. Ciuchini et al., Nucl. Phys. B 415 Ž1994. 403; S. Herrlich, U. Nierste, Nucl. Phys. B 455 Ž1995. 39. w9x T. Das, V.S. Mathur, S. Okubo, Phys. Rev. Lett. 19 Ž1967. 859. w10x T. Das, G.S. Guralnik, V.S. Mathur, F.E. Low, J.E. Young, Phys. Rev. Lett. 18 Ž1967. 759. w11x J.F. Donoghue, E. Golowich, Phys. Rev. D 49 Ž1994. 1513.
w12x R. Gupta, T. Battacharya, S. Sharpe, Phys. Rev. D 55 Ž1997. 4036. w13x A. Soni, private communication; T. Blum et.al. heplatr9908025. w14x A. Donini, V. Gimenez, L. Giusti, G. Martinelli, heplatr9910017. w15x V. Cirigliano, J.F. Donoghue, E. Golowich, work in progress.
Physics Letters B 478 Ž2000. 172–184
Dispersive calculation of B7Ž3r2. and B8Ž3r2. in the chiral limit John F. Donoghue a
a,b
, Eugene Golowich
a
Department of Physics and Astronomy, UniÕersity of Massachusetts, Amherst, MA 01003, USA b TH DiÕision, CERN, GeneÕa, Switzerland Received 2 December 1999; accepted 16 February 2000 Editor: R. Gatto
Abstract We show how the isospin vector and axialvector current spectral functions r V,3 and rA,3 can be used to determine in leading chiral order the low energy constants B7Ž3r2. and B8Ž3r2.. This is accomplished by matching the Operator Product Expansion to the dispersive analysis of vacuum polarization functions. The data for the evaluation of these dispersive integrals has been recently enhanced by the ALEPH measurement of spectral functions in tau decay, and we update our previous phenomenological determination. Our calculation yields in the NDR renormalization scheme and at renormalization scale m s 2 GeV the values B7Ž3r2. s 0.55 " 0.07 " 0.10 and B8Ž3r2. s 1.11 " 0.16 " 0.23 for the quark mass values ms q m ˆ s 0.1 GeV. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The recent KTeV and NA48 findings that e Xre , 20 P 10y4 raise the important question whether a value so large can be consistent with Standard Model expectations w1x. One of the key quantities upon which the Standard Model prediction is based is the B-factor B8Ž3r2.. In this paper, we work in the chiral limit to obtain an analytic expression for B8Ž3r2. Žand also for B7Ž3r2. .. Our results take the form of sum rules involving the difference r V y rA of isospin-one spectral functions. The constants B7Ž3r2. Ž m . and B8Ž3r2. Ž m . are defined in terms of the matrix elements ²2p < QiŽ3r2. < K :m ' BiŽ3r2. Ž m . ²2p < QiŽ3r2. < K :mvac
Ž i s 7,8 . , Ž 1 .
where m is the renormalization scale, Q7Ž3r2. and Q8Ž3r2. are the D I s 3r2 electroweak penguin operators Q7Ž3r2. ' s a G Lm d a u b GmR u b y d b GmR d b
ž
/
q s a G Lm u a u b GmR d b , Q8Ž3r2. ' s a G Lm d b u b GmR u a y d b GmR d a
ž
q s a G Lm u b u b GmR d a ,
/ Ž 2.
a,b are color labels and G Lm ' g m Ž1 q g 5 ., G Rm ' g m Ž1 y g 5 .. Analogous B-factors are defined for the other weak operators. The most important contributions to e Xre are the matrix elements of the penguin operator Q6 and the electroweak penguin operator
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 2 3 9 - 2
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
Q83r2 . As expressed in terms of the B-factors this is seen in the approximate numerical relation,
eX e
s 10 = 10y4 = Ž 2.5B6 y 1.3 B8Ž 3r2 . .
ž
100 MeV ms q md
/
,
Ž 3.
when evaluated at m s 2 GeV in the MS-NDR renormalization scheme. Alternatively, in terms of the operator matrix elements themselves one has
eX e
s 10 = 10y4 y3.1 GeVy3 P ² Q6 : 0 y0.51 GeVy3 P ² Q8Ž 3r2 . : 2 ,
Ž 4.
where O1 , O8 are the local four-quark operators t3 t3 t3 t3 O1 ' qgm q qg m q y qgm g 5 q qg mg 5 q , 2 2 2 2 t t 3 3 O8 ' qgm l A q qg ml A q 2 2 t t3 3 y qgmg 5 l A q qg mg 5 l A q . Ž 7. 2 2 In the above, q s u,d, s, t 3 is a Pauli Žflavor. matrix, l A 4 are the Gell Mann color matrices and the subscripts on O1 , O8 refer to the color carried by their currents. 1 A chiral evaluation of B7Ž3r2. and B8Ž3r2. is thus seen Žcf. Eq. Ž6.. to depend upon ² O1 :m and ² O8 :m . The K 0 ™ pp matrix elements are likewise recoverable in the chiral limit from the vacuum matrix elements,
where ² Q6 : 0 ' ² Ž pp . Is0 < Q6 < K
0:
lim ² Ž pp . Is2 < Q7Ž3r2. < K 0 :m s y
and
² Q8Ž3r2. : 2 ' ² Ž pp . Is2 < Q8Ž3r2. < K 0 : ,
ps0
Ž 5.
again defined in the NDR scheme at m s 2 GeV. A kaon-to-pion weak matrix element can be analyzed by chiral methods and expressed as an expansion in momenta and quark masses w2x. Here we shall calculate the leading term in such an expansion, valid in the limit of exact chiral symmetry where the Goldstone bosons become massless. In fact, our analysis yields values for the relevant K-to-p matrix elements themselves. However, we shall transcribe this information into the equivalent form of B-factors in order to express our results in the more conventional form and thus allow comparison with other techniques. When considered in the chiral limit, the content of Eq. Ž1. reduces to lim B7Ž3r2. Ž m .
ps0
3 sy
Fp4
P
mu q md mp2
P
mu q ms m2K
² O1 :m ,
lim B8Ž3r2. Ž m .
ps0
1 sy =
Fp4 1 3
P
mu q md mp2
P
173
4 Fp3
² O1 :m ,
lim ² Ž pp . Is2 < Q8Ž3r2. < K 0 :m
ps0
4 sy
Fp3
1 3
² O1 :m q 12 ² O8 :m .
Ž 8.
The rest of this paper describes a calculational procedure to obtain analytic expressions for the vacuum matrix elements. In Section 2, we show how to extract ² O1 :m and ² O8 :m from the isospin-one vector and axialvector correlators by deriving sum rules for ² O1 :m and ² O8 :m in a momentum cutoff scheme. This renormalization is especially well suited for comparing theory to experiment. In Section 3 we introduce a four-quark operator ODSs1 , distinct from the familiar nonleptonic hamiltonian HDSs1 , whose kaon-to-pion matrix element is related to ² O1 :m and ² O8 :m in the chiral limit. We demonstrate consistency of this information with the sum rules of Section 2. Section 4 describes a procedure for obtaining ² O1 :m and ² O8 :m in MS renormalization, which is commonly used in lattice-theoretic simulations. Our final numerical results and concluding statements appear, respectively, in Section 5 and Section 6.
mu q ms m2K
² O1 :m q 12 ² O8 :m ,
1
Ž 6.
Throughout we denote vacuum expectation values as ²0 < O <0:m ' ² O :m .
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
174
2. Dispersive analysis of vacuum polarization functions In seeking values for ² O1 :m and ² O8 :m , it is natural to study the vacuum polarization functions as these are also defined in terms of vacuum matrix elements of four-quark operators Žbut generally not all at the same spacetime point.. Thus we consider the combination P V ,3 y P A,3 Žthe subscript ‘3’ denotes the isospin flavor., i d 4 x e i qP x ²0 < T Ž V3m Ž x . V3n Ž 0 . y A 3m Ž x . An3 Ž 0 . . <0:
H
Ž r V ,3 y rA ,3 . Ž s . ;
1
Ž 9.
; ln
Q2
ž / 2
H0L ds s
q
w P V ,3 y P A ,3 x Ž Q . s
Q4
2
2
w r V ,3 y rA ,3 x Ž s . ,
2
² a s2 O1 :m y ² a s2 O8 :m
w r V ,3 y rA ,3 x Ž s . q O Ž Qy2 . .
Ž 10 . 2.1. Correlators in d-dimensions
2
where Q ' yq . In writing this spectral relation, we have made use of the first and second Weinberg sum rules w3x, which are both valid in the chiral limit Due to the complexity of QCD, there exist no analytic expressions for the correlators and spectral functions that are valid over the entire energy domain. However, some crucial information is available. At low energies, r V ,3 and rA,3 are determined from t-lepton decays and from eqey scattering. As one proceeds from the resonance region of nonperturbative physics to larger energies, the effect of individual channels becomes indistinguishable and perturbative QCD ŽpQCD. becomes operative. The boundary between nonperturbative and perturbative regions defines a scale L ; 2 ™ 3 GeV. In the pQCD domain, the leading-log behavior of Ž P V ,3 y P A,3 .Ž Q 2 . is given by w4x Q 6 Ž P V ,3 y P A ,3 . Ž Q 2 .
Q
m2
q... .
4 d i qP x mddy .r . i d x e
H
=²0 < T Ž V3m Ž x . V3n Ž 0 . y A 3m Ž x . An3 Ž 0 . . <0: s Ž q mq n y q 2 g mn . Ž P V ,3 y P A ,3 . Ž q 2 . 2 y q mq nP AŽ0. ,3 Ž q . .
²0 < T Ž V3m Ž x . Vm ,3 Ž 0 . y A 3m Ž x . Am ,3 Ž 0 . . <0:
Ž d y 1 . md4y.r .d
s
dr2
`
H0 dQ
8 3
Ž 14 .
The energy scale md.r. Ž‘d.r.’ denotes dimensional regularization. has been introduced to maintain the proper dimensions away from d s 4. It is straightforward to invert Eq. Ž14. and we find
=
2
ž /
Consider the definition of P V ,3 y P A,3 as expressed in d-dimensions,
Ž 4p .
; 2p ² a s O8 :m q ln
Ž 12.
Ž 13 .
s2
H0 ds s q Q
8 3
L2
2
`
² a s2 O1 :m y ² a s2 O8 :m q . . . .
Together, the spectral relations of Eqs. Ž10. and Ž12. imply
Associated with this correlator is the difference of spectral functions r V ,3 y rA,3 ,
1
8 3
s3
Q 6 Ž P V ,3 y P A ,3 . Ž Q 2 .
s Ž q mq n y q 2 g mn . Ž P V ,3 y P A ,3 . Ž q 2 . 2 y q mq nP AŽ0. ,3 Ž q . .
This asymptotic relation will be of special value to our determination of B8Ž3r2. since it contains information on the vacuum matrix element ² O8 :m Žcf. Eq. Ž8... The large-s behavior of the spectral functions can be inferred from the logarithmic term in Eq. Ž11. via continuation to the real q 2-axis,
2
G Ž dr2 . eyi qP x Q d Ž P V ,3 y P A ,3 . Ž Q 2 . .
Ž 15 .
² a s2 O1 :m y ² a s2 O8 :m
Ž 11.
Up to this point the procedure is well defined, as all quantities are finite-valued.
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
To obtain a relation for ² O1 :m , we need to evaluate Eq. Ž15. in the limit of x ™ 0. However, the asymptotic condition of Eq. Ž11. implies that unless the integral on the right-hand-side of Eq. Ž15. is regularized, it will diverge as x ™ 0. There are a number of ways to perform the regularization, and we shall consider two particularly useful approaches – first a momentum space cutoff directly below and then MS renormalization in Section 4. We shall distinguish vacuum matrix elements in the two schemes by means of the superscripts ‘Žc.o..’ for momentum-cutoff and ‘ŽMS.’ for modified minimal subtraction. 2.2. Two sum rules in momentum-space cutoff renormalization Let us remove the divergence which occurs for d s 4 in Eq. Ž15. by cutting off the Q 2-integral at Q 2 s m2 , where m is the renormalization scale and for convenience we set md.r.s m. It is valid to take d s 4 in this case since the integral is finite. We find ² O1 :mŽc .o .. s
3 16p
2
2
H0m dQ
2
Q 4 Ž P V ,3 y P A ,3 . Ž Q 2 . .
175
must be true for the scale m. Then by combining Eq. Ž18. with Eq. Ž10., we obtain the sum rule 2p ² a s O8 :mŽc .o .. `
s I8 '
H0 ds s
2
m2 s q m2
w r V ,3 y rA ,3 x Ž s . .
Ž 19 .
Despite their apparent similarity, it is important to understand that there is a basic difference between the sum rules for ² O1 :mŽc.o.. and ² a s O8 :mŽc.o... The former is obtained rather directly by taking the x ™ 0 limit of Eq. Ž15. and using a cutoff in momentum to regularize the procedure. However, the latter rests upon assuming the dominance of the leading Qy6 term in the OPE of Eq. Ž11.. This assumption becomes increasingly questionable as m is lowered to energies just above the resonance region. It leads to an uncertainty in the value of ² a s O8 :mŽc.o.. which is not present in ² O1 :mŽc.o... We postpone discussion to Section 5 regarding numerical evaluation of the integrals I1 , I8 appearing in Eqs. Ž17. and Ž19..
3. Kaon-to-pion matrix elements of a left-right operator
Ž 16 . Using Eq. Ž10. to express this relation in terms of spectral functions, we arrive immediately at the following sum rule, 16p 2 3
² O1 :mŽc .o .. `
s I1 '
H0
ds s 2 ln
ž
s q m2 s
/
w r V ,3 y rA ,3 x Ž s . . Ž 17 .
It is equally straightforward to derive a sum rule for ² a s O8 :mŽc.o... We first set Q 2 s m2 in Eq. Ž11. to obtain ² a s O8 :mŽc .o .. s
m6 2p
A distinct but equivalent path to learn about ² O1 :m and ² O8 :m is to perform a chiral analysis of the kaon-to-pion matrix elements themselves. However, the usual ŽV y A. = ŽV y A. weak hamiltonian HDSs1 would be of no help in the chiral limit since its K-to-pi matrix elements vanish there. Instead we introduce a ŽV y A. = ŽV q A. nonleptonic operator ODSs1 defined as ODSs1 '
g 22 8
Hd
4
x Dmn Ž x , MW2 . J mn Ž x . ,
J mn Ž x . ' 12 T d Ž x . g m Ž 1 q g 5 . u Ž x . = u Ž 0. g n Ž 1 y g 5 . s Ž 0. m m s 12 T Ž V1y i2 Ž x . q A1yi2 Ž x . .
Ž P V ,3 y P A ,3 . Ž m2 . .
Ž 18 .
Because the variable Q 2 is constrained in Eq. Ž11. to lie in the range where pQCD makes sense, the same
n n = Ž V4q i5 Ž 0 . y A 4qi5 Ž 0 . . ,
Ž 20 .
where Dmn is the W-boson propagator and Vam, A am Ž a s 1, . . . ,8. are the flavor-octet vector, axialvector
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
176
currents. Operators similar to ODSs1 have received some previous attention in the literature w5,6x. The LR chiral structure of ODSs1 ensures the survival of the K-to-pi matrix element M Ž p . s ²pyŽ p .< ODSs1 < KyŽ p .: in the p ™ 0 limit, where we obtain
where a s ' g 32r4p is the QCD fine structure constant and m - MW . Using standard techniques w2x, we use the renormalization group ŽRG. to provide a summation of the leading-log dependence over the range from MW down to m ,
M ' lim M Ž p .
M,
ps0
s
g 22 16 Fp2
Hd
= ²0 < T Ž
4
Ž x . Vm ,3 Ž 0 .
y A 3m
Ž x . Am ,3 Ž 0 . . <0: . Ž 21 .
In the following, we perform a leading-log calculation of QCD corrections to the chiral matrix element M . This leads naturally to renormalization group equations ŽRGE. for the quantities ² O1 :m and ² a s O8 :m . Since the W-boson propagator in Eq. Ž21. acts as a cutoff for contributions with < x < G My1 W , we consider the leading term of the following operator product expansion, V3m Ž x . V3n Ž 0 . y A 3m Ž x . An3 Ž 0 .
Ž 0.
V3n
Ž 0.
y A 3m
Ž 0.
An3
c1 Ž m . s
c8 Ž m . s
3.1. Leading-log analysis of QCD corrections
s V3m
c1 Ž m . ² O1 :m q c8 Ž m . ² O8 :m ,
2'2 Fp2
Ž 0. q O Ž x . .
1
ž ž
9 1 6
8r9
asŽ m.
/ ž / ž q8
a s Ž MW .
8r9
asŽ m.
y
a s Ž MW .
y1r9
asŽ m. a s Ž MW . asŽ m.
a s Ž MW .
/ y1r9
/
asŽ m. s 1 q 9
asŽ m. 4p
ln
MW2
ž / m2
a s Ž MW . .
ž /
0 16r3
3r2 7
O1 , O8
Ž 25 .
Ž 28 .
An expansion of Eq. Ž26. through second order in a s Ž m . gives GF
3
2'2 Fp2 3 q 32p
² O1 :m q
ln2 2
8p
ln
MW2
8
2
3
ž /ž m
MW2
ž / m2
² a s O8 :m
² a s2 O1 :m y ² a s2 O8 :m
/
.
Ž 29 .
H
In order to express this vacuum matrix element at some lower energy m , we must take QCD radiative corrections into account. The effect of these will be to mix O1 with O8 . The result of mixing at one-loop order is
,
with
Ž 22 .
so that the matrix element specified at energy scale MW becomes GF ² O1 :M W . M, Ž 24 . 2'2 Fp2
,
Ž 27 .
M,
Evaluation of the spacetime integral in Eq. Ž21. is straightforward, gmn d 4 x Dmn Ž x , MW2 . s 2 , Ž 23 . MW
as MW2 O1 O1 ™ q ln O8 O8 4p m2
Ž 26 .
where
x D Ž x , MW2 .
V3m
GF
Let us gain some feeling for the numbers involved. The minimum value of renormalization scale considered in this paper is m 0 s 2 GeV. Adopting this scale and taking a s Ž MW . s 0.119 and a s Ž m 0 . s 0.334 w7x, we find for the RG coefficients in Eq. Ž26., M,
GF 2'2 Fp2
1.071² O1 :m 0 q 0.268² O8 :m 0 ,
Ž 30 .
whereas the coefficients in the perturbative expression of Eq. Ž29. become M,
GF 2'2 Fp2
² O1 :m 0 q 0.88² a s O8 :m 0
y0.52 Ž ² a s2 O8 :m 0 y 83 ² a s2 O1 :m 0 . .
Ž 31 .
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
Finally, the condition of scale independence for the matrix element M , E m2 2 M s 0 , Ž 32 . Em leads directly to the renormalization group equations E 3 ² a s O8 :m , m2 2 ² O1 :m s Ž 33 . 8p Em E 1 16 ² a s2 O1 :m y 2² a s2 O8 :m . m2 2 ² a s O8 :m s 4p 3 Em Ž 34 . To summarize – the above operator-product analysis involves computing radiative corrections perturbatively to one-loop order in QCD Žcf. Eq. Ž25.. and retaining only the dependence on leading logarithms in the evolution from scale MW down to scale m. The value of m cannot be taken too small, otherwise the perturbative framework breaks down. 3.2. Verification of the operator product expansion Despite the explicit difference between the procedures of Section 2 and that carried out directly above, the two are equivalent. In particular, we can show that the ² O1 :m sum rule of Eq. Ž17. and the ² O8 :m sum rule of Eq. Ž19. reproduce the OPE to the leading log level. This verifies both the derivations that we provided and gives a direct insight into the workings of the OPE. Consider a partition of M characterized by the scale m , M s M- Ž m . q M) Ž m . , Ž 35 . where M- Ž m . and M) Ž m . are dependent respectively on contributions with Q - m and Q ) m. Also, in addition to maintaining the requirement that m lie in the pQCD domain, we further constrain it to obey m < MW . We then obtain M- Ž m . s
3GF MW2
m2
32'2 p 2 Fp2
H0
2
dQ 2
Q4 Q 2 q MW2
= P V ,3 Ž Q . y P A ,3 Ž Q 3GF m2 s dQ 2 Q 4 2 2 32'2 p Fp 0
2
and M) Ž m . s
M) Ž m . s
3
'2 Fp2
8p
3 32p
Q4
2
2
Q 2 q MW2
ln2 2
MW2
ž / ž /
ln
m2
Ž 37 .
² a s O8 :m
MW2
m2
= Ž ² a s2 O8 :m y 83 ² a s2 O1 :m . .
Ž 38 .
Comparison of Eq. Ž29. with Eq. Ž38. yields the relation, M- Ž m . s
GF 2'2 Fp2
² O1 :m .
Ž 39 .
We see that the operators that we originally defined independently of the weak interaction are in fact the ones that appear in the Operator Product Expansion. Of course, this is to be expected, but it proves an explicit pedagogical demonstration of the nature of the OPE. 3.3. Sum rules and RG relations To complete the chain of logic, we demonstrate consistency of the spectral function sum rules for ² O1 :mŽc.o.. and ² O8 :mŽc.o.. with the corresponding renormalization group relations obtained previously from the operator product expansion Žcf. Eqs. Ž33. and Ž34... The RG equation for ² O1 :mŽc.o.. is immediately recovered upon differentiating the sum rule of Eq. Ž17. and making use of Eq. Ž19.,
Em2
² O1 :mŽc .o .. 3
s 8p 3
Ž 36 .
GF
y
= P V ,3 Ž Q 2 . y P A ,3 Ž Q 2 . q O Ž m2rM W2 .
32'2 p 2 Fp2
`
Hm dQ
Upon inserting the large-Q form of Eq. Ž11. into Eq. Ž37., we obtain
E
H
3GF MW2
= P V ,3 Ž Q 2 . y P A ,3 Ž Q 2 . .
m2
.
177
s 8p
m2 P
`
s2
H ds s q m 2p 0
² a s O8 :mŽc .o .. .
2
w r V ,3 y rA ,3 x Ž s . Ž 40 .
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
178
To derive the RG relation for ² O8 :mŽc.o.. we start with the sum rule of Eq. Ž19.,
E m
2
Em
below m2 , we can let d ™ 4 and recover exactly the cutoff integrand of Eq. Ž16., ² O1 :mŽd .r .. s ² O1 :m
² O8 :mŽc .o .. 2
Ž d y 1 . m4y d
q
m2
E
s 2p Em
m2
` 2
ds
H0
sqm
2
Ž 4p .
s 2 w r V ,3 y rA ,3 x Ž s . .
`
=
Hm dQ
Ž 41 . The integral in the above is seen to be m6 Ž P V ,3 Ž m2 . y P A ,3 Ž m2 . . . We replace it using the asymptotic expression of Eq. Ž13. to obtain
E m2
Em2
² O8 :mŽc .o ..
m2 s
2p Em2
=
8 3
ln
ž / L2
2
2
2
G Ž dr2 .
Q d Ž P V ,3 y P A ,3 . Ž Q 2 . .
Ž 43 . The asymptotic tail can be analysed in d dimensions. This introduces scheme dependence depending on which method is used to define Dirac algebra away from four dimensions, e.g. the naive dimensional regularization ŽNDR. and ’t Hooft-Veltman ŽHV. schemes in which g 5 is respectively anticommuting and commuting w8x. We find 1 3
² a s2 O1 :mŽ c .o . . y ² a s2 O8 :mŽ c .o . .
H0L ds s
q
m2
E
2
dr2
Ž d y 1 . Q d P Ž P V ,3 y P A ,3 . Ž Q 2 . s 2pa s² O8 :m Q dy 6 1 q Ž d s q 14 . e q O Ž a s2 . . Ž 44 .
w r V ,3 y rA ,3 x Ž s . q O Ž my2 . , Ž 42 .
where e ' 4 y d and d s ' d scheme is associated with the loop integration and scheme-dependence. The values of d s in the NDR and HV schemes are
from which the RG relation of Eq. Ž34. follows directly.
ds s
4. MS renormalization
The Q 2 ) m2 integral can then be performed with the result
½
y5r6 1r6
Ž NDR. Ž HV. .
Ž 45 .
² O1 :mŽd .r .. The work of the preceding sections was based on a momentum-space cutoff renormalization scheme, which is useful in yielding sum rules directly related to experimental data. At the same time, however, it is distinct from the more standard MS prescription. In this section, we demonstrate how to relate the two approaches. 4.1. Short distance analysis Let us reconsider Eq. Ž15.. We can show how the cutoff renormalization is related to dimensional regularization by keeping the high-Q 2 part of the integral in Eq. Ž15. and analyzing it in terms of an e-expansion. We divide the integral into integration ranges below and above m2 . For the part of integral with Q 2
s ² O1 :m 3
2
q 8p 4 y d ² = a s O8 :m .
y g q ln4p q 32 q 2 d s
Ž 46 .
The MS prescription is a subcase of dimensional regularization in which the terms 2rŽ4 y d . y g q ln4p in Eq. Ž46. are removed in the renormalization procedure. This gives our desired relation in a given scheme, 3as 3 Žc .o .. ² O1 :MS q Ž q 2 d s . ² O8 :m . Ž 47. m s ² O1 :m 8p 2 To derive an analogous relation between ² O8 :m MS and ² O8 :mŽc.o.. we employ the leading behavior of correlators and spectral functions in the MS renor-
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
179
Fig. 1. Some QCD corrections: full theory Ža.-Žb., effective theory Žc.-Žd..
malization prescription, which has been calculated using the NDR scheme w4x, Q 6 Ž P V ,3 y P A ,3 . Ž Q 2 . ;2p ² a s O8 :mŽ MS. 8
q 2q
3
119
ln
Q2
ž / ž / m2
Q2
y ln
q 12
² a s2 O1 :mŽ MS.
m2
² a s2 O8 :mŽ MS.
Ž 48 .
and Ž r V ,3 y rA ,3 . Ž s . 1 ; 3 83 ² a s2 O1 :mŽMS . y ² a s2 O8 :mŽMS . s q... . Ž 49. Then by setting Q s m in Eq. Ž48. and combining the result with Eq. Ž49. we find
m
2
H0 ds s q m s 2pa s q
as p
ž
GF
2
1q
119a s 24p
/
² qq < O1 < qq :tree
2'2
w r V ,3 y rA ,3 x Ž s .
3as 8p
ž
ln
MW2 m2
/
y 1 ² qq < O8 < qq :tree
O Ž a s2 . . qO
² O8 :mŽMS .
² O1 :mŽMS . .
Ž 50 .
Comparison with Eq. Ž47. then implies that the NDR matrix element is given to first order in a s by 119a s as ² O8 :mŽ MS. s 1 y ² O8 :mŽc .o .. y ² O1 :m . 24p p Ž 51 .
ž
Ms
q
s2
`
Determination of the coefficients c1 and c8 proceeds in two steps: first calculate QCD radiative corrections in both the full and effective theories, and then ‘match’ the two calculations. We shall carry out this procedure at one-loop order of the QCD radiative corrections. Since c1 s 1 q O Ž a s2 . and c8 s O Ž a s ., this will yield a determination of c8 . For definiteness, we consider the free scattering of zero momentum quarks and adopt a common quark mass m to serve as the infrared cutoff. In the full theory, evaluations of the one-loop radiative corrections like those displayed in Fig. 1Ža., Fig. 1Žb. are finite and yield
/
4.2. MS matching at one loop In this section we perform the matching at one loop and verify the scheme independence of the . result. The effective operator ODŽeff Ss1 , is expressed in O O terms of local operators 1 and 8 , GF . ODŽeff c1 Ž m . O1 q c8 Ž m . O8 . Ž 52 . Ss1 s 2'2
Ž 53 .
The analogous calculation in the effective theory is divergent and must be regularized. We employ dimensional regularization which introduces the scheme dependence mentioned above. Our calculation of amplitudes like those in Fig. 1Žc., Fig. 1Žd. gives Ms
GF
² qq < O1 < qq :tree
2'2 q
=
3as 8p 2
e
Ž1 q ds e q d l e .
y g q ln4p y ln
qc8 ² qq < O8 < qq :tree ,
m2
m2
² qq < O8 < qq :tree
Ž 54 .
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
180
After the removal of the divergent term in MS renormalization, we compare the full theory and the effective theory to identify the coefficient function as c8 Ž m . s
3as
ln
8p
MW2
m2
y 32 y 2 d s .
Ž 55 .
. When ODŽeff Ss1 is applied to our problem of vacuum matrix elements, we have the amplitude
Ms
GF 2'2 Fp2
MS c1 Ž m . ² O1 :MS . m q c 8 Ž m . ² O8 :m
Given our previous identification of the scheme dependent operator O 1 in Eq. Ž47., it can be seen that the scheme dependence cancels between that of the matrix element and of the coefficient in the operator product expansion.
5. Numerical analysis and uncertainties We base our numerical determination of ² O8 :mŽc.o.. and ² O1 :mŽc.o.. respectively on the sum rules in Eq. Ž19. and Eq. Ž17.. This involves calculation of the integrals I8 and I1 , which contain the combination of spectral functions r V ,3 y rA,3 , `
2
H0 ds K Ž s, m . w r i
V ,3 y r A ,3
x Ž s . Ž i s 1,8 . , Ž 57 .
with K8 s s2
m2 s q m2
,
K 1 ' s 2 ln
ž
s q m2 s
/
.
Ž 58 .
As such, I8 and I1 belong to the family of spectral integrals which include the DMO sum rule w9x, the two Weinberg sum rules w3x and the sum rule for the pion electromagnetic mass splitting w10x. The kernels occurring in these ‘classical’ sum rules are K DM O s 1rs , K W2 s s ,
K W1 s 1 ,
K em s sln s .
I8 s y Ž 0.30 " 0.04 . P 10y2 , I1 s y Ž 0.42 " 0.06 . P 10y2 ,
Ž 60 .
At the higher renormalization scale, m s 4 GeV, we obtain I8 s y Ž 0.43 " 0.06 . P 10y2 ,
Ž 56 .
Ii s
set of constraints for any evaluation of I8 and I1. Using an updated form of our earlier study w11x of chiral sum rules, we find for renormalization scale m s 2 GeV the values
Ž 59 .
This happenstance is most fortunate as the integrals defined by the kernels in Eq. Ž59. form a powerful
I1 s y Ž 0.97 " 0.12 . P 10y2 .
Ž 61 .
5.1. Uncertainties from data analysis The error bars quoted above correspond to our estimate of the uncertainty in the sum rules due to imprecision in our knowledge of the spectral functions. Before proceeding let us describe how these were arrived at, and assess other sources of uncertainty. The data at lower values of s are extremely well known, and they introduce very little uncertainty compared to other sources which we are concerned with. The high energy tail of r V y rA is small above s s 5 GeV 2 . In the m s 2 GeV integrals, there remains essentially no sensitivity to this high energy tail once the constraints are imposed 2 . It is in the matching of these two regions that one encounters the greatest uncertainties. Fortunately, the four integral constraints described above are very stringent and allow us to limit the uncertainties in this region. We have used several methods to construct spectral functions which match the data within error bars and yet satisfy our sum rule constraints. These give variations in our integrals of under 6%. In addition, we have considered the situation where
2 At higher values of m there occurs more sensitivity to the asymptotic tail, and it is the tail that describes the logarithmic running of the O 8 matrix element. While we have a good handle on the size of the 1r s 3 component of the tail, we know little about the 1r s 4 component. However as long as the 1r s 4 portion is not much larger than the 1r s 3 behavior for s) 5 GeV 2 , its effect is within our quoted error bars.
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
the asymptotic form of the spectral function is reached only on the average, with a damped oscillating term that provides deviations from the average. Since our sum rules are equivalent to transforming back to euclidean Q 2 , these oscillations give exponentially suppressed effects at large m2 once integrated. Again the constraints are very powerful in further limiting this effect, and our studies lead us to increase the uncertainty in the fit to 10% to account for this form of variation. We also must account for the fact that the data and the input into the constraints are measured in a world where mp2 is not zero, yet we are interested in the result in the chiral limit. This introduces corrections of order mp2 rmr2 which is of order 3%. In fact, since we know some of the physics involved in passing to the chiral limit, we could attempt to make a realistic correction for the extrapolation to the chiral limit. However, since this would appear to introduce some model dependence into our procedure, we prefer to simply include the uncertainty as an error bar. In practice, the effect which has the most sensitivity for our results is the constraint of the pion electromagnetic mass difference, since the kernel K em bears the greatest resemblance to K 1 and K 8 . Work on the pion and kaon electromagnetic mass differences indicates that the quark mass corrections are somewhat larger than average. Therefore, to be conservative we triple the canonical error estimate, leading us to quote a 9% uncertainty for the extrapolation to the chiral limit. We have added this in quadrature to the statistical uncertainty to arrive at the error bars cited above. 5.2. Uncertainty from the operator product expansion Finally, we need to address the fact that it has become common to cite matrix elements at a scale m s 2 GeV, which is a rather low scale for perturbative QCD to be fully in the asymptotic region. In fact, our method can be used for any m , and we can check if the asymptotic QCD behavior is obtained. For example, the renormalization group equations relate the m-dependence to the magnitudes of the operator matrix elements. Although one of the relations ŽEqs. Ž33. and Ž40.. is automatically satisfied,
181
we explicitly showed above that the second holds only if m is large enough, i.e. that it is well into the region where the asymptotic tail of the spectral functions becomes applicable. It is easy to see from the data alone that this is not the case at m s 2 GeV. Another way to state the same result is that there remain power corrections in the sum rule, although the renormalization group states that the running with m should be only logarithmic. We believe Žbecause of the generality of our framework. that this issue must also be present in the lattice results, and we urge the evaluation of lattice matrix elements at larger values of m. We do see such non-asymptotic behavior in our results. Actually, for O1 our method of cutting off the high frequency modes of the current-current product is in accord with Wilson’s original idea of the definition of a scale-dependent matrix element. Therefore, our sum rule for ² O1 :mŽc.o.. can be treated as a definition of this amplitude at any scale m , even if that scale is not yet asymptotic. For O8 , however, there is some uncertainty as to an ideal definition in the non-asymptotic region. For example, the RG relation of Eqs. Ž33. and Ž40. requires that we use exactly our definition, yet this only is foolproof if the RGEs are fully valid. Equivalently, if this matrix element is defined via the coefficient of Qy6 in the vacuum polarization, there can be order Qy8 corrections remaining if one works in the non-asymptotic region. We see evidence of such power corrections. Moreover, attempting to discard the Qy8 effects leads to a larger matrix element. At m s 4 GeV, the corrections are rather modest, in line with other uncertainties that we have described. However, at m s 2 GeV, these non-asymptotic effects represent a significant intrinsic uncertainty. We haÕe taken these into account by combining two eÕaluations, one obtained by eÕaluating the sum rule at m s 4 GeV and using the RGE to transform down to m s 2 GeV, and the other by direct eÕaluation of the sum rule at the lower scale. We aÕerage these two and assign the difference as an independent error bar. The error bar is chosen such that a one-sigma variation reproduces the full range between the two methods of evaluation. We do this for both matrix elements. The quoted error bar at m s 4 GeV is scaled down from the m s 2 GeV values by a factor of four, as appropriate for quadratic power corrections.
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
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5.3. ConÕersion to MS renormalization
In the NDR scheme with m s 2 GeV this translates into the following B-factor determinations,
We now transform to the MS matrix elements. The results of our direct evaluation at m s 2 GeV leads to the matrix elements
B7Ž3r2.
w NDR, m s 2 GeVx
ž
2GeV
Ž 62 .
B8Ž3r2.
w NDR, m s 2 GeVx
ž
s 1.11 " 0.16 " 0.23 . where we have taken a s Ž2 GeV. , 0.334. When we evaluate the integrals at m s 4 GeV and use the RGE to rescale back to m s 2 GeV, we instead obtain ² O8 : ŽMS. s y Ž 1.29 " 0.15 . P 10y3 GeV 6 , 2GeV
² O1 : ŽMS. s y Ž 1.02 " 0.10 . P 10y4 GeV 6 , 2GeV
Ž 63 .
which is a measure of the potential non-asymptotic corrections found at low values of m. As explained above, this leads us to quote our result as ² O8 : ŽMS. s y Ž 0.98 " 0.13 " 0.23 . P 10y3 GeV 6 , 2GeV
² O1 : ŽMS. s y Ž 0.86 " 0.10 " 0.16 . P 10y4 GeV 6 , 2GeV
Ž 64 . The first error bar corresponds to the uncertainty in the evaluation of the sum rule whereas the second is the potential non-asymptotic uncertainty defined above. Note that the two matrix elements differ by an order of magnitude. The related K 0 ™ pp matrix elements are then ² Ž pp . Is2 < Q7Ž3r2. < K 0 : 2 GeV s Ž 0.43 " 0.05 " 0.08 . GeV 3 ,
/
0.1 GeV ms q m ˆ
2
/ Ž 66 .
Note that the combination of B-factor and quark masses is ‘physical’, appearing in the formula for e Xre . The comparison of these results with some lattice evaluations is hampered by the fact that our evaluation is of the full matrix elements, while most lattice calculations are of the B-factors directly w12,13x. If large values of quark masses are used, our results are larger that other estimates, yet for the currently favored smaller quark masses the results are not inconsistent. There is one recent lattice evaluation which provides absolute matrix elements which we can compare to. The Rome group w14x quotes the matrix element in in the the quenched approximation using K ™ p matrix elements plus the chiral relation between K ™ p and K ™ pp . When the meson masses are taken as the kaon mass they find, in the NDR scheme at m s 2 GeV, ² Ž pp . Is2 < Q7Ž3r2. < K 0 : 2 GeV s Ž 0.22 " 0.04 . GeV 3 , ² Ž pp . Is2 < Q8Ž3r2. < K 0 : 2 GeV s Ž 1.02 " 0.10 . GeV 3 . Ž 67 . The quoted error does not include estimates of the effect of quenching nor the extrapolation to the continuum limit. Their results seem to be systematically smaller than ours. Finally at m s 4 GeV we have the vacuum matrix elements, y3 ² O8 :ŽMS. GeV 6 , 4G eV s y Ž 1.63 " 0.20 " 0.06 . P 10
² Ž pp . Is2 < Q8Ž3r2. < K 0 : 2 GeV s Ž 2.58 " 0.37 " 0.47 . GeV 3 .
ms q m ˆ
2
s 0.55 " 0.07 " 0.10 ,
² O8 : ŽMS. s y Ž 0.67 " 0.09 . P 10y3 GeV 6 , ² O1 : ŽMS. s y Ž 0.70 " 0.10 . P 10y4 GeV 6 , 2GeV
0.1 GeV
Ž 65 .
y4 ² O1 :ŽMS. GeV 6 . 4G eV s y Ž 1.71 " 0.20 " 0.04 . P 10 Ž 68 .
J.F. Donoghue, E. Golowichr Physics Letters B 478 (2000) 172–184
The corresponding K ™ pp matrix elements are ² Ž pp . Is2 < Q7Ž3r2. < K 0 :4 GeV s Ž 0.85 " 0.10 " 0.02 . GeV 3 , ² Ž pp . Is2 < Q8Ž3r2. < K 0 :4 GeV s Ž 4.34 " 0.56 " 0.15 . GeV 3
Ž 69 .
and for the B-factors we obtain, B7Ž3r2. w NDR, m s 4 GeV x
ž
0.1 GeV ms q m ˆ
2
/
s 1.10 " 0.13 " 0.03 , B8Ž3r2.
w NDR, m s 4 GeVx
ž
s 1.87 " 0.25 " 0.07 .
0.1 GeV ms q m ˆ
2
/
O 8 matrix element. This raises the concern that when one is working at such a low value of m , there may be significant corrections even in lattice evaluations. Certainly, use of m - 1 GeV, as occurs in many model dependent evaluations, appears extremely dubious. The values displayed above are based on working in the chiral limit of massless quarks. One must, however, add to these the chiral corrections. Work has begun on this important problem w15x. Nevertheless, it is interesting to look at the phenomenological consequences of the results reported in this paper.. While we cannot give a full evaluation of e Xre because we have not evaluated the contribution of B6 , we can give the contribution arising from the electroweak penguin,
eX
Ž 70 .
That B7Ž3r2. has a large variation with m is expected from the RGE of Eq. Ž34., given our previous result that the vacuum matrix element of O8 is much larger than that of O1.
183
že/
B8
s Ž y12 " 3 . P 10y4 .
Ž 71 .
The effect of B6 is expected to be positive, and needs to be almost three times larger than that of B8 if the Standard Model is to account for the experimental value.
6. Concluding comments Acknowledgements The method that we have described has the virtue of being a fully rigorous framework. Moreover, the input data is largely taken from experiment, and hence represents an evaluation that is model independent. Besides the direct comparison with the results with lattice calculations, there may also be other lessons in this calculation. Since in our method the matrix elements are evaluated by constructing the Euclidean vacuum polarization function, lattice calculations may also be able to directly follow many of the steps in our procedure, and thereby test their methods in more detail. By explicitly studying the product of currents at non-zero values of the spatial separation, the matrix elements can be evaluated without some of the operator mixing problems that occur on the lattice when using local operators. Moreover, by studying vacuum matrix elements as well as hadronic matrix elements, the chiral relations can be checked on the lattice. Finally, we recall the lesson, discussed above, that power-law corrections still appear to exist at m s 2 GeV, especially in the
This work was supported in part by the National Science Foundation. We thank Guido Martinelli for useful comments.
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w8x See e.g. A.J. Buras, P.H. Weisz, Nucl. Phys. B 333 Ž1990. 66; M. Ciuchini et al., Nucl. Phys. B 415 Ž1994. 403; S. Herrlich, U. Nierste, Nucl. Phys. B 455 Ž1995. 39. w9x T. Das, V.S. Mathur, S. Okubo, Phys. Rev. Lett. 19 Ž1967. 859. w10x T. Das, G.S. Guralnik, V.S. Mathur, F.E. Low, J.E. Young, Phys. Rev. Lett. 18 Ž1967. 759. w11x J.F. Donoghue, E. Golowich, Phys. Rev. D 49 Ž1994. 1513.
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