B0d,s–B0d,s mass-differences from QCD spectral sum rules

Physics Letters B 540 (2002) 233–240 www.elsevier.com/locate/npe

0 Bd,s –B 0d,s mass-differences from QCD spectral sum rules

Kaoru Hagiwara a , Stephan Narison b , Daisuke Nomura c a KEK Theory Group, Tsukuba, Ibaraki 305-0801, Japan b Laboratoire de Physique Mathématique, Université de Montpellier II Place Eugène Bataillon,

34095, Montpellier Cedex 05, France c Institute for Particle Physics Phenomenology, University of Durham, Durham DH1 3LE, UK

Received 12 May 2002; received in revised form 10 June 2002; accepted 14 June 2002 Editor: G.F. Giudice

Abstract We present the first QCD spectral sum rules analysis of the SU(3) breaking parameter ξ and an improved estimate of the
B both entering into the B 0 –B 0 mass-differences. The averages of renormalization group invariant (RGI) bag constant B q d,s d,s    
B
(247 ± 59) MeV and ξ ≡ fB B
B fB B
B
the results from the Laplace and moment sum rules to order αs are fB B s s (1.18 ± 0.03), in units where fπ = 130.7 MeV. Combined with the experimental data on the mass-differences Md,s , one obtains the constraint on the CKM weak mixing angle |Vt s /Vt d |2
20.0(1.1). Alternatively, using the weak mixing angle from the analysis of the unitarity triangle and the data on Md , one predicts Ms = 18.6(2.2) ps−1 in agreement with the present experimental lower bound and within the reach of Tevatron 2.  2002 Elsevier Science B.V. All rights reserved.

×

1. Introduction 0 B(s) and B 0(s) are not eigenstates of the weak Hamiltonian, such that their oscillation frequency is governed by their mass-difference Mq . The measurement by the UA1 Collaboration [1] of a large value of Md was the first indication of the heavy top quark mass. In the SM, the mass-difference is approximately given by [2]

Mq


 2  G2F 2 mt ∗ 2 M |V V | S ηB CB (ν) tq tb 0 2 4π 2 W MW

E-mail addresses: [email protected] (K. Hagiwara), [email protected] (S. Narison), [email protected] (D. Nomura).

  1  0  B q Oq (x)Bq0 , 2MBq

(1)

where the B = 2 local operator Oq (x) is defined as ¯ µ Lq)(bγ ¯ µ Lq), Oq (x) ≡ (bγ

(2)

with L ≡ (1 − γ5 )/2 and q ≡ d, s. S0 , ηB and CB (ν) are short distance quantities calculable perturbatively. (Here we are following the notation of Ref. [2].) On the other hand, the matrix element B 0q |Oq |Bq0  requires non-perturbative QCD calculations, and is usually parametrized for as    4 fB2q 2 MBq BBq . B 0q Oq Bq0 = (3) 3 2 fBq is the Bq decay constant normalized as fπ = 130.7 MeV, and BBq is the so-called bag parameter 

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 2 ) 0 2 1 3 3 - 0

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which is BBq = 1 if one uses a vacuum saturation of the matrix element and equal to 3/4 in the large Nc limit. From Eq. (1), it is clear that the measurement of

Md provides one of the CKM mixing angles |Vt d | if one uses |Vt b |
1. One can also extract this quantity from the ratio  

Ms  Vt s 2 MBd B 0s |Os |Bs0  =

Md  Vt d  MBs B 0d |Od |Bd0     Vt s 2 MBd 2   ≡ (4) ξ , Vt d  MBs since in the SM with three generations and unitarity constraints, |Vt s |
|Vcb |. Here

fBs BBs gs ξ≡ (5) ≡ . √ gd fB BB The great advantage of Eq. (4) compared with the former relation in Eq. (1) is that in the ratio, different systematics in the evaluation of the matrix element tends to cancel out, thus providing a more accurate prediction. However, unlike Md = 0.479(12) ps−1 , which is measured with a good precision [3], the determination of Ms is an experimental challenge due to the rapid oscillation of the Bs0 –B 0s system. At present, only a lower bound of 13.1 ps−1 is available at the 95% CL from experiments [3], but this bound already provides a strong constraint on |Vt d |.

2. Two-point function sum rule Ref. [4] has extended the analysis of the K 0 –K 0 systems of Ref. [5], using two-point correlator of the four-quark operators into the analysis of the quantity √ fB BB which governs the B 0 -B 0 mass difference. The two-point correlator defined as

 †  2 ψH q ≡ i d 4 x eiqx 0|T Oq (x) Oq (0) |0, (6) is built from the B = 2 weak operator given in Eq. (2). The two-point function approach is very convenient due to its simple analytic properties which are not the case of approach based on three-point functions.1 However, it involves non-trivial QCD calculations which become technically complicated when 1 For detailed criticisms, see [6].

one includes the contributions of radiative corrections due to non-factorizable diagrams. These perturbative (PT) radiative corrections due to factorizable and nonfactorizable diagrams have been already computed in Ref. [7] (referred as NP), where it has been found that the factorizable corrections are large while the nonfactorizable ones are negligibly small. NP analysis has confirmed the estimate in Ref. [4] from lowest order calculations, where under some assumptions on the contributions of higher mass resonances to the spectral function, the value of the bag parameter BB has been found to be  BBd 4Mb2
(1 ± 0.15).

(7)

This value is comparable with the one BBd = 1 from the vacuum saturation estimate, which is expected to be a quite good approximation due to the relative high-scale of the B-meson mass. Equivalently, the corresponding RGI quantity is
Bd
(1.5 ± 0.2), B

(8)

where we have used the relation     αs 5165 γ0 /β1
, 1+ BBq = BBq (ν)αs 12696 π

(9)

with γ0 = 1 being the anomalous dimension of the operator Oq and β1 = −23/6 for 5 flavours. ν is the subtraction point. The NLO corrections have been obtained in the MS scheme [2]. We have also used the value of the bottom quark pole mass [6,8] Mb = (4.66 ± 0.06) GeV.

(10)

In the following, we study (for the first time), from the QCD spectral sum rules (QSSR) method,2 the SU(3) breaking effects on the ratio ξ defined previously in Eq. (5), where a similar analysis of the ratios of the decay constants has given the values [10] fDs
1.15 ± 0.04, fD

fBs
1.16 ± 0.04. fB

(11)

We shall also improve the previous result of Ref. [4, 7] on BBd by the inclusion of the Bq Bq∗ and Bq∗ Bq∗ resonances into the spectral function. 2 Our preliminary results have been presented in [9].

K. Hagiwara et al. / Physics Letters B 540 (2002) 233–240

3. Inputs for the sum rule analysis We shall be concerned here with the two-point correlator defined in Eq. (6). The hadronic part of the spectral function can be conveniently parametrized using the effective realization [4] 1 Oqeff = gq ∂µ Bq0 ∂ µ Bq0 + · · · , 3

The lowest order perturbative contribution for  0 to the two-point correlator is ms =

×

(13)

below the QCD continuum threshold tc . The function λ(x, y, z) is a phase space factor, (14)

We have used, to a first approximation, the large Mb and vacuum saturation relations: gq ≡ fB2q BBq
gq ∗ ≡ fB2q∗ BBq∗

4. SU(3) breaking contributions

1 pert Im ψs (t) π  = θ t − 4(Mb + ms )2

1 Im ψqeff (t) π   2  2 2 4MB2 q 2M 2 gq Bq t2 1 − 1− = 9 8π t t   × θ t − 4MB2 q

λ(x, y, z) ≡ x 2 + y 2 + z2 − 2xy − 2yz − 2zx.

(OPE) including non-perturbative condensates [11].3 The massless (mq = 0) expression for the lowest perturbative and gluon condensate contributions has been obtained in Ref. [4]. Radiative factorizable and non-factorizable corrections to the perturbative graphs in the massless light quark case have been obtained in NP [7].

(12)

where · · · indicates higher resonances and gq ≡ fB2q × BBq . Retaining the BB, BB ∗ and B ∗ B ∗ resonances and parametrizing the higher resonances with the QCD continuum contribution, it gives

 MB2 ∗ MB4 ∗  q q + 1−4 + 12 2 t t  4MB2 ∗  q θ t − 4MB2 q∗ × 1− t  M2 M2∗  Bq Bq , + 2λ3/2 1, t t   2 × θ t − (MBq + MBq∗ )

235

(15)

among the couplings. The results fB ≈ fB ∗ have been also obtained from QCD spectral sum rules [6], while the vacuum saturation relation BBq∗
1
BBq is a posteriori expected to be a good approximation as indicated by the result obtained later on in this Letter The short distance expression of the spectral function is obtained using the Operator Product Expansion

t4 1536π 6

√ √ (1− δ− δ  )2

√ 2 (1−

z)

dz √ √ ( δ+ δ  )2

du zu

√ √ ( δ+ δ  )2   δ  1/2

  δ δ δ × λ1/2 (1, z, u)λ1/2 1, , 1, , λ z z u u        δ δ δ δ , , × 4f f z z u u     δ δ δ δ g , , − 2f z z u u     δ δ δ δ − 2g , f , z z u u     (1 − z − u)2 δ δ δ δ + (16) g , g , . zu z z u u Here δ ≡ Mb2 /t and δ  ≡ m2s /t, respectively. The functions f (x, y) and g(x, y) are defined by f (x, y) ≡ 2 − x − y − (x − y)2 ,

(17)

g(x, y) ≡ 1 + x + y − 2(x − y) .

(18)

2

We include the O(αs ) correction from factorizable diagrams by using the results in the MS scheme for the two-point correlators of currents [15].4 This can 3 We shall not include here the effects of tachyonic gluon mass [12] or some other renormalon-like terms which give small effects in various examples [13,14]. 4 We shall neglect the nonfactorizable corrections in our analysis to the results in NP [7] obtained in a slight variant of the MS scheme.

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K. Hagiwara et al. / Physics Letters B 540 (2002) 233–240

be done using the convolution formula,

5. The sum rule analysis

1 Im ψsαs (t) π  = θ t − 4(Mb + ms )2

For the sum rule analysis, we shall work like [4,6] with the moments

t2 × 6π 4

√ √ (1− δ− δ  )2

dz √ √ ( δ+ δ  )2

(q)

tc

√ 2 (1−

z)

M(n) q =

du

4(Mb +mq

√ √ ( δ+ δ  )2

× λ1/2 (1, z, u)  0 × Im Πµν (zt) Im Π αs µν (ut)  αs + Im Πµν (zt) Im Π 0µν (ut) .

(q)

(19)

q12

1 Im ψq (t). π

(26)

)2

In so doing, in addition to the pQCD input parameters given previously, we shall need the values of the QCD condensates and SU(3) breaking parameters, which we give in Table 1. Weshow in Fig. 1 the moment sum
B(s) for different values of rq rules analysis of fB(s) B and n. As one can see from Fig. 1a, b, the stability regions of the quantity, 


Bd (27) gˆd ≡ fBd B

(d)

   

Mb2 q12 2 × λ0 λ1 1 + 2 − 2 q1 q q    Mb2 q12  2 2 2 . (21) + f1 1 − 2 + 2 q − Mb − q1 q q

(Mb2 − m2s )2 m2s − 2 . q12 q14

4(Mb +mq

rd ≡

(Mb +ms )2

+

d e−t τ

L(τ )q =

versus the number n of moments and the continuum threshold

1 ms ¯s s 384π 3 √ 2 ( t−M 

b)

 2 2 ∂ dq1 λ1 4 + 2q × ∂q 2

Mb2

(25)

)2

tc

1 Im ψs ¯s s (t) = π  θ t − 4(Mb + ms )2

f1 ≡ 1 +

1 Im ψq (t), π

and with the Laplace sum rule

0 (q 2 ) and Π αs (q 2 ) are, respectively, the Here Πµν µν lowest and the next-to-leading order QCD contribution to the two point correlator Πµν (q 2 ) defined by

 Πµν q 2 ≡ i d 4 x eiqx   ¯ × 0|T b(x)γ µ Ls(x) s¯ (0)γν Lb(0) |0. (20) The quark condensate contribution reads

Here λ0 , λ1 , and f1 are defined by   q12 Mb2 λ0 ≡ λ 1, 2 , 2 , q q   Mb2 m2s λ1 ≡ λ 1, 2 , 2 , q1 q1

dt t n

tc , 4Mb2

(28)

are obtained for large ranges ending with an extremum for n
−30,

rd
1.13.

(29)

These range of values of the sum rule parameters are in good agreement with previous results in Ref. [4] and are NP [7]. Analogous values of n and rs stabilities 


Bs (see also obtained in the analysis of gˆs ≡ fBs B Fig. 1c, d), with

(22) (23) (24)

n
−26,

rs
1.17.

(30)

One can notice that the stabilities in the continuum for gˆd and gˆs differ slightly as a reflection of the SU(3) breakings, which one can parametrize numerically as   √ ms 2 rs
(31) rd + ≈ rd + 0.05. Mb

K. Hagiwara et al. / Physics Letters B 540 (2002) 233–240

237

Table 1 
B and ξ in units where fπ = 130.7 MeV. The box marked with – means that the error is Different sources of errors in the estimate of fB B zero or negligible 
B ) [MeV] Sources

(fB B

ξ × 102 Moments

Laplace

Moments

Laplace

−n
(30–10) τ
(0.1–0.31) GeV−2 rd
1.06–1.17 Λ5 = (216+25 −24 ) MeV [3,16] ν = Mb –2Mb αs2 : geometric PT series Mb = (4.66 ± 0.06) GeV [8]

αs G2  = (0.07 ± 0.01) GeV4 [17]

uu(2) ¯ = −(254 ± 15)3 MeV3 [6,8,18]

¯s s/ uu ¯ = 0.7 ± 0.2 [6,8,18] m ¯ s (2) = (117 ± 23) MeV [6,8,18]

8.3 – 7.9 0.4 8.7 43.0 34.6 1.3 – – –

– 12.4 7.8 0.4 9.1 45.0 37.1 1.6 – – –

1.5 – 1.0 0.1 0.2 0.6 0.9 – 0.2 0.3 1.5

– 1.7 2.5 0.1 0.2 0.7 1.2 – 0.3 0.4 1.7

Total

57.1

60.8

2.6

3.8

Similar analysis is done with the Laplace  sum rules.
B(s) for We show in Fig. 2 the predictions of fB(s) B different values of rq and τ , where an extremum is obtained for τ
0.3 GeV

−2

.

(32)

6. Results and implications on |Vt s /Vt d |2 and Ms We take as a conservative result for gˆd , from the moments sum rule analysis, the one from a large range of n = −10 to −30 and for rd = 1.06 to 1.17. Adding quadratically the different sources of errors in Table 1, we obtain 


B
(245 ± 57) MeV, fB B

(33)

in units where fπ = 130.7 MeV. The most relevant errors given in Eq. (33) come from Mb and the truncation of the PT series. We have estimated the latter by assuming that the coefficient of the αs2 contribution comes from a geometric growth of the PT coefficients. ¯ αs G2  and ν The other parameters n, rd , Λ, qq, (subtraction point) induce smaller errors as given in Table 1. We proceed in a similar way for gˆs . Then, we take the range n = −10 to −26 and rs = 1.10 to 1.21,

and deduce the ratio

fBs BBs


1.174 ± 0.026, ξ≡ fB BB

(34)

where the errors come almost equally from n, rq , ms , Mb and the αs2 term. As expected, we have smaller errors for the ratio ξ due to the cancellation of the systematics. We proceed in the same way with the Laplace sum rules where we take the range of τ values from 0.1 to 0.37 GeV−2 (see Fig. 2a, b) in order to have a conservative result. Then, we deduce 
B
(249 ± 61) MeV, fB B ξ
1.187 ± 0.038.

(35)

As a final result, we take the arithmetic average from the moments and Laplace sum rules results. Then, we deduce 
B
(247 ± 59) MeV, fB B ξ
1.18 ± 0.03,

(36)

in the unit where fπ = 130.7 MeV. These results  can
B
be compared with different lattice results fB B (230 ± 32) MeV, ξ
1.14 ± 0.06, and  global-fit of
B
(231 ± the CKM mixing angles giving fB B 15) MeV quoted in Ref. [19,21]. By comparing our

238

K. Hagiwara et al. / Physics Letters B 540 (2002) 233–240

 
B for different values of rq and n: fB B
B versus: (a) rd at n = −30, (b) n at rd = 1.13; Fig. 1. Moment sum rules analysis of fB(s) B (s) 
B versus: (c) rs at n = −26, (d) versus n at rs = 1.17. Dotted curve: lowest order perturbative contribution; dashed curve: lowest order fB B s

s

perturbative +ms ¯s s [only for (c) and (d)] + αs G2  condensates; solid curve: total contribution to order αs .

results Eq. (36) with the one of fBs /fBd in Eq. (11),5 one can conclude (to a good approximation) that
Bd
(1.65 ± 0.38)
Bs ≈ B B  ⇒ BBd,s 4Mb2
(1.1 ± 0.25),

(37)

indicating a negligible SU(3) breaking for the bag parameter. For a consistency, we have used the estimate to order αs [22] fB
(1.47 ± 0.10)fπ
(192 ± 19) MeV,

and we have assumed that the error from fB compensates the one in Eq. (36). The result is in excellent agreement with the previous result of Ref. [7] in Eqs. (7) and (8), and agrees within the errors with the lattice estimates [19,20]. Using the experimental values

Md = 0.479(12) ps−1 ,

Ms
13.1 ps−1 (95% CL),

(39)

(38) one can deduce from Eq. (4)

5 One can notice that similar strengths of the SU(3) breakings have been obtained for the B → K ∗ γ and B → Klν form factors [21].

ρsd

   V t s 2 
20.0(1.1).  ≡ V  td

(40)

K. Hagiwara et al. / Physics Letters B 540 (2002) 233–240

239

 
B for different values of rq and τ : fB B
B versus: (a) rd at τ = 0.31 GeV−2 , (b) τ at Fig. 2. Laplace sum rules analysis of fB(s) B (s) 
B versus: (c) rs at τ = 0.26 GeV−2 , (d) versus τ at rs = 1.20. The curves are the same as in Fig. 1. rd = 1.13; fBs B s

Alternatively, using ρsd


1
28.4(2.9) λ2 [(1 − ρ) ¯ 2 + η¯ 2 ]

(41)

in agreement with the present experimental lower bound and within the reach of Tevatron run 2 experiments.

with [19] λ
0.2237(33),   λ2
0.223(38), ρ¯ ≡ ρ 1 − 2   λ2 η¯ ≡ η 1 −
0.316(40), 2

7. Conclusions

(42)

λ, ρ and η being the Wolfenstein parameters, we deduce

Ms
18.6(2.2) ps−1 ,

(43)

We have applied QCD spectral sum rules for extracting ( for the first time) the SU(3) breaking parameter  ξ , and for improving the estimate of the quantity
B . Our predictions are given in Eq. (36). The fB B phenomenological consequences of our results for the 0 –B 0 mass-differences CKM mixing angle and Bd,s d,s are given in Eqs. (40) and (43).

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K. Hagiwara et al. / Physics Letters B 540 (2002) 233–240

Acknowledgements This work has been initiated when one of the authors (S.N.) has been a visiting professor at the KEK theory group from January to July 2000.

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