Higher threshold parameters in ππ scattering

25 December 1997

Physics Letters B 415 Ž1997. 402–410

Higher threshold parameters in pp scattering B. Ananthanarayan a , P. Buttiker ¨ a

b,1

Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560 012, India b Institut fur ¨ Theoretische Physik, UniÕersitat ¨ Bern, CH-3012 Bern, Switzerland Received 14 July 1997 Editor: R. Gatto

Abstract A family of threshold parameters which probe the stability of chiral predictions is considered. The relevant criteria for the choice of threshold parameters are discussed. Sum rules for these quantities are derived from dispersion relations and evaluated from effective range formulae. Good agreement with two-loop chiral estimates for many of these quantities is found and interesting discrepancies are discussed. q 1997 Elsevier Science B.V. PACS: 12.39.Fe; 13.75.Lb; 11.55.Fv; 25.80.Dj Keywords: Sum rules; pp scattering; Chiral perturbation theory

1. Dispersion relations for pp scattering amplitudes with two subtractions have been rigorously established in axiomatic field theory w1x. It is convenient to consider dispersion relations for s-channel amplitudes of definite iso-spin, TsI Ž s,t,u., I s 0, 1, 2, where s, t, u are the Mandelstam variables. Furthermore, each of the amplitudes may be written down in terms of a unique function as Ts0 Ž s,t,u. s 3 AŽ s,t,u. q AŽ t,u, s . q AŽ u, s,t ., Ts1 Ž s,t,u . s AŽ t,u, s . y AŽ u, s,t ., Ts2 Ž s,t,u. s AŽ t,u, s . q AŽ u, s,t .. Unitarity, analyticity and crossing symmetry have been used extensively to study this fundamental process of elementary particle physics. Introducing a partial wave expansion for these amplitudes via TsI Ž s,t,u. s 32p ÝŽ2 l q 1. f lI Ž s . Pl ŽŽ t y u.rŽ s y 4.., elastic unitarity implies above threshold and below the fourpion threshold that the partial wave amplitudes may

1

E-mail: [email protected].

be described in terms of the phase shifts d lI Ž s . by f lI Ž s . s sr Ž s y 4 . expŽ i d lI Ž s ..sin d lI Ž s ., where we have set the pion mass Ž mp s 139.6 MeV. to unity. Note the threshold expansion Re f lI Ž n . s n l Ž a lI q b lIn q c lIn 2 q d lIn 3 q . . . ., n s Ž s y 4.r4 ) 0 being the square of the three momentum in the centre of mass frame, which defines the scattering lengths a lI, the effective ranges b lI and the higher threshold parameters c lI and d lI, etc. Sum rules have been established for Žcombinations of. scattering lengths and effective ranges in the past employing analyticity and crossing symmetry constraints. One culmination of the dispersion relation approach to pp scattering has been the Roy equations w2,3x which is a system of coupled integral equations for partial wave amplitudes which further trades the two unknown t-dependent functions in fixed-t dispersion relations for the scattering lengths a 00 and a02 . Roy equation fits to phase shift data w4,5x have been extensively studied. Best fits to data give a 00 s 0.26 " 0.05 w6x.

0370-2693r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 2 3 1 - 8

(

B. Ananthanarayan, P. Buttikerr Physics Letters B 415 (1997) 402–410 ¨

2. Chiral perturbation theory w7x is the effective low energy theory of the standard model and describes processes involving pionic degrees of freedom, viewed as the approximate Goldstone bosons of spontaneously broken axial-vector symmetry of massless QCD. At leading order the pp scattering amplitude is given by the Weinberg result from PCAC: AŽ s,t,u. s Ž s y 1.rFp2 w8x. ŽIn chiral perturbation theory this amplitude has now been computed to one-loop w7x and even two-loop order w9,10x.. Furthermore this implies that the only non-vanishing threshold parameters are a 00 s 7rŽ32p Fp2 ., a 20 s y1rŽ16p Fp2 ., a11 s 1rŽ24p Fp2 ., b 00 s 1rŽ4p Fp2 . and b 02 s y1rŽ8p Fp2 .. It is important to note the well known result that the set of functions: t 0 s Ž2 b Ž s y 4r3. q 5 ar3.rŽ Fp2 ., t 1 s b Ž t y u.rŽ Fp2 ., t 2 s Žyb Ž s y 4r3. q 2 ar3.rŽ Fp2 ., where a and b are arbitrary real constants, verifies dispersion relations with two subtractions and vanishing absorptive parts. The Weinberg amplitude is a special case of this general linear amplitude, with a s b s 1. It may be noted that a generalized version of chiral perturbation theory that is motivated by considerations of a small quark condensate in the QCD vacuum allows a to vary over a range between unity and as much as three, and reorganizes the chiral power counting w11x. In this discussion we confine ourselves to the more predictive standard chiral perturbation theory with a s 1 for much of our discussion. Dispersion relations with two subtractions have been used to write down sum rules for Žcombinations. of some of these threshold parameters, for example the Wanders sum rules which may be written down as:

6 b 02 s y2 a 00 q 5a 02 q

403

1 4p

` 2

H0

dn

(Ž n Ž n q 1. .

3

= 2 s 0 Ž n . q 3ns 1 Ž n . q Ž 7n q 6 . s 2 Ž n . y6 Ž 2 n q 1 . s 2 Ž 0 . , where s I Ž n . ' Ž4pr n Ž n q 1 . . P ÝŽ2 l q 1. = Im f lI Ž n . are the cross-sections. The Weinberg predictions for the quantities involved in these sum rules obey these relations identically with all the cross-sections set to zero, since the original amplitude obeys dispersion relations with vanishing absorptive parts. Furthermore, it may be noted that since these are relations at leading order in the chiral expansion, it would be fair to expect that the relevant dispersion integrals make numerically less significant contributions to 18 a11 , 3b 00 and 6 b 02 compared to 2 a 00 y 5a02 , even at higher orders in the chiral expansion. The presence of two subtractions in the dispersion relations where a 00 and a02 play effectively the role of subtraction constants, render them difficult to pin down on the basis of dispersion relation phenomenology alone: Õital chiral inputs are required to make sharp predictions for these quantities.

(

3. At one-loop order, chiral perturbation theory requires the introduction of several coupling constants in L Ž4., which account for the non-renormalizable character of the non-linear sigma model Lagrangian, L Ž2., which is the basis of the Weinberg result. Four of these constants l i , i s 1, 2, 3, 4 enter pp scattering and in particular l 1 and l 2 enter predictions for c 00 , c 02 , b 11 , a20 and a22 , while all higher threshold parameters will receive no contribu` tions from the trees generated by L Ž4. at this order. 1 d n 18 a11 s 2 a00 y 5a20 y In the past experimentally known values for a 02 and 4p 2 0 Ž n Ž n q 1 . . 3 a22 w6x were used to fix these quantities w7x, while no values have been reported in the literature for c lI and = 2 ns 0 Ž n . y 3 Ž 3n q 2 . s 1 Ž n . y 5ns 2 Ž n . , b 1. For completeness we note the one-loop formulas 1 for c lI since they have been reported earlier: ` 1 dn 3b 00 s 2 a 00 y 5a 20 q 2 1 4p 0 Ž n Ž n q 1 . . 3 c 00 s Ž y295 q 88l1 q 112 l 2 . 2304p 3 Fp4 = Ž 4n q 3 . s 0 Ž n . y 3 Ž 2 n q 1 . s 0 Ž 0 . 1 2 c s Ž 1. 1 2 Ž y193 q 40 l1 q 160l 2 . 0 y3ns Ž n . q 5ns Ž n . , 5760p 3 F 4

H

(

H

(

p

B. Ananthanarayan, P. Buttikerr Physics Letters B 415 (1997) 402–410 ¨

404

Recently, rapidly converging sum-rules were written down w12x in order to estimate b 11 which in principle could be used to fix the values of the two chiral coupling constants of interest, indeed, as one could from the values of c 0I should they be known. However, systematic ambiguities inherent to one-loop predictions for these quantities have also been discussed recently. l 1 and l 2 were fixed instead by rewriting the chiral and axiomatic representations of the scattering amplitudes to the appropriate order in the momentum expansion w13x. This method is now being extended to the two-loop case in order to fix the coupling constants that enter at that order in CHPT. Note that the trees generated by L Ž6. will now contribute to the threshold parameters d 0I , c11, b 2I and a13 . At the one-loop level they are given by the expressions: d 00 s y1643rŽ40320p 3 Fp4 ., d 02 s y893rŽ40320p 3 Fp4 ., c11 s y23rŽ13440p 3 Fp4 ., b 20 s y 4 8 1 r Ž 2 0 1 6 0 0 p 3 Fp4 . , b 22 s y 2 7 7 r Ž201600p 3 Fp4 . and a13 s 11rŽ94080p 3 Fp4 .. To summarize, we note that the trees generated by L Ž2., contribute to a0I

5. We work in the approximation that the absorptive parts are modeled entirely by the S- and P-waves ŽIn the numerical analysis we further assume that the contribution to the dispersion integrals from the Sand P-waves also above the KK threshold may be neglected as in Ref. w12x.. which has been found to be justified phenomenologically in the past and is supported today by chiral power counting. In this approximation it is particularly convenient to represent the iso-spin amplitudes as w3,14x: T I Ž s,t ,u . 2

s 32p

I s0

Ž 2.

a11

1 q

while those generated by L b 0I a11

Ž4.

c 0I

q

H4

X

q3

dx

c 0I b 11

c11 b 2I

q

d 0I

x Ž x y 4. C stII

X

q

½

II u Ž u y 4 . C su

X

q

II C su

xyu

Ž 4.

a13 . 4. The main purpose of this letter is precisely to provide estimates to those quantities appearing in Eq. Ž3. and Eq. Ž4. for which no information is available in the literature and to compare whenever possible with the predictions of CHPT. Such a consistency check may be viewed as a probe into the range of validity in energy of chiral predictions. Indeed, when Roy equation fits to the planned precision experi-

X

xys X

xyu

s Ž t y u . I II

uŽ t y s .

X

s Ž s y 4 . I II

t Ž s y u . C stII

xys

b 0I

X

Ž sI II q tCstII q uCsuII . a0I

`

t Ž t y 4.

Ž 3.

and those of L Ž6. contribute to

a2I

ž

X

1 4

xyt

b 11

a11

p

contribute to

a2I

a 0I

Ý X

b 0I

a 0I

ments are performed, all the quantities discussed here may be evaluated afresh, which would then amount to a high precision experimental determination of these numbers. Such Roy equation fits may then be employed to evaluate chiral parameters determined from dispersion relation phenomenology, and such a consistency check may then be performed again.

X

Im f 0I Ž x . X

xyt X X

Im f 1I Ž x .

5/

.

Ž 5.

Projecting these amplitudes onto partial waves will yield the Roy equations for each of the waves in the S- and P-wave approximation. wThis approximation is equivalent to setting the so-called driÕing terms of the Roy equations to zero. However, in the numerical work described below, we evaluate the contribution of the f 2 Ž1270. resonance w15x to the threshold parameters of interest as a measure of the contribution of the driving terms.x The most convenient manner in which the sum rules of interest may be computed is to consider the Roy equations for each of the partial waves f 0I , f 11, f 2I and f 31 in the neighborhood of the threshold, as power series in n .

B. Ananthanarayan, P. Buttikerr Physics Letters B 415 (1997) 402–410 ¨

Furthermore, in order to isolate the quantities of interest we need to consider the Cauchy Principal Value of the relevant integrals. The Principal Value singularity occurs due to self-coupling of the waves and must be removed during the process of computing the power series of the real parts of the waves. An example is worked out for the I s 0 S-wave in Appendix A and the other waves may be treated analogously. The complete set of sum rules for the quantities of interest that have not been published in the past is listed in Appendix B.

Table 1 List of inputs for the effective range formula

6. The spirit of this work will closely follow that of Ref. w14x wherein a modified effective range formula was employed in order to model the absorptive parts of the amplitudes. Note that the dispersion integrals we encounter for Žmost of. the quantities of interest are very rapidly converging which implies that it is the near threshold region that needs to be modeled accurately, the region where the effective range formula is applicable. The modified effective range formula for the S- and P-wave phase shifts is of the type first proposed by Schenk w16x and is:

(

tan d 0I Ž n . s

=

tan d 11 Ž n . s

n nq1

b 0I y a 0I rn 0I q Ž a 0I . n

 a11 q

b 11 y a11rn 11 n 4

n 0I n 0I y n

(

=

I 0

3

½a q

n3 nq1 n 11

n 11 y n

Ž 6.

7. Our numerical work requires inputs to the effective range formulae. Much of these inputs are guided by one-loop chiral perturbation theory. We also perform a sample computation with a set of input parameters with a00 corresponding to the best fits to the experimental data of K e4 of 0.26 and the rest of the quantities computed for that optimal Roy equation fit. In Table 1 the complete set of inputs is tabulated, besides the conventional choice n 00 s 8.5, n 11 s 6.6, n 02 s y5.0. In Table 2 we present the computed values of the threshold parameters of interest for the inputs of

405

5

a

a00

b 00

a 20

b 02

a11

b11

I II III IV

0.19 0.20 0.21 0.26

0.237 0.236 0.236 0.231

y0.04 y0.036 y0.035 y0.021

y0.074 y0.074 y0.075 y0.076

0.035 0.035 0.035 0.036

0.006 0.006 0.006 0.006

Table 1. The contribution of the f 2 resonance w15x in the narrow width approximation to the quantities of interest, calculated by plugging in the appropriate absorptive part into the relevant Roy equation and computed in the appropriate limit, yields the results tabulated along f 2 in Table 2. This should be considered as setting the scale of the corrections arising from all the higher waves and the higher energy tail that would be described in terms of Regge parameterization of the absorptive parts and possible Pomeron contributions. It may be judged from this that the bulk of the contribution is received from the S- and P-wave low energy absorptive parts. In Table 3 we tabulate the one-loop and two-loop predictions for these quantities whenever available w9,17x. Let us first consider the one loop predictions for c 0I which receive contributions from l 1 and l 2 . The latter computed from optimal Roy equation fits of Ref. w13x have been inserted into Eq. Ž1. to produce the entries in the first two columns of Table 3. The rest of the quantities are of the pure one-loop variety and their values numerically tabulated. Alternatively, one may take the values of the c 0I and solve for l 1,2 via Eq. Ž1.. Taking the extreme values wincluding the contribution of the f 2 x for y0.0049 F c 00 F 0.010 and 0.012 F c 02 F 0.015 we obtain y4.7 F l 1 F 0.08 and 3.8 F l 2 F 5.7. While these numbers are not to be taken literally since the contributions of the high energy tail are completely neglected and the effective range formula is not a real substitute for a Roy equation solution of the lowest partial waves, they continue to provide an important consistency check on the values of these coupling constants. These are consistent with several prior determinations for these quantities. The comparison of the values of c 0I with the two-loop predictions continue to be encouraging as we observe from Tables 2 and 3. Some attention may be paid to the quantity c11 : at one-loop order it is

B. Ananthanarayan, P. Buttikerr Physics Letters B 415 (1997) 402–410 ¨

406

Table 2 Higher threshold parameters computed from sum rules for the inputs of Table 1. The contribution of f 2 is also listed a

c 00

c 02

d 00

d 02

c11

b 20

b 22

a13

I II III IV f2

0.0084 0.0061 0.0041 y0.0049 0.0016

0.012 0.013 0.013 0.015 0.00024

y0.016 y0.016 y0.016 y0.017 y0.000030

y0.0045 y0.0047 y0.0050 y0.0061 y0.0000304

0.00080 0.00084 0.00087 0.00096 5.7 P 10y6

y0.00031 y0.00031 y0.00032 y0.00036 0.000016

y0.00028 y0.00029 y0.00030 y0.00035 2.2 P 10y6

0.000049 0.000051 0.000053 0.000060 3.2 P 10y7

negative where as the two-loop prediction is positive and larger in magnitude. This value creeps up towards the value provided by the sum rules. It could be that the presence of the r as a non-perturbative feature of hadron dynamics is responsible for the mismatch between even the two-loop prediction and the sum rule result for c11. Furthermore, the Taylor series of Re f lI Ž n . s n qn 1 sinŽ2 d lI Ž n ..r2 computed with the effective range formulae for the phase shifts, itself yields coefficients c lI and d 0I which are comparable with the quantities computed from the sum-rules. This proves to be a check on the effective range parameterization itself. For instance, for the choice I in Table 1, we find c 00 s 0.0092, c 02 s 0.011, c11 s 0.00092, d 00 s y0.013, d 02 s y0.0012, whereas for that of choice IV, we have c 00 s y0.0025, c 02 s 0.011, c11 s 0.00084, d 00 s y0.015, d 02 s y0.0013. Thus we see that detailed considerations of unitarity, analyticity and crossing in the near threshold region is in excellent agreement with chiral predictions – these, however, do not suffice to discriminate between the standard and generalized scenarios of chiral perturbation theory.

(

8. In order to come to grips with these numbers, we have also chosen to compare the predictions for

the P-wave threshold parameters with numbers arising from resonance saturation with the r . In particular, the narrow width formula for the absorptive part generated by the r : x Im f 11 Ž x . s pGr mr d Ž x y mr2 . Ž 7. xy4 gives rise to an amplitude AŽ s,t,u. by inverting Eq. Ž5. via AŽ s,t,u. s ŽT 0 Ž s,t,u. y T 2 Ž s,t,u..r3,

(

A r Ž s,t ,u . s

Ž 48p . Gr mr2 Ž mr2 y 4 .

=

ž

t Ž s y u. mr2 y t

3r2

q

uŽ s y t . mr2 y u

/

.

Ž 8.

This is identical to the result in Eq. ŽC.9. of Ref. w7x obtained from effective Lagrangian techniques with the appropriate identification of the relevant coupling constant when we take into account the identities, 1rŽ mr2 y t . s 1rmr2 P Ž1 q trŽ mr2 y t .. and w t Ž s y u. q uŽ s y t .x s wy2Ž s y 2. 2 q 1r2Ž s 2 q Ž t y u. 2 .x. This is a consequence of two subtractions in the dispersion relations used here. The formula Eq. Ž8. when inserted into the I s 1 amplitude and projected onto the P-wave yields for a11 y Ž2 a 00 y 5a20 .r18 s 0.0073, b 11 s 0.0077, c11 s 0.0006, when parameters of the resonance w15x are inserted into the formulas

Table 3 Table of values of quantities computed at one- and two-loop Žstandard and generalized. order. For the ClI the quantities are computed from l1 and l 2 listed in Table 1 of Ref. w13x. The two-loop quantities are from w9x and w17x a

c 00

c 02

d 00

d 02

c11

b 20

b 22

a13

O-L

0.0075 0.0081 0.0087 0.0068

0.015 0.015 0.015 0.013

y0.0066

y0.0036

y0.00028

y0.00039

y0.00022

0.000020

y0.017

y0.0049

0.00032 0.00046

y0.00031 y0.00034

y0.00031 y0.00035

0.000050 0.000058

T-LŽS. T-LŽG.

B. Ananthanarayan, P. Buttikerr Physics Letters B 415 (1997) 402–410 ¨

for these quantities. Note the r contribution to l 1 and l 2 are yp mrrŽ2 Gr . and p mrrŽ4 Gr . respectively wsee, e.g., Eq. ŽC.10. in w7xx. Another treatment of resonance saturation w18x with the r is based on writing an unsubtracted dispersion relation for the I s 1 t-channel amplitude divided by Ž s y u.. This is expected to converge under the assumption of the validity of the Pomeranchuk theorem, supported by the behavior of the relevant Regge trajectory. The r contribution to l 1 and l 2 are, however, yp mrrŽ4 Gr . and p mrrŽ4 Gr . respectively was read off from Eqs. Ž28. and Ž32. of w18xx. This is in variance with Eq. ŽC.10. of w7x. This must result from considering dispersion relations without a sufficient number of subtractions that guarantee the absence of crossing constraints on the absorptive parts due to the P-wave. wAn analogous situation arises with the resonance saturation with states of l G 2 of fixed-t dispersion relations with two subtractions; this does not yield a crossing symmetric amplitude and must be treated with care.x

Acknowledgements It is a pleasure to thank H. Leutwyler for discussions and insights. BA thanks B. Moussallam and PB thanks J. Gasser for discussions.

Appendix A. Derivation of the sum rules The three lowest partial waves can be written as: n Re f 00 Ž n . s a00 q Ž 2 a 00 y 5a 02 . 3 `

2

q4

`

X

H0 dn K

P

Ý Ý X

X

I s0 l s0

X X

l I 00

X

Ž n ,n X . Im f lIX Ž n X . Ž A.1 .

Re f 11 Ž n . s Ž 2 a00 y 5a20 . `

2

q4 9. In the preceding sections we have established a program of comparison between families of threshold parameters for which sum rules have been derived and the quantities numerically estimated, and then compared with one- and two-loop predictions for these quantities. The role of the number of subtractions in dispersion relations is crucial and has been emphasized in a variety of ways. Other recent efforts in this confrontation and comparison between data, dispersion relations and chiral perturbation theory use extrapolation to subthreshold regions reaching analogous conclusions regarding the interplay of analyticity and unitarity w19x. Our work is also in keeping with the expectation expressed in Ref. w20x that sensitive tests of QCD in low energy pp scattering should use all theoretical constraints and pertinent low energy observables. It should, however, be emphasized that we have considered only quantities that do not require vital chiral inputs such as a 0I : work is in progress to extract sharp predictions for these quantities. After this work was completed, we have received a preprint w22x where several O Ž p 6 . coupling constants have been evaluated. It would be important to study the implications of this evaluation to the program discussed here and vice versa.

407

`

X

X

H0 dn K

P

Ý Ý X

I s0 l s0

n 18 X X

l I 11

X

Ž n ,n X . Im f lIX Ž n X . Ž A.2 .

Re f 02 Ž n . s a02 y Ž 2 a 00 y 5a02 . `

2

q4

`

X

X

H0 dn K

P

Ý Ý X

I s0 l s0

X X

l I 02

n 6 X

Ž n ,n X . Im f lIX Ž n X . Ž A.3 .

As an example we give the derivation of the higher threshold parameters for I s 0, l s 0. Working in the S- and P-wave approximation, the summation over the angular momentum l runs from zero to one. Besides, this is equivalent to projecting Eq. Ž5. on the relevant partial wave. The contributions of the three lowest partial waves to f 00 Ž n . are explicitly written down as: f 0Žn . '

1

p

`

H0 dn

P

X

½

2n

1 X

n yn

y

X

3n Ž 1 q n X .

3 q 5n X y

3n X Ž 1 q n X . 2

q

3n

ln

nqnXq1 nXq1

5

Im f 00 Ž n X . ,

B. Ananthanarayan, P. Buttikerr Physics Letters B 415 (1997) 402–410 ¨

408

1

f 1Žn . '

`

P

H0

p

3 Ž 3n q 2 n X q 4 .

½

dn X y

X

We note that the integrals not involving G in Eq. ŽA.8. are free of singularities. Furthermore GŽ n . y GŽ0. is also free of singularities as we show below. Consider

X

n Ž1qn .

X

6 Ž 2 n q n q 2.

q

nn X nqnXq1

=ln 1

f 2Žn . '

nXq1 `

H0 dn

P

p

10 q 3n

ln

X

½

5

G Ž n . y G Ž 0. s

X

Im f 11 Ž n . ,

H0

3n X Ž 1 q n X .

nqnXq1

5

nXq1

Im f 02 Ž n X . ,

Ž A.4 .

such that Re

f 00

Žn .

s a00 q

Ž

2 a 00 y 5a 02

P

H0

.3

qf 0Žn . qf 1Žn . qf 2 Žn . .

Ž A.5 .

The singularity of the integrand resides in the first term of f 0 Ž n . . A dding and subtracting Im f 00 Ž n X .rŽp Ž n q n X q 1.. does not change the integral and we may write f 0 Ž n . s GŽ n . q

1

`

H p 0

dn X

½

2 3n

3n X Ž 1 q n X . 1

y

nqnXq1

5

y

Im

dt X

Ž A.10 .

and t X s n X Ž n X q 1 . and t s n Ž n q 1 . we can write

(

`

(

2n X q 1

X

H0 dn (n Ž n q 1. Ž n y n . Ž n q n q 1. s 0 .

P

X

X

X

X

Ž A.11 .

X

n q1

Furthermore, in the S- Žand P-.wave approximation, assuming normal threshold behavior, we have:

3n X Ž 1 q n X . f 00

s0

t X2 yt 2

nqnXq1

3 q 5n X

2 y

ln

P

which still contains the Principal Value singularity. With `

n

p

Ž 2 n X q 1. Ž n q n 2 . X 0 X X X X Im f 0 Ž n . Ž n y n . Ž n q n q 1. Ž n q 1. n Ž A.9 .

`

5Ž n y 2n X .

1

X

Žn . ,

Im f 00 Ž n . s

Ž A.6 .

1 4p

(n Ž n q 1. s

0

Žn .

Ž A.12 .

and therefore we can write

where GŽ n . s `

H0 dn

1

p

G Ž n . y G Ž 0. P

1 s

X

2n q 1

X X

X

Ž n y n . Ž n q n q 1.

Im f 00 Ž n X . .

Ž A.7 .

`

=

H0

Consider the difference f 00 Ž n . y f 00 Ž 0 . s Ž 2 a 00 y 5a 02 .

n 3

q G Ž n . y G Ž 0.

q f 0 Ž n . y f 0 Ž 0. q f 1 Ž n . y f 1 Ž 0. q f 2 Ž n . y f 2 Ž 0.

Ž A.8 . f 0 Ž n . ' f 0 Ž n . y GŽ n .

4p 2 dn X

Ž 2 n X q 1. Ž n q n 2 . (n X Ž n X q 1. Ž n X y n . Ž n q n X q 1.

= s 0 Ž n X . y s 0 Ž n . 4

Ž A.13 .

which is seen to be free from the Principal Value singularity. The higher threshold parameters are now given by the Taylor coefficients of the difference f 00 Ž n . y f 00 Ž0..

B. Ananthanarayan, P. Buttikerr Physics Letters B 415 (1997) 402–410 ¨

409

Appendix B. Sum rules for the higher threshold parameters

Similar one gets the threshold parameters for other waves:

For the higher threshold parameters considered in the text and which have not been considered in the literature w21x, we find

c11 s

64

c 00 s

H0 dn

p

288 Ž 1 q n .

p

y 32 n Ž 1 q n .

3

3

1024 Ž 1 q n .

Im f 02 Ž n .

Im f 11 Ž n . y

q

ž

1

1

64n 3

q 288 Ž 1 q n .

(n Ž n q 1.

y

3

256pn Ž 1 q n .

3

/

Im f 00 Ž n .

½

s 0 Ž n . < ns0

dn

p

5

`

H0 dn

y 1536 Ž 1 q n .

2 q 5n 512 n Ž 1 q n .

ž

4

5

256n

1 y 1536 Ž 1 q n .

(n Ž 1 q n .

y

½

1qnqn 2 d 2 n 2 Ž 1 q n . dn

1 q 2n q 2n 2 q

n 3Ž1qn . 1 q

3

d

Im f 11 Ž n .

1 q 2n d 2

s 1 Ž n . < ns0

1

`

H dn p 0

288 Ž 1 q n .

q 64n Ž 1 q n . q

ž

3

3

Im f 00 Ž n .

Im f 11 Ž n .

1

1

64n

3

q 576 Ž 1 q n .

(n Ž n q 1. 3

3

/

Im f 02 Ž n .

Ž 1 q 2n .

½

= Ž1qnqn 2 . s 2 Ž n .

/

Im f 00 Ž n .

qŽ n q n 2 .

d 02 s

256

p

dn

s 2 Ž n . < ns0

y 1536 Ž 1 q n .

1024n Ž 1 q n . q

5

Ž B.2 .

y

ž

4

Ž B.4 .

1 4

Im f 00 Ž n .

4

Im f 11 Ž n .

1 256n

5

1

`

H0 dn

y

2

s 0 Ž n . < ns0

d

2 q 5n

s 0 Ž n . < ns0

s 0 Ž 0.

2 Ž n q n . dn 2 2

4

/

4

3072pn Ž 1 q n . n q n 2 dn 2

y

Ž 1 q 2n .

1024pn Ž n q 1 . =

5120 Ž 1 q n .

(n Ž 1 q n .

Im f 02 Ž n .

Im f 11 Ž n .

1 4

256p

256pn 3 Ž 1 q n .

q

q

4

y

1 q 2n

d

Im f 02 Ž n .

2 q 11n

1

64

c 02 s

Ž B.1 . 256

ž

4

Im f 00 Ž n .

4

Ž B.3 .

Ž 1 q 2n .

3

= s 0 Ž 0. Ž 1 q n q n 2 . q

d 00 s

2560 Ž 1 q n .

1

q 1 q 2n

1

`

H0 dn

y

5

`

256

y 3072 Ž 1 q n .

(n Ž 1 q n . 1024pn Ž 1 q n .

4

/

Ž 1 q 2n .

Im f 02 Ž n .

B. Ananthanarayan, P. Buttikerr Physics Letters B 415 (1997) 402–410 ¨

410

=

½

1qnqn 2 d 2

n 2 Ž 1 q n . dn

1 q 2n q 2n 2 q

n 2 Ž1qn . 1 q

3

References

s 2 Ž n . < ns0

2

s Ž 0.

d2

2 Ž n q n 2 . dn 2

s 2 Ž n . < ns0

5

Ž B.5 .

In the same way one gets the threshold parameters for I s 0, l s 2, I s 2, l s 2: b 20 s

1

`

1

H dn n Ž 1 q n . 30p 0

4

yn Im f 00 Ž n .

q Ž 3n y 6 . Im f 11 Ž n . y 5n Im f 02 Ž n . b 22 s

1

`

1

H dn n Ž 1 q n . 60p 0

4

Ž B.6 .

y2 n Im f 00 Ž n .

y Ž 3n y 6 . Im f 11 Ž n . y n Im f 02 Ž n . d 0I ,

c11

Ž B.7 . b 2I

A check on the sum rules for and is to saturate the right hand sides with the lowest order chiral phase shifts which should then yield the pure one-loop formulae for these quantities due to the perturbative unitarity of the chiral expansion. The sum rules for c 0I do not converge fast enough; one may consider appropriate linear combinations with say the D-wave scattering lengths a 2I in which the l 1,2 dependence drop out and to saturate the corresponding sum rules with the lowest order chiral phase shifts and reproduce the one-loop formulae for these combinations.

w1x For a comprehensive review, see, G. Wanders, Springer ŽEd... Tracts in Modern Physics 57 Ž1971. 22, G. Hohler ¨ w2x S.M. Roy, Phys. Lett. B 36 Ž1971. 353. w3x J.-L. Basdevant, J.C. Le Guillou, H. Navelet, Nuo. Cim. A 7 Ž1972. 363. w4x J.-L. Basdevant, C.D. Froggatt, Nucl. Phys. B 72 Ž1974. 413. w5x M.R. Pennington, S.D. Protopopescu, Phys. Rev. D 7 Ž1973. 1429. w6x M.M. Nagels et al., Nucl. Phys. B 147 Ž1979. 189. w7x J. Gasser, H. Leutwyler, Ann. Phys. 158 Ž1984. 142. w8x S. Weinberg, Phys. Rev. Lett. 17 Ž1966. 616. w9x M. Knecht et al., Nucl. Phys. B 457 Ž1995. 513. w10x J. Bijnens et al., Phys. Lett. B 374 Ž1996. 210. w11x M. Knecht, J. Stern, Contribution to the 2nd DAFNE Handbook, p. 169; M. Knecht, B. Moussallam, J. Stern, Contribution to the 2nd DAFNE Handbook, p. 221. w12x B. Ananthanarayan, D. Toublan, G. Wanders, Phys. Rev. D 53 Ž1996. 2362. w13x B. Ananthanarayan, P. Buttiker, Phys. Rev. D 54 Ž1996. ¨ 1125. w14x G. Mahoux, S.M. Roy, G. Wanders, Nucl. Phys. 70 Ž1974. 297. w15x Particle Data Group, M.R. Barnett et al., Phys. Rev. D 54 Ž1996. 1. w16x A. Schenk, Nucl. Phys. B 363 Ž1991. 97. w17x B. Moussallam, unpublished. w18x M.R. Pennington, J. Portoles, ` Phys. Lett. B 344 Ž1995. 399. w19x M.R. Pennington, J. Portoles, ` Phys. Rev. D 55 Ž1997. 3082. w20x M. Girlanda, M. Knecht, B. Moussallam, J. Stern, preprint, hep-phr9703448. w21x B. Ananthanarayan, P. Buttiker, Phys. Rev. D 54 Ž1996. ¨ 5501. w22x G. Wanders, hep-phr9705323.