- Email: [email protected]
Ke5 decays in chiral perturbation theory
16 February 1995
PHYSICS
LETTERS B
Physics Letters B 345 ( 1995) 287-290
K,, decays in chiral perturbation theory Stefan Blaser
’
lnstitutflir theoretische Physik, Universitiit Bern, Sidlerstrasse 5. CH-3012 Bern, Switzerland
Received 28 October 1994 Editor: R. Gatto Abstract We evaluate the branching
ratios for the decays K + aaneu
at leading order in chiral perturbation theory and give an
isospin relation for the decay rates. Keywords: Kaon; Semileptonic decay; Chiral perturbation theory; Phase space volume; Chiral anomaly; DA@NE
1. We discuss
the Kes decays
K+ -4
z-+7i--7r”e+Ir et
Ko -
7r07ron--e+v e7
K+ Ko -
r”ro?roe+v e7 77fr-7i-TT-e+u,
in the framework of chiral perturbation theory (CHF’T) [ 1,2]. For low momenta relevant in the present case, the transition amplitude for K ---fmrre+v, reduces in the standard model to the current times current form - ~5)4pe)(V~
i”‘= GFV*%~,)~p(l Jz
us
- AFL),
(1)
where vcL - AN = (‘r(!‘l 2. To calculate order, C = $
)r(k’2h(p3)
out
h’“(
the hadronic matrix elements
tr (d,UPU+
+ XV+ +
1 -
YS)UIK(P)).
VP and Ap, we use the effective Lagrangian
x+U),
(2)
of QCD at leading
(3)
where F = 93.2 MeV is the pion decay constant in the chiral limit. Furthermore, we work in the isospin limit, i.e., nrU = md G riz, and set x = 2Ba diag (rit, P?Z,m,), where Ba is related to the quark condensate in the chiral -’ E-mail address: [email protected]. 0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDfO370-2693(94)01627-5
288
S.Blaser/PhysicsLettersB345(1995)287-290
Fig. 1. Tree level diagrams for Ke5 decays.
limit [2]. The unitary 3 x 3 matrix U incorporates the fields of the eight pseudoscalar convenient parametrization is U = exp( &Q/F) with
Goldstone
bosons.
@f~~_&ZJ. In addition, v4-i5 c1
A
(4)
the vector current relevant in the present case reads =
z
tr ([ A4 - ih5] [ lM,U+ + U+d,Cl]) ,
(5)
where A4 and h5 denote Gell-Mann matrices. The corresponding because it is odd under the transformation Cp + -@. 3. The relevant Feynman
diagrams
(~+(P,>~-(P&r”(Ps)
at leading order in CHIT
out ]V,u]K+(P))
(rococo
out IW,#+(P,)
= -&(1,2,3) =A,(l,2,3)
(~“(P&r-(pz)~o(~s)
out ISrpu(@?P))
(~+(Pt>~-(P~)~-(P3)
out I~y,uI~
axial vector current
does not contribute,
are shown in Fig. 1. Their contribution
(6)
+B,(1,2,3), +A,(1,3,2)
+A,(3,2,
l),
(7) (8)
=fiA,(l,3,2), =-&{A,(1,2,31
is
fA,U,3,2)},
(9)
where
Afi(1.2.3)=g(M2 P~~~~~!p,2[Pl-P*+P3-PlpfMz ~~~~~~~)P)2,pI-p*-p3+P K
I
WC+ 2PlP2) 'P1+P2-P3-p1~+~2_~p,+p2+p3~2[pl+p2+p3+p1~ ?r
P(PI +P2)
-
M2,-(p,+p2-p)2
+21Pl
+p2
-P3lp
8,(1’2’3)=$(M,
:;;;;;~p)2[PI+P2-P3-Plr+M2 I
-P3)
(Pi fP3
(10)
7
>
K
P(PI + M”K -
K
-Pj2
[PI -P2
+P3
K p~~~~~~p)2(p,-p2-p3+plp
-PI,
>
.
(11)
289
S. Blaser / Physics Letters B 345 (1995) 287-290 Table 1 Rates and branching
ratios of K,5 decays, evaluated
from the leading order term in
Decay K+ -
4. Defining
7r+r-n-“e+ve
the Lorentz invariant
CHPT
Decay rate in s-’
Branching
2.4 2.0 2.4 6.5
3.0 2.5 12 33
x x x x
1O-4 10-4 1O-4 1O-4
x x x x
ratio
lo-‘2 1O-12 10-12 lo-‘*
measure
(12) the differential
decay rate is given by
dI- =
C]
(13)
T2d~~ps(p;p,,p,,p,.p2,p3). 1
SPilH
The rates and branching ratios which follow from Eqs. (6) -( 13) are displayed in Table 1. The smallness of the decay rates is due to the suppression of phase space. Indeed, consider the ratio of the four- and five-dimensional phase space volumes
M:,~dLrPs(p;Pe.p~,p,,pz) 2! (27r)‘Z
3! (2%-)‘5
x
SdLIPs(p;pe,pY~p~,p2,p~)
where we have inserted Mi for dimensional sponding rates at tree level in CHPT r(K+
+ 7r07roe+v e ) me
I( Kf -+ n%%‘e+v,)m,
M 3.4 x 106,
reasons.
z 2*3 x lo”
(14)
On the other hand, we find for the ratio of the corre-
(15)
which is of the same order of magnitude. 5. Turning now to the corrections at next-to-leading order, we note that the matrix element of the axial current receives a contribution from the chiral anomaly [3,4]. Besides the local term of the Wess-ZuminoWitten action, also the nonlocal part, which contains at least five meson fields, gives a contribution. However, an explicit calculation shows that it is suppressed by the factor m, in the matrix element and therefore undetectable in the near future. We expect from experience with other calculations in CHPT that the remaining contributions to the matrix element (2) at this order enhance the tree-level results for the decay rates at most by a factor of two to three. 6. Due to isospin symmetry, the relations which are given in Eqs. (6)-(9) are valid to all order in CHFT, with Ap( 1,2,3) symmetric in p1 and ~2, and BN( 1,2,3) totally antisymmetric in ~1, ~2, and p3. The matrix elements of the axial current ?yPysu have an analogous decomposition, with the same symmetry properties of the reduced matrix elements. From this follows the isospin relation
S. Blaser/ Physics Letters B 345 (1995) 287-290
290
7. Hitherto, only poor experimental the upper bound IY(K+ -+ ?r’r’n-‘e+v r total
e
)
data on Kes decays are available.
The Particle Data Group
< 3.5 x 10-6,
[5] quotes
(17)
which is six orders of magnitude bigger than our result. DAQNE, which will produce K* and tiL with an annual rate of 9 x lo9 and 1.1 x 109, respectively [6], may improve the upper bounds for K,5 decays considerably. To summarize, we have evaluated the rates and branching ratios of K,5 decays at leading order in CHPT (IQ. (6)-( 13) ) and given the isospin relation ( 16) for the decay modes. We have furthermore seen that Kp5 decays, in particular any effects from chiral anomaly, will be invisible at DA@NE. However, the upper bounds for the branching ratios can be improved significantly. I thank J&g Gasser for useful discussions
and for a critical reading of the manuscript.
References ] I] [2] [ 31 [4] [ 51 [6]
S. Weinberg, Physica A 96 ( 1979) 327. J. Gasser and H. Leutwyfer, Ann. Phys. 158 (1984) 142; Nucl. Phys. B 250 (1985) 465. J. Wess and B. Zumino, Phys. Lea. B 37 (1971) 95. E. Witten, Nucl. Phys. B 223 (1983) 422. Review of Particle Properties, Phys. Rev. D 50 (1994) 1173. L. Maiani, G. Pancheri and N. Paver, Eds., The DA@NE Physics Handbook (INFN, Frascati, 1992).
PHYSICS
LETTERS B
Physics Letters B 345 ( 1995) 287-290
K,, decays in chiral perturbation theory Stefan Blaser
’
lnstitutflir theoretische Physik, Universitiit Bern, Sidlerstrasse 5. CH-3012 Bern, Switzerland
Received 28 October 1994 Editor: R. Gatto Abstract We evaluate the branching
ratios for the decays K + aaneu
at leading order in chiral perturbation theory and give an
isospin relation for the decay rates. Keywords: Kaon; Semileptonic decay; Chiral perturbation theory; Phase space volume; Chiral anomaly; DA@NE
1. We discuss
the Kes decays
K+ -4
z-+7i--7r”e+Ir et
Ko -
7r07ron--e+v e7
K+ Ko -
r”ro?roe+v e7 77fr-7i-TT-e+u,
in the framework of chiral perturbation theory (CHF’T) [ 1,2]. For low momenta relevant in the present case, the transition amplitude for K ---fmrre+v, reduces in the standard model to the current times current form - ~5)4pe)(V~
i”‘= GFV*%~,)~p(l Jz
us
- AFL),
(1)
where vcL - AN = (‘r(!‘l 2. To calculate order, C = $
)r(k’2h(p3)
out
h’“(
the hadronic matrix elements
tr (d,UPU+
+ XV+ +
1 -
YS)UIK(P)).
VP and Ap, we use the effective Lagrangian
x+U),
(2)
of QCD at leading
(3)
where F = 93.2 MeV is the pion decay constant in the chiral limit. Furthermore, we work in the isospin limit, i.e., nrU = md G riz, and set x = 2Ba diag (rit, P?Z,m,), where Ba is related to the quark condensate in the chiral -’ E-mail address: [email protected]. 0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDfO370-2693(94)01627-5
288
S.Blaser/PhysicsLettersB345(1995)287-290
Fig. 1. Tree level diagrams for Ke5 decays.
limit [2]. The unitary 3 x 3 matrix U incorporates the fields of the eight pseudoscalar convenient parametrization is U = exp( &Q/F) with
Goldstone
bosons.
@f~~_&ZJ. In addition, v4-i5 c1
A
(4)
the vector current relevant in the present case reads =
z
tr ([ A4 - ih5] [ lM,U+ + U+d,Cl]) ,
(5)
where A4 and h5 denote Gell-Mann matrices. The corresponding because it is odd under the transformation Cp + -@. 3. The relevant Feynman
diagrams
(~+(P,>~-(P&r”(Ps)
at leading order in CHIT
out ]V,u]K+(P))
(rococo
out IW,#+(P,)
= -&(1,2,3) =A,(l,2,3)
(~“(P&r-(pz)~o(~s)
out ISrpu(@?P))
(~+(Pt>~-(P~)~-(P3)
out I~y,uI~
axial vector current
does not contribute,
are shown in Fig. 1. Their contribution
(6)
+B,(1,2,3), +A,(1,3,2)
+A,(3,2,
l),
(7) (8)
=fiA,(l,3,2), =-&{A,(1,2,31
is
fA,U,3,2)},
(9)
where
Afi(1.2.3)=g(M2 P~~~~~!p,2[Pl-P*+P3-PlpfMz ~~~~~~~)P)2,pI-p*-p3+P K
I
WC+ 2PlP2) 'P1+P2-P3-p1~+~2_~p,+p2+p3~2[pl+p2+p3+p1~ ?r
P(PI +P2)
-
M2,-(p,+p2-p)2
+21Pl
+p2
-P3lp
8,(1’2’3)=$(M,
:;;;;;~p)2[PI+P2-P3-Plr+M2 I
-P3)
(Pi fP3
(10)
7
>
K
P(PI + M”K -
K
-Pj2
[PI -P2
+P3
K p~~~~~~p)2(p,-p2-p3+plp
-PI,
>
.
(11)
289
S. Blaser / Physics Letters B 345 (1995) 287-290 Table 1 Rates and branching
ratios of K,5 decays, evaluated
from the leading order term in
Decay K+ -
4. Defining
7r+r-n-“e+ve
the Lorentz invariant
CHPT
Decay rate in s-’
Branching
2.4 2.0 2.4 6.5
3.0 2.5 12 33
x x x x
1O-4 10-4 1O-4 1O-4
x x x x
ratio
lo-‘2 1O-12 10-12 lo-‘*
measure
(12) the differential
decay rate is given by
dI- =
C]
(13)
T2d~~ps(p;p,,p,,p,.p2,p3). 1
SPilH
The rates and branching ratios which follow from Eqs. (6) -( 13) are displayed in Table 1. The smallness of the decay rates is due to the suppression of phase space. Indeed, consider the ratio of the four- and five-dimensional phase space volumes
M:,~dLrPs(p;Pe.p~,p,,pz) 2! (27r)‘Z
3! (2%-)‘5
x
SdLIPs(p;pe,pY~p~,p2,p~)
where we have inserted Mi for dimensional sponding rates at tree level in CHPT r(K+
+ 7r07roe+v e ) me
I( Kf -+ n%%‘e+v,)m,
M 3.4 x 106,
reasons.
z 2*3 x lo”
(14)
On the other hand, we find for the ratio of the corre-
(15)
which is of the same order of magnitude. 5. Turning now to the corrections at next-to-leading order, we note that the matrix element of the axial current receives a contribution from the chiral anomaly [3,4]. Besides the local term of the Wess-ZuminoWitten action, also the nonlocal part, which contains at least five meson fields, gives a contribution. However, an explicit calculation shows that it is suppressed by the factor m, in the matrix element and therefore undetectable in the near future. We expect from experience with other calculations in CHPT that the remaining contributions to the matrix element (2) at this order enhance the tree-level results for the decay rates at most by a factor of two to three. 6. Due to isospin symmetry, the relations which are given in Eqs. (6)-(9) are valid to all order in CHFT, with Ap( 1,2,3) symmetric in p1 and ~2, and BN( 1,2,3) totally antisymmetric in ~1, ~2, and p3. The matrix elements of the axial current ?yPysu have an analogous decomposition, with the same symmetry properties of the reduced matrix elements. From this follows the isospin relation
S. Blaser/ Physics Letters B 345 (1995) 287-290
290
7. Hitherto, only poor experimental the upper bound IY(K+ -+ ?r’r’n-‘e+v r total
e
)
data on Kes decays are available.
The Particle Data Group
< 3.5 x 10-6,
[5] quotes
(17)
which is six orders of magnitude bigger than our result. DAQNE, which will produce K* and tiL with an annual rate of 9 x lo9 and 1.1 x 109, respectively [6], may improve the upper bounds for K,5 decays considerably. To summarize, we have evaluated the rates and branching ratios of K,5 decays at leading order in CHPT (IQ. (6)-( 13) ) and given the isospin relation ( 16) for the decay modes. We have furthermore seen that Kp5 decays, in particular any effects from chiral anomaly, will be invisible at DA@NE. However, the upper bounds for the branching ratios can be improved significantly. I thank J&g Gasser for useful discussions
and for a critical reading of the manuscript.
References ] I] [2] [ 31 [4] [ 51 [6]
S. Weinberg, Physica A 96 ( 1979) 327. J. Gasser and H. Leutwyfer, Ann. Phys. 158 (1984) 142; Nucl. Phys. B 250 (1985) 465. J. Wess and B. Zumino, Phys. Lea. B 37 (1971) 95. E. Witten, Nucl. Phys. B 223 (1983) 422. Review of Particle Properties, Phys. Rev. D 50 (1994) 1173. L. Maiani, G. Pancheri and N. Paver, Eds., The DA@NE Physics Handbook (INFN, Frascati, 1992).