On the lowest resonance of QCD

H. Leutwylera

aInstitute for Theoretical Physics, University of Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland

I report on recent work done in collaboration with Irinel Caprini and Gilberto Colangelo. We observe that

the Roy equations lead to a representation of the ππ scattering amplitude that exclusively involves observable

quantities, but is valid for complex values of s. At low energies, this representation is dominated by the con-

tributions from the two subtraction constants, which are known to remarkable precision from the low energy

theorems of chiral perturbation theory. Evaluating the remaining contributions on the basis of the available data,

we demonstrate that the lowest resonance carries the quantum numbers of the vacuum and occurs in the vicinity

of the threshold. Although the uncertainties in the data are substantial, the pole position can be calculated quite

accurately, because it occurs in the region where the amplitude is dominated by the subtractions. The calculation

neatly illustrates the fact that the dynamics of the Goldstone bosons is governed by the symmetries of QCD.

The present talk concerns the remarkable theo-retical progress made in low energy pion physicsin recent years. Pions play a crucial role when-ever the strong interaction is involved at low en-ergies – the Standard Model prediction for themuon magnetic moment provides a good illustra-tion. I will concentrate on work based on disper-sion theory [1], but will also show some lattice re-sults, obtained with quark masses that are smallenough for a controlled extrapolation to the val-ues of physical interest to be within reach. Theexisting low energy precision measurements [2–4] are consistent with the theoretical predictionsand even offer a stringent test for one of these.

From the point of view of dispersion theory, ππscattering is particularly simple: the s-, t- andu-channels represent the same physical process.As a consequence, the real part of the scatteringamplitude can be represented as a dispersion in-tegral over the imaginary part and the integralexclusively extends over the physical region [5].The representation involves two subtraction con-stants which may be identified with the S-wavescattering lengths a0

0, a20. The projection of the

amplitude on the partial waves leads to a disper-sive representation for these, the Roy equations.

The pioneering work on the physics of the Royequations was carried out more than 30 years ago[6]. The main problem encountered at that timewas that the two subtraction constants occurring

in these equations were not known. These con-stants dominate the dispersive representation atlow energies, but since the data available at thetime were consistent with a very broad range ofS-wave scattering lengths, the Roy equation ana-lysis was not conclusive.

The insights gained by means of chiral per-turbation theory thoroughly changed the situa-tion. The corrections to Weinberg’s low energytheorems for a0

0, a20 (left dot in figure 1, ref. [7])

have been worked out to first nonleading order(middle dot, ref. [8]) and those of next-to-next-toleading order are also known (dot on the right,ref. [9]). As shown in [10], very accurate pre-dictions for the scattering lengths are obtainedby matching the chiral and dispersive representa-tions in the interior of the Mandelstam triangle(small ellipse). The lattice results of the MILCcollaboration [11] also yield an estimate for thescattering lengths. Using their values for the cou-pling constants L4, L5, L6, L8 of the effective chi-ral lagrangian and neglecting two-loop effects, wearrive at the one standard deviation contour indi-cated by the dashed ellipse. The horizontal bandrepresents the value of the scattering length a2

0

obtained by the NPLQCD collaboration [12].The plot shows that ππ scattering is one of

the very rare hadronic processes where theory isahead of experiment: The two large ellipses andthe tilted band indicate the results obtained on

Nuclear Physics B (Proc. Suppl.) 162 (2006) 139–143

0920-5632/$ – see front matter © 2006 Elsevier B.V. All rights reserved.

www.elsevierphysics.com

doi:10.1016/j.nuclphysbps.2006.09.076

0.16 0.18 0.2 0.22 0.24 0.26

a00

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

a20

Universal bandtree (1966), one loop (1983), two loops (2000)Prediction (ChPT + dispersion theory, 2001)E 865 (2003)DIRAC (2005)NA48 (2005)MILC (2004)NPLQCD (2005)

Figure 1. S wave scattering lengths.

the basis of experiments done at Brookhaven [2]and CERN [3,4]. Possibly, the error estimatesattached to the lattice results are too optimisticand the analysis of the K → 3π data also re-quires further study, but it is fair to say that allof the available informations confirm the theore-tical predictions.

In the remainder of this report, I focus on thelowest resonance in the spectrum of QCD. Forrecent reviews of the situation, I refer to [13–15].The current knowledge relies almost exclusivelyon phenomenological analyses, but these do notagree. In fact, until ten years ago, the informa-tion about the σ = f0(600) was so shaky thatthis resonance was banned from the data tables.The work of Tornqvist and Roos [16] resurrectedit, but the estimate of the Particle Data Group[17], Mσ − i

2Γσ = (400 − 1200) − i (300 − 500)

MeV, indicates that it is not even known for surewhether the lowest resonance of QCD carries thequantum numbers of the σ or those of the ρ.

The positions of the poles represent universalproperties of the strong interactions which areunambiguous even if the width of the resonanceturns out to be large, but they concern the non-perturbative domain, where an analysis in termsof the local degrees of freedom of QCD – quarksand gluons – is not in sight.

One of the reasons why the values for the poleposition of the σ quoted by the Particle DataGroup cover a very broad range is that all ofthese rely on the extrapolation of hand made

parametrizations: the data are represented interms of suitable functions on the real axis andthe position of the pole is determined by contin-uing this representation into the complex plane.If the width of the resonance is small, the ambi-guities inherent in the choice of the parametriza-tion do not significantly affect the result, but thewidth of the σ is not small.

We have found a method that does not invokeparametrizations of the data. It relies on the fol-lowing two observations:

1. The S-matrix has a pole on the second sheetif and only if it has a zero on the first sheet. In or-der to determine the pole position it thus sufficesto have a reliable representation of the scatteringamplitude on the first sheet.

2. The Roy equations hold not only on the realaxis, but in a limited region of the first sheet.Since the pole from the σ occurs in that region,we do not need to invent a parametrization, butcan rely on the explicit representation of the am-plitude provided by these equations.

The first statement is a consequence of uni-tarity. The S-matrix element connecting the in-coming and outgoing ππ states with I = � = 0is of the form S0

0(s) = η00(s) exp 2 i δ0

0(s), whereη00(s) and δ0

0(s) represent the elasticity and thephase shift, respectively. The function S0

0(s) isreal-analytic, S0

0(s)� = S00(s�), with a cut on

the left (s < 0) as well as one on the right(s > 4M2

π). Unitarity implies that, in the re-gion 4M2

π < x < 16M2π, the elasticity is equal

to 1, so that S00(x ± iε) = exp±2 i δ0

0(x). Thesecond sheet is reached from the first via ana-lytic continuation from the upper rim of the elas-tic cut into the lower half plane, S0

0(x − iε)II =S0

0(x + iε)I . On the elastic interval, the valueson the first and second sheets are thus related byS0

0(x−iε)II = 1/S00(x−iε)I . Since the function is

analytic in s, this relation holds throughout thes-plane, S0

0(s)II = 1/S00(s)I . Hence there is a

pole on the second sheet if and only if there is azero on the first sheet.

In order to explain the second observation, Ifirst express the S-matrix element in terms of thepartial wave amplitude t00(s):

S00(s) = 1 + 2 i ρ(s) t00(s) , (1)

H. Leutwyler / Nuclear Physics B (Proc. Suppl.) 162 (2006) 139–143140

with ρ(s) ≡√

1 − 4M2π/s. The Roy equation for

the isoscalar S-wave represents t00(s) as a sum ofdispersion integrals over the imaginary parts ofall of the partial waves:

t00(s) = a + (s − 4M 2π) b (2)

+

∫∞

4M2π

ds′K0(s, s′) Im t00(s

′)

+

∫∞

4M2π

ds′K1(s, s′) Im t11(s

′) + . . .

The representation involves two subtractions.Crossing symmetry implies that these can be ex-pressed in terms of the S-wave scattering lengths:a = a0

0, b = (2 a00−5 a2

0)/(12M2π). The kernels are

given by elementary functions, for instance

K0(s, s′) =

1

π (s′ − s)− 5 s′ + 2 s − 16M2

π

3 π s′ (s′ − 4M2π)

+2

3 π (s − 4M2π)

ln

(s + s′ − 4M2

π

s′

).

The first term on the right is the familiar Cauchykernel that accounts for the discontinuity acrossthe right hand cut. The remainder of K0(s, s

′)describes those contributions from the left-handcut that are also due to Im t00, while those fromIm t11 (P -wave) and from the remaining partialwaves are booked separately.

As shown by Roy [5], eq. (2) holds on the realaxis, for −4M 2

π < s < 60M2π. Using the meth-

ods described in [18], we have demonstrated thateq. (2) also holds for complex values of s, in theintersection of the relevant Lehmann-Martin el-lipses [1]. The dash-dotted curve in fig. 2 showsthe domain of validity that follows from axiomaticfield theory, while the full line depicts the slightlylarger domain obtained under the assumptionthat the scattering amplitude obeys the Mandel-stam representation. I emphasize that the bound-ary does not represent a singularity of the ampli-tude, but merely limits the region where eq. (2)holds in the form given. Modified representa-tions with a much larger domain of validity can befound in [19] and in the references quoted therein.

For our analysis, it is essential that the dis-persion integrals in eq. (2) are dominated by thecontributions from the low energy region: becausethe Roy equations involve two subtractions, the

0 20 40 60

Res in units of Mπ2

-40

-20

0

20

40

Ims

σ

σ

f0(980)

f0(980)

ρ

ρ

Figure 2. Domain of validity of the Roy equa-tions.

kernels Kn(s, s′) fall off in proportion to 1/s′ 3

when s′ becomes large. The contributions fromthe left-hand cut play an important role here: ifthese are dropped, then K0(s, s

′) reduces to thefirst term, which falls off only with the first powerof s′. Taken by itself, the contribution from theright hand cut is sensitive to the poorly knownhigh energy behaviour of Im t00(s

′), but taken to-gether with the one from the left-hand cut, thehigh energy tails cancel.

The Roy equations thus provide us with anexplicit representation of the function S0

0(s) forcomplex values of s, in terms of rapidly conver-gent dispersion integrals over the imaginary partsof the partial waves. In connection with the de-termination of the pole from the σ, the most im-portant contribution is the one from the subtrac-tion term. The dispersion integrals over the S-P - and D-waves generate a correction which canbe evaluated with available phase shift analyses– in particular with the one obtained by solvingthe Roy equations [10]. As discussed in detailin ref. [20], the contributions from high energiesand high angular momenta can be estimated bymeans of the Regge representation of the scatter-ing amplitude – these contributions barely affectthe pole position. For a brief account of the nu-merics, in particular also for a discussion of theerror analysis, I refer to [1] – a more detailed re-port is in preparation.

For the central solution of the Roy equations,

H. Leutwyler / Nuclear Physics B (Proc. Suppl.) 162 (2006) 139–143 141

the function S00(s) contains two pairs of zeros in

the domain of interest:

sσ = (6.2 ± i 12.3)M 2π , sf0

= (51.4 ± i 1.4)M 2π .

These are indicated in fig. 2, which may also beviewed as a picture of the second sheet – the dotsthen represent poles rather than zeros. For com-parison, the figure also indicates the position ofthe zeros in S1

1(s), which characterize the ρ.The higher one of the two pairs of zeros rep-

resents the well-established resonance f0(980),which sits close to the threshold of the transi-tion ππ → KK. The corresponding pole gener-ates a spectacular interference phenomenon withthe branch point singularity, which gives rise toa sharp drop in the elasticity. Our analysis addslittle to the detailed knowledge of that structure.

The lower pair of zeros corresponds to a polein the lower half of the second sheet at [1]

√sσ = 441

+16

− 8 − i 272+9

−12.5 MeV . (3)

The error bar in eq. (3) is calculated by (a) esti-mating the uncertainties in the input used whensolving the Roy equations and (b) following errorpropagation to determine the uncertainty in theresult for the pole position. The errors quoted ineq. (3) account for all sources of uncertainty. Fordetails, I refer to [1].

The reason why the pole position of the σ canbe calculated rather accurately is that (a) the poleoccurs at low energies and (b) there, the represen-tation (2) is dominated by the subtraction term.This is illustrated in fig. 3, where the real partof t00(s) is plotted versus s. The graph demon-strates that, in the region shown, the full ampli-tude closely follows the subtraction term. Thefinal state interaction does generate curvature –in particular, the cusp at s = 4M 2

π is visible – butsince Goldstone bosons of low momentum interactonly weakly, the contributions from the dispersionintegrals amount to a small correction. In the lan-guage of chiral perturbation theory, the disper-sion integrals only show up at NLO. Moreover, atleading order, the subtraction constants are de-termined by the pion decay constant. Droppingthe dispersion integrals and inserting the lowestorder predictions for the scattering lengths, the

-4 -2 0 2 4 6 8

s in units of M2π

-0.2

0

0.2

0.4

σ

threshold

Ret00

Subtraction term

Weinberg 1966

Adler zero

Figure 3. Dominance of the subtraction term.

Roy equation (2) reduces to the well-known for-mula, which Weinberg derived 40 years ago [7],

t00(s) =2 s− M2

π

32 π F 2π

, (4)

and which is shown as a dash-dotted line in fig. 3.The main feature at low energies is the occur-

rence of an Adler zero, t00(sA) = 0. In the aboveapproximation, the zero occurs at sA = 1

2M2

π .The higher order corrections generate a smallshift, which can be evaluated from our solutionof the Roy equations. The uncertainties in theresult, sA = (0.41 ± 0.06)M 2

π, are dominated bythose in our predictions for the scattering lengths.While the Adler zero sits below the threshold, ata real value of s, the lowest zeros of the S-matrix,S0

0(sσ) = 0, occur in the region Resσ > 4M2π,

with an imaginary part that is about twice aslarge as the real part.

The Weinberg formula (4) explains why the S-matrix has a zero in the vicinity of the threshold:Inserting it in eq. (1), the relation S0

0(s) = 0 takesthe form of a cubic equation for s, which pos-sesses a complex pair of zeros on the first sheet,at

√s = 365 ± i 291 MeV and a real zero on the

second sheet at√

s = 12 MeV, very close to theorigin. The latter is what becomes of the kine-matic singularity at the origin in this approxi-mation: the full S-matrix contains a zero on thesecond sheet at s = 0. The number obtained forthe complex zero in the lower half plane differs

H. Leutwyler / Nuclear Physics B (Proc. Suppl.) 162 (2006) 139–143142

from the “exact” result in eq. (3) by about 20percent. The dispersion integrals are essential forthe partial wave to obey unitarity, but they onlyrepresent a correction. In view of this, the preci-sion claimed in eq. (3) is rather modest.

One of the reasons why many of the attemptsat a determination of the mass and width of theσ have failed is that the left-hand cut was ne-glected1. In the language of eq. (2), this amountsto replacing the kernel K0 by the first term ineq. (3) and dropping the remaining dispersion in-tegrals. The value of the subtraction constant bmay be adjusted, but the operation discards thecurvature generated by the left-hand cut. Thepole is not sufficiently far away from this cut forsuch an approximation to be justified.

I conclude that the same theoretical frameworkthat leads to incredibly sharp predictions for thethreshold parameters of ππ scattering [10] alsoshows that the lowest resonance of QCD carriesthe quantum numbers of the vacuum. The polesits in the vicinity of the threshold, but quite farfrom the real axis: the width of the σ is largerthan the width of the ρ by a factor of 3.7.

The physics of the σ is governed by the dynam-ics of the Goldstone bosons: the properties of theinteraction among two pions are relevant [22]. Inquark model language, the wave function containsan important tetraquark component [23]. Theproperties of the resonance f0(980) are also gov-erned by Goldstone boson dynamics – two kaonsin that case. It would be of considerable inter-est to apply the above analysis to the Kπ S-wavewith I = 1

2. This should clarify the situation with

the κ.It is a pleasure to thank the organizers for a

stimulating and enjoyable stay at Novosibirsk.

REFERENCES

1. I. Caprini, G. Colangelo and H. Leutwyler,Phys. Rev. Lett. 96 (2006) 132001.

2. S. Pislak et al. [BNL-E865 Collaboration],

1A notable exception is ref. [21], where the two-loop rep-resentation of chiral perturbation theory is used to obtainan estimate for the left-hand discontinuity. The result ofthis calculation is in good agreement with our analysis.

Phys. Rev. Lett. 87 (2001) 221801; Phys.Rev. D 67 (2003) 072004.

3. B. Adeva et al. [DIRAC Collaboration], Phys.Lett. B 619 (2005) 50.

4. J. R. Batley et al. [NA48/2 Collaboration],Phys. Lett. B 633, 173 (2006).

5. S. M. Roy, Phys. Lett. B 36 (1971) 353.6. J. L. Basdevant, C. D. Froggatt and J. L. Pe-

tersen, Nucl. Phys. B 72 (1974) 413.7. S. Weinberg, Phys. Rev. Lett. 17 (1966) 616.8. J. Gasser and H. Leutwyler, Phys. Lett. B

125 (1983) 325; Annals Phys. 158 (1984) 142.9. J. Bijnens, G. Colangelo, G. Ecker, J. Gasser

and M. E. Sainio, Phys. Lett. B 374 (1996)210; Nucl. Phys. B 508 (1997) 263; ibid. B517 (1998) 639 (E).

10. G. Colangelo, J. Gasser and H. Leutwyler,Nucl. Phys. B 603 (2001) 125.

11. C. Aubin et al., [MILC Collaboration],Phys. Rev. D 70 (2004) 114501.

12. S. R. Beane, P. F. Bedaque, K. Orginos andM. J. Savage [NPLQCD Collaboration], Phys.Rev. D 73 (2006) 054503.

13. P. Minkowski and W. Ochs, AIP Conf. Proc.814 (2006) 52 [arXiv:hep-ph/0511126].

14. D. V. Bugg, AIP Conf. Proc. 814 (2006) 78[arXiv:hep-ex/0510021].

15. M. R. Pennington, arXiv:hep-ph/0604212.16. N. A. Tornqvist and M. Roos, Phys. Rev.

Lett. 76, 1575 (1996).17. S. Eidelman et al. [Particle Data Group],

Phys. Lett. B 592 (2004) 1.18. A. Martin, Scattering Theory: Unitarity,

Analyticity and Crossing, Lecture Notes inPhysics, Vol. 3, (Springer-Verlag, Berlin,1969).

19. S. M. Roy and G. Wanders, Nucl. Phys. B141 (1978) 220.

20. B. Ananthanarayan, G. Colangelo, J. Gasserand H. Leutwyler, Phys. Rept. 353 (2001)207.

21. Z. Y. Zhou, G. Y. Qin, P. Zhang, Z. G. Xiao,H. Q. Zheng and N. Wu, JHEP 0502 (2005)043.

22. V. E. Markushin and M. P. Locher, FrascatiPhysics Series, 15 (1999) 229.

23. R. L. Jaffe, Phys. Rev. D 15 (1977) 267, 281.

H. Leutwyler / Nuclear Physics B (Proc. Suppl.) 162 (2006) 139–143 143