Table Of Content{ 1{
PSEUDOSCALAR-MESON DECAY CONSTANTS
Revised March 1998 by M. Suzuki (LBNL).
Charged mesons
The decay constant f for a charged pseudoscalar meson P
P
is de(cid:12)ned by
h0jA (0)jP(q)i = if q ; (1)
(cid:22) P (cid:22)
where A is the axial-vector part of the charged weak cur-
(cid:22)
rent after a Cabibbo-Kobayashi-Maskawa mixing-matrix ele-
ment Vqq0 has been removed. The state vector is normalized
by hP(q)jP(q0)i = (2(cid:25))3 2E (cid:14)(q−q0), and its phase is chosen
q
to make f real and positive. Note, however, that in many
P p
theoretical papers our f = 2 is denoted by f .
P P
In determining f experimentally, radiative corrections
P
must be taken into account. Since the photon-loop correction
introducesaninfrareddivergencethat iscanceledbysoft-photon
emission, we can determine f only from the combined rate for
P
P(cid:6) ! ‘(cid:6)(cid:23) and P(cid:6) ! ‘(cid:6)(cid:23) γ. This rate is given by
‘ ‘
Γ(P ! ‘(cid:23) +‘(cid:23) γ) =
‘ ‘
(cid:18) (cid:19)
G2FjVqq02jf2 m2 m 1− m2‘ 2[1+ ((cid:11))] : (2)
8(cid:25) P ‘ P m2 O
P
Here m and m are the masses of the lepton and meson.
‘ P
Radiative corrections include inner bremsstrahlung, which is
independent of the structure of the meson [1{3], and also a
structure-dependent term [4,5]. After radiative corrections are
made, there are ambiguities in extracting f from experimental
P
measurements. In fact, the de(cid:12)nition of f is no longer unique.
P
CITATION: C. Caso et al., The European Physical Journal C3, 1 (1998) and 1999 o(cid:11)-year partial update for the 2000 edition (URL: http://pdg.lbl.gov/)
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It is desirable to de(cid:12)ne f such that it depends only on the
P
properties of the pseudoscalar meson, not on the (cid:12)nal decay
products. The short-distance corrections to the fundamental
electroweak constants like GFjVqq0j should be separated out.
Following Marciano and Sirlin [6], we de(cid:12)ne f with the
P
following form for the ((cid:11)) corrections:
O
(cid:20) (cid:21)(cid:20) (cid:21)
(cid:16) (cid:17)
2(cid:11) m (cid:11)
Z
1+ ((cid:11)) = 1+ ln 1+ F(x)
O (cid:25) m(cid:26) (cid:25)
!
(cid:26) (cid:20) (cid:21)(cid:27)
(cid:16) (cid:17)
(cid:11) 3 m m2 m2 m2
(cid:2) 1− ln (cid:26) +C +C ‘ ln (cid:26) +C ‘ +::: ;
(cid:25) 2 m 1 2m2 m2 3m2
P (cid:26) ‘ (cid:26)
(3)
where m and m are the masses of the (cid:26) meson and Z boson.
(cid:26) Z
Here
13−19x2 8−5x2
F(x) = 3lnx+ − x2lnx
8(1−x2) 2(1−x2)2
(cid:16) (cid:17) (cid:16) (cid:17)
1+x2 1+x2
−2 lnx+1 ln(1−x2)+2 L(1−x2) ;
1−x2 1−x2
with Z
z ln(1−t)
x (cid:17) m =m ; L(z) (cid:17) dt : (4)
‘ P
t
0
The (cid:12)rst bracket in the expression for 1+ ((cid:11)) is the short-
O
distanceelectroweakcorrection. Aquarterof(2(cid:11)=(cid:25)) ln(m =m )
Z (cid:26)
is subject to the QCD correction (1−(cid:11) =(cid:25)), which leads to a
s
reduction of the total short-distance correction of 0.00033 from
the electroweak contribution alone [6]. The second bracket to-
gether with the term −(3(cid:11)=2(cid:25))ln(m =m ) in the third bracket
(cid:26) P
corresponds to the radiative corrections to the point-like pion
decay ((cid:3) (cid:25) m ) [2]. The rest of the corrections in the
cuto(cid:11) (cid:26)
third bracket are expanded in powers of m =m . The expansion
‘ (cid:26)
coe(cid:14)cients C , C , and C depend on the hadronic structure
1 2 3
of the pseudoscalar meson and in most cases cannot be com-
puted accurately. In particular, C absorbs the uncertainty in
1
the matching energy scale between short- and long-distance
strong interactions and thus is the main source of uncertainty
in determining f accurately.
(cid:25)+
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With the experimental value for the decay (cid:25)+ ! (cid:22)+(cid:23) +
(cid:22)
(cid:22)+(cid:23) γ, one obtains
(cid:22)
f = 130:7(cid:6)0:1(cid:6)0:36 MeV ; (5)
(cid:25)+
where the (cid:12)rst error comes from the experimental uncertainty
on jV j and the second comes from the uncertainty on C (=
ud 1
0 (cid:6) 0:24) [6]. Similarly, one obtains from the decay K+ !
(cid:22)+(cid:23) +(cid:22)+(cid:23) γ the decay constant
(cid:22) (cid:22)
f = 159:8(cid:6)1:4(cid:6)0:44 MeV ; (6)
K+
where the (cid:12)rst error is due to the uncertainty on jV j.
us
For the heavy pseudoscalar mesons, uncertainties in the
experimental values for the decay rates are much larger than
the radiative corrections. For the D+, only an upper bound can
be obtained from the published data:
f < 310 MeV (CL = 90%) : (7)
D+
For the D+, the decay constant has been extracted from both
s
the D+ ! (cid:22)+(cid:23) and the D+ ! (cid:28)+(cid:23) branching fractions. Two
s (cid:22) s (cid:28)
values have been reported since the last edition [7,8]:
f = 194(cid:6)35(cid:6)20(cid:6)14 MeV from D+ ! (cid:22)+(cid:23) ;
D+ s (cid:22)
s
f = 309(cid:6)58(cid:6)33(cid:6)38 MeV from D+ ! (cid:28)+(cid:23) :
D+ s (cid:28)
s
There are now altogether (cid:12)ve reported values for f spread
D+
s
over a wide range,
f = 194 MeV (cid:24) 430 MeV (8)
D+
s
with large uncertainties attached. We must wait for better data
before giving a meaningful value for f . (See the measure-
D+
s
ments of the D+ ! ‘+(cid:23) modes in the Particle Listings for the
s ‘
numbers quoted by individual experiments.)
There have been many attempts to extract f from spec-
P
troscopy and nonleptonicdecaysusing theoreticalmodels. Since
it is di(cid:14)cult to estimate uncertainties for them, we have listed
here only values of decay constants that are obtained directly
from the observation of P(cid:6) ! ‘(cid:6)(cid:23) .
‘
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Light neutral mesons
The decay constants for the light neutral pseudoscalar
mesons (cid:25)0, (cid:17), and (cid:17)0 are de(cid:12)ned by
p
h0jA (0)jP0(q)i = i(f = 2)q ; (9)
(cid:22) P (cid:22)
where A is a neutral axial-vector current of octet or singlet.
(cid:22)
However f for the neutral mesons cannot be extracted directly
p
from the data.
In the limit of m ! 0, the Adler-Bell-Jackiw anomaly
P
determines f through the matrix element of the two-photon
P
decay P0 ! γγ [9,10]. The extrapolation to the mass shell is
needed to extract the physical value of f . In the case of f ,
P (cid:25)0
the extrapolation is small and the experimental uncertainty in
the (cid:25)0 lifetime dominates in the uncertainty of f :
(cid:25)0
f = 130(cid:6)5 MeV ; (10)
(cid:25)0
which is consistent with isospin symmetry.
0
For the(cid:17) and (cid:17) , the extrapolation tothemass shellislarger
and therefore the dominance of the anomaly on the mass shell
0 0
is questionable, particularly for the (cid:17) ; and (cid:17){(cid:17) mixing adds to
the uncertainty. If the corrections are computed for the octet
with the chiral Lagrangian [11], one obtains f (cid:25) 1:3f for the
8 (cid:25)
decay constant of the I = 0 octet state. For the singlet state, if
the (cid:17) ! γγ and (cid:17)0 ! γγ decay rates are (cid:12)tted with the same
form as the anomaly indicates, f (cid:25) f would give a viable
1 (cid:25)
(cid:12)t for f (cid:25) 1:3f and the (cid:17){(cid:17)0 mixing angle of (cid:18) (cid:25) −20(cid:14).
8 (cid:25) P
However,because of the arbitrariness even in de(cid:12)ning the decay
constants, we do not quote numbers for f(cid:17) or f(cid:17)0 here.
References
1. S. Berman, Phys. Rev. Lett. 1, 468 (1958).
2. T. Kinoshita, Phys. Rev. Lett. 2, 477 (1959).
3. A. Sirlin, Phys. Rev. D5, 436 (1972).
4. M.V. Terent’ev, Yad. Fiz. 18, 870 (1973) [Sov. J. Nucl.
Phys. 18, 449 (1974)].
5. T. Goldman and W.J. Wilson, Phys. Rev. D15, 709
(1977).
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6. W.J. Marciano and A. Sirlin, Phys. Rev. Lett. 71, 3629
(1993).
7. K. Kodama et al., Phys. Lett. B382, 299 (1996).
8. M. Acciarri et al., Phys. Lett. B396, 327 (1997).
9. S.L. Adler, Phys. Rev. 177, 2426 (1969).
10. J.S. Bell and R. Jackiw, Nuovo Cimento 60A, 46 (1969).
11. J.Gasser and H.Leutwyler,Nucl.Phys.B250, 465 (1985).
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