arXiv:0711.4618v3 [hep-ph] 16 May 2008
The electromagnetic effects in Ke4decay.
S.R.Gevorkyan∗, A.N.Sissakian, A.V.Tarasov,
May 16, 2008
Joint Institute for Nuclear Research, 141980 Dubna, Russia
The final state interaction of pions in Ke4decay allows to obtain
the value of the isospin and angular momentum zero ππ scattering
0. We take into account the electromagnetic interaction of
pions and isospin symmetry breaking effects caused by different masses
of neutral and charged pions and estimate the impact of these effects
on the procedure of scattering length extraction from Ke4decays.
For many years the decay
was considered as the cleanest method to determine the isospin and angular
momentum zero scattering length a0
predicted by Chiral Perturbation Theory (ChPT) with high precision  and
its measurement with relevant accuracy can provide useful constraints on the
ChPT Lagrangian. The appearance of new precise experimental data [3, 4,
5, 6] requires approaches which can take into account the effects neglected
up to now in extracting the scattering length from experimental data on Ke4
The common way to get the scattering length a0
.At present the value of a0
0from the decay probability
∗On leave of absence from Yerevan Physics Institute
†On leave of absence from Siberian Physical Technical Institute
is based on the classical works [7, 8].The transition amplitude for decay (1)
can be written as the product of the lepton and hadronic currents:
The leptonic part of this matrix element is known exactly,while the hadronic
part can be described by four hadronic form factors1F,G,R,H . Making
the partial-wave expansion of the hadronic current with respect to the angular
momentum of the dipion system the hadronic form factors can be written in
the following form:
G = gpeiδp(s);
= fseiδs(s)+ fpeiδp(s)cosθπ
H = hpeiδp(s)
Here s = M2
the pion in the dipion rest frame measured with respect to the flight direction
of dipion in the K meson rest frame. The coefficients fs,fp,gp,hp can be
parameterized as functions of pion momenta q in the dipion rest system and
of the invariant mass of lepton pair seν in the known way . It is widely
accepted that the s and p- wave phases δs,δpcoincide with the corresponding
phases in elastic ππ scattering (Fermi—Watson theorem
related to the scattering lengths using the set of Roy equations .
Nevertheless the different masses of charged and neutral pions lead to the
isospin symmetry breaking [11, 12, 13] and require the new approach to
connect the phases with scattering lengths.
Another isospin symmetry breaking effect is the electromagnetic interaction
in the dipion system [13, 14, 15], which would has impact on the value of
scattering length extracted from Ke4decay rates. In the present work we
develop the approach allows one to take into account the electromagnetic
interaction in the dipion system and estimates its impact on the value of
scattering lengths extracted from Ke4decay.
ππis the square of dipion invariant mass; θπis the polar angle of
) and can be
2Isospin symmetry breaking due to pions
The s-wave phase shift δshas an impact only on axial form factor F, whereas
the axial form factors G and vector form factor H depend only on p-wave
1The form factor R is proportional to the electron mass and thus it can not be extracted
phase shift δp. If one confins by s and p waves, the inelastic process π0π0→
π+π−and the reversed one are forbidden due to identity of neutral pions in
l=1 state. Thus inelastic transitions can change only the first term in the
form factor F, relevant to production of pions in s-wave.
In one loop approximation of nonperturbative effective field theory (see e.g.
) the decay amplitude relevant to dipion in the state with I=l=0 reads:
T = T1(1 + ikcac(s)) + iknax(s)T2.(4)
Here T1,T2are the so called “unperturbed” amplitudes [17, 18] corresponding
to the decays with charged and neutral dipions in the final state.
are the pion momenta in the π0π0and π+π−
systems with the same invariant mass s= M2
are relevant to elastic scattering π+π−→ π+π−and charge exchange reaction
In the case of isospin symmetry they can be expressed through the s-wave
”amplitudes” with certain isospin a0(s),a2(s),which at threshold are equal to
relevant scattering lengths a0
we adopt the relations followed from ChPT :
ππ. The real functions ac(s),ax(s)
0. In the case of isospin symmetry breaking
2a0(s) + a2(s)
3(a0(s) − a2)(1 +η
(1 + η);
In the isospin symmetry limit (kc= kn= k;η = 0) a simple relation takes
place between the “unperturbed” amplitudes T1=√2T2, which follows from
the rule ∆I = 1/2 for semi-leptonic decays. In this limit it is easy to obtain:
T = T1(1 + ika0(s)) = T1
1 + k2a0(s)2eiδ0
This equation is nothing else than the Fermi-Watson theorem  for the ππ
interaction in the final states.
In the general case using the expressions (4),(5) and relations between the
s-wave ”amplitudes” and relevant phases :
tanδs(s) = kcac(s);tanδ0
after a bit algebra it is easy to obtain:
δs = arctan(Astanδ0
2(1 + η) + λ(1 +η
;Bs=(1 + η) − λ(1 +η
Another isospin breaking effect which can be important in the procedure of
the scattering lengths extraction from the experimental data on Ke4decay,
is the Coulomb interaction between the charged pions [13, 14, 15] The widely
spread wisdom is that in order to take this effect into account it is sufficient
to multiply the square of matrix element (2) by Gamov factor
1 − e−2πξ;ξ =α
√1 − 4β
1 − 2β;β =2kc
Here υ is the relative velocity in the dipion system and α =
Later on we show that besides this multiplier the electromagnetic interaction
between pions also change the expression (8) for the strong phase and add
the proper Coulomb phase.
4πis the fine
3 Electromagnetic interaction in ππ system
In order to take into account the electromagnetic interactions between pions,
we take an advantage of the trick successfully used in
the electromagnetic interaction, we replace the charged pion momenta kcin
(7) by a logarithmic derivative of the pion wave function in the Coulomb
potential at the boundary of the strong field r0:
.To switch on
ikc→ τ =dlog[G0(kr) + iF0(kr)]
Here F0,G0are the regular and irregular solutions of the Coulomb problem.
In the region kr0≪ 1, where the electromagnetic effects are significant, this
expression can be simplified:
τ= ik − αm[log(−2ikr0) + 2γ + ψ(1 − iξ)] = Re τ + i Im τ
= −αm[log(2kr0) + 2γ + Reψ(1 − iξ)];
Here γ = 0.5772 is Euler constant and ψ(z) =
Using the above relations one can express the modified phase for π+π−state
(I=l=0) through the known  phases δ0
Representing the modified s-wave phase as a sum of strong δstrand electro-
magnetic δemterms, we obtain:
˜δs = δstr+ δem
Bem=G(1 + η) − λ(1 +η
2G(1 + η) + λ(1 +η
Let us note that, whereas the electromagnetic phase δemhas a common text-
book form , the strong phase is essentially modified by electromagnetic
effects (the Gamov factor G in δstr) as well as by isospin symmetry breaking
effects provided by pions mass difference.
Using the same approach one can show that the modified p-wave phase reads:
differences δ =˜δs−˜δpas a function of the invariant mass of dipion Mππ.
The dashed line on Fig.1 corresponds to exact isospin symmetry limit m0=
mc;α = 0. The solid line gives the dependence of modified phases differ-
ence accounting for all isospin breaking effects. The experimental data are
The considered above isospin breaking effects change remarkably δ and would
have impact on the values of scattering lengths extracted from experimental
In table 1 we cite δ as a function of dipion invariant mass Mππin respect to
different isospin breaking corrections. This allows one to estimate separately
the contribution of considered above effects.
1from Appendix D of , we calculated the modified phases
0= −0.03706mc−1and using the relevant
Table 1: The impact of considered corrections on phase difference δ = δs−δp:
1) Standard case  with a0
charge exchange process λ =
With electromagnetic interaction. 5) With the additional Coulomb phase
kc3) With parameter η (expression (5)) 4)
0= −0.03706mc−12) With
The isospin symmetry breaking corrections considered above increase the
phase difference δ. Their contribution is the largest near the threshold, but
they are essential even far from it.
The Ke4decay amplitude in the real world with isospin symmetry breaking
depends on two scattering lengths a0
proposed approach allows one to extract the values of scattering lengths with
higher accuracy than in standard approximation.
The authors thank V.D. Kekelidze and D.T. Madigozhin for useful discus-
sions and support. We are grateful to J.Gasser and A.Rusetsky for useful
comments and friendly criticism, which assist to improve the present work
and our understanding of considered above problems.
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Figure 1: The dependence of phases difference δ = δs− δp
on dipion invariant mass in the exact isospin symmetry
case (dashed line) and with all isospin symmetry breaking
corrections taken into account (solid line).