# Unitarity and Interfering Resonances in

Scattering and in Pion Production .

###### Abstract

Additivity of Breit-Wigner phases has been proposed to describe interfering resonances in partial waves in scattering. This assumption leads to an expression for partial wave amplitudes that involves products of Breit-Wigner amplitudes. We show that this expression is equivalent to a coherent sum of Breit-Wigner amplitudes with specific complex coefficients which depend on the resonance parameters of all contributing resonances. We use analyticity of partial wave amplitudes to show that they must have the form of a coherent sum of Breit-Wigner amplitudes with complex coefficients and a complex coherent background. The assumption of additivity of Breit-Wigner phases is a new constraint on partial wave amplitudes independent of partial wave unitarity. It restricts the partial waves to analytical functions with very specific form of residues of Breit-Wigner poles. Since there is no physical reason for such a restriction, we argue that the general form provided by the analyticity is more appropriate in fits to data to determine resonance parameters. The partial wave unitarity can be imposed using the modern methods of constrained optimization. We discuss the production amplitudes in reactions and use analyticity in the dipion mass variable to justify the common practice of writing the production amplitudes in production processes as a coherent sum of Breit-Wigner amplitudes with free complex coefficients and a complex coherent background in fits to mass spectra with interfering resonances. The unitarity constraints on partial wave amplitudes with resonances determined from fits to mass spectra of production amplitudes measured in reactions can be satisfied with an appropriate choice of complex residues of contributing Breit-Wigner poles.

###### pacs:

## I Introduction

In 1930’s, Breit and Wigner introduced[1, 2] a parametrization of resonances observed in the energy dependence of integrated and differential cross-sections of nuclear reactions. The original Breit-Wigner formula was only a one-resonance approximation and its justification was initially only phenomenological. A theoretical justification for Breit-Wigner formula later emerged from quantum collision theory[3]. The evident existence of multiple and overlapping resonances in nuclear reactions led to two distinct generalizations of the Breit-Wigner formula for an isolated resonance to multiresonance description of the scattering process.

One generalization was undertaken by Feshbach[4, 5], Humblet[6] and McVoy[7] who used the analyticity properties of the -matrix to show that the transition matrix can be written as a coherent sum of Breit-Wigner terms with complex coefficients and a coherent background. Since the transition matrix must satisfy unitarity, the parameters and coefficients of this multiresonance parametrization are not independent[5, 8]. In principle it is possible to use the methods of nonlinear programming[9, 10] and constrained optimization with computer programs such as MINOS developed at Stanford University[11] to impose the conditions of unitarity in fitting the experimental data.

Another approach to multiresonance description of scattering process was proposed by Hu in 1948[12]. He observed that the Breit-Wigner contribution of an isolated resonance to the -matrix is unitary and proposed to describe the multiresonance contributions in the -matrix by the product of isolated Breit-Wigner contributions for each resonance. Since each term is unitary, the product also satisfies unitarity. The partial wave phase shift is then a sum of Breit-Wigner phases of contributing resonances and a background phase. As a result, the expressions for partial wave amplitudes involve products of Breit-Wigner amplitudes. This method has been recently used by Bugg et al[13] and by Ishida et al[14] in their analyses of phase shift data.

Up to now the connection between these two descriptions of multiresonance contributions (interfering resonances) has not been clarified. In this work we show that the Hu description is a special case of a more general description based on analyticity. We show that the Hu method also leads to a coherent sum of Breit-Wigner amplitudes with complex coefficients and a complex coherent background for any partial wave as expected from the analyticity of the -matrix. However, the complex coefficients have a very specific form in terms of resonance parameters of all contributing resonances. The assumption of additivity of Breit-Wigner phases is a new constraint that restricts the partial waves to analytical functions with these specific residues of Breit-Wigner poles. Furthermore we show that the additivity of Breit-Wigner phases is an assumption entirely independent of the unitarity property of partial wave amplitudes which is a condition imposed on their inelasticity.

Since there is no physical reason why the physical partial waves must have the form of a coherent sum of Breit-Wigner amplitudes with the specific complex coefficients required by the additivity of Breit-Wigner phases, we conclude that the general form imposed by the analyticity is more appropriate for fits to data to determine resonance parameters. This conclusion is particularly relevent for analysis of interfering resonances in the mass spectra in production processes such as or . Using analyticity in the invariant mass variables we justify the common practice of parametrizing the production amplitudes in terms of a coherent sum Breit-Wigner amplitudes with free complex coefficients and a complex coherent background[15, 16, 17, 18, 19, 20, 21, 22, 23].

The paper is organized as follows. In Section II we briefly review the unitarity and the problem of interfering resonances in potential scattering since it motivates the analysis in hadronic reactions. In Section III we review the two-body partial wave unitarity in scattering and its relation to the general form of isospin partial waves. In Section IV, we introduce the assumption of additivity of Breit-Wigner phases in the scattering and show that it leads to partial waves in a form of a coherent sum of Breit-Wigner amplitudes with specific complex coefficients and a coherent background. In Section V we generalize dispersion relations for partial wave amplitudes in scattering to Breit-Wigner poles and show that the form obtained from the additivity of Breit-Wigner phases is a special case. In Section VI we focus the discussion of the two methods to a finite energy interval and argue that the addition of Breit-Wigner phases imposes an unjustified constraint on fits to data. In Section VII we formulate unitarity for production amplitudes in reaction and contrast it with partial wave unitarity in scattering. In Section VIII we show that the method of addition of Breit-Wigner phases can be generalized to production amplitudes. We also use analyticity in the invariant mass to obtain a more general form for production amplitudes in terms of a coherent sum of Breit-Wigner amplitudes with free complex coefficients (pole residues) and a complex coherent background. We argue that this general form is more appropriate in fits to measured mass spectra. Although the discussion is confined to pion production amplitudes in , the conclusions have general validity. We also comment on determination of partial wave amplitudes from resonance parameters determined in measurements of production amplitudes in reactions. The paper closes with a summary in Section IX.

## Ii Unitarity and interfering resonances in potential scattering.

### ii.1 Unitarity

We will consider the scattering of a spinless particle of mass by a real, central potential [24]. In the asymptotic form of the stationary scattering wave function, the outgoing wave is characterized by the scattering amplitude where is the wave number of the particle related to its energy by

(1) |

and is the scattering angle. In the units the wave number has the meaning of momentum . The scattering amplitude can be written in the form

(2) |

The partial wave amplitudes are given by

(3) |

where is called -matrix. For elastic scattering

(4) |

where the phase-shifts describe the interaction and are related to the potential . For elastic scattering which is the condition of elastic unitarity.

When a particle collides with a target, non-elastic processes are possible and particles are removed from the incident (elastic) channel. Since the interaction can alter only the outgoing part of the wave function, we require that the amplitude of the outgoing wave be reduced if non-elastic processes occur. The reduction of scattering amplitudes leads to conditions of inelastic unitarity

(5) |

This suggests that we write

(6) |

where is called inelasticity and has values

(7) |

The partial wave then has a general form

(8) |

¿From (2.8) it follows that

(9) |

This equation expresses the unitarity condition on the partial waves .

### ii.2 Interfering Resonances.

In the following we will work with partial wave amplitudes

(10) |

and the energy instead of . A detailed study of the potential scattering[24] shows that the phase shift may be decomposed as where is the background phase which does not depend on the shape and depth of the interaction potential while the part does depend on the details of the potential. Near resonant energy

(11) |

where is the width of the resonance. We introduce a Breit-Wigner resonance phase

(12) |

such that in the energy interval centered about we have and

(13) |

From (2.12) it follows that

(14) |

where

(15) |

is the Breit-Wigner amplitude of the resonance . For an isolated resonance we then obtain

(16) |

If resonances contribute over an interval then, following Hu[12] and references[13, 14], we can write

(17) |

The prescription (2.17) clearly satisfies unitarity but seems to lead to a complicated expression for partial waves in terms of Breit-Wigner amplitudes .

On the other hand, analyticity of -matrix was used by Feshbach[4, 5], Humblet[6] and McVoy[7] to derive a general form for [5]

(18) |

where is a background term and are complex coefficients. The sum in (2.18) can be written as a coherent sum of Breit-Wigner amplitudes

(19) |

In Section IV we show that the prescription (2.17) leads to the analytical form (2.19) with specific expressions for the coefficients and the background .

## Iii Isospin amplitudes and unitarity in scattering.

Hadron resonances have definite values of spin and isospin. It is therefore necessary to express the amplitudes for charged pion processes in terms of isospin amplitudes with definite isospin and work with partial wave amplitudes [25]. At first we will work with the center-of-mass energy to pursue the analogy with the potential scattering.

The partial wave amplitudes satisfy partial wave unitarity equations[25, 26]

(20) |

where are the contributions from inelastic channels, such as , , , and is c.m. momentum where is the pion mass.. Let us write in the form

(21) |

Then the unitarity equation (3.1) has the same form as (2.9) and the partial waves can be written as

(22) |

where the are phase shifts and the inelasticity

(23) |

is given by the inelastic unitarity contributions . In analogy with potential scattering we expect that . As we shall see later, the descriptions of interfering resonances in scattering do not depend on the condition that .

The positivity of inelasticity in (3.4) imposes a constraint

(24) |

We can now show that the unitarity equation (3.1) admits no solution for . If the inelastic unitarity contributions satisfy this condition we can write

(25) |

where . Setting we get from the unitarity equation (3.1)

(26) |

which is not possible. Thus the conditions (3.5) represent genuine constraints on the inelastic unitarity contributions and the parameterization (3.3) of partial wave amplitudes with (3.4) is the most general solution of the unitarity equation (3.1) for all .

Since the values of inelastic terms in the partial wave unitarity equations (3.1) are not known, we constrain the partial waves by inequalities imposed by the unitarity. From the positivity of and the condition (3.5) we obtain

(27) |

If we add the requirement that , then from (3.2), and we also have the usual unitarity constraint

(28) |

The inequality (3.9) implies positivity

(29) |

at all energies.

Finally we note the following observation. Let be any complex function. Then is a complex function that can be written as

(30) |

where and is real. Thus any complex function can be written in the form

(31) |

and satisfies the equation

(32) |

We see that the unitarity equations (3.1) are a special case of (3.13) with given by (3.4).

## Iv Interfering resonances in scattering using the addition of Breit-Wigner phases.

The general form of phase shift parametrization of partial wave amplitudes is

(33) |

with inelasticity determined by unitarity via (3.4). In the following we will omit the indices and for simplicity. In analogy with potential scattering, we decompose the phase shifts into two parts

(34) |

where is the nonresonant background phase and is the phase due to physical particle resonances occurring in the partial wave . The phase of a single isolated resonance is given by the Breit-Wigner formula

(35) |

Let us consider that resonances contribute to the partial wave amplitude . Following ref. [12–14] we now assume, that the resonant phase shift is given by the sum of the Breit-Wigner phases of the contributing resonances

(36) |

We assume that is finite. Then

(37) |

We can write for each Breit-Wigner phase

(38) |

where is the Breit-Wigner amplitude

(39) |

Then we can write

(40) |

where is given in terms of products of Breit-Wigner amplitudes The partial wave amplitude then has a general form

(41) |

Let us consider the case . Then the resonant part of the amplitude is

(42) |

where the interference term

(43) |

With a notation

(44) |

we write

(45) |

The requirement that this equality holds leads to relation

(46) |

Next we require that to eliminate the dependent term, and get . Then (4.13) has the form of a sum

(47) |

and we can write the resonant part (4.10) of the partial wave amplitude as the sum of two Breit-Wigner amplitudes

(48) |

where the complex coefficients

(49) |

are exactly such that the unitarity condition

(50) |

is satisfied for all . The energy dependence of the widths introduces energy dependence in .

Consider now the case of three interfering resonances . Then

(51) |

We can write the last term as a sum

(52) |

Requiring that the terms proportional to and in the numerator on r.h.s. of (4.20) vanish, we obtain a sum

(53) |

where

(54) |

with

(55) |

The resonant part of the partial wave amplitude is again a coherent sum of the Breit-Wigner terms with complex coefficients

(56) |

where

(57) |

This procedure is general and valid for any finite . Assuming that the resonant phase can be separated from the phase shift and is given by the sum of Breit-Wigner phases, we will always get the resonant part of the partial wave amplitudes in (4.9) as a sum of Breit-Wigner amplitudes

(58) |

In (4.26) the complex coefficients have an explicit form in terms of resonance parameters such that satisfies the unitary condition (4.18). The form of coefficients depends on the number of resonances contributing to the partial wave .

As the result of (4.26) we can conclude that the multiresonance parametrization of partial wave amplitudes based on additivity of Breit-Wigner phases has a general form of a coherent sum of Breit-Wigner amplitudes with complex coefficients and a complex coherent background

(59) |

where

(60) |

Comparing (4.27) with expression (2.19), we see that the description of multiresonance contributions using the addition of Breit-Wigner phases leads to the same form of partial wave amplitudes as the analyticity of the -matrix. However, the complex background and the complex coefficients in (4.27) have the explicit form (4.28) imposed by the additivity of Breit-Wigner phases.

Note that in the derivation of (4.26) for , and in the resultant form (4.27) with (4.28), we have not needed or used the assumption that inelasticity . The Hu method is based on the unitarity of and is not related to the unitarity of the whole partial wave amplitude .

Finally we give a relativistic form for the multiresonance description of partial wave amplitudes. The relativistic form of Breit-Wigner amplitudes (4.7) is given by

(61) |

where we have used instead of to emphasize that is the mass of the resonance. To obtain the corresponding coefficients , we make replacements in (4.17) or (4.25)

(62) |

The partial wave amplitudes then have the relativistic form

(63) |

where and are still given by (4.28) with replacements (4.30) to satisfy the unitarity of .

## V Generalized dispersion relations for partial wave amplitudes and interfering resonances in scattering.

In this section we shall relate the multiresonance parametrization (4.31) of partial wave amplitudes with a multiresonance parametrization obtained from analyticity. To this end we shall use generalized dispersion relations for the amplitudes

(64) |

where is the Mandelstam energy variable.

Our starting point is the well-known[27] dispersion respresentation of a complex function with simple poles at in the complex plane , a branch cut along a positive real axis from to and with asymptotic property as . We shall also assume that the function is a real function . Using Cauchy’s integral theorem and the process of contour deformation, it can be shown[27] that

(65) |

A remarkable feature of the proof of (5.2) is that it takes place for a fixed value of [27]. As the result, the dispersion relation (5.2) is also valid for moving poles for which . In such a case the residues in (5.2) also depend on , i.e. . Furthermore, the dispersion relation (5.2) is easily generalized to include a left-hand cut and for functions that are not real. In the latter case in (5.2) is replaced by a discontinuity function along the cut(s).

In scattering, the partial wave amplitudes have a right-hand cut for (where is the pion mass), and a left-hand cut for due to Mandelstam analyticity[28]. Let us assume that the amplitude has a finite number of complex poles

(66) |

corresponding to the resonances in . Note that the imaginary part of the poles depends on the energy variable . In principle, the mass could also depend on the energy . This possibility has been recently considered by Pennington[29]. Omitting the indices and , the generalized dispersion relations for the partial wave amplitude read

(67) |

where are the dispersion integrals over the left-hand and right-hand cuts[28] and are the pole residues. It is convenient to rewrite (5.4) in a form using Breit-Wigner amplitudes

(68) |

where we have redefined the pole residues with

(69) |

The representation (5.5) is valid for all . The representation (5.5) of partial waves coincides with the parametrization (4.31) provided that

(70) |

We see that the multiresonance parameterization based on additivity of Breit-Wigner phases (4.4) imposes a special form on the dispersion integrals and pole residues given by (5.7).

In general, a partial wave can be written in two forms

(71) |

Apart from the partial wave unitarity equations (3.1) and (3.4) and the analyticity assumptions, there are no constraints on the partial waves. The assumption of additivity of Breit-Wigner phases (4.4) is a new constraint that restricts the partial waves to analytical functions that satisfy the conditions (5.7). We find no physical justification for such a restriction and no advantage in using it in phenomenological fits to data to determine resonance parameters.

## Vi Interfering resonances in a finite energy interval.

In the previous two sections we have assumed that is the total number of resonances contributing to a partial wave. The parametrizations (4.31) and (5.5) were valid for all energies . In practice is not known and fits to data are done in a finite energy interval. Such is the case e.g. of analyses [13, 14]. In this Section we develop parametrizations of partial waves appropriate for analyses in a finite energy interval where only a few resonances contribute. The parametrizations will be based both on additivity of Breit-Wigner phases and analyticity and we shall compare their use in practical fits to data. The results will be used in the Section VIII.

Let us consider energy interval where resonances contribute. In the framework of the assumption of additivity of Breit-Wigner phases we will assume that the resonant phases of resonances outside of this energy interval are absorbed in the background phase. The total phase shift then is

(72) |

where

(73) |

(74) |

The partial wave then takes the form

(75) |

where the resonant part

(76) |

is unitary

(77) |

Alternatively we can rewrite (4.31) in the form

(78) |

where

(79) |

is the background term. Note that the sum in (6.7) is not unitary. We cannot compare the coefficients of Breit-Wigner amplitudes in (6.4) with those in (6.7) since in (6.4) contains the terms but the sum in (6.7) does not.

If we look at the general form (5.5) from analyticity, then for we can write

(80) |

where the background term

(81) |

In (6.9) the residues are not constrained by the conditions (5.7).

In fitting data using the parametrization (6.4) we explicitly make use of the assumption of additivity of Breit-Wigner phases. This is also the case when we use (6.7) if is known and the coefficients in can be calculated. In general is not known, and the background and residues are free parameters. Then there is no difference in using (6.7) or the general form (6.9) from analyticity alone, since in (6.9) the background and residues are not constrained except for unitarity. In all cases we use constrained optimization of the function. In the case of (6.4) we require that inelasticity function . In the case of (6.7) or (6.9) we require that and use programs such as MINOS[11] for constrained optimization.

It is not obvious that the use of parametrization (6.4) from additivity of Breit-Wigner phases and the parametrization (6.9) from analyticity alone, will lead to the same resonance parameters in both cases. The use of parametrization (6.4) confers no phenomenological or computational advantage over the parametrization (6.9). The assumption of additivity of Breit-Wigner phases restricts the background and the complex coefficients multiplying the Breit-Wigner amplitudes in the parametrization (6.4) to specific forms. Since there is no physical justification for such a restriction and the parametrization (6.9) is free from such constraints, we suggest that the use of parametrization (6.9) is more appropriate in determining resonance parameters in scattering.

## Vii Unitarity in pion production

It is a common misconception to identify the partial wave production amplitudes in reaction with partial waves in scattering and demand that the partial wave production amplitudes also satisfy the partial wave unitarity (3.1). In this Section we clarify the distinction between the two kinds of amplitudes and the associated unitarity relations.

The production process is described by production amplitudes[25, 30, 31]

(82) |

where and are proton and neutron helicities, s is the c.m.s. energy squared, t is the momentum transfer between the incident pion and the dipion system , is the dipion mass squared, and is the solid angle of the final pion in the dipion rest frame. The dipion state does not have a definite spin. The production amplitudes (7.1) can be expressed in terms of partial wave production amplitudes corresponding to definite dimeson spin using the angular expansion[25, 30, 31]

(83) |

where is the spin and the helicity of the dimeson system.

It is evident from (7.2) that the partial wave production amplitudes cannot be identified with the partial wave amplitudes . The amplitudes can be thought of as two-body helicity amplitudes for a process where the ”particle” has spin J and mass .

The production amplitudes satisfy the unitarity condition [26]

(84) |

where is the -body Lorentz invariant phase space of the intermediate state . Since the initial state in is a two-body state and the final state is a three-body state, the amplitude enters the unitarity integral only linearly. This occurs only when the intermediate state is or . However the three-body intermediate state involves a amplitude and a three-body phase space integral. Separating the two-body intermediate states and , we can write (7.3) in the form

(85) |

where and are helicity amplitudes of reactions and , respectively. The amplitude corresponds to process . The inelastic unitarity contribution can be expanded in the form analogous to (7.2)

(86) |

Using expansions (7.2) and (7.5) in (7.4) we get unitarity relations for partial wave production amplitudes

(87) |

Using time-reversal relations for two-body helicity amplitudes[44]

(88) |

we see that the left hand side of the partial wave unitarity relation (7.6) does not simplify to as is the case for the partial waves in scattering. The right hand side of (7.6) involves only linearly and not quadratically as is the case in scattering. Futhermore, the right hand side of unitarity relation (7.6) includes (linearly) partial wave production amplitudes for the process . We conclude that the unitarity relations (7.6) for partial wave production amplitudes are complex relations that do not have the simple form

(89) |

of the partial wave unitarity relations in scattering.

For brevity let us define where stands for the helicities. The amplitude is a complex function and so is the function . In analogy with (3.12) and (3.3) we can write

(90) |

where is the ”inelasticity” and is the ”phase shift”. The amplitude satisfies relation similar to (3.13)

(91) |

Unlike in scattering, the form of the equation (7.10) does not coincide with the form of partial wave unitarity (7.6) and the ”inelasticity” cannot be related to the inelastic unitarity contributions , in contrast to (3.4).

## Viii Interfering resonances in production processes.

The amplitudes describing the production processes such as , or are far more complex than the isospin amplitudes in the scattering. As an example, consider pion production in . The angular distribution of the dipion state is described by partial wave production amplitudes defined in the previous Section with eq. (7.2). The measurements of on polarized target actually determine the moduli of nucleon transversity amplitudes [30, 31] which are linear combinations of helicity amplitudes