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Comment on the prediction of two-loop standard chiral perturbation theory for low-energy ππ scattering
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18 September 1997
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t!!a /
PHYSICS
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ELSlWIER
LETTERS I3
Physics Letters B 409 (1997) 461468
Comment on the prediction of two-loop standard chiral perturbation theory for low-energy vv scattering L. Girlandaa,
M. Knecht b, B. Moussallam a, J. Sterna
a Division de Physique Thkorique, Institut de Physique Muclkaire, F-91406 Orsay Cedex, France ’ b Centre de Physique Thtzforique , CNRS Luminy Case 907, F-13288 Marseitle Cedex 9, France2 Received 2 April 1997; revised manuscript received 20 June 1997 Editor: R. Gatto
Abstract Four of the six parameters defining the relations. Combining this information with phases and the 0(p4) constants I;, 1;. The the correct D-waves but it is incompatible 1997 Elsevier Science B.V.
two-loop rq scattering amplitude have been determined using Roy dispersion the Standard @I expressions, we obtain the threshold parameters, low-energy result Zg(MP) = (1.6 =t 0.4 f 0.9) x 10m3 (1; = 4.17 f 0.19 f 0.43) reproduces with existing Standard xF’T analyses of K14form factors beyond one loop. @
1. During the last few years there has been a noticeable revival of interest in the high precision analysis of low-energy ~7r scattering [ l-131. There are at least two reasons for this. First, it has been shown [ 13,1,3] and repeatedly emphasized [ 141 that the GT~ scattering amplitude in the threshold region is particularly sensitive to the strength of quark anti-quark pair condensation in the QCD vacuum: the smaller the condensate, the stronger the isoscalar S-wave rr interaction. The accurate measurement of S-wave scattering lengths would, indeed, provide the first experimental evidence in favour of, or against, the standardly admitted hypothesis according to which the mechanism of spontaneous chiral symmetry breaking is dominated by the formation of a large (qq) condensate. Within QCD, this hypothesis is by no means a ’ Unit6 de Recherche des Universitis Paris XI et Paris VI associte au CNRS. ‘Laboratoire propre du Centre National de la Recherche Scientifique.
logical necessity and its experimental test might well become an important step towards a non-perturbative understanding of the quark-gluon dynamics. The second reason which makes detailed Z-G- studies topical, is that there are two new high precision experiments currently under preparation: i) The phase shift difference 4 (E) - 6; (E) at low energies (E < 400MeV) will be extracted from a new K,4-decay experiment [ 151 performed with the KL,OE detector at the Frascati #-factory DAQtNE [ 161. ii) At CERN, the project DIRAC [ 171 aims at the measurement of the lifetime of r+n-atoms to lo%, implying the determination of the combination of scattering lengths lug ui[ with a 5% accuracy. On the theoretical side an even better precision can be reached by a.systematic use of chiral perturbation theory [ 18,191 ( ,@T) . The low-energy expansion of the rr scattering amplitude A(slt,u) starts at order O(p*) given by Weinberg more than 30 years ago [ 201. Subsequently, the oneloop 0(p4) contribution to A(sl t, u) has been cal-
0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00872-l
462
L. Girlanda et al/Physics
culated by Gasser and Leutwyler [ 21,191. It is given by four low-energy constants 11, 12, 13, 14 besides the (charged) pion mass M, and the decay constant F,. The present state of the art involves the two-loop O(p6) order and the present letter concerns this degree of accuracy. 2. The 0(p6) amplitude A(slt,u) given in Ref. [ 1] in the form
has been first
(1) The function A~sF, which depends on the Mandelstam variables s, t, u and on the six parameters (Y, 6, At, . . .h4, is explicitly displayed in [ 11. Here, p denotes the characteristic pion momentum and Au is the mass scale of bound states not protected by the chiral symmetry, Au N 4rF, N 1 GeV. The result ( 1) holds independently of the strength of the quark condensate. The latter merely shows up in the size of the constant CY:for standard, large values3 (&) Y - (250 MeV) 3 one has LYN 1 and its value increases up to cr N 4 for I(@)] decreasing down to zero. The parameter p is less sensitive to the value of the condensate, remaining always close to unity. It has been shown [ 31 that the remaining four constants At, . . . A4 can be rather accurately determined from the existing vr scattering data [22] in the intermediate energy range 0.5 GeV < E < 1.9 GeV using the Roy dispersion relations [ 231. The latter explicitly incorporate crossing symmetry and consequently they strongly constrain the HIT amplitude at low energies. Equating the perturbative formula (1) with the Roy dispersive representation in a whole low-energy region of the Mandelstam plane, one infers the values of At, . . .A4 whereas the parameters ff and p remain essentially undetermined. The resulting Ai’s are almost independent of LYand p. Here we quote and use the central values corresponding to Q = 1.04, /3 = 1.08 [ 31, A1 = (-5.7
f 2.2) x 10-3,
A2 = (9.3 i 0.5) x 10-3, 3 (qq) denotes the single flavour condensate in the W(2) x SU( 2) chiral limit mu = rnd = 0 at the QCD scale Y = 1 GeV.
Letters B 409 (1997) 461-468
A3 = (2.2 f 0.6) x 10-4, A4 = (-1.5
&0.12)
x 10-4.
(2)
The quoted errors include experimental uncertainties on 7r7r phase-shifts and inelasticities in the medium energy region and an estimate of the systematic error arising from neglected higher orders in the low-energy representation ( 1). The errors due to the uncertainty in the high-energy behaviour of the rr scattering amplitude are negligible. 3. With the constants Ai determined, Eq. ( 1) allows one to convert new high-precision experimental information on low-energy ~7r phase shifts and/or threshold parameters into a measurement of (Yand p and finally, into an experimental determination of the quantity (m, + m4) (44) (the detailed relation between (Y and /3 and the condensate can be found in Ref. [ 1] ) . Conversely, Eq. ( 1) can be used to predict, for each value of the condensate, all the low-energy observables. It is of particular importance to assess with as much accuracy as possible the prediction concerning the standard alternative of a large (@) condensate. The strength of the (qq) condensate is conveniently described by the deviation from the Gell-Mann-OakesRenner relation, i.e. by the parameter F2M2 :=5$&-l.
(3)
Here, m = $ (m, + md) is the running quark mass and me is a mass scale characteristic of 4q condensation. The standard alternative of a large condensate corresponds to ma 2 hu. In this special case the ratio (3) can be treated as an expansion parameter, m/m0 = 0 ( p2 /A&) and the general low-energy expansion becomes the standard chiral perturbation theory ( SxPT) [ 191. The complete SXPT two-loop calculation of the rr-scattering amplitude has been recently completed by Bijnens et al. [2]. Not surprisingly, this calculation recovers the formula ( 1) giving, in addition, the expressions of the six parameters ry, p, At,. . . , A4 in terms of i) M,, F,, ii) four 0(p4) constants If&), E;(p), I;(p) and Ei(,u) and finally iii) six 0(p6> constants Y; ( pu> , . . . , t-k
463
L. Girlanda et al./Physics Letters B 409 (1997) 461-468
23 A3 = -18 +
37 277 k1 - 36 k2 + 1990656+
( 19 + 5368 L - 13056 1; - 9600 I;) 414721 z-z
+ r\ - -Jjri , kl + g
A4 = ;
+
(8) kz +
3311 3981312+’
’ (-2-257L+3361’,+8401’,) 10368 rr2 4 - - rr 3 6’
(9)
with d 1; ~~=--~ n=$
3/i
72=$
B=-;r
y4=2,
(10)
and
ki(P)
At=--
1
iL+21;--
f
36~~ 79
-;kt-;k2-;k4+81;1;+-
9216 7p
[ 1 + 2272 L - 2496 1;
+&(-
- 2160 1; - 384 1:) M?r F2) lr
+ r; - t-i + 6r; - 2rk I
A2=-;L+l;--
5 288 z-2
- ; kl - ; kz - ; k4 + 4 1; 1; +
+
(6)
[
1223 331776+
17 + 752 L + 3840 1’1+ 1536 1; + 276481 r2 (
-
1920 1;) + 2J4
1g F;
’
(7)
= (41:(p)
- YiL)L.
(11)
(These expressions are obtained from the expansions of the parameters bt, . . ., b6 originally given in [ 21, which are in one-to-one correspondence with LY,p, Al, . . .. &. We prefer to work with the latter set of parameters for the reader’s convenience: explicit formulae for low-energy observables in terms of a, p, Ai are given in Ref. [ 11,whereas similar expressions in terms of the hi’s are at present not available in the literature). A few points are worth recalling. i) The parameters LY,p, Ai are p-independent. This fact, together with Eq. ( 10) fixes the scale dependence of the low-energy constants I-[( ,u) . ii) Eqs. (4) -( 9) fix the expansion of the parameters CY,j3, Ai in powers of A4: and log Mi (and/or in powers of the quark mass m), sinceI{(p),. Zi ( ,z) and the r[ ( p) are quark mass independent. Contributions of successive chiral orders to cz, /3, Ai can be identified by counting the powers of Mi/F$ Notice that a and p start by an order 0(p2) contribution (a~ = 1, p = 1) followed by 0(p4) and O(p6) corrections. The expansions of Al. A2 consist of 0(p4) and 0(p6) contributions, whereas A3 and A4 are entirely of order O(#). iii) The 0(p4) constants 13 and 14 belong to the explicit symmetry breaking sector of the effective lagrangian. They represent
L. Girlanda et al./Physics
464
Letters B 409 (1997) 461-468
the fine tuning of the (@I) condensate to its presumed large value: in SxPT, the deviation from the GellMann-Oakes-Renner relation (3) is given by [ 191
I( ?T > 2
m -=
m0
[
$
2w+2z;(4L
+.__,
(12)
Similarly, Zi controls the deviation of p from 1. On the other hand, the hi’s are independent of 1; (and only very weakly dependent on Ii) reflecting the fact that they are only marginally sensitive to the size of the (&I) condensate. In the sequel, we complete our de$nition of the standard xPT by adopting the standardly used central values of 1; and lf; [ 19,2] : Z;(M,)
=0.82
x 10-3,
Z;(M,)
= 5.6 x 10-3.
(13)
Finally, the constants 1; and 15 do not describe explicit symmetry breaking effects (they are coefficient of four-derivative terms in Cc4)) and they are insensitive to the size of the quark condensate. They control the parameters A1 and &. 4. Eqs. (4) -( 9) can be used to predict the parameters a, p, At, . . ., A4 and consequently, all low-energy rr scattering observables, provided the low-energy constants II, . . . ,Z4 and rr , . . . , r,j are determined from the analysis of different processes. This is a path advocated by the authors of Ref. [ 21. In the present letter this kind of analysis will be confronted with additional experimental information contained in Eq. (2). Bijnens et al. [2] have used the values (13) for 13 and 14; for 1’1and 15 they have taken the central values obtained from the SxPT analysis of Kr4 form factors [24]: If(&)
= (-5.4%
I;(&)
= (5.7f
1.1) x 10-3, 1.1) x 10-3.
(14)
As for the 0(p6) constants rf (,x), the authors of [ 21 take rf( 1GeV) = 0 and they check that this approximation confronted with a resonance saturation model produces a negligible error. With the values ( 14)) and r-f = 0 at ,X = 1 GeV one obtains (in this letter we always use F, = 93.2 MeV and M, = 139.6 MeV): a=
1.074,
p=
1.105,
Fig. 1. The phase shift difference C$ - 8: in the energy region of K14 decays. The dashed curve is obtained with the values of Rq. (15) and it coincides with the curve displayed in Fig. 1 of Ref. 121. The solid line is obtained with the values of Eqs. (2) and ( 18) while the shaded area results adding the corresponding error bars quadratically. The experimental points are from Ref. [ 321,
A1 = -8.91
x 10-3,
A3 = 2.04 x 10-4,
A:! = 14.5 x 10-3, & = -1.79
x 10-4.
(15)
For these values of the parameters a, p, At,. . . , Ah. one obtains the S-wave scattering lengths a! = 0.218, ai - a: = 0.259 corresponding4 to the predictions given in Ref. [ 21. The resulting phase shift difference 8 -c$ (measurable in Kr4 decays) is shown as a function of the center of mass energy as the dashed line in Fig. 1. It reproduces the curve displayed in Fig. 1 of Ref. [ 21. Finally, a few remaining threshold parameters not discussed in Ref. [ 21 are collected in the first column of Table 1, using the expressions displayed in Appendix D of Ref. [ 11. The numbers in parentheses are obtained keeping in higher orders only those components of a, /3. ht and A:! that actually contribute at most to the order O(p6). These exactly coincide with the corresponding predictions one would obtain using the amplitude given in [ 21. Among the latter it is worth noticing the value predicted for the isoscalar D-wave scattering length ai = 26.3 x 10e4, which is substantially above the value extracted from the analysis of Roy equations [ 251. This disagreement reflects the fact that the 4 Actually these have to be compared with the numbers given in Rq. (4) of Ref. [2] in parentheses (rf( 1 GeV) = 0). The small difference provides an estimate of 0(p8) effects: it is entirely due to the fact that the amplitude A~SF of Ref. [ 11 coincides with the amplitude calculated in Ref. [2] only modulo c3(ps) contributions.
L. Girlanda et al./Physics
Letters B 409 (1997) 461-468
465
Table 1 Threshold parameters of burr scattering (in units of M,+) in the standard framework using the two-loop expressions of Ref. [ 11, App. D. The first column results from the values of Eq. (15) (see the text for the numbers in parentheses). The second column is obtained in the same way but taking the values of Eqs. (2) and ( 18) as input
no 8 b0
-IOU2 -lob!
u; - a; loa; 10’b’ 10%~ lO%lS 10%:3
Bijnens et al. [2]
KMSF
Experiment [ 251
0.218 (0.2156) 0.273 (0.271) 0.411 (0.4094) 0.709 (0.704) 0.259 (0.2565) 0.395 (0.3956) 0.785 (0.784) 0.263 (0.267) 0.237 (0.2356) 0.428 (0.478)
0.209 f 0.004 0.255 i 0.010 0.44 rfI 0.01 0.80 ztO.02 0.254 Jr 0.004 0.373 f 0.008 0.55 f 0.07 0.16 f0.02 0.09 ho.13 0.49 f 0.07
0.26 f 0.05 0.25 + 0.03 0.28 zt 0.12 0.82 f 0.08 0.29 zt 0.05 0.38 f 0.02 0.17 f 0.03 0.13 * 0.30 0.6 f0.2
value ( 1.5) of A2 is significantly above the value (2) inferred from experimental phase shifts in Ref. [ 31. We would like to stress that both the canonical value CZ$= (17 f 3) x 10e4 and the determination of the constant A2 = (9.3 f 0.5) x 10e3 are based on the Roy dispersion relations [23] using the experimental vrr data above 500 MeV as input. Furthermore, in both cases, the dominant contribution comes from the P-wave in the p( 770) region, which explains the relatively small error bars. In order to make the predictions of SXPT [ 21 agree with the values (2) of the parameters Al,. . . , A4 and with the standard value of ai, we proceed as follows: fixing 1; and l$ according to Eq. (13), we solve Eqs. (6) and (7) for Z\(M,), &(MJ’ Ii
= (-4.0f + [-1.1
1.0) x 1o-3
Y; + l.Ori
- 6.3~; +2.1
r~]p=IGeV (16)
Z;(M,)
= (1.6kO.4)
x 1O-3
+ [0.1J,-3.5r;+0.5r;-0.2r~]~10~”
(17)
where the values and errors (2) have been used for ht and AT. Eqs. (16) and (17) are then inserted back into the formulae (4) and (5) for LXand p. Keeping in mind that IX and p are sensitive to 11 and 12 only at next-to-next-to-leading level, the unknown constants Y;( 1 GeV) are viewed as a source of uncertainty in cx and p. Inspired by ndive dimensional analysis [ 261 we
400
E,,, WV)
Fig. 2. The isospin 2, S-wave phase shifts at low energies. The different curves are obtained with the values ( 15) (dashed) and the values (2) and ( 18) (solid). In the latter case tbe shaded area shows the corresponding error band. The experimental points are taken from Ref. [22].
take in the expressions for cr and /?, r-f( 1 GeV) = ( O+ 2) x 10m4. Adding the corresponding uncertainties quadratically, we obtain CY= 1.07 f 0.01 )
p = 1.105 f 0.015.
(18)
It should be stressed that the error in Eq. ( 18) does not include the uncertainty in the low-energy constants Ij and Zi. As in the case of the chiral condensate itself, the constant Zj has not yet been determined experimentally and for this reason it is hard to associate an error bar with it. The values ( 18) have to be viewed as corresponding to the “standard alternative” of a large condensate defined by the values (13) of lj and 1;. We now use the formulae given in Ref. [ 1 ] to generate the predictions for threshold parameters and phase shiftsthatcorrespondtocu,P(18)andAt,...,&(2). Adding the errors quadratically, the resulting threshold parameters are summarized in the second column of Table 1. Observe that we obtain deviations of LZ~ and ug - ui from their centruE experimental values that are larger than those predicted in Ref. [ 21. Notice that now, the D-wave scattering lengths perfectly agree with their Roy-equation “experimental” values as expected from the manner the values (2) of the constants At, . . . , A4 have been obtained. A similar conclusion holds for the phase shift difference @ - St, shown as the solid curve in Fig. 1 with the error band indicated by the shaded area: the curve displayed in Ref. [ 21 is significantly higher, i.e. closer to the experimental central-value points. For illustration, the phase 8: is also shown in Fig. 2.
L. Girlandaet al./ PhysicsLettersB 409 (1997) 461-468
466
5. The origin of the mismatch described in the previous paragraph clearly appears upon comparing Eqs. (16) and (17) with the values (14) of the constants Z;,2( MP) extracted in Ref. [ 241 from the “unitarized” one-loop SXPT KM form factors (Eq. (5.10) of [ 241) . The question is how close the expressions ( 16) and ( 17) can be brought to these values keeping at the same time the S(p6) constants , ri( 1 GeV) at a reasonable magnir;(l GeV),... tude. If one proceeds as before treating the ri’s at 1 GeV as randomly distributed around zero with a standard deviation f2 x 10W4, one gets Z’l(Mp) = (-4.0& Z;(M,)
l.O&l.S)
= (1.6&0.4~0.9)
x 10-3, x 10-3,
(19)
or I; = -0.37
rt 0.95 i 1.71 ,
iz=4.17*o.19f0.43,
(20)
where the first error has its origin in At and A2 (Eq. (2)), whereas the second error arises from the presumed uncertainties in the individual ri’s added quadratically. Two checks of the size of the constants ri are conceivable. i) First, one can make full use of the information contained in Eq. (2) determining the parametersZi,2(Mp) andrj(lGeV),...,r:(lGeV) by a simultaneous fit to Eqs. (6)-( 9) and to the constraints ri( 1 GeV) = 0 f 2 x 10-4. The resulting x2/d.o.f. is 1.9/2 and one obtains 1; (Mp)
= (-4.0
Z$(M,)
= (2.0 i 0.3) x 10-3,
compatible
f 0.5) x 10-3,
with (19),
(21)
whereas for the ri’s one gets
rj ( 1 GeV) = (-0.3
f 2.0) x 10-4,
T-:( 1 GeV) = (-0.7
f 0.9) x 10W4,
r1 = -0.61
x 10-4,
r2 = 1.3 x 10-4,
r3 = -1.70
X 10e4,
r4 = -1.0
f-5 = 1.14 x 10m4,
(22)
This result turns out to be rather stable: if one increases the uncertainties in the Yi’s to f3 x 10B4, the new X2/d.o.f. = 1.24/2, the values (21) become (-4.8 f 0.5) x 10W3 and (2.1 f 0.3) x low3 respectively, and the changes in the ri’s also remain rather modest. On
x 10d4,
rg = 0.3 x 10m4.
(23)
Estimating low-energy constants by resonance saturation does not, in principle, fix the renormalization scale ,X at which the estimate is supposed to hold. Actually, if a constant exhibits a strong scale dependence, its resonance saturation estimate is subject to caution. Interpreting Eqs. (23) as values of 6 ( p) at ZL= 1 GeV, one observes a striking coherence with the preceding analysis: (23) is, indeed, consistent not only with dimensional analysis or with the assumption 161 < 2 x lop4 but, moreover it agrees with the fit (22). One can even repeat the fit to Eqs. (6)-(9) constraining rz( 1 GeV) to the values (23) allowing for a 100% error: the fit is excellent (X2/d.o.f. = 0.91/2) and it yields Zt(Mp) = (-4.0 f 0.5) x 10W3, E;(Mp) = (2.1 f 0.3) x 10m3, again compatible with ( 19) and (2 1) . On the other hand, one finds that between p = M, and ,u = 1 GeV, only the constants ~4, t-5 and rg show a moderate scale dependence: had we assumed that the values (23) concern the scale ,U = Mp (as suggested in Ref. [ 271) , the comparison with our previous analysis would be less favourable as far as the constant t-3 is concerned, fi3( 1 GeV) = -4.9 x 10e4 in this case. Notice however that according to Eq. ( 17) the correction to the “critical” constant Zs( Mp) is dominated by t-i whose scale dependence is rather weak: ri( 1 GeV) = ri(Mp)
t-5( 1 GeV) = ( 1.5 f 0.5) x 10m4, I$( 1 GeV) = (0.4 f 0.9) x 10m4.
the other hand, the errors obtained by this procedure (increase of x2 by one unit) and shown in Eqs. (21) and (22) are probably heavily underestimated. ii) Next, it is instructive to confront the previous discussion with the estimate of the constants ri by resonance saturation as quoted recently by Hannah [ 271:
- 7 x 10e6.
(24)
In order that the constant I;( MP) ( 17) differ from the Kt4 value (14) by at most two standard deviations, the constant ri( 1 GeV) would have to be t-i ( 1 GeV) N -5 x 10M4. This cannot be excluded but it looks unlikely in the light of the present analysis. 6. The constants It and 12 ( 14) have not been obtained from a full two-loop analysis of KM form factors F and G, which is not yet available. Instead, their
L. Girlanda et al/Physics
determination is based on matching a dispersive representation for the form factor F with the one-loop SXPT expressions, the latter merely serving to fix the subtraction constants. This method of “improving” oneloop xPT calculations has been often used in the past [28] and it suffers from a basic ambiguity: one has to assume that the one-loop and two-loop amplitudes practically coincide in a particular kinematical point M. Even if one admits the very existence of such a matching point M, the results can still depend on its choice. In Ref. [ 241 the matching point has been chosen at the threshold s, = 4Mi of the Swave amplitude z-n --f K + axial current, where s, stands for the dipion invariant mass squared. We have repeated the analysis of Ref. [24] for other choices of the matching point between sV = 4M$ and the left-hand-cut branch point s, = 0. We reproduce the result ( 14) and find that it is actually rather insensitive to the matching point except in the vicinity of the singular point So = 0, where the outcome for 11 (but not 12) becomes less stable. For instance, with the matching point at s, = 2M$ we obtain
Z'l(Mp) = (-4.8
-+ 2.1) x 10-3,
Z',(M,) = (5.3 f 1.0) x 10-3.
(25)
Given the present state and quality of Kt4 experimental data, it seems hard to ascribe the discrepancy described above to the SXPT analysis performed in Ref. [ 241. On the other hand, it should be kept in mind that outside the standard framework, i.e. for low values of the condensate (&), the constants Ii and 12 extracted from Kt4 data will be modified already at the one-loop level: since in GxPT the loop contributions are more important, the resulting central values of 1111and 1121 are expected to come out somewhat smaller [ 291. 7. A few concluding remarks are in order. The past determinations [ 19,30,6,7] of the constants Z; and Zi have operated within the 0(p4) order of xPT. They have shown an apparent coherence and compatibility with the Ki4 analyses of Ref. [ 241. This compatibility might be lost at 0(p6> order and we have to understand why. The resonance saturation models are the only ones that determine the constants Et,2 directly, integrating out the resonance degrees of freedom from an extended effective lagrangian Leg. However, incorporating resonances into L:,R is not free of ambigu-
Letters B 409 (1997) 461-468
467
ities, especially if one aims at the 0(p6) accuracy. On the other hand, less model-dependent sources of information, such as TV D-waves [ 191 and/or sum rules [ 6,7] primarily determine the physical parameters Ai, AZ. It turns out that this determination is rather stable and barely affected by switching from order S(p4) to S(p6). At the O(p4) level, i.e. neglecting in Eq. ( 1) the two-loop effects and setting As = A4 = 0, one would get from the u; and ai experimental central values At = -6.4 x 10e3 and A2 = 10.8 x 10V3, to be compared with Eq. (2). In other words, the relationship between D-wave scattering lengths and the parameters At, A2 is almost unaffected by O(p6) effects. The latter however become rather important in the relationship between A2 and I$. Rewriting Elq. (7) to make the dependence on I$_(M,,) appear explicitly, one obtains A2 =
{Z;(M,) + 5.45 x 10-3}
+ (0.32 x
Z;(M,)+ 1.7 x 10-3},
(26)
where the first (second) curly brackets collect all 0(p4) (S(#)) contributions (r4 has been neglected). The 0(p6) contribution is as large as 30% and it is dominated by double logs, whose importance has been anticipated by Colangelo [4]. It follows that for a given A2 (D-waves), the resulting value of I;( M,,)caneasily differ by a factor N 2 depending whether in Eq. (26) one includes the 0(p6) term or not. Whether the consistency with KITform factors can be understood within the large condensate hypothesis remains to be clarified. It might be, for instance, that at 0(p6) level the Kt4 form factors also receive an important contribution from double logs, which the unitarization procedure would not take into account [ 3 11. Independently of this issue, the main conclusion of this letter is the following: within the framework of SXPT we obtain predictions for ai, ug - ai and ?$ - 8: which are systematically below those given in Ref. [ 21, as shown in Fig. 1 and Table 1 of the present paper. A closely related fact is the failure of the values of 5 and Ii used in Ref. [2] to describe the D-waves in agreement with Roy equations analyses. This shows, once more, that a sensible and sensitive test of QCD in low-energy rr scattering should be based on a global analysis making use of all theoretical constraints and all pertinent low-energy observables.
468
L. Girlanda et al./Physics
References
Letters B 409 (1997) 461-468
[ 151 M. Baillargeon and PJ. Fmnzini, in [ 161. Handbook, L. Maiani, G. Pancheri and N. Paver, dds., INFN-LNF (May 1995). atoms to [I71 B. Adeva et al., Lifetime measurement of &rtest low-energy QCD predictions, CERN/SPSLC 95-l. [I81 S. Weinberg, Physica A 96 (1979) 327. 1191 J. Gasser and H. Leutwyler, Phys. L&t. B 125 (1983) 321; Ann. Phys. (NY) 158 (1984) 142. [201 S. Weinberg, Phys. Rev. Lett. 17 (1966) 616. 1211 J. Gasser and H. Leutwyler, Phys. Lett. B 125 (1983) 325. [221 B. Hyams et al., Nucl. Phys. B 64 (1973) 134; G. Grayer et al., Nucl. Phys. B 75 (1974) 189; W. Hoogland et al., Nucl. Phys. B 126 (1977) 109. [231 S.M. Roy, Phys. L&t. B 36 (1971) 353; Helv. Phys. Act. 63 (1990) 627. [241 J. Bijnens, G. Colangelo and J. Gasser, Nucl. Phys. B 427 (1994) 427. [251 0. Dumbrajs et al., Nucl. Phys. B 216 (1983) 277. [261 A. Manohar and H. Georgi, Nucl. Phys. B 234 (1984) 189; H. Georgi and L. Randall, Nucl. Phys. B 276 (1986) 241; H. Georgi, Phys. Lett. B 298 (1993) 187. t271 T. Hanah, preprint IFA-97-04, hep_ph/9701389. 1281 J.F. Donoghue, J. Gasser and H. Leutwyler, Nucl. Phys. B 343 (1990) 341; J.F. Donoghue and B.R. Holstein, Phys. Rev. D 48 (1993) 137; M. Knecht, B. Moussallam and J. Stem, Nucl. Phys. B 429 (1994) 125; A.V. Anisovich and H. Lcutwyler, Phys. Lett. B 375 (1996) 335. [291 M. Knecht et al., to be published; M. Knecht, in: 2nd Euroda@ne Collaboration meeting Frascati (1994), LNF-94/033. [30] G. Ecker, J. Gasser, A. Pith and E. de Rafael, Nucl. Phys. B 321 (1989) 311. [ 311 H. Leutwyler, private communication. [32] L. Rosselet at al., Phys. Rev. D 15 (1977) 574; P. Estabrooks and A.D. Martin, Nucl. Phys. B 79 (1974) 301.
[ 161 The Second Da@ne Physics
[l] M. Knecht, B. Moussallatn, J. Stem and N.H. Fuchs, Nucl. Phys. B 457 (1995) 513. [2] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser and M. Sainio, Phys. Lett. B 374 (1996) 210. [3] M. Knecht, B. Moussallam, J. Stem and N.H. Fuchs, Nucl. Phys. B 471 (1996) 445. [4] G. Colangelo, Phys. Lett. B 350 (1995) 85; B 351 (1995) 234 (E). Y51 B. Ananthanarayan, D. Toublan and G. Wanders, Phys. Rev. D 51 (1995) 1093; D 53 (1996) 2362; G. Wanders, Helv. Phys. Act. 70 (1997) 287. [61 B. Ananthanarayan and P. Biittiker, Phys. Rev. D 54 (1996) 1125; D 54 (1996) 5501. [71 M.R. Pennington and J. PortoKs, Phys. Lett. B 344 (1995) 399. [81 M.R. Pennington and J. Portok%, Phys. Rev. D 55 (1997) 3082. 191 D. Morgan and M.R. Pennington, in [ 161; M. Knecht, B. Moussallam and J. Stem, in [ 161; J. Gasser, in [ 161. 1101 M.V. Polyakov and V.V. Vereshagin, Phys. Rev. D 54 (1996) 1112. [Ill E Sannino and J. Schechter, Phys. Rev. D 52 (1995) 96; M. Harada, F. Sannino and J. Schechter, Phys. Rev. D 54 (1996) 1991. Cl21 A.A. Bolokhov et al., preprint SPBU-IP-96-39, hepphl9612473; M.V. Polyakov and G. Weidl, hep-ph/9612486. [I31 N.H. Fuchs, H. Sazdjian and J. Stem, Phys. Lett. B 269 (1991) 183; J. Stem, H. Sazdjian and N.H. Fuchs, Phys. Rev. D 47 (1993) 3814. El41 J. Stem, in: Workshop on Physics and Detectors for Da@ne ‘96, R. Baldini, E Bossi, G. Cafon, G. Pancheri, eds., Frascati Physics Series 4 (1996), hep-ph/9510318; M.R. Pennington, in: DAPHCE workshop on hadron dynamics with the new Da@ne and TJNAF facilities, Frascati (1996), hep-phI9612417; H. Leutwyler, in: DAPHCE workshop on hadron dynamics with the new Da@ne and TJNAF facilities, Frascati (1996).
18 September 1997
mm
t!!a /
PHYSICS
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ELSlWIER
LETTERS I3
Physics Letters B 409 (1997) 461468
Comment on the prediction of two-loop standard chiral perturbation theory for low-energy vv scattering L. Girlandaa,
M. Knecht b, B. Moussallam a, J. Sterna
a Division de Physique Thkorique, Institut de Physique Muclkaire, F-91406 Orsay Cedex, France ’ b Centre de Physique Thtzforique , CNRS Luminy Case 907, F-13288 Marseitle Cedex 9, France2 Received 2 April 1997; revised manuscript received 20 June 1997 Editor: R. Gatto
Abstract Four of the six parameters defining the relations. Combining this information with phases and the 0(p4) constants I;, 1;. The the correct D-waves but it is incompatible 1997 Elsevier Science B.V.
two-loop rq scattering amplitude have been determined using Roy dispersion the Standard @I expressions, we obtain the threshold parameters, low-energy result Zg(MP) = (1.6 =t 0.4 f 0.9) x 10m3 (1; = 4.17 f 0.19 f 0.43) reproduces with existing Standard xF’T analyses of K14form factors beyond one loop. @
1. During the last few years there has been a noticeable revival of interest in the high precision analysis of low-energy ~7r scattering [ l-131. There are at least two reasons for this. First, it has been shown [ 13,1,3] and repeatedly emphasized [ 141 that the GT~ scattering amplitude in the threshold region is particularly sensitive to the strength of quark anti-quark pair condensation in the QCD vacuum: the smaller the condensate, the stronger the isoscalar S-wave rr interaction. The accurate measurement of S-wave scattering lengths would, indeed, provide the first experimental evidence in favour of, or against, the standardly admitted hypothesis according to which the mechanism of spontaneous chiral symmetry breaking is dominated by the formation of a large (qq) condensate. Within QCD, this hypothesis is by no means a ’ Unit6 de Recherche des Universitis Paris XI et Paris VI associte au CNRS. ‘Laboratoire propre du Centre National de la Recherche Scientifique.
logical necessity and its experimental test might well become an important step towards a non-perturbative understanding of the quark-gluon dynamics. The second reason which makes detailed Z-G- studies topical, is that there are two new high precision experiments currently under preparation: i) The phase shift difference 4 (E) - 6; (E) at low energies (E < 400MeV) will be extracted from a new K,4-decay experiment [ 151 performed with the KL,OE detector at the Frascati #-factory DAQtNE [ 161. ii) At CERN, the project DIRAC [ 171 aims at the measurement of the lifetime of r+n-atoms to lo%, implying the determination of the combination of scattering lengths lug ui[ with a 5% accuracy. On the theoretical side an even better precision can be reached by a.systematic use of chiral perturbation theory [ 18,191 ( ,@T) . The low-energy expansion of the rr scattering amplitude A(slt,u) starts at order O(p*) given by Weinberg more than 30 years ago [ 201. Subsequently, the oneloop 0(p4) contribution to A(sl t, u) has been cal-
0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00872-l
462
L. Girlanda et al/Physics
culated by Gasser and Leutwyler [ 21,191. It is given by four low-energy constants 11, 12, 13, 14 besides the (charged) pion mass M, and the decay constant F,. The present state of the art involves the two-loop O(p6) order and the present letter concerns this degree of accuracy. 2. The 0(p6) amplitude A(slt,u) given in Ref. [ 1] in the form
has been first
(1) The function A~sF, which depends on the Mandelstam variables s, t, u and on the six parameters (Y, 6, At, . . .h4, is explicitly displayed in [ 11. Here, p denotes the characteristic pion momentum and Au is the mass scale of bound states not protected by the chiral symmetry, Au N 4rF, N 1 GeV. The result ( 1) holds independently of the strength of the quark condensate. The latter merely shows up in the size of the constant CY:for standard, large values3 (&) Y - (250 MeV) 3 one has LYN 1 and its value increases up to cr N 4 for I(@)] decreasing down to zero. The parameter p is less sensitive to the value of the condensate, remaining always close to unity. It has been shown [ 31 that the remaining four constants At, . . . A4 can be rather accurately determined from the existing vr scattering data [22] in the intermediate energy range 0.5 GeV < E < 1.9 GeV using the Roy dispersion relations [ 231. The latter explicitly incorporate crossing symmetry and consequently they strongly constrain the HIT amplitude at low energies. Equating the perturbative formula (1) with the Roy dispersive representation in a whole low-energy region of the Mandelstam plane, one infers the values of At, . . .A4 whereas the parameters ff and p remain essentially undetermined. The resulting Ai’s are almost independent of LYand p. Here we quote and use the central values corresponding to Q = 1.04, /3 = 1.08 [ 31, A1 = (-5.7
f 2.2) x 10-3,
A2 = (9.3 i 0.5) x 10-3, 3 (qq) denotes the single flavour condensate in the W(2) x SU( 2) chiral limit mu = rnd = 0 at the QCD scale Y = 1 GeV.
Letters B 409 (1997) 461-468
A3 = (2.2 f 0.6) x 10-4, A4 = (-1.5
&0.12)
x 10-4.
(2)
The quoted errors include experimental uncertainties on 7r7r phase-shifts and inelasticities in the medium energy region and an estimate of the systematic error arising from neglected higher orders in the low-energy representation ( 1). The errors due to the uncertainty in the high-energy behaviour of the rr scattering amplitude are negligible. 3. With the constants Ai determined, Eq. ( 1) allows one to convert new high-precision experimental information on low-energy ~7r phase shifts and/or threshold parameters into a measurement of (Yand p and finally, into an experimental determination of the quantity (m, + m4) (44) (the detailed relation between (Y and /3 and the condensate can be found in Ref. [ 1] ) . Conversely, Eq. ( 1) can be used to predict, for each value of the condensate, all the low-energy observables. It is of particular importance to assess with as much accuracy as possible the prediction concerning the standard alternative of a large (@) condensate. The strength of the (qq) condensate is conveniently described by the deviation from the Gell-Mann-OakesRenner relation, i.e. by the parameter F2M2 :=5$&-l.
(3)
Here, m = $ (m, + md) is the running quark mass and me is a mass scale characteristic of 4q condensation. The standard alternative of a large condensate corresponds to ma 2 hu. In this special case the ratio (3) can be treated as an expansion parameter, m/m0 = 0 ( p2 /A&) and the general low-energy expansion becomes the standard chiral perturbation theory ( SxPT) [ 191. The complete SXPT two-loop calculation of the rr-scattering amplitude has been recently completed by Bijnens et al. [2]. Not surprisingly, this calculation recovers the formula ( 1) giving, in addition, the expressions of the six parameters ry, p, At,. . . , A4 in terms of i) M,, F,, ii) four 0(p4) constants If&), E;(p), I;(p) and Ei(,u) and finally iii) six 0(p6> constants Y; ( pu> , . . . , t-k
463
L. Girlanda et al./Physics Letters B 409 (1997) 461-468
23 A3 = -18 +
37 277 k1 - 36 k2 + 1990656+
( 19 + 5368 L - 13056 1; - 9600 I;) 414721 z-z
+ r\ - -Jjri , kl + g
A4 = ;
+
(8) kz +
3311 3981312+’
’ (-2-257L+3361’,+8401’,) 10368 rr2 4 - - rr 3 6’
(9)
with d 1; ~~=--~ n=$
3/i
72=$
B=-;r
y4=2,
(10)
and
ki(P)
At=--
1
iL+21;--
f
36~~ 79
-;kt-;k2-;k4+81;1;+-
9216 7p
[ 1 + 2272 L - 2496 1;
+&(-
- 2160 1; - 384 1:) M?r F2) lr
+ r; - t-i + 6r; - 2rk I
A2=-;L+l;--
5 288 z-2
- ; kl - ; kz - ; k4 + 4 1; 1; +
+
(6)
[
1223 331776+
17 + 752 L + 3840 1’1+ 1536 1; + 276481 r2 (
-
1920 1;) + 2J4
1g F;
’
(7)
= (41:(p)
- YiL)L.
(11)
(These expressions are obtained from the expansions of the parameters bt, . . ., b6 originally given in [ 21, which are in one-to-one correspondence with LY,p, Al, . . .. &. We prefer to work with the latter set of parameters for the reader’s convenience: explicit formulae for low-energy observables in terms of a, p, Ai are given in Ref. [ 11,whereas similar expressions in terms of the hi’s are at present not available in the literature). A few points are worth recalling. i) The parameters LY,p, Ai are p-independent. This fact, together with Eq. ( 10) fixes the scale dependence of the low-energy constants I-[( ,u) . ii) Eqs. (4) -( 9) fix the expansion of the parameters CY,j3, Ai in powers of A4: and log Mi (and/or in powers of the quark mass m), sinceI{(p),. Zi ( ,z) and the r[ ( p) are quark mass independent. Contributions of successive chiral orders to cz, /3, Ai can be identified by counting the powers of Mi/F$ Notice that a and p start by an order 0(p2) contribution (a~ = 1, p = 1) followed by 0(p4) and O(p6) corrections. The expansions of Al. A2 consist of 0(p4) and 0(p6) contributions, whereas A3 and A4 are entirely of order O(#). iii) The 0(p4) constants 13 and 14 belong to the explicit symmetry breaking sector of the effective lagrangian. They represent
L. Girlanda et al./Physics
464
Letters B 409 (1997) 461-468
the fine tuning of the (@I) condensate to its presumed large value: in SxPT, the deviation from the GellMann-Oakes-Renner relation (3) is given by [ 191
I( ?T > 2
m -=
m0
[
$
2w+2z;(4L
+.__,
(12)
Similarly, Zi controls the deviation of p from 1. On the other hand, the hi’s are independent of 1; (and only very weakly dependent on Ii) reflecting the fact that they are only marginally sensitive to the size of the (&I) condensate. In the sequel, we complete our de$nition of the standard xPT by adopting the standardly used central values of 1; and lf; [ 19,2] : Z;(M,)
=0.82
x 10-3,
Z;(M,)
= 5.6 x 10-3.
(13)
Finally, the constants 1; and 15 do not describe explicit symmetry breaking effects (they are coefficient of four-derivative terms in Cc4)) and they are insensitive to the size of the quark condensate. They control the parameters A1 and &. 4. Eqs. (4) -( 9) can be used to predict the parameters a, p, At, . . ., A4 and consequently, all low-energy rr scattering observables, provided the low-energy constants II, . . . ,Z4 and rr , . . . , r,j are determined from the analysis of different processes. This is a path advocated by the authors of Ref. [ 21. In the present letter this kind of analysis will be confronted with additional experimental information contained in Eq. (2). Bijnens et al. [2] have used the values (13) for 13 and 14; for 1’1and 15 they have taken the central values obtained from the SxPT analysis of Kr4 form factors [24]: If(&)
= (-5.4%
I;(&)
= (5.7f
1.1) x 10-3, 1.1) x 10-3.
(14)
As for the 0(p6) constants rf (,x), the authors of [ 21 take rf( 1GeV) = 0 and they check that this approximation confronted with a resonance saturation model produces a negligible error. With the values ( 14)) and r-f = 0 at ,X = 1 GeV one obtains (in this letter we always use F, = 93.2 MeV and M, = 139.6 MeV): a=
1.074,
p=
1.105,
Fig. 1. The phase shift difference C$ - 8: in the energy region of K14 decays. The dashed curve is obtained with the values of Rq. (15) and it coincides with the curve displayed in Fig. 1 of Ref. 121. The solid line is obtained with the values of Eqs. (2) and ( 18) while the shaded area results adding the corresponding error bars quadratically. The experimental points are from Ref. [ 321,
A1 = -8.91
x 10-3,
A3 = 2.04 x 10-4,
A:! = 14.5 x 10-3, & = -1.79
x 10-4.
(15)
For these values of the parameters a, p, At,. . . , Ah. one obtains the S-wave scattering lengths a! = 0.218, ai - a: = 0.259 corresponding4 to the predictions given in Ref. [ 21. The resulting phase shift difference 8 -c$ (measurable in Kr4 decays) is shown as a function of the center of mass energy as the dashed line in Fig. 1. It reproduces the curve displayed in Fig. 1 of Ref. [ 21. Finally, a few remaining threshold parameters not discussed in Ref. [ 21 are collected in the first column of Table 1, using the expressions displayed in Appendix D of Ref. [ 11. The numbers in parentheses are obtained keeping in higher orders only those components of a, /3. ht and A:! that actually contribute at most to the order O(p6). These exactly coincide with the corresponding predictions one would obtain using the amplitude given in [ 21. Among the latter it is worth noticing the value predicted for the isoscalar D-wave scattering length ai = 26.3 x 10e4, which is substantially above the value extracted from the analysis of Roy equations [ 251. This disagreement reflects the fact that the 4 Actually these have to be compared with the numbers given in Rq. (4) of Ref. [2] in parentheses (rf( 1 GeV) = 0). The small difference provides an estimate of 0(p8) effects: it is entirely due to the fact that the amplitude A~SF of Ref. [ 11 coincides with the amplitude calculated in Ref. [2] only modulo c3(ps) contributions.
L. Girlanda et al./Physics
Letters B 409 (1997) 461-468
465
Table 1 Threshold parameters of burr scattering (in units of M,+) in the standard framework using the two-loop expressions of Ref. [ 11, App. D. The first column results from the values of Eq. (15) (see the text for the numbers in parentheses). The second column is obtained in the same way but taking the values of Eqs. (2) and ( 18) as input
no 8 b0
-IOU2 -lob!
u; - a; loa; 10’b’ 10%~ lO%lS 10%:3
Bijnens et al. [2]
KMSF
Experiment [ 251
0.218 (0.2156) 0.273 (0.271) 0.411 (0.4094) 0.709 (0.704) 0.259 (0.2565) 0.395 (0.3956) 0.785 (0.784) 0.263 (0.267) 0.237 (0.2356) 0.428 (0.478)
0.209 f 0.004 0.255 i 0.010 0.44 rfI 0.01 0.80 ztO.02 0.254 Jr 0.004 0.373 f 0.008 0.55 f 0.07 0.16 f0.02 0.09 ho.13 0.49 f 0.07
0.26 f 0.05 0.25 + 0.03 0.28 zt 0.12 0.82 f 0.08 0.29 zt 0.05 0.38 f 0.02 0.17 f 0.03 0.13 * 0.30 0.6 f0.2
value ( 1.5) of A2 is significantly above the value (2) inferred from experimental phase shifts in Ref. [ 31. We would like to stress that both the canonical value CZ$= (17 f 3) x 10e4 and the determination of the constant A2 = (9.3 f 0.5) x 10e3 are based on the Roy dispersion relations [23] using the experimental vrr data above 500 MeV as input. Furthermore, in both cases, the dominant contribution comes from the P-wave in the p( 770) region, which explains the relatively small error bars. In order to make the predictions of SXPT [ 21 agree with the values (2) of the parameters Al,. . . , A4 and with the standard value of ai, we proceed as follows: fixing 1; and l$ according to Eq. (13), we solve Eqs. (6) and (7) for Z\(M,), &(MJ’ Ii
= (-4.0f + [-1.1
1.0) x 1o-3
Y; + l.Ori
- 6.3~; +2.1
r~]p=IGeV (16)
Z;(M,)
= (1.6kO.4)
x 1O-3
+ [0.1J,-3.5r;+0.5r;-0.2r~]~10~”
(17)
where the values and errors (2) have been used for ht and AT. Eqs. (16) and (17) are then inserted back into the formulae (4) and (5) for LXand p. Keeping in mind that IX and p are sensitive to 11 and 12 only at next-to-next-to-leading level, the unknown constants Y;( 1 GeV) are viewed as a source of uncertainty in cx and p. Inspired by ndive dimensional analysis [ 261 we
400
E,,, WV)
Fig. 2. The isospin 2, S-wave phase shifts at low energies. The different curves are obtained with the values ( 15) (dashed) and the values (2) and ( 18) (solid). In the latter case tbe shaded area shows the corresponding error band. The experimental points are taken from Ref. [22].
take in the expressions for cr and /?, r-f( 1 GeV) = ( O+ 2) x 10m4. Adding the corresponding uncertainties quadratically, we obtain CY= 1.07 f 0.01 )
p = 1.105 f 0.015.
(18)
It should be stressed that the error in Eq. ( 18) does not include the uncertainty in the low-energy constants Ij and Zi. As in the case of the chiral condensate itself, the constant Zj has not yet been determined experimentally and for this reason it is hard to associate an error bar with it. The values ( 18) have to be viewed as corresponding to the “standard alternative” of a large condensate defined by the values (13) of lj and 1;. We now use the formulae given in Ref. [ 1 ] to generate the predictions for threshold parameters and phase shiftsthatcorrespondtocu,P(18)andAt,...,&(2). Adding the errors quadratically, the resulting threshold parameters are summarized in the second column of Table 1. Observe that we obtain deviations of LZ~ and ug - ui from their centruE experimental values that are larger than those predicted in Ref. [ 21. Notice that now, the D-wave scattering lengths perfectly agree with their Roy-equation “experimental” values as expected from the manner the values (2) of the constants At, . . . , A4 have been obtained. A similar conclusion holds for the phase shift difference @ - St, shown as the solid curve in Fig. 1 with the error band indicated by the shaded area: the curve displayed in Ref. [ 21 is significantly higher, i.e. closer to the experimental central-value points. For illustration, the phase 8: is also shown in Fig. 2.
L. Girlandaet al./ PhysicsLettersB 409 (1997) 461-468
466
5. The origin of the mismatch described in the previous paragraph clearly appears upon comparing Eqs. (16) and (17) with the values (14) of the constants Z;,2( MP) extracted in Ref. [ 241 from the “unitarized” one-loop SXPT KM form factors (Eq. (5.10) of [ 241) . The question is how close the expressions ( 16) and ( 17) can be brought to these values keeping at the same time the S(p6) constants , ri( 1 GeV) at a reasonable magnir;(l GeV),... tude. If one proceeds as before treating the ri’s at 1 GeV as randomly distributed around zero with a standard deviation f2 x 10W4, one gets Z’l(Mp) = (-4.0& Z;(M,)
l.O&l.S)
= (1.6&0.4~0.9)
x 10-3, x 10-3,
(19)
or I; = -0.37
rt 0.95 i 1.71 ,
iz=4.17*o.19f0.43,
(20)
where the first error has its origin in At and A2 (Eq. (2)), whereas the second error arises from the presumed uncertainties in the individual ri’s added quadratically. Two checks of the size of the constants ri are conceivable. i) First, one can make full use of the information contained in Eq. (2) determining the parametersZi,2(Mp) andrj(lGeV),...,r:(lGeV) by a simultaneous fit to Eqs. (6)-( 9) and to the constraints ri( 1 GeV) = 0 f 2 x 10-4. The resulting x2/d.o.f. is 1.9/2 and one obtains 1; (Mp)
= (-4.0
Z$(M,)
= (2.0 i 0.3) x 10-3,
compatible
f 0.5) x 10-3,
with (19),
(21)
whereas for the ri’s one gets
rj ( 1 GeV) = (-0.3
f 2.0) x 10-4,
T-:( 1 GeV) = (-0.7
f 0.9) x 10W4,
r1 = -0.61
x 10-4,
r2 = 1.3 x 10-4,
r3 = -1.70
X 10e4,
r4 = -1.0
f-5 = 1.14 x 10m4,
(22)
This result turns out to be rather stable: if one increases the uncertainties in the Yi’s to f3 x 10B4, the new X2/d.o.f. = 1.24/2, the values (21) become (-4.8 f 0.5) x 10W3 and (2.1 f 0.3) x low3 respectively, and the changes in the ri’s also remain rather modest. On
x 10d4,
rg = 0.3 x 10m4.
(23)
Estimating low-energy constants by resonance saturation does not, in principle, fix the renormalization scale ,X at which the estimate is supposed to hold. Actually, if a constant exhibits a strong scale dependence, its resonance saturation estimate is subject to caution. Interpreting Eqs. (23) as values of 6 ( p) at ZL= 1 GeV, one observes a striking coherence with the preceding analysis: (23) is, indeed, consistent not only with dimensional analysis or with the assumption 161 < 2 x lop4 but, moreover it agrees with the fit (22). One can even repeat the fit to Eqs. (6)-(9) constraining rz( 1 GeV) to the values (23) allowing for a 100% error: the fit is excellent (X2/d.o.f. = 0.91/2) and it yields Zt(Mp) = (-4.0 f 0.5) x 10W3, E;(Mp) = (2.1 f 0.3) x 10m3, again compatible with ( 19) and (2 1) . On the other hand, one finds that between p = M, and ,u = 1 GeV, only the constants ~4, t-5 and rg show a moderate scale dependence: had we assumed that the values (23) concern the scale ,U = Mp (as suggested in Ref. [ 271) , the comparison with our previous analysis would be less favourable as far as the constant t-3 is concerned, fi3( 1 GeV) = -4.9 x 10e4 in this case. Notice however that according to Eq. ( 17) the correction to the “critical” constant Zs( Mp) is dominated by t-i whose scale dependence is rather weak: ri( 1 GeV) = ri(Mp)
t-5( 1 GeV) = ( 1.5 f 0.5) x 10m4, I$( 1 GeV) = (0.4 f 0.9) x 10m4.
the other hand, the errors obtained by this procedure (increase of x2 by one unit) and shown in Eqs. (21) and (22) are probably heavily underestimated. ii) Next, it is instructive to confront the previous discussion with the estimate of the constants ri by resonance saturation as quoted recently by Hannah [ 271:
- 7 x 10e6.
(24)
In order that the constant I;( MP) ( 17) differ from the Kt4 value (14) by at most two standard deviations, the constant ri( 1 GeV) would have to be t-i ( 1 GeV) N -5 x 10M4. This cannot be excluded but it looks unlikely in the light of the present analysis. 6. The constants It and 12 ( 14) have not been obtained from a full two-loop analysis of KM form factors F and G, which is not yet available. Instead, their
L. Girlanda et al/Physics
determination is based on matching a dispersive representation for the form factor F with the one-loop SXPT expressions, the latter merely serving to fix the subtraction constants. This method of “improving” oneloop xPT calculations has been often used in the past [28] and it suffers from a basic ambiguity: one has to assume that the one-loop and two-loop amplitudes practically coincide in a particular kinematical point M. Even if one admits the very existence of such a matching point M, the results can still depend on its choice. In Ref. [ 241 the matching point has been chosen at the threshold s, = 4Mi of the Swave amplitude z-n --f K + axial current, where s, stands for the dipion invariant mass squared. We have repeated the analysis of Ref. [24] for other choices of the matching point between sV = 4M$ and the left-hand-cut branch point s, = 0. We reproduce the result ( 14) and find that it is actually rather insensitive to the matching point except in the vicinity of the singular point So = 0, where the outcome for 11 (but not 12) becomes less stable. For instance, with the matching point at s, = 2M$ we obtain
Z'l(Mp) = (-4.8
-+ 2.1) x 10-3,
Z',(M,) = (5.3 f 1.0) x 10-3.
(25)
Given the present state and quality of Kt4 experimental data, it seems hard to ascribe the discrepancy described above to the SXPT analysis performed in Ref. [ 241. On the other hand, it should be kept in mind that outside the standard framework, i.e. for low values of the condensate (&), the constants Ii and 12 extracted from Kt4 data will be modified already at the one-loop level: since in GxPT the loop contributions are more important, the resulting central values of 1111and 1121 are expected to come out somewhat smaller [ 291. 7. A few concluding remarks are in order. The past determinations [ 19,30,6,7] of the constants Z; and Zi have operated within the 0(p4) order of xPT. They have shown an apparent coherence and compatibility with the Ki4 analyses of Ref. [ 241. This compatibility might be lost at 0(p6> order and we have to understand why. The resonance saturation models are the only ones that determine the constants Et,2 directly, integrating out the resonance degrees of freedom from an extended effective lagrangian Leg. However, incorporating resonances into L:,R is not free of ambigu-
Letters B 409 (1997) 461-468
467
ities, especially if one aims at the 0(p6) accuracy. On the other hand, less model-dependent sources of information, such as TV D-waves [ 191 and/or sum rules [ 6,7] primarily determine the physical parameters Ai, AZ. It turns out that this determination is rather stable and barely affected by switching from order S(p4) to S(p6). At the O(p4) level, i.e. neglecting in Eq. ( 1) the two-loop effects and setting As = A4 = 0, one would get from the u; and ai experimental central values At = -6.4 x 10e3 and A2 = 10.8 x 10V3, to be compared with Eq. (2). In other words, the relationship between D-wave scattering lengths and the parameters At, A2 is almost unaffected by O(p6) effects. The latter however become rather important in the relationship between A2 and I$. Rewriting Elq. (7) to make the dependence on I$_(M,,) appear explicitly, one obtains A2 =
{Z;(M,) + 5.45 x 10-3}
+ (0.32 x
Z;(M,)+ 1.7 x 10-3},
(26)
where the first (second) curly brackets collect all 0(p4) (S(#)) contributions (r4 has been neglected). The 0(p6) contribution is as large as 30% and it is dominated by double logs, whose importance has been anticipated by Colangelo [4]. It follows that for a given A2 (D-waves), the resulting value of I;( M,,)caneasily differ by a factor N 2 depending whether in Eq. (26) one includes the 0(p6) term or not. Whether the consistency with KITform factors can be understood within the large condensate hypothesis remains to be clarified. It might be, for instance, that at 0(p6) level the Kt4 form factors also receive an important contribution from double logs, which the unitarization procedure would not take into account [ 3 11. Independently of this issue, the main conclusion of this letter is the following: within the framework of SXPT we obtain predictions for ai, ug - ai and ?$ - 8: which are systematically below those given in Ref. [ 21, as shown in Fig. 1 and Table 1 of the present paper. A closely related fact is the failure of the values of 5 and Ii used in Ref. [2] to describe the D-waves in agreement with Roy equations analyses. This shows, once more, that a sensible and sensitive test of QCD in low-energy rr scattering should be based on a global analysis making use of all theoretical constraints and all pertinent low-energy observables.
468
L. Girlanda et al./Physics
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