FIELD THEORY, GRAVITATION AND PARTICLE PHYSICS

OMNÈS REPRESENTATIONS WITH INELASTIC EFFECTSFOR HADRONIC FORM FACTORS

IRINEL CAPRINI

National Institute of Physics and Nuclear EngineeringPOB MG 6, Bucharest, R-077125 Romania

Received December 10, 2004

We derive a generalized Omnès representation for the hadronic form factors,which satisfies Watson theorem in the elastic region and includes the effects ofinelastic channels. As an application we discuss the behaviour of the scalar form

factors of the pion near the KK threshold. The results are useful also for the calculationof the phases produced by the strong final state interactions in the nonleptonic decaysof the K and B mesons.

1. INTRODUCTION

The hadronic form factors, defined as matrix elements of operators bilinearin the quark fields among hadronic states, play an important role both inperturbative and nonperturbative quantum chromodynamics (QCD). PerturbativeQCD predicts the behaviour of the form factors only at large momentum transferin the space-like region, where asymptotic freedom holds and the hadronicthresholds are absent [1]. On the other hand, at low energies, chiral perturbationtheory (ChPT) provides a systematic expansion of these quantities in powers ofthe momenta and the quark masses. The form factors of the light pseudoscalarmesons were calculated to one-loop in [2] and beyond this order in [3]. Thecomplete evaluation to two loops is given in [4].

Dispersion theory provides also a powerful tool for studying the formfactors and relating their low and the high energy behaviours. In the completetheory of QCD, which includes confinement, the form factors are analyticfunctions of real type in the complex energy plane cut along the real axis fromthe threshold imposed by unitarity to infinity. The most convenient dispersiverepresentation is the so-called Omnès representation [5], which expresses theform factors in terms of their phase along the cut. This representation allows aneasy implementation of Watson theorem [6], which states that in the elasticregion the phase of the form factor is equal to the phase of the elastic final statescattering. Many applications of the Omnès representation and its mathematicalgeneralizations for various weak and electromagnetic form factors exists in the

Rom. Journ. Phys., Vol. 50, Nos. 1–2 , P. 7–17, Bucharest, 2005

8 Irinel Caprini 2

literature(see for instance [7] where dispersive and chiral symmetry constraintson the light meson form factors were derived).

The inclusion of the inelastic channels in the Omnès representation isnecessary for calculating quantities of interest for ChPT like the quadratic radiusof the pion. In particular, the effect of the KK channel on the scalar form factorsof the pion was recently a controversial subject. The problem was investigated inthe frame of a two-channel generalization of the Omnès representation (the so-called Mushkhelishvili-Omnès (M-O) equations [8]) in [9], [10], and morerecently in [11]. The conclusion of these works is that the opening of the KKchannel can have important effects on the phase of the scalar form factors around1 GeV, the behaviour depending strongly of the quark structure of thecorresponding operator. In Ref. [12], on the other hand, it is claimed that theeffect of inelasticity is negligible. As this conclusion is based on a single-channelOmnès formalism, it is of interest to include the effect of inelasticity in thisformalism. In the present paper we address this problem and write down asingle-channel Omnès representation which includes explicitly the influence ofthe inelastic channels. We stress that the complete solution is provided only bysolving the coupled-channels M-O equations. The formulae which we derive areuseful however since they offer a rather transparent picture of the inelasticeffects, allowing us to understand in a qualitative way the different behaviour ofthe various form factors. Also, understanding the inelastic channels is crucial forpredicting the effect of the final state interactions in nonleptonic weak decayslike K → ππ or B → ππ.

2. DISPERSIVE REPRESENTATION IN TERMS OF THE PHASE

We consider the scalar form factors of the pion

( ) ( ) ( ) 0u ds p p m uu m dd′Γ =< π π | + | > (1)and

( ) ( ) ( ) 0ss p p m ss′∆ =< π π | | >, (2)

where 2( ) ,s p p′= + u, d, s are the quark fields and mu, md, ms their currentmasses. For simplicity, we denote generically the above form factors by F(s).The function F(s) is analytic in the s-plane cut from the elastic threshold at

24s mπ= to infinity. The phase ( )F sδ of F(s) on the cut is defined by theboundary condition

2 ( ) 2( ) e ( ) 4Fi sF s i F s i s mδ π+ = − , > .ε ε (3)

This relation represents a Riemann boundary value problem [8], with the generalsolution [5]:

3 Omnès representations for hadronic form factors 9

24

( )d( ) ( )exp

( )F

n

m

s ssF s P ss s s

π

∞⎡ ⎤′ ′δ⎢ ⎥= ,′ ′π −⎢ ⎥⎣ ⎦∫ (4)

where ( )nP s is a polynomial of degree n.Perturbative QCD predicts the asymptotic behaviour [1]

( )( ) ~ s

sF s s

sα −

, | |→∞,−

(5)

where 2( ) 4 9 ln( )s s sα − = π/ − /Λ is the QCD running coupling. From the Omnèsrepresentation (4), this implies the asymptotic behaviour of the phase

2

( ) ~ ( 1)ln

F s n ssπδ + π + , →∞.

Λ

(6)

Chiral expansions suggest that the form factor ( )sΓ has no zeros in the complexplane, which means that n = 0, the polynomial in (4) reduces to a constant and

2

( ) ~ln

s ssΓπδ π + , →∞.

Λ

(7)

On the other hand, for the form factor ∆(s), ChPT indicates a zero close to s = 0,which means that in this case n = 1 and the asymptotic behaviour of the phase is

2

( ) ~ 2ln

s ss∆πδ π + , →∞.

Λ

(8)

3. UNITARITY RELATION AND WATSON THEOREM

Along the cut 24 ,s mπ> the form factor ( )F s satisfies the unitarity relation

(0)0Im ( ) ( ) ( )[ ( )] ( )inF s s F s f s s

∗= σ + σ , (9)

where 2( ) 1 4s m sπσ = − / and the isoscalar S-partial wave (0)0 ( )f s is

parametrized as(0)0(0) 2

(0) 00

e 1( )

2 ( )

i

f si s

δη −= ,

σ(10)

in terms of the elasticity (0)0 1η ≤ and the phase-shift (0)0δ . All the functions in

(9) are evaluated on the upper edge of the cut. Neglecting the small contribution

10 Irinel Caprini 4

of four pions, the inelastic term ( )in sσ can be approximated at low energies by

the contribution of the KK channel

2( ) ( 4 ) ( ) ( ) ( )in K K K Ks s m s F s T s∗πσ = θ − σ , (11)

where 21 4K Km sσ = − / is the phase space, ( )KF s is the kaon scalar form factor

defined by replacing in Eqs. (1) or (2) the pion pair by a KK pair, and KTπdenotes the KKππ→ S-wave amplitude.

Eq. (9) can be written as

( )(0)0( ) 1 2 ( ) ( ) ( ) 2 ( )inF s i i s f s F s i i s∗⎡ ⎤+ − σ − − = σ ,⎣ ⎦ε ε (12)or, using (10), as:

(0)0(0) 2 ( )

0( ) ( )e ( ) 2 ( )i s

inF s i s F s i i s− δ+ η − − = σ .ε ε (13)

In the elastic region 24 ,Ks m< where (0)0 1η = and ( ) 0,in sσ = Eq. (13) reduces to

(0)02 ( ) 2( ) e ( ) 4i s KF s i F s i s mδ+ = − , < ,ε ε (14)

which, compared to (3), yields Watson theorem (0)0( ) ( )F s sδ = δ for 24 Ks m< .

Above the KK threshold the phase δF is no longer equal to the phase shift.

4. OMNÈS REPRESENTATIONS WITH INELASTICITY

Equation (13) has the form of an nonhomogeneous Riemann boundaryvalue problem, whose general solution is given in [8]. Following [16], we lookfor solutions F which satisfy elastic unitarity and time reversal invariance, whichrequires that the form factor is real-analytic in the cut plane, ( ) ( ).F s F s∗ ∗= Welook for solutions of the form

( ) ( ) ( )F s G s O s= , (15)

where ( )O s is an Omnès function defined in terms of a certain phase, and ( )G sa residual function which accounts for the inelasticity. To satisfy Watson

theorem, the phase of ( )O s must be equal to (0)0 ( )sδ below the inelastic threshold,but above it the phase is arbitrary. For every choice of the phase of ( )O s wecalculate the remaining function G from the unitarity relation.

5 Omnès representations for hadronic form factors 11

4.1. OMNÈS FUNCTION DEFINED WITH THE PHASE SHIFT

A natural choice is to take the phase of O equal to the phase shift (0)0δalong the whole cut up to infinity. So we write the form factor as

1 1( ) ( ) ( )F s G s O s= (16)where

2

(0)0

14

( )( ) exp d

( )m

ssO s ss s s

π

∞⎡ ⎤′δ⎢ ⎥′= .′ ′π −⎢ ⎥⎣ ⎦∫ (17)

In order to calculate the residual function G1 appearing in (16), we first noticethat by Watson theorem it is real below the inelastic threshold, so it is analytic inthe s-plane cut only for 24 .Ks m> From (16) we have

(0) (0)1 0 0

1

(0) (0)1 0 0

1

1Re ( ) [Re cos Im sin ]( )

1Im ( ) [Im cos Re sin ]( )

G s F FO s

G s F FO s

= δ + δ ,| |

= δ − δ .| |

(18)

On the other hand, by multiplying both sides of (13) with (0)0 ( )ei sδ and taking the

real and imaginary part we have

(0)0

(0)0

(0) (0)0 0 (0)

0

(0) (0)0 0 (0)

0

2Re cos Im sin Im e1

2Im cos Re sin Re e1

iin

iin

F F

F F

δ

δ

⎡ ⎤δ + δ = σ⎣ ⎦− η

⎡ ⎤δ − δ = σ .⎣ ⎦+ η

(19)

By comparing with (18) we have

(0)0

(0)0

1 (0)10

1 (0)10

Im[ e ]2Re ( )( )1

Re[ e ]2Im ( )( )1

iin

iin

G sO s

G sO s

δ

δ

σ= ,

| |− η

σ= .

| |+ η

(20)

Therefore, the function 1( )G s in the representation (16) satisfies (up to apolynomial real on the cut) the Omnès representation

2

11

4

( )( ) exp d

( )Km

ssG s ss s s

∞⎡ ⎤′ψ⎢ ⎥′= ′ ′π −⎢ ⎥⎣ ⎦∫ (21)

12 Irinel Caprini 6

where 1( )sψ is the argument of 1 1Re ( ) Im ( ) :G s i G s+

{ }{ }

(0 )0

(0)0

(0)0

1 (0)0

Re e1( ) Arctg

1 Im e

iin

iin

sδ

δ

⎡ ⎤σ− η⎢ ⎥ψ = .

+ η⎢ ⎥σ⎣ ⎦(22)

From (16), (17) and (21) it follows that the phase Fδ of the form factor is given by

(0)10( ) ( )F s sδ = δ + ψ . (23)

4.2. OMNÈS FUNCTION DEFINED WITH THE PHASEOF THE PARTIAL WAVE AMPLITUDE

An alternative choice is to take the phase of the Omnès function O in (15)equal to the phase δt of the partial wave amplitude, defined by

(0) (0) (0)0 0 0(0) (0) (0)0 0 0

Im ( ) 1 cos2( ) Arctg

Re ( ) sin2t

f ss

f s

⎡ ⎤ − η δδ = = .⎢ ⎥

η δ⎢ ⎥⎣ ⎦(24)

So, we take

2 2( ) ( ) ( )F s G s O s= (25)where

2

24

( )( ) exp d

( )t

m

ssO s ss s s

π

∞⎡ ⎤′δ⎢ ⎥′= .′ ′π −⎢ ⎥⎣ ⎦∫ (26)

In the elastic region (0)0 ,tδ = δ but above the KK threshold this equality is no

longer valid1. In fact, the phase-shift (0)0δ has the peculiarity that it raises rapidly

near the KK threshold reaching the value π above (and close to) it. FromEq. (24) it folows that when (0)0 1η < and

(0)0δ passes through π, the phase δt has

a dip, which becomes steeper when the elasticity is close to 1. So, above theinelastic threshold the phase δt of the amplitude is quite different from the phase

shift (0)0δ .

1 Actually, if (0)0 0( )sδ = π for 20 4 ,Ks m< then at this point phase δt makes a jump by –π,(0)0 0 0( ) ( ) ,ts sδ − δ = π in order to preserve the positivity of the imaginary part

(0) (0)20 0Im sin .f = δ

We shall assume that (0)0δ reaches π only above the KK threshold, as indicate most experimental

parametrizations, so that (0)0δ = δ in the whole elastic region.

7 Omnès representations for hadronic form factors 13

In order to calculate the residual function 2G defined in (25) we notice that

for 24 Ks m> the two terms in the r.h.s. of Eq. (9) are not separately real, but theirsum must be real. Taking the real and the imaginary parts of (9) we have:

{ }{ }

(0)0

(0)0

Re Im Re [ ]

Im Im ( )[ ]

in

in

F F f

F s f

∗

∗

σ = − σ ,

σ = − σ .(27)

Inserting in these relations the representation (25) and noting that with the choice(26) the product (0) (0)2 20 0[ ]O f O f

∗ =| || | is real, we obtain

2 (0)2 0

2 22 (0)

0

ImIm ( )

( )

Re Im ( )cosRe ( )

sin

in

in t

t

G sO s f

O G sG s

f

σ= −

σ | || |

σ / | | − δ= .

δ − σ | |

(28)

Denoting by 2 ( )sψ the argument of 2 2Re ( ) Im ( )G s i G s+

22

2

Im ( )( ) Arctg

Re ( )G s

sG s

⎡ ⎤ψ = ,⎢ ⎥⎣ ⎦(29)

we write the function G2, up to a polynomial, as

2

22

4

( )( ) exp d

( )Km

ssG s ss s s

∞⎡ ⎤′ψ⎢ ⎥′= ′ ′π −⎢ ⎥⎣ ⎦∫ (30)

From (25), (26) and (30) it follows that the phase Fδ of the form factor is given by

2( ) ( )F ts sδ = δ + ψ . (31)

5. COMMENTS ON THE SCALAR FORM FACTORS OF THE PION

The two approaches described above are equivalent and must lead to identicalresults. We checked this equivalence fot the numerical solutions of the two-channel M–O equations calculated in [11] using the experimental data from [13].

More exactly, we evaluated the right hand sides of the relations (23) and(31) using as input the corresponding quantities calculated in [11] and checkedthat they lead to identical results, which moreover coincide with the phase of theform factor obtained in [11]. The results are shown in Fig. 1 (reproduced from[11]), where we indicate the phase shift (0)0δ , the phase δt of the partial wave,and the phases of the form factors Γ and ∆. Below the opening of the inelastic

14 Irinel Caprini 8

channels all the phases depicted are equal. Above the KK threshold, Γδ has apronounced dip and then follows closely the phase δt of the scattering amplitude,

staying below the phase shift (0)0δ by approximately –π. This behaviour of theform factor Γ is confirmed by the experimental data on the central production ofpion pairs in pp collisions [17]. In the notations of the previous section, thismeans that for the form factor Γ the additional phase 1ψ from (23) is negativeand approaches rapidly the value −π , while the additional phase 2ψ from (31)is close to zero. On the other hand, the phase of the form factor ∆ follows closelythe phase shift also above the KK threshold, which means that the additionalphase ψ1 from (23) is small, while the phase ψ2 from (31) is large and close to π.

Fig. 1. – The phase δΓ of the pion form factor Γ(s) calculated from the two-channel M-O equations [11] (solid line). The dashed and dotted lines describe

the phase shift (0)0δ and the phase δt of the partial wave amplitude, respectively. The dash-dotted line depicts the phase δ∆ of the form factor ∆(s).

This behaviour, obtained numerically in [11], can be understoodqualitatively using the expressions derived in Section 4. For illustration, weconsider the representation given in subsection 4.1, where the Omnès function isexpressed in terms of the phase shift. As we mentioned, the experimental data onππ scattering [13] indicate that the phase shift (0)0 ( )sδ raises very rapidly and

reaches the value π just above the inelastic KK threshold. Moreover, just abovethis threshold the elasticity (0)0η has a sharp decrease, indicating a very strong

inelasticity, and then rather quickly approaches again the elastic value (0)0 1η = .

9 Omnès representations for hadronic form factors 15

From the expression (22) it follows that, if the elasticity (0)0 ( )sη is close to 1, theadditional phase 1ψ appearing in (23) is equal to 0 modulo ±π. Denoting by inσδ

the phase of the complex quantity exp( )inin in

i σσ =| σ | δ and omitting the

irrelevant positive factors, we notice from (18) that the phase 1ψ depends on thesigns of the quantities

(0) (0)1 0 0Re ~ sin( ) (1 )inG σδ + δ / − η (32)

(0)1 0Im ~ cos( )inG σδ + δ . (33)

In the vicinity of the threshold the phase inσδ is suppressed by the phase space.

Since, as we mentioned above, the phase shift (0)0δ is close to π, the quantity(0)0cos( ),inσδ + δ which determines the sign of the imaginary part of G1, is

negative. Moreover, just above the threshold, where (0)0δ is still less than π, thereal part 1ReG defined in (32) is positive. This means that the point associatedto the complex quantity G1 is situated in the fourth quadrant of the trigonometriccircle, and 1 0ψ < . From (23) it follows therefore that the inelasticity has theeffect of lowering the phase of the form factor. The evolution of the phase 1ψ athigher energies depends on the sign of the real part 1ReG . If

(0)0sin( ) 0inσδ + δ > (34)

then 1Re 0G > and the point associated to 1G remains in the fourth quadrant.

Hence, when the inelasticity (0)0η approaches again the value 1, the phase 1ψtends to 0 through negative values, and the phase of the form factor approaches

(0)0δ from below. But if

(0)0sin( ) 0inσδ + δ < , (35)

then the point associated to G1 enters the third quadrant, and 1ψ → −π when(0)0η becomes close to 1. The decisive role is played therefore by the phase of the

inelastic term ~ ( ) ( )in K KF s T s∗πσ defined in (11). This quantity can be

understood by noticing that the two-channel unitarity equations given in [9] aresatisfied by the following expressions:

1 2

1 2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )K

K K KK

F s c s T s c s T s

F s c s T s c s T sππ π

π

= += + ,

(36)

16 Irinel Caprini 10

where (0)0 ,T fππ = KTπ and KKT denote the S-wave projections of the ππ→ ππ ,

KKππ→ and KK KK→ amplitudes, respectively, and the functions 1( )c s and

2 ( )c s are real for 24s mπ> . If the coefficients c1 and c2 are positive, the relations

(36) imply, by the parallelogram rule for vector addition, that the phase of thepion form factor Fπ is larger than the phase of Tππ and smaller than the phase ofTπK, while the phase of the kaon form factor KF is larger than the phase of TKKand smaller than the phase of TπK. We recall that by unitarity the phase of thenondiagonal amplitude TπK is the sum of the phase shifts of the diagonal elements

[9]. The experimental data [13]–[15] indicate that the phase shift δK of the KK →

KK→ transition is negative. Using a relation similar to (10) (with (0)0δ replaced

by Kδ ), we obtain for the phase of KKT positive values in the second quadrant.Let us consider first the form factor ( )sΓ defined in (1). The coefficients c1

and c2 take values consistent with the asymptotic condition (7). The explicitcalculation with data from [13] indicate that 1( ) ~ ( ) ( ),s c s T sππΓ which impliesthat ~ .tΓδ δ The Omnès formalism is consistent with this result: inserting

1( ) ~ ( ) ( )K Ks c s T sπΓ in the expression (11), it follows that the inσδ is close to 0

and the relevant quantity in Eq. (32) is (0) (0)0 0sin( ) ~ sin .inσδ + δ δ Since above the

KK threshold (0)0δ becomes rapidly greater than π, the inequality (35) holds,

which means that the difference (0)0Γδ − δ tends to −π , as shown in Fig. 1. Thisresult is quite stable with respect to the parametrizations of the unitary S-matrix.Indeed, even if the first term in (36) is not dominant and the phase of inσ is

negative, the correction is not very large, and still leads to (0)0sin( ) 0inσδ + δ < ,

due to the large values of (0)0δ .

In the case of the form factor ∆ defined in (2), the asymptotic condition (8)select a different pattern for the coefficients c1 and c2, which may be negative. It

follows that inσδ is a large negative phase, so that the sum

(0)0inσδ + δ becomes

less than π and the inequality (34) holds. Therefore, when the elasticity (0)0η

approaches 1, the difference (0)0∆δ − δ tends to 0 .

6. CONCLUSIONS

In the present paper we derived single channel Omnès representations forthe hadronic form factors, which include explicitly the effects of the inelastic

11 Omnès representations for hadronic form factors 17

channels in the unitarity sum. As we discussed in Section 4, the results provide aqualitative understanding of the scalar form factors of the pion in the vicinity ofthe inelastic KK threshold. We mention that the results are useful also forincluding the effects of final state rescattering in the nonleptonic decays likeK →ππ and B →ππ , which are of interest for the CP-violation parameters inthe Standard Model. Dispersion relations and Omnès representations for theamplitudes of these decays, including the effects of initial and final stateinteractions, were derived in [18], [19]. The function G1 describing the inelasticchannels in K →ππ decay was expanded in a power series based on a conformalmapping [19]. In the case of B nonleptonic decays, the additional phase (22)produced by the final state interactions can be evaluated in terms of the weakdecay amplitudes into intermediate pseudoscalar and vector mesons, usingRegge theory for the strong rescattering amplitudes [20].

Acknowledgments. This work was supported by a grant of the Romanian Academy, underthe contract 20/2004.

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