Table Of ContentCERN-TH/98-369
TUM-T31-337/98
December 1998
The Rare Decays K πνν¯, B Xνν¯ and B l+l :
−
→ → →
An Update
9
9
9
1
n
a
J Gerhard Buchallaa and Andrzej J. Burasb
3
1
aTheory Division, CERN, CH-1211 Geneva 23, Switzerland
1
v bPhysik Department, Technische Universit¨at Mu¨nchen,
8 D-85748 Garching, Germany
8
2
1
0
9
9 Abstract
/
h
p We update the Standard Model predictions for the rare decays K+ π+νν¯
→
- and K π0νν¯. In view of improved limits on B –B¯ mixing we derive a strin-
p L s s
→
e gent and theoretically clean Standard Model upper limit on B(K+ π+νν¯),
h ¯ ¯ →
which is based on the ratio of B –B to B –B mixing, ∆M /∆M , alone. This
: d d s s d s
v
methodavoidsthelargehadronicuncertainties present intheusualanalysisofthe
i
X CKM matrix. We find B(K+ π+νν¯) < 1.67 10−10, which can be further im-
r → ·
proved in the future. In addition we consider the extraction of V from a future
a td
| |
measurement of B(K+ π+νν¯), discussing the various sources of uncertainties
→
involved. We also investigate theoretically clean constraints on B(K π0νν¯).
L
→
We take the opportunity to review the next-to-leading order (NLO) QCD cor-
rections to K πνν¯, K µ+µ−, B Xνν¯ and B l+l−, including a
L
→ → → →
small additional term that had been missed in the original publications. The
phenomenological impact of this change is negligible, the corresponding numeri-
cal shift being essentially within the small perturbative uncertainties at the NLO
level.
1 Introduction
The rare decays K+ π+νν¯, K π0νν¯, B X νν¯ and B l+l− are
L s,d s,d
→ → → →
very promising probes of flavour physics. They are sensitive to the quantum
structure of Standard Model flavour dynamics and can at the same time be com-
puted theoretically to an exceptionally high degree of precision. In particular the
kaon modes K+ π+νν¯ and K π0νν¯ have received considerable interest in
L
→ →
recent years. The theoretical status of these decays has been reinforced by the
calculation of next-to-leading order QCD corrections [1]–[4]. The corrections to
the top-quark dominated modes K π0νν¯, B X νν¯ and B l+l− [1, 2]
L s,d s,d
→ → →
have recently been recomputed in [6]. The results of [6] confirm the calculations
of [1,2] upto asmall term arising fromthebox diagramthat hadbeen overlooked
in [2]. We are in agreement with [6] that this term needs to be included to obtain
the complete NLO correction. As will be explained in more detail below, the
missing piece follows from a simple one-loop calculation, which can be performed
separately, and it can be added to the old result. The latter was obtained from
the two-loop matching calculation in [2], which remains unaffected.
In contrast to K π0νν¯, where only the top-quark contribution is relevant,
L
→
also the charm sector is important for K+ π+νν¯. This sector, which has not
→
been considered in [6], will also be discussed in the present article.
The numerical impact of the modification is small. The changes are within
the perturbative error of the NLO calculation, both for the top and the charm
contribution. All qualitative conclusions, regarding the NLO error analysis from
residual scale dependence and the theoretical precision that can be achieved in
the calculation of these decays, remain unchanged.
The purpose of this paper is twofold. We will first present the complete NLO
effective hamiltonians for K+ π+νν¯ (section 2), the short-distance part of
K µ+µ− (section 3) and the→top-quark dominated modes K π0νν¯, B
L L
X →νν¯ and B l+l− (section 4), including the additional NLO→contributio→n
s,d s,d
→
from the box diagram. The origin and the derivation of this contribution are
explained in the Appendix. Secondly, we update the Standard Model predictions
for B(K+ π+νν¯) and B(K π0νν¯) and discuss some phenomenological
L
→ →
aspects of these decays that have not been emphasized before. In particular, we
derive in section 5 a clean correlation between the ratio of B –B¯ and B –B¯
d d s s
mixing and the K+ π+νν¯ branching fraction. In view of recent experimental
progress, bothforK+→ π+νν¯andB–B¯ mixing, thiscouldprovide aninteresting
→
Standard Model test in the near future. We also analyse the determination of
V from B(K+ π+νν¯) and investigate the theoretical uncertainties involved.
td
| | →
Section 6 explores the impact of ∆M /∆M and sin2β, as obtained from the
d s
CP asymmetry in B (B¯ ) J/ΨK , on the neutral mode K π0νν¯. The
d d S L
→ →
combination of these three theoretically clean observables offers another stringent
consistency check on the Standard Model. We conclude in section 7.
1
2 Effective Hamiltonian for K+ π+νν¯
→
The effective hamiltonian for K+ π+νν¯ can be written as
→
G α
Heff = √F22πsin2Θ Vc∗sVcdXNl L +Vt∗sVtdX(xt) (s¯d)V−A(ν¯lνl)V−A. (1)
W l=Xe,µ,τ(cid:16) (cid:17)
The index l=e, µ, τ denotes the lepton flavour. The dependence on the charged
lepton mass, resulting from the box-graph, is negligible for the top contribution.
In the charm sector this is the case only for the electron and the muon, but not
for the τ-lepton.
The function X(x), relevant for the top part, reads to (α ) and to all orders in
s
O
x = m2/M2
W
α
s
X(x) = X (x)+ X (x) (2)
0 1
4π
with [7]
x 2+x 3x 6
X (x) = + − lnx (3)
0
8 "−1 x (1 x)2 #
− −
and the QCD correction
29x x2 4x3 x+9x2 x3 x4
X (x) = − − − − lnx
1 − 3(1 x)2 − (1 x)3
− −
8x+4x2 +x3 x4 4x x3
+ − ln2x − L (1 x)
2(1 x)3 − (1 x)2 2 −
− −
∂X (x)
0
+ 8x lnx (4)
µ
∂x
where x = µ2/M2 with µ = (m ) and
µ W O t
x lnt
L (1 x) = dt (5)
2
− 1 t
Z1 −
The µ-dependence in the last term in (4) cancels to the order considered the
µ-dependence of the leading term X (x(µ)).
0
The expression corresponding to X(x ) in the charm sector is the function Xl .
t NL
It results from the RG calculation in NLLA and is given as follows:
Xl = C 4B(1/2) (6)
NL NL − NL
C and B(1/2) correspond to the Z0-penguin and the box-type contribution,
NL NL
respectively. One has
CNL = x(3m2)Kc2245 "(cid:18)478K+ + 1214K− − 67976K33(cid:19) αs4(πµ) + 115827152(1−Kc−1)!
µ2 1176244 2302 3529184
+ 1−ln m2!(16K+ −8K−)− 13125 K+ − 6875K− + 48125 K33
56248 81448 4563698
+ K K+ K− + K33 (7)
4375 − 6875 144375
(cid:18) (cid:19)(cid:21)
2
where
α (M ) α (µ)
s W s
K = K = (8)
c
α (µ) α (m)
s s
6 −12 −1
K+ = K25 K− = K 25 K33 = K 25 (9)
B(1/2) = x(m)K2254 3(1 K ) 4π + 15212(1 K−1)
NL 4 c " − 2 αs(µ) 1875 − c !
µ2 rlnr 77 15212 4364
ln + K + KK (10)
− m2 − 1 r − 3 625 2 1875 2#
−
Here K = K−1/25, m = m , r = m2/m2(µ) and m is the lepton mass. We will
2 c l c l
at times omit the index l of Xl . In (7) – (10) the scale is µ = (m ). For the
NL O c
charm contribution we need the two-loop expression for α (µ) given by
s
4π β lnln µ2
α (µ) = 1 1 Λ2 (11)
s β ln µ2 − β2 ln µ2
0 Λ2 0 Λ2
2 38
β = 11 f β = 102 f (12)
0 1
− 3 − 3
The effective number of flavours is f = 4 in the expressions above for the charm
(4)
sector. The QCD scale in (11) is Λ = Λ = (325 80)MeV.
MS ±
Again – to the considered order – the explicit ln(µ2/m2) terms in (7) and (10)
cancel the µ-dependence of the leading terms.
Forphenomenologicalapplicationsitisusefultodefineforthetopcontribution
a QCD correction factor η by
X
X(x ) = η X (x ) η = 0.994 (13)
t X 0 t X
·
With the MS definition of the top-quark mass, m m¯ (m ), the QCD factor η
t t t X
≡
is practically independent of m . This is the definition of m that we will employ
t t
throughout this paper.
For the charm sector it is useful to introduce
1 2 1
P (X) = Xe + Xτ (14)
0 λ4 3 NL 3 NL
(cid:20) (cid:21)
with the Wolfenstein parameter [8] λ = 0.22. Results for the charm functions are
collected in tables 1 and 2, where m stands for the MS mass m¯ (m ).
c c c
3 Effective Hamiltonian for (K µ+µ )
L − SD
→
The analysis of (K µ+µ−) proceeds in essentially the same manner as for
L SD
→
K+ π+νν¯. The only difference is introduced through the reversed lepton line
→
in the box contribution. In particular there is no lepton mass dependence, since
3
Table 1: The functions Xe and Xτ for various Λ(4) and m .
NL NL MS c
Xe /10−4 Xτ /10−4
NL NL
(4)
Λ [MeV] m [GeV] 1.25 1.30 1.35 1.25 1.30 1.35
c
MS \
245 10.73 11.60 12.50 7.34 8.06 8.81
285 10.44 11.31 12.20 7.05 7.77 8.52
325 10.14 11.00 11.90 6.75 7.47 8.21
365 9.82 10.69 11.58 6.44 7.15 7.89
405 9.49 10.35 11.24 6.10 6.81 7.55
(4)
Table 2: The function P (X) for various Λ and m .
0 c
MS
P (X)
0
(4)
Λ [MeV] m [GeV] 1.25 1.30 1.35
MS \ c
245 0.410 0.445 0.481
285 0.397 0.432 0.468
325 0.385 0.419 0.455
365 0.371 0.406 0.442
405 0.357 0.391 0.427
only massless neutrinos appear as virtual leptons in the box diagram.
The effective hamiltonian in next-to-leading order can be written as follows:
G α
F ∗ ∗
Heff = −√22πsin2Θ (VcsVcdYNL +VtsVtdY(xt))(s¯d)V−A(µ¯µ)V−A +h.c. (15)
W
The function Y(x) is given by
α
s
Y(x) = Y (x)+ Y (x) (16)
0 1
4π
where [7]
x 4 x 3x
Y (x) = − + lnx (17)
0 8 "1 x (1 x)2 #
− −
and
10x+10x2 +4x3 2x 8x2 x3 x4
Y (x) = − − − lnx
1
3(1 x)2 − (1 x)3
− −
2x 14x2 +x3 x4 2x+x3
+ − − ln2x+ L (1 x)
2(1 x)3 (1 x)2 2 −
− −
∂Y (x)
0
+ 8x lnx (18)
µ
∂x
4
Similarly to (13) one may write
Y(x ) = η Y (x ) η = 1.012 (19)
t Y 0 t Y
·
The RG expression Y representing the charm contribution reads
NL
(−1/2)
Y = C B (20)
NL NL − NL
where C is the Z0-penguin part given in (7) and B(−1/2) is the box contribution
NL NL
in the charm sector, relevant for the case of final state leptons with weak isospin
T = 1/2. One has
3
−
Table 3: The functions Y and P (Y) = Y /λ4 for various Λ(4) and m .
NL 0 NL c
MS
Y /10−4 P (Y)
NL 0
(4)
Λ [MeV] m [GeV] 1.25 1.30 1.35 1.25 1.30 1.35
c
MS \
245 2.75 2.94 3.13 0.117 0.125 0.134
285 2.80 2.99 3.19 0.119 0.128 0.136
325 2.84 3.04 3.24 0.121 0.130 0.138
365 2.89 3.09 3.29 0.123 0.132 0.140
405 2.92 3.13 3.33 0.125 0.133 0.142
B(−1/2) = x(m)K2245 3(1 K ) 4π + 15212(1 K−1)
NL 4 c " − 2 αs(µ) 1875 − c !
µ2 317 15212 23081
ln + K + KK (21)
− m2 − 12 625 2 7500 2#
(1/2)
Note the simple relation to B in (10) (for r = 0)
NL
B(−1/2) B(1/2) = 3 x(m)K2254(KK 1) (22)
NL − NL 16 c 2 −
4 Effective Hamiltonians for K π0νν¯, B
L
→ →
X νν¯ and B l+l
s,d s,d −
→
With the above results it is easy to write down also the effective hamiltonians for
K π0νν¯, B X νν¯ and B l+l−. Since only the top contribution is
L s,d s,d
→ → →
important in these cases, we have
G α
F ∗ ′
Heff = √22πsin2Θ VtnVtn′X(xt)(n¯n)V−A(ν¯ν)V−A +h.c. (23)
W
5
for the decays K π0νν¯, B X νν¯ and B X νν¯, with (n¯n′) = (s¯d), (¯bs),
L s d
¯ → → →
(bd), respectively. Similarly
G α
F ∗ ′ ¯
Heff = −√22πsin2Θ VtnVtn′Y(xt)(n¯n)V−A(ll)V−A +h.c. (24)
W
for B l+l− and B l+l−, with (n¯n′) = (¯bs), (¯bd). The functions X, Y are
s d
→ →
given in (2) and (16).
5 Phenomenology of K+ π+νν¯
→
5.1 General Aspects and Standard Model Prediction
The branching fraction of K+ π+νν¯ can be written as follows
→
2 2
Imλ Reλ Reλ
B(K+ π+νν¯) = κ tX(x ) + cP (X)+ tX(x ) (25)
→ + · λ5 t ! λ 0 λ5 t !
3α2B(K+ π0e+ν)
κ = r → λ8 = 4.11 10−11 (26)
+ K+ 2π2sin4Θ ·
W
Here x = m2/M2 , λ = V∗V and r = 0.901 summarizes isospin breaking
t t W i is id K+
corrections in relating K+ π+νν¯ to the well measured leading decay K+
→ →
π0e+ν [9]. In the standard parametrization λ is real to an accuracy of better
c
than 10−3. We remark that in writing B(K+ π+νν¯) in the form of (25) a
→
negligibly small term (Xe Xτ )2 has been omitted (0.2% effect on the
∼ NL − NL
branching ratio).
A prediction for B(K+ π+νν¯) in the Standard Model can be obtained
→
using information on kaon CP violation (ε ), V /V and B B¯ mixing to
K ub cb
| | −
constrain the CKM parameters Reλ and Imλ in (25). This standard analysis
t t
of the unitarity triangle is described in more detail in [5]. Here we present an
updated prediction using new input from the 1998 Vancouver conference [10]. We
take
m f √B
B = 0.80 0.15 Bs Bs Bs < 1.2 P (X) = 0.42 0.06 (27)
K 0
± smBd fBd BBd ±
q
for the (scheme-invariant) kaon bag parameter B , the SU(3) breaking among
K
the matrix elements of B –B¯ and B –B¯ mixing, and P (X) in (25), respectively.
s s d d 0
Next we use [10]
m = m¯ (m ) = (166 5)GeV V = 0.040 0.003 V /V = 0.091 0.016
t t t cb ub cb
± ± | | ±
(28)
and the experimental results on B B¯ and B B¯ mixing [10]
d d s s
− −
∆M = (0.471 0.016)ps−1 ∆M > 12.4ps−1 (29)
d s
±
6
Scanning all parameters within the above ranges one obtains
B(K+ π+νν¯) = (0.82 0.32) 10−10 (30)
→ ± ·
where the error is dominated by the uncertainties in the CKM parameters. Eq.
(30) may be compared with the result from Brookhaven E787 [11]
B(K+ π+νν¯) = (4.2+9.7) 10−10 (31)
→ −3.5 ·
Clearly, within thelargeuncertainties, thisresult iscompatiblewiththeStandard
Model expectation.
5.2 Upper bound on B(K+ π+νν¯) from ∆M /∆M
d s
→
Anticipating improved experimental results, the question arises of how large a
branching fraction could still be accomodated by the Standard Model. In other
words, how large would B(K+ π+νν¯) have to be, in order to unambiguously
exp
→
signalNewPhysics. Inthiscontext we recallthatthecleannatureofK+ π+νν¯
→
implies a relation between the branching ratio and CKM parameters with very
good theoretical accuracy. However, in order to constrain the poorly known
CKM quantities to predict (30), one introduces theoretical uncertainties (related
to V /V or B ) that are not intrinsic to K+ π+νν¯ itself. We would
ub cb K
| | →
therefore like to investigate to what extent an upper bound can be derived on
B(K+ π+νν¯), without relying on V /V or the constraint from kaon CP
ub cb
→ | |
violation (ε ) involving B . For this purpose we will make use of the ratio
K K
∆M /∆M . This is motivated by the theoretically fairly clean nature of this
d s
ratio and the improved lower bound on ∆M (29).
s
In terms of Wolfenstein parameters λ, A, ̺ and η [8] one has [5]
V 2 R2
td = λ2 t R2 = (1 ̺¯)2 +η¯2 (32)
V 1+λ2(2̺¯ 1) t −
(cid:12) ts(cid:12)
(cid:12) (cid:12) −
(cid:12) (cid:12)
where [12] (cid:12) (cid:12)
λ2 λ2
̺¯= ̺ 1 η¯= η 1 (33)
− 2 ! − 2 !
A measurement of ∆M /∆M determines R according to [5]
d s t
r ∆M λ2 m f √B
R = sd d 1 (1 2̺¯) r = Bs Bs Bs (34)
t sd
λ s∆Ms − 2 − ! smBd fBd BBd
q
The ratio of hadronic matrix elements r has been studied in lattice QCD. As
sd
discussed in [13]–[15], the current status can be summarized by
f √B
Bs Bs = 1.14 0.08 (35)
f B ±
Bd Bd
q
7
This result is based on the quenched approximation, but the related uncertainties
are expected to be moderate for the ratio (35). In the following we shall use
r = 1.2 0.2 (36)
sd
±
Sincer = 1intheSU(3)-flavoursymmetrylimit,itisthedifferencer 1thatis
sd sd
−
a priori unknown and has to be determined by non-perturbative calculations. To
be conservative, we have assigned a 100% error on the SU(3)-breaking correction
in (36).
We may next rewrite (25) using an improved Wolfenstein parametrization [12]
as [5]
1
B(K+ π+νν¯) = κ A4X2(x ) (ση¯)2 +(̺ ̺¯)2 (37)
+ t 0
→ σ −
h i
2
1 P (X)
0
σ = ̺ = 1+ (38)
1 λ2! 0 A2X(xt)
− 2
Eq. (37) defines an (almost circular) ellipse in the (̺¯,η¯) plane, centered at (̺ ,0).
0
Now, for fixed R , the maximum possible branching ratio occurs for η¯= 0 and is
t
given by
2
κ r ∆M
B(K+ π+νν¯) = + P (X)+A2X(x ) sd d (39)
max 0 t
→ σ " λ s∆Ms#
from(32),(34)and(37). Thisequationprovidesasimpleandtransparentrelation
for the maximal B(K+ π+νν¯) that would still be consistent with a given value
→
of ∆M /∆M in the Standard Model. In particular, a lower bound on ∆M
d s s
immediately translates into an upper bound for B(K+ π+νν¯). We stress that
→
(39) is theoretically very clean. All necessary input is known and does not involve
uncontrolled theoretical uncertainties. At present the largest error comes from
r , but a systematic improvement is possible within lattice gauge theory.
sd
For these reasons (39) can serve as a clearcut test of the Standard Model and
has the potential to indicate the presence of New Physics in a clean manner.
Using
∆M
d
< 0.2 A < 0.89 P (X) < 0.48 X(x ) < 1.57 r < 1.4
0 t sd
s∆M
s
(40)
one finds
B(K+ π+νν¯) = 1.67 10−10 (41)
max
→ ·
This limit could be further strengthened with improved input. (For central values
of parameters the number 1.67 in (41) would change to 0.94.) However, even with
present knowledge, represented by (41), the bound is strong enough to indicate
a clear conflict with the Standard Model if B(K+ π+νν¯) should be measured
at 2 10−10. →
·
In table 4 we illustrate how the upper bound on B(K+ π+νν¯) (bound A)
→
improves when the maximal possible value for r is assumed to be smaller. The
sd
8
outcome is compared with the result from the standard analysis of the unitarity
triangle (bound B). We find that as long as r is higher than 1.4, the r bound
ds ds
has no impact on the maximal value in the standard analysis. We also observe
that bound A, which is theoretically very reliable, is only slightly weaker than
the less clean result of bound B.
Table 4: B(K+ π+νν¯) 1010 from ∆M /∆M alone (bound A), compared
max d s
→ ·
with the maximum value from the standard analysis (bound B), where also the
information from V /V , ε and ∆M is used. The bounds are shown for
ub cb K d
| |
various r , the maximum of the SU(3) breaking parameter r .
sd,max sd
r Bound A Bound B
sd,max
1.40 1.67 1.32
1.30 1.49 1.21
1.25 1.40 1.17
1.20 1.32 1.14
5.3 V from B(K+ π+νν¯)
td
| | →
Eventually, a precise experimental determination of B(K+ π+νν¯), in particu-
→
lar if compatible with Standard Model expectations, can be used to extract V
td
| |
directly from (25) [4, 16]. We would like to illustrate such an analysis here by
detailing the sources of uncertainty and their impact on the final result. Our
findings are summarized in table 5. We remark that the sensitivity of V to
td
| |
variations in the input is fairly linear for the parameters B(K+ π+νν¯), and
→
(4)
V through Λ , so that the effect of other choices for the errors can be easily
cb
MS
infered from this table.
6 Phenomenology of K π0νν¯
L
→
The rare decay mode K π0νν¯ is a measure of direct CP violation [17] and
L
→
thereforeofparticular interest. Using theeffective hamiltonian(23)andsumming
over three neutrino flavours one finds [4, 5]
2
Imλ
B(K π0νν¯) = κ tX(x ) (42)
L → L · λ5 t !
r τ(K )
κ = κ KL L = 1.80 10−10 (43)
L +r τ(K+) ·
K+
with r = 0.944 the isospin breaking correction from [9] and κ given in (26).
KL +
Using the improved Wolfenstein parametrization [12] we can rewrite (42) as [5]
B(K π0νν¯) = κ η2A4X2(x ) (44)
L L t
→
9
