Decay Spectrum of $K^+ \to e^+ ν_{e}γ$

0.18 MB·

by  C. H. Chen

February 2, 2008 16:46 WSPC/INSTRUCTION FILE Lih-ICFP07
International JournalofModernPhysicsA
(cid:13)c WorldScientificPublishingCompany
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0
0 DECAY SPECTRUM OF K+ → e+νeγ
2

n
a
J C.H.CHEN1,C.Q.GENG2 andC.C.LIH3

7 1Department of Physics, National Cheng-Kung University,Tainan 701, Taiwan
2Department of Physics, National Tsing-HuaUniversity,Hsinchu 300, Taiwan

] 3General Education Center,Tzu-Chi College of Technology, Hualien 970, Taiwan
h
p
- TheformfactorsoftheK+→γtransitionarestudiedinthelight-frontquarkmodeland
p
e chiral perturbation theory of O(p6). The decay spectrum of K+ → e+νeγ, dominated
h bythestructuredependent contribution, isillustratedinbothmodels.
[

1 It is known that the decay of K+ e+ν γ receives two types of contributions:
e
v “inner bremsstrahlung” (IB) and “st→ructure-dependent” (SD) 1,2. The former is
4
7 helicity suppressed and contains the electromagnetic coupling constant α, while
0 the latter gives the dominant contribution to the decay rate as it is free of the
1
helicity suppression. In the standard model (SM), the decay amplitude of the SD
.
1 partinvolves vectorand axial-vectorhadronic currents,which canbe parametrized
0
interms ofthe vectorformfactor F andaxial-vectorformfactorF ,respectively.
8 V A
0 However, the experimental determinations on these form factors are poorly given
: andmodel-dependent3,4,5.Inparticular,theexperimentalresultsonthedecayrate
v
i ofK+ e+νeγinRef.3,4,5werebasedontheassumptionofFV andFAbeingsome
X constan→tvaluesinthechiralperturbationtheory(ChPT)atO(p4)6.Intheongoing
r
a data analysisofthe E949experimentatBNL, moreprecisionmeasurementsonthe
decay of K+ e+ν γ are expected 7 andthus, the model-independent extractions
e
→
of the SD form factors are possible. Theoretical calculations of F and F in the
V A
K+ γ transition have been previously done in the ChPT at O(p4) 6 and O(p6)
8,9 a→s ell as the light-frontquark model (LFQM) 10. However,the results have not
been fully applied to the decay of K+ e+ν γ yet.
e
→ 11
In this talk, we will present our recent results on the transition form factors
of K+ γ in the ChPT of O(p6) and light-front quark model (LFQM). We will
→
show the spectrum of the differential decay branching ratio of K+ e+ν γ as a
e
→
function of x=2E /m .
γ K
WestartwiththeamplitudeofthedecayK+ e+ν γintheSM,givenby2,6,12
e
→
M = M +M ,
IB SD
G G
M = ie Fsinθ F m ǫ∗Kα, M = ie Fsinθ ǫ∗L Hµν, (1)
IB √2 c K e α SD − √2 c µ ν
where Kα = u¯(p )(1 + γ ) pαK 2pαe+q6γα v(p ), L = u¯(p )γ (1 γ )v(p ),
ν 5 pK·q − 2pe·q e ν ν ν − 5 e
(cid:16) (cid:17)
1
February 2, 2008 16:46 WSPC/INSTRUCTION FILE Lih-ICFP07
2 Chen, Geng and Lih
Hµν = FA ( gµνp q+pµqν)+iFV ǫµναβq p ǫ is the photon polarization
mK − K · K mK α Kβ α
vector, p , p , p , and q are the four-momenta of K+, ν , e+, and γ, and F and
K ν e e K
F are the K meson decay constant and the axial-vector (vector) form factor
A(V)
corresponding to the axial-vector (vector) part of the weak currents, respectively,
defined by
F
0s¯γµγ uK+(p ) = iF pµ, γ(q)u¯γµsK(p ) =ie V εµαβνǫ q p ,
h | 5 | K i − K K h | | K i m ∗α β ν
K
F
γ(q)u¯γµγ sK(p ) =e A [(p q)ǫ∗µ (ǫ∗ p)qµ], (2)
5 K
h | | i m · − ·
K
with p = p q being the transfer momentum. We note that in Eq. (1)
K IB
− M
is suppressed due to the small electron mass m . In the decay of K+ e+ν γ,
e e
→
the form factors F in Eq. (2) are the analytic functions of p2 = (p q)2 in
A,V K
−
the physical allowed region of m2 p2 m2 . The relation between the transfer
e ≤ ≤ K
momentum p2 and x is given by p2 =m2 (1 x).
At O(p6) in the ChPT, one obtains tKhat−11
m 256 64
F (p2)= K 1 π2m2 Cr+256π2(m2 m2)Cr + π2p2Cr
V 4√2π2F − 3 K 7 K − π 11 3 22
K(cid:26)
1 3 m2 7 m2 m2
m2ln η + m2 ln π +3m2 ln K
−16π2(√2FK)2(cid:20)2 η µ2 ! 2 π (cid:18)µ2 (cid:19) K (cid:18) µ2 (cid:19)
xm2 +(1 x)m2 x(1 x)p2
2 xm2 +(1 x)m2 x(1 x)p2 ln π − K − − dx
− π − K − − µ2
Z (cid:18) (cid:19)
(cid:2) (cid:3) xm2 +(1 x)m2 x(1 x)p2
2 xm2 +(1 x)m2 x(1 x)p2 ln η − K − − dx
− η − K − − µ2 !
Z
(cid:2) (cid:3)
m2
4 m2 ln π dx , (3)
− π µ2
Z (cid:18) (cid:19) (cid:21)(cid:27)
4√2m m
F (p2)= K(Lr+Lr )+ K [142.65(m2 p2) 198.3]
A F 9 10 6F3(2π)8 K − −
K K
m m2 m2
K (4Lr+7Lr+7Lr )m2 ln π +3(Lr+Lr )m2ln η
−4√2FK3π2 ( 3 9 10 π (cid:18)m2ρ(cid:19) 9 10 η m2ρ!
m2
+2(8Lr 4Lr+4Lr+7Lr+7Lr )m2 ln K
1− 2 3 9 10 K m2
(cid:18) ρ (cid:19)(cid:27)
4√2m
K 2m2(18yr 2yr 6yr +2yr +3yr yr +6yr )
− 3F3 π 18− 81− 82 83 84− 85 103
K
+2m2 (18(cid:8)yr +36yr 4yr 12yr +4yr +6yr +4yr 3yr
K 17 18− 81− 82 83 84 85− 100
3
+6yr +12yr 6yr +3yr )+ (m2 p2)(2yr 4yr +yr ) , (4)
102 103− 104 109 2 K − 100− 109 110
(cid:27)
where Cr, Lr and yr are the renormalized coupling constants. Note that the first
i i i
terms in Eqs. (3) and (4) correspond to F and F at O(p4) 6, respectively.
V A
February 2, 2008 16:46 WSPC/INSTRUCTION FILE Lih-ICFP07
Decay Spectrum of K+→e+νeγ 3
10 11
In the framework of the LFQM , we obtain
dzd2k 1 2m Ak2Θ 1m +Bk2Θ
F (p2)= 4m ⊥Φ z′,k2 u− ⊥ + s ⊥ ,
A K 2(2π)3 ⊥ 1 z′ 3 m2 +k2 3 m2+k2
Z − (cid:26) u ⊥ s ⊥ (cid:27)
dzd2k (cid:0) (cid:1) 1
F (p2)= 8m ⊥Φ z′,k2
V K 2(2π)3 ⊥ 1 z′
Z −
2m z′(m (cid:0)m )k2(cid:1)Θ 1m +(1 z′)(m m )k2Θ
u− s− u ⊥ s − s− u ⊥ , (5)
3 m2 +k2 − 3 m2+k2
(cid:26) u ⊥ s ⊥ (cid:27)
11
where the parameters and variables are defined in Ref. .
The numerical values of F (p2) in the ChPT of O(p6) are plotted in Fig. 1.
A,V
Inthesefigures,wehavealsoincludedthe results inthe ChPTatO(p4).Explicitly,
we find that F (p2 = 0) = 0.0945 (0.0425), 0.082 (0.034) and 0.106 (0.036) in
V(A)
the ChPT at O(p4), ChPT at O(p4) and LFQM, respectively.
The differential decay rate as a function of x is given by
dΓ m5
= K αG2 sin2θ A(x) (6)
dx 64π2 F c
11
where the function of A(x) is given in Ref. . By integrating out the variable x
in Eq. (6), in Table 1 we give the decay branching ratio of K+ e+ν γ. Here, as
e
→
the IB term diverges at the limit of x 0 corresponding to p2 p2 = m2 , we
→ → max K
have used the cuts of x = 0.01 and 0.1, respectively. With the cuts, from Table 1
we see that the IB contributions are much smaller than the SD± ones, which are
insensitivetothecut.InFig.2,wealsodisplaythespectrumofthedifferentialdecay
branching ratio in the ChPT at both O(p4) and O(p6) and the LFQM. From Fig.
2,we see that in the regionof x<0.7 or E <173 MeV, the decay branchingratio
γ
in the LFQMis much samllerthan that in the ChPT atO(p6). On the other hand,
in the region of x > 0.7 the statement is reversed. However, if we only consider
the contributions in the ChPT at O(p4), the conclusion is weaker. It is clear in
7
the future data analysis such as the one at the experiment BNL-E949 , one could
concentrate on these two regions to find out which model is preferred.
We have studied the axial-vector and vector form factors of the K+ γ tran-
→
sition in the LFQM and ChPT of O(p6). Based on these form factors, we have
Fig.1. FV,A(p2)asfunctions ofthetransfermomentump2.
February 2, 2008 16:46 WSPC/INSTRUCTION FILE Lih-ICFP07
4 Chen, Geng and Lih
Table1. ThedecaybranchingratioofK+→e+νeγ (inunitsof10−5).
Model xCut IB SD+ SD− Total
ChPTatO(p4) 0.01 1.65×10−1 1.34 1.93×10−1 1.70
0.1 0.69×10−1 1.34 1.93×10−1 1.60
ChPTatO(p6) 0.01 1.65×10−1 1.15 2.58×10−1 1.57
0.1 0.69×10−1 1.15 2.58×10−1 1.47
LFQM 0.01 1.65×10−1 1.12 2.59×10−1 1.54
0.1 0.69×10−1 1.12 2.59×10−1 1.44
calculated the decay branching ratio of K+ e+ν γ. We have demonstrated that
e
→
the SD parts give the dominant contributions to the decay in the whole allowed
regionofthephotonenergyexceptthe lowendpoint.Future precisionexperimental
7
measurements on the decay spectrum should give us some useful information to
determine the SD contributions as well as the vector and axial-vectorform factors.
References
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40 4
ChPT O(p)
35 ChPT O(p6)
) / dx 30 LFQM
e 25
610 d Br ( K 112050
5
0
0.0 0.2 0.4 0.6 0.8 1.0
x (2E / mK)
Fig.2. Thedifferentialdecay branchingratioasafunctionofx=2Eγ/mK.