CPT-2001/P.4228

Isospin Breaking Corrections to Low-Energy Scattering

[40pt]
A. Nehme^{*}^{*}* and
P. Talavera

Centre de Physique Théorique, CNRS–Luminy, Case 907

F-13288 Marseille Cedex 9, France.

[10pt]

Abstract

We evaluate the matrix elements
for the processes
and
in the presence of isospin breaking terms at leading and next-to-leading
order. As a direct application the relevant combination of the
S-wave scattering lengths involved in the pion-kaon atom lifetime is determined.
We discuss the sensitivity of the
results with respect to the input parameters.

PACS: 14.40 Aq; 13.40 Ks; 13.75 Lb; 12.39 Fe

Keywords : Electromagnetic correction; Threshold parameters;
Pion Kaon scattering; Chiral perturbation theory.

###### Contents

## 1 Introduction

Chiral perturbation theory [1] has become one of the most
used tools in exploring QCD low-energy dynamics. It applies in the
non-perturbative
regime of QCD.
The initial QCD lagrangean, ,
is then replaced by an
effective one which contains the same
symmetries as the fundamental ,
composed by a string of higher and higher
dimension operators involving derivatives and quark masses.
Technically, the new lagrangean is not renormalizable
(in the Wilsonian sense) but
fortunately at a given order in the momenta (quark mass) expansion
the number of needed counter-terms is finite. Thus *assuming* that
the chiral series converges [2] one can truncate it at a given order and deal
with a finite number of unknown constants. Restricting the analysis to the
next-to-leading order in the mesonic sector (see below) one can obtain
the unknown constants
from the existing data and large-N arguments. To this respect
one makes use of the experimental knowledge on the
pseudoscalar masses and decay constants, pion vector form-factor
and
K form-factors. Therefore none of those data
informations can be used to claim any theoretical *predictability*.
To exhibit the consistency of the theory
one has to use other processes where the low-energy
constants are given as mere inputs and confront the
theoretical
results with the experimental data.
To this aim the scattering lengths have deserved a careful
examination [3] but unfortunately they only bring information about
the SU(2) sector. In line with the previous general argument,
scattering
stems for the simplest meson-meson scattering process that involves
strangeness
and can be used as an independent
source of information on the validity of the extra assumptions that common
wisdom
assesses to hold in chiral perturbation theory,
as for instance large-N arguments.
This will, hopefully, bring some insight on the role of the strange
quark mass inside the chiral expansion.
Recently it has been noticed that some observables depend
strongly on the number of light sea quarks [17].
This fact can cast doubts on the validity of the chiral expansion
in the SU(3) sector where
m is treated as a small parameter.
Before any judgment is taken
it is necessary to make more accurate experiments and precision calculations
on *testing processes*. In that sense the next proposal for
the measurement of the lifetime and lamb-shift in atoms
() at CERN
constitutes a major step from the experimental side [6].
Experiments on this
system will constitute one of the most stringent test on chiral symmetry
breaking existing up to the moment. Even though we
want to stress that our treatment will not allow to deal with bound states, and a more
refined analysis in the line of the one performed for the atom is needed
[7, 8]. In a very brief way
the lifetime ()
of the A atom,
is given in terms of [9]

Thus any theoretical insight on the shift for the scattering lengths due to the isospin breaking terms constitutes a key role for a real estimate of A atom lifetime.

In this paper our aim is
to incorporate some of the theoretical effects that were not
taken into account in a previous analysis [5].
We shall deal first with the
more *academic* process where there is no
presence of an explicit
virtual soft-photon, but electromagnetic effects will appear in the
expressions of the scattering lengths as differences
of charged and neutral pseudoscalar masses.
We proceed the analysis considering the
transition where
in addition explicit exchange of virtual photons should be considered.

The paper is organized as follows: in sec. 2 we review briefly the inclusion of electromagnetic corrections inside the framework of effective lagrangeans, emphasizing the role of the low-energy constants. We continue with the isospin decomposition for the scattering amplitudes, analyzing the scattering lengths at leading order in the isospin limit and comparing them with the isospin breaking corrections in sec. 3. Next, in sec. 4 we proceed with the analysis of the process at next-to-leading order, reviewing first the kinematics and carefully explaining how to deal with isospin breaking effects, strong and electromagnetic ones, in order to have an expression consistent with the chiral power counting. In sec. 5 we turn to the evaluation of the experimental mode emphasizing the soft-photon contribution, the extraction of the Coulomb pole at threshold and the proper definition of observables once isospin breaking corrections are taken into account. In a more technical section, sec. 6, we explain how to perform the threshold expansion of the non-Coulomb part of the scattering amplitude. In sec. 7 we review the experimental and theoretical status of the S-wave scattering lengths and we present our results discussing them in terms of all input parameters. We make use of our findings to determine the lifetime of in sec. 8. Sec. 9 summarizes our results. Finally, for not interrupting the discussion we have collected in the appendices all relevant expressions.

## 2 The effective lagrangean to lowest order

This section covers briefly the inclusion of electromagnetic corrections in a systematic way in the low-energy theory describing hadron interactions [11, 12]. Due to the smallness of the electromagnetic constant, , these effects have been theoretically neglected so far in the scattering process, but is well known that near threshold isospin breaking effects can enhance considerably some observables.

In presence of electromagnetism it is convenient to split the lowest order effective lagrangean in three terms

(2.1) |

The foregoing lagrangean possesses the same symmetry restrictions as
the one in the strong sector. Additionally one has to impose an *extra* symmetry,
charge conjugation, affecting only the photon fields and the spurions (see
below). The first term in eq. (2.1) corresponds to the usual Maxwell
lagrangean containing the classical
photon field, , and the field strength
tensor,

(2.2) |

The second term formally describes the dynamics of the strong interaction sector [14] and is given at lowest order by

(2.3) |

As usual brackets, , stand for trace over flavour. The field , parametrises the dynamics of the low-energy modes in terms of elements of the Cartan subalgebra [15]. The covariant derivative is slightly modified with respect to the pure QCD interaction expression to accommodate the electromagnetic field

(2.4) |

and are the aforementioned spurions fields, containing the sources for the electromagnetic operators and . Furthermore, from now on we set them to their constant value

(2.5) |

While, as usual, and stand for the axial and vector sources respectively. The scalar and pseudoscalar sources are contained inside the SU(3) matrix as

(2.6) |

At lowest order the last term in eq. (2.1), , determines the masses of the mesons in the chiral limit, which are of purely electromagnetic origin. This means that even at tree level the pole of the two-point Green function is shifted from its QCD value modifying therefore the kinematics of the low-energy region

(2.7) |

Obviously the coupling is taken to be universal, thus we expect the same contribution to the masses for charged pions and kaons. Furthermore, for simplification, the results will be presented in terms of

(2.8) |

The inclusion of isospin breaking
terms can be seen in a very naïve way as coming through two different
sources in eq. (2.1). First a pure *strong* isospin breaking,
i.e. . And second an *electromagnetic*
interaction, , eq. (2.7). In view of the seemingly different role of both
contributions one has to relate them in a consistent way. For instance, the
lagrangean eq. (2.1) involves an expansion in several parameters: and . Where refers to the external momenta, to the
quark-masses, to the electric charge and finally . For
being consistent, eq. (2.1) should contain operators with the same
chiral order in the series expansion, thus a possible solution can be the
choice . At the next-to-leading order,
, none of the following terms are impeded to appear by
chiral power counting: , , , , , ,
, , and , although there is quantitative
support to the assumption, often used in phenomenological discussions, that
the and contributions are tiny and can therefore be
safely disregarded in front of the rest.

Hitherto we have listed all possible electromagnetic, lowest
order operators. Once quantum fluctuations are considered using vertices from
the functional (2.1) results are ultraviolet divergent. Those
divergences depend of the regularisation method employed in the loops
diagrams. As is customary we shall adopt the
modified
subtraction scheme. In order to remove these
ultraviolet divergences, higher order operators, modulated by simple constants,
should be incorporated into the theory with the guidance of the previously
mentioned symmetry requirements. This allows to deal with a theory which is
ultraviolet finite order by order in the parameter expansion and hopefully it
adequately describes several observables.

These modulated constants are order parameters of the effective theory
(low-energy constants) and in the case at hand are determined by the underlying
low-energy dynamics of QCD and QED. For instance at lowest order the order
parameters are given by (eq. (2.3)), (eq. (2.6)) and
(eq. (2.7)). Describing the lowest pseudoscalar decay constant, the
vacuum condensate parameter and the electromagnetic pion mass in the
chiral limit respectively. For the strong SU(3) sector, at next chiral
order, there are ten new low-energy constants [14],
and two *high*-energy constants. At this level of
accuracy the ten low-energy constants may be extracted almost independently
one from each other by matching some observables with the
corresponding experimental determination.
In
the electromagnetic SU(3) sector there is also need of
higher order operators, up to modulated via , to
render any observable free of ultraviolet divergences.
To gain some information
on these low-energy constants
one has to resort to models [16], to sum-rules
[17] or to a simple crude order estimates.
All in all, one can see that the inclusion of electromagnetic corrections to
hadronic processes increases enormously the number of low-energy constants thus
washing out any predictability.

## 3 Isospin breaking corrections at leading order

Under strong interactions symmetry
pions are assigned to a triplet of states,
“*isotriplet* states”,
whereas kaons can be collected into doublets. This means that the
pion and the kaon are isospin eigenstates with
eigenvalues and respectively. Therefore, the amplitudes for the
scattering processes are solely described in terms of two independent
isospin-eigenstates amplitudes and

(3.1) |

It is obvious that under crossing the last two matrix elements are related. In particular one finds

(3.2) |

thus in the isospin limit it is sufficient to compute one of the processes. It is convenient to use, instead of the invariant amplitudes , the partial wave amplitudes defined in the -channel by

(3.3) |

or by the inverse expression

(3.4) |

where is the total angular momentum, the scattering angle in the center of mass frame and are the Legendre polynomials with . Near threshold the partial wave amplitudes can be parametrized in terms of the scattering lengths, and slope parameters, . In the normalization (3.3) the real part of the partial wave amplitude reads

(3.5) |

with being the center of mass three-momentum.

Let us estimate the scattering lengths in the isospin limit at tree level.
Using for instance the first two processes in (3) one can disentangle
the values of each scattering length separately.
In order to match the prescription of the scattering lengths
given in [18] we define them in terms of the
charged pion and kaon masses. They read^{†}^{†}†We use MeV. See Sec. (7)
for the rest of values.

(3.6) |

where the first quoted number refers to the choice while the
second is for ^{‡}^{‡}‡In the sequel we shall use
.

When switching-on the isospin-breaking effects, we are not allowed anymore to refer to the scattering lengths in a given isospin-state and new terms in addition to the previous ones arise. Furthermore in principle relations as (3.2) do not hold anymore. This failure can be seen already at tree level in and terms. The modified scattering lengths in presence of isospin breaking can be split as

where and denote the strong (isospin limit) S-wave scattering lengths and represents the leading correction to the corresponding combination of scattering lengths due to the isospin breaking effects. The evaluation of these corrections is straitforward and can be read from the scattering amplitude at leading order

with Notice that whereas in the isospin limit the combination for the scattering lengths cancels in the process this is no longer true when isospin breaking terms are considered. In the process the isospin breaking in the combination of the scattering lengths is roughly a couple of orders of magnitude smaller than the leading isospin limit quantity. Even though the future experimental bounds on the combination of scattering lengths for this process is quite restrictive being worthwhile to control higher order corrections. For the rest of processes the isospin effects are also rather small, roughly two order of magnitude less than the isospin limit counter parts.

## 4 process

Let us start considering the following process

(4.1) |

Our aim in this section will be to compute its amplitude taking into account all possible isospin breaking effects. Even if the process has not the same experimental interest as the reaction it is worth to be considered because both processes share almost the same features and complications, exception of the one photon exchange contribution (see sec. 5.1).

### 4.1 Kinematics

The amplitude for the process, eq. (4.1), can be studied on general grounds in terms of the Mandelstam variables

(4.2) |

In the isospin limit and at lowest order of perturbation theory (corresponding to the PCAC results), see diagram (a) in fig. 1, the off-shell amplitude is given by [18, 19]

(4.3) |

It is worth to review briefly the kinematics of the process that will be needed subsequently. In the center of mass frame the Mandelstam variables are defined in terms of and by

### 4.2 General framework

Hitherto we have considered the process eq. (4.1) at leading order. In this section we shall sketch the role of isospin breaking at next-to-leading order. Indeed, as has been mentioned in the introduction, the isospin violating terms that are retained in the scattering differ from the ones in the reaction due to the inclusion of the s-quark. The later process can be described fully in terms of SU(2) quantities where, for instance, the difference between the charged and neutral pion masses are order and thus can be disregarded, whereas in the former there exist intermediate strangeness states like that give contributions of order and .

To be consistent with the chiral power counting one has to take into account all possible scenarios. For instance, given a generic reaction mediated via the diagram (b) in fig. 1 there are two different possibilities of incorporating isospin violating terms : (i) consider that one of the vertices breaks isospin through the terms or via the quark-mass difference, thus at the order we shall work the other vertex and the two propagators are taken in the isospin limit, or (ii) if both vertices are taken in the isospin limit that forces to consider the splitting between charged and neutral masses (in a given channel) in the same triplet for the pions or in the doublet for the kaons in the propagators. Thus within this prescription we shall consider in the chiral series terms up to including and corrections besides the usual at next-to-leading order.

As a matter of fact, the previous distinction, disentangling *strong* and
*electromagnetic* contributions to the isospin breaking terms, is quite
artificial as one can realize when the pseudoscalar masses are
rewritten in terms of bare quantities. Even though it constitutes
a great conceptual help because ultraviolet divergences involving and
terms do not mix at this order allowing to keep track of each
term independently.

For the case we are interested in, involving only neutral particles, there is no direct contribution from virtual photon loops, therefore the amplitude is safe of infrared singularities and all dependence is due to the e.m. mass difference of mesons or by the integration of hard photon loops. Hence to obtain the amplitude including all corrections one needs to restrict the evaluation to the one-particle-irreducible diagrams depicted in fig. 1 corresponding to the Born amplitude, (a), unitary contribution, (b), tadpole, (c) and finally the counter-term piece (d). For the precise expressions of this last contribution we refer the reader to the original literature [14, 11].

To ascertain the correctness of our expression we look at the scale independence of the result once all contributions of the one-particle-irreducible diagrams are added, the wave function renormalization for the external field are taken into account and the mixing is treated correctly (see below). Furthermore, when restricting the expressions to the isospin limit we recover the results given in [5].

Once the amplitude is finite in terms of bare quantities we have to renormalize the coupling constant, , and the masses appearing at lowest order. For the latter contribution we obtain agreement with the results quoted in [14] for the terms up to including corrections and with [11] for the electromagnetic ones. While for the former we shall use two choices: the first one is to fully renormalize as and for comparison purposes as the combination . To this end we use the isospin limit quantities [14]

(4.4) |

and

(4.5) |

Being the finite part of the well-known tadpole integral. We refrain to use in the numerical estimates of because experimentally is quite poorly known and instead we shall make use of the charged decay constant value.

The latest contribution enters through the mixing. At lowest order the mixing angle is given by

(4.6) |

Notice that given the order of accuracy we are considering, does not suffice and the next-to-leading order contribution, , to the mixing angle needs to be considered. We shall use the same approach as in [22] where we refer for a more detailed explanation. It consists essentially in diagonalyzing the mixing matrix at the lowest order redefining in that way the and fields. While higher order terms in the mixing are treated by direct computation of the S-matrix off-diagonal elements. This procedure is equivalent to the one outlined in [23].

Taking into account all mentioned contributions we obtain the renormalized S-matrix element, , for the transition that is gathered in app. A where we refer the reader for a detailed exposition.

## 5 process

In this section we shall consider the more relevant process

(5.1) |

with the following Mandelstam variables

(5.2) |

In the center of mass frame these variables read

(5.3) |

with

(5.4) |

and being the three-momentum of the charged (neutral) particles.

As has been pointed out earlier, the relevance of this process is intimately related to the lifetime of the system. Even though we want to stress that our formalism only allows us to deal with free, on-shell external particles, in clear contrast with the atom where the states are bounded and off-shell [7, 8]. We shall not pursue here the more complete approach.

For the construction of most of the graphs (those equivalent to fig. 1) we shall use the same arguments presented in the preceding section. For the remaining ones we shall sketch their treatment in the next section.

### 5.1 Soft photon contribution

In the case of the process, besides the corrections due to the mass difference of the up and down quarks and those generated by the integration of hard photons one has to consider corrections due to virtual photons. At order these corrections arise from the wave function renormalization of the charged particles just as from the one-photon exchange diagrams depicted in fig. 2. The result of diagram (a) in this figure reduces at threshold to combinations of polynomials and logarithms. Whereas the second, (b), needs a closer consideration. It develops at threshold (see below) a singular behaviour. This singularity is issued from the ultraviolet finite three-point function defined by

(5.5) |

with the on-shell conditions

(5.6) |

where

and finally the dilogarithm function is defined as

The contribution of the function via the diagram (b) in fig. 2 to the amplitude is [c.f. eq. (B.1)]

Expanding the real part of the preceding function in the vicinity of the threshold by the use of eqs. (5.6) and (5.4) one obtains a Coulomb type behaviour, i.e. . Then schematically the threshold expansion of the real part of the amplitude takes the following form

(5.7) |

with

(5.8) |

being the reduced mass of the system.

A remarkable feature of the threshold expansion of the scattering amplitude is that neither nor the long-range force of the photon exchange is affected by the infrared singularity, which only contributes to or to higher orders terms. In fact, the scattering amplitude contains two infrared divergent contributions. The first one comes from the wave function renormalization of the charged particles and can be read from the Born-type amplitude (B.3). The second infrared divergent piece is due to diagram (b) in fig. 2 and is contained in the function (5.6). Adding both contributions one has the following infrared piece for the real part of the amplitude

As is expected when evaluated at threshold, this expression vanishes rendering as an infrared finite quantity. Although the scattering lengths defined in this way will be infrared finite the slope parameters will not. In order to define an infrared finite observable one should notice that the cancellation of infrared divergences takes place order by order in . This requires that besides the virtual photon corrections, to take into account real soft photon emission from the external particles. Notice that the experimental result will include this bremsstrahlung effect. In our case is just sufficient to consider one single photon emission, which amplitude reads

with and being the momenta of the photon and its polarization vector respectively. As a result, one can write the infrared finite cross-section including all but neglecting terms as

(5.9) |

where corresponds to the detector resolution. Once this is done, the corrected scattering length might be defined from the threshold expansion of the infrared finite cross-section, eq. (5.9), by subtracting the Coulomb pole term and excluding the corrections due to the mass squared differences in the phase-space in the following way [25]

(5.10) |

We have checked that the corrections due to the real, soft photon emission are negligible and thus we expect that the corrected scattering length will only differ beyond our accuracy from the one obtained using eq. (5.7), that is, from the infrared finite real part of the scattering amplitude at threshold.

## 6 Threshold expansion

Let us explain how we obtain the scattering lengths from the scale invariant amplitudes eqs. (A) and (B.1). Since we are only interested in the S-wave threshold parameters it is just sufficient to expand the scattering amplitude around the threshold values. Even though, this step is not quite straightforward because in order to match the prescription given in [18] for the scattering lengths we need to shift isospin limit and neutral masses to the corresponding charged ones. The procedure is rather cumbersome and we shall use the following substitutions for the masses

(6.1) |

where are small quantities that in the case of pions contain at leading order pieces while for kaons contain both and terms. It is therefore sufficient to expand all quantities up to first order in . For instance in the charged neutral transition we obtain for the kinematical variables

(6.2) |

Once this step is performed and in order to book-keep properly the power counting any charged mass multiplying or pieces is settled to its isospin limit

(6.3) |

Only for estimating higher order corrections we shall eventually keep charged masses in the ratios introduced in eq. (6).

Even if the outlined procedure for shifting the neutral masses is rather involved it allows to expand all one loop integrals in an analytic form with quite compact expressions. For instance in the s-channel for the neutral neutral transition we obtain after performing the mentioned steps the following expansion

For loop functions involving the mass the use of Gell-Mann–Okubo relation reduces its expression considerably.

In the neutral neutral process, although the kinematics allows at threshold, there is no need to consider the expansion of the function in powers of . This is the case because all channels contributions behave like without any inverse power of kinematical variables ( in this case). This does not turn out to be the case in the charged neutral transition. There, one deals with terms like , where in the isospin limit . Expanding near threshold we shall obtain contributions from the terms linear in . Besides this last remark there is no more differences in the treatment of the two processes.

## 7 Results and discussion

Let us first point out that due to the hight threshold of production, MeV, it is not necessary true that a single one-loop calculation is enough to approach the physical values for the scattering lengths. Even though and due to the fuzzy existing data we consider that this can only be answered once the size of the next-to-next–to-leading order is computed. Also there are opening of intermediate particle productions, for instance in the t-channel, that presumably affects strongly the chiral series convergence.

### 7.1 Input parameters

Before presenting our results we want to stress the relevance of the
scattering process. Its importance goes beyond the determination
of some threshold quantities, but is a touchstone in the knowledge of
spontaneous symmetry breaking.
Up to nowadays chiral perturbation theory has been used to *parametrize*
the low-energy QCD phenomenology.
Lacking of enough processes to determine
all low-energy constants one has to resort in theoretical inputs (or prejudices). For instance
the determination of the low-energy constants in the
electromagnetic sector have in general a quite mild impact on the results and
therefore have been relegated to a secondary place and only recently
received some attention [16, 17] due to the increasing precision in the
experiment. But very little is known about them
exception of model estimates. In our treatment
these constants can play an important role, and
we include them as given in [16],
where they are estimated by means of resonance saturation.
(Hereafter all our results are given at the scale ).

If instead we use a naïve dimensional analysis the value assigned to each of them would have been restricted to be inside the range

which is taken as a crude indication on the error. Notice that the central values quoted in [26, 17] lie inside this error band.

Contrary to the previous case the low-energy constants in the strong sector are better known. In a series of works [14, 27] most of the next-to-leading low-energy constants were pinned down. In addition to the experimental data a large-N arguments were used to settle the marginal relevance of some operators (those entering together with and ). The use of data in the channel can disentangle (in principle) the value of , due to its product with which enhances its sensitivity to the role of [28].

In order to have a more complete control over our results
we use two different set of constants [22].
The first one was obtained by fitting simultaneously
the next-to-leading expressions of
the meson masses, decay constants and the threshold values of the
form-factors to their experimental values^{§}^{§}§
Notice that in the decay
the s quark is involved. Thus although if in principle it can be used to obtain
the value of the form-factors and turn out
to be rather insensitive to its
actual value
[22]..We shall refer
to it as set I and is given by^{¶}^{¶}¶Quantities with a star are theoretical inputs.

The second set (set II) is obtained with the same inputs and under the same assumptions as the previous one, but this time the fitted expressions are next-to-next-to-leading quantities