aa r X i v : . [ h e p - ph ] A p r Model independent determination of the s pole H. Leutwyler
Institute for Theoretical Physics, University of Bern,Sidlerstrasse 5, CH-3012 Bern, Switzerland email
Abstract.
The first part of this report reviews recent developments at the interface between latticework on QCD with light dynamical quarks, effective field theory and low energy precision exper-iments. Then I discuss how dispersion theory can be used to analyze the low energy structure ofthe pp scattering amplitude in a model independent manner. This leads to an exact formula for themass and width of the lowest few resonances, in terms of observable quantities. As an application, Iconsider the pole position of the s , paying particular to error propagation in the numerical analysis.The report is based on work done in collaboration with Irinel Caprini and Gilberto Colangelo [1].Contribution to the proceedings of the Workshop on Scalar Mesons and Related Topics,honouring the 70th birthday of Michael Scadron, Lisbon, Portugal, Feb. 11-16, 2008. Keywords:
QCD, dispersion theory, mesons
PACS:
MOTIVATION
QCD with massless quarks is the ideal of a theory: it does not contain a single dimen-sionless free parameter. At high energies, the degrees of freedom occurring in the La-grangian are suitable for a description of the phenomena and the interaction among thesedegrees of freedom can be treated as a perturbation. At low energies, on the other hand,QCD reveals a rich spectrum of hadrons, the understanding of which is beyond the reachof perturbation theory. In my opinion, one of the main challenges within the StandardModel is to understand how an intrinsically simple beauty like QCD can give rise to theamazing structures observed at low energy.The progress achieved in understanding the low energy properties of QCD has beenvery slow. A large fraction of the papers written in this field does not concern QCDas such, but models that resemble it in one way or the other: constituent quarks, NJL-model, linear s model, hidden local symmetry, AdS/CFT and many others. Some ofthese may be viewed as simplified versions of QCD that do catch some of the salientfeatures of the theory at the semi-quantitative level, but none provides a basis for acoherent approximation scheme that would allow us, in principle, to solve QCD.This talk concerns the model independent approach to the problem based on disper-sion theory. More precisely, I would like to show that the low energy structure of thesector with the quantum numbers of the vacuum, I = ℓ =
0, can quantitatively be under-stood on the basis of the symmetries of QCD, without invoking any model, but insteadrelying on the available, rather crude experimental information at energies above 1 GeV.At low energies, the main characteristic of QCD is that the energy gap is very small, M p ≃
140 MeV. More than 10 years before the discovery of QCD, Nambu [2] found
Model independent determination of the s pole October 26, 2018 1 ut why that is so: the gap is small because the strong interactions have an approximatechiral symmetry. Indeed, QCD does have this property: for yet unknown reasons, two ofthe quarks happen to be very light. The symmetry is not perfect, but nearly so: m u and m d are tiny. The mass gap is small because the symmetry is “hidden” or “spontaneouslybroken”: for dynamical reasons, the ground state of the theory is not invariant underchiral rotations, not even approximately. The spontaneuous breakdown of an exact Liegroup symmetry gives rise to strictly massless particles, “Goldstone bosons”. In QCD,the pions play this role: they would be strictly massless if m u and m d were zero, sothat the symmetry would be exact. The only term in the Lagrangian of QCD that isnot invariant under the group SU(2) × SU(2) of chiral rotations is the mass term of thetwo lightest quarks, m u uu + m d dd . This term equips the pions with a mass. Althoughthe theoretical understanding of the ground state is still poor, we do have very strongindirect evidence that Nambu’s conjecture is right – we know why the energy gap ofQCD is small. LATTICE RESULTS RELEVANT FOR LOW ENERGY QCD
As pointed out by Gell-Mann, Oakes and Renner [3], the square of the pion mass isproportional to the strength of the symmetry breaking, M p (cid:181) ( m u + m d ) . This propertycan now be checked on the lattice, where – in principle – the quark masses can be variedat will. In view of the fact that in these calculations, the quarks are treated dynamically,the quality of the data is impressive. The masses are sufficiently light for c PT to allowa meaningful extrapolation to the quark mass values of physical interest. The resultsindicate that the ratio M p / ( m u + m d ) is nearly constant out to values of m u , m d that areabout an order of magnitude larger than in nature.The Gell-Mann-Oakes-Renner relation corresponds to the leading term in the expan-sion in powers of the quark masses. At next-to-leading order, this expansion contains alogarithm: M p = M (cid:26) + M p F p ln M L + O ( M ) (cid:27) , M ≡ B ( m u + m d ) . (1)Chiral symmetry fixes the coefficient of the logarithm in terms of the pion decay constant F p , but does not determine the scale L . A crude estimate was obtained more than 20years ago [4], on the basis of the SU(3) mass formulae for the pseudoscalar octet. Theresult is indicated at the bottom of the left panel in Fig.1. The other entries representrecent lattice results for this quantity [5]-[10], which are considerably more accurate.The right panel shows the results for the coupling constant ¯ ℓ , which determines thequark mass dependence of the pion decay constant at NLO of the chiral expansion. Inthat case, we obtained a rather accurate result in 2001, from a dispersive analysis of thescalar pion form factor [11]. The lattice determinations of ¯ ℓ have reached comparableaccuracy and corroborate the outcome of our analysis.The hidden symmetry not only controls the size of the energy gap, but also determinesthe interaction of the Goldstone bosons at low energies, among themselves, as well aswith other hadrons. In particular, as pointed out by Weinberg [12], the leading term in thechiral expansion of the S-wave pp scattering lengths (tree level of the effective theory) Model independent determination of the s pole October 26, 2018 2 Gasser & L. 1984MILC 2007Del Debbio et al. 2006, N f = 2ETM 2007, N f = 2RBC/UKQCD 2007JLQCD 2007, N f = 2PACS-CS (preliminary) PS fr a g r e p l ace m e n t s ¯ ℓ = ln L M p ¯ ℓ = l n L M p
33 44 55
Gasser & L. 1984Colangelo, Gasser & L. 2001MILC 2007ETM 2007, N f = 2JLQCD 2007, N f = 2RBC/UKQCD 2007PACS-CS (preliminary) PS fr a g r e p l ace m e n t s ¯ ℓ = l n L M p ¯ ℓ = ln L M p FIGURE 1.
Determinations of the effective coupling constants ℓ and ℓ . is determined by the pion decay constant. The corresponding numerical values of a and a are indicated by the leftmost dot in Fig. 2, while the other two show the resultobtained at NLO and NNLO of the chiral expansion, respectively. The exotic scatteringlength a is barely affected by the higher order corrections, but the shift seen in a isquite substantial. The physics behind this enhancement of the perturbations generatedby m u and m d is well understood: it is a consequence of the final state interaction, whichis attractive in the S -wave, rapidly grows with the energy and hence produces chirallogarithms with large coefficients.Near the center of the Mandelstam triangle, the contributions from higher ordersof the chiral expansion are small [11]. Using dispersion theory to reach the physicalregion, we arrived at the remarkably sharp predictions for the two scattering lengthsindicated on the left panel of Fig. 2. Our analysis also shows that the corrections a -0.05 -0.05-0.04 -0.04-0.03 -0.03-0.02 -0.02-0.01 -0.01 a Universal bandtree (1966), one loop (1983), two loops (1996)Prediction ( c PT + dispersion theory, 2001) l from low energy theorem for scalar radius (2001)NPLQCD (2005, 2007) l and l from MILC (2004, 2006) l from Del Debbio et al. (2006) l and l from ETM (2007) l and l from RBC/UKQCD (2007) l and l from PACS-CS (preliminary) a -0.05 -0.05-0.04 -0.04-0.03 -0.03-0.02 -0.02-0.01 -0.01 a Universal bandtree (1966), one loop (1983), two loops (1996)Prediction ( c PT + dispersion theory, 2001)E865 Ke4 (2003) isospin correctedDIRAC (2005)NA48 K3 p (2006)NA48 Ke4 (preliminary) isospin corrected FIGURE 2.
Comparing the theoretical predictions for the pp S-wave scattering lengths with latticeresults (left) and with experiment (right).Model independent determination of the s pole October 26, 2018 3 o Weinberg’s low energy theorem for a , a are dominated by the effective couplingconstants ¯ ℓ , ¯ ℓ discussed above – if these are known, the scattering lengths can becalculated within small uncertainties. Except for the horizontal band, which representsa direct determination of a based on the volume dependence of the levels [13], all ofthe lattice results for the scattering lengths shown on the left panel of Fig. 2 are obtainedin this way from the corresponding results for ℓ and ℓ . The figure neatly demonstratesthat the lattice results confirm the predictions for a , a . PRECISION EXPERIMENTS AT LOW ENERGY
The right panel of Fig. 2 compares the predictions for the scattering lengths with recentexperimental results. While the K e data of E865 [14], the DIRAC experiment[15] andthe NA48 data on the cusp in K → p [16] all confirm the theoretical expectations, themost precise source of information, the beautiful K e data of NA48 [17], gives rise toa puzzle. The Watson theorem implies that – if the electromagnetic interaction and thedifference between m u and m d are neglected – the relative phase of the form factorsdescribing the decay K → e npp coincides with the difference d − d of scatteringphase shifts. At the precision achieved, the data on the form factor phase do not agreewith the theoretical prediction for the phase shifts.The origin of the discrepancy was identified by Colangelo, Gasser and Rusetsky [18].The problem has to do with the fact that neutral kaons may first decay into a pair ofneutral pions, which then undergoes scattering and winds up as a charged pair. The massdifference between the charged and neutral pions affects this process in a pronouncedmanner: it pushes the form factor phase up by about half a degree – an isospin breakingeffect, due almost exclusively to the electromagnetic interaction. Fig. 3 shows that thediscrepancy disappears if the NA48 data on the relative phase of the form factors arecorrected for isospin breaking. Accordingly, the range of scattering lengths allowed bythese data, shown on the right panel of Fig. 2, is in perfect agreement with the prediction.The intersection of this range with the band from the low energy theorem for the scalarradius (left panel) yields a = . ( ) .In the mass range M pp >
350 MeV, Fig. 3 indicates a marginal disagreement betweenNA48/2 and E865. While E865 collects all events in this region in a single bin, theresolution of NA48/2 is better. The fit to all K e data is therefore dominated by NA48/2.For a detailed discussion of these issues, I refer to the talks by B. Bloch-Devaux, G.Colangelo and J. Gasser at KAON 2007. I conclude that the puzzle is gone: K e confirmsthe theory to remarkable precision. ROY EQUATIONS
As mentioned above, the straightforward evaluation of the chiral perturbation seriesfor the pp scattering amplitude is useful only in a very limited range of the kinematicvariables – definitely, the resonance poles are outside this range. The domain of validitycan be extended considerably by means of dispersion theory: analyticity, unitarity andcrossing symmetry essentially determine the low energy properties of the scatteringamplitude in terms of the two S-wave scattering lengths. I now wish to show that the Model independent determination of the s pole October 26, 2018 4 .28 0.3 0.32 0.34 0.36 0.38 0.4 GeV -505101520 d - d theoretical prediction (2001)Geneva-Saclay (1977)E865 (2003) isospin correctedNA48/2 (2006) isospin corrected FIGURE 3.
Comparison of K e data with the prediction for d − d properties of the lowest resonance of QCD can be worked out on this basis.From the point of view of dispersion theory, pp scattering is particularly simple:the s -, t - and u -channels represent the same physical process. As a consequence, thescattering amplitude can be represented as a dispersion integral over the imaginary partand the integral exclusively extends over the physical region [19]. The projection ofthe amplitude on the partial waves leads to a dispersive representation for these, theRoy equations. I denote the S-matrix elements by S I ℓ = h I ℓ exp 2 i d I ℓ and use the standardnormalization for the corresponding partial wave amplitudes t I ℓ : S I ℓ ( s ) = + i r ( s ) t I ℓ ( s ) , r ( s ) = q − M p / s . (2)The S-matrix elements and the partial wave amplitudes are analytic in the cut s -plane.There is a right hand cut (4 M p < s < ¥ ) as well as a left hand cut ( − ¥ < s < I = ℓ = t ( s ) = a + ( s − M p ) b + (cid:229) I = ¥ (cid:229) ℓ = Z ¥ M p ds ′ K I ℓ ( s , s ′ ) Im t I ℓ ( s ′ ) . (3)The equation contains two subtraction constants. As a consequence of crossing symme-try, these can be expressed in terms of the S-wave scattering lengths: a = a , b = ( a − a ) / M p . (4)The kernels K II ′ ℓℓ ′ ( s , s ′ ) are explicitly known algebraic expressions which only involve thevariables s , s ′ and the mass of the pion. The integrals on the right hand side thus onlyinvolve observable quantities: the imaginary parts of the partial waves.As demonstrated by Roy, his equations rigorously follow from general principles ofquantum field theory. They are valid for real values of s in the interval − M p < s < M p (the upper end is pushed up to 68 M p if the scattering amplitude obeys Mandelstamanalyticity). Using known results of general quantum field theory [20, 21], we haveshown that these equations also hold for complex values of s , in the intersection of therelevant Lehmann-Martin ellipses [1]. Model independent determination of the s pole October 26, 2018 5 he pioneering work on the physics of the Roy equations was carried out more than 30years ago [22]. The main problem encountered at that time was that the two subtractionconstants were not known. These dominate the dispersive representation at low energies,but since the data available at the time were consistent with a very broad range of S -wavescattering lengths, the Roy equation analysis was not conclusive.The insights gained by means of c PT thoroughly changed the situation. As discussedin detail in the first part of this report, the S-wave scattering lengths are now known veryaccurately. The main limitation in the numerical evaluation of equation (3) arises fromthe accuracy to which the imaginary parts can be pinned down. In this connection, it isessential that the Roy equations involve two subtractions, so that the kernels fall off withthe third power of the variable of integration. This ensures that the contributions fromthe low energy region dominate. Note that the left hand cut plays an important role here:taken by itself, the part of the kernel that accounts for the right hand cut falls off onlywith the first power of the variable of integration, but the high energy tail is cancelled bythe contribution from the left hand cut.At low energies, the S- and P-waves dominate. In [23], we solved the Roy equationsfor these waves only below 800 MeV, relying on the literature to estimate the contri-butions from higher energies and from the partial waves with ℓ ≥
2. In the meantime,we have extended our analysis and now solve the Roy equations on their full range ofvalidity, not only for the S- and P-waves, but also for the D- and F-waves. Treating theS-wave scattering lengths, the imaginary parts above s max = M p and the elasticitiesas input, the Roy equations admit a two-parameter family of solutions. We identify thetwo free parameters with the values of the phase shifts d and d at 800 MeV. In viewof the excellent experimental information about the vector form factor of the pion, thevalue of d (
800 MeV) is reliably known, but the phenomenological information about d (
800 MeV) is comparatively meagre – this currently represents the main source ofuncertainty in low energy pp scattering and will be discussed in detail below. POLE FORMULA
The positions of the poles represent universal properties of QCD, which are unambigu-ous even if the width of the resonance turns out to be large, but they concern the non-perturbative domain, where an analysis in terms of the local degrees of freedom – quarksand gluons – is not in sight. One of the reasons why the values for the pole position ofthe s quoted by the Particle Data Group cover a very broad range is that all but one ofthese either rely on models or on the extrapolation of simple parametrizations: the dataare represented in terms of suitable functions on the real axis and the position of thepole is determined by continuing this representation into the complex plane. If the widthof the resonance is small, the ambiguities inherent in the choice of the parametrizationdo not significantly affect the result, but the width of the s is not small. For a thoroughdiscussion of the sensitivity of the pole position to the freedom inherent in the choice ofthe parametrization, I refer to [24].The determination of the s pole provides a good illustration for the strength of thedispersive method and for the relative importance of the various terms on the right handside of the Roy equations. The representation of the S-matrix element given above holds Model independent determination of the s pole October 26, 2018 6 n a limited region of the first sheet. The pole sits on the second sheet, which is reachedfrom the first by analytic continuation from the upper half plane into the lower half plane,crossing the real axis in the interval 4 M p < s < M p , where the scattering is elastic.Now, unitarity implies that the values of the S-matrix element on the first and secondsheets are related by S ( s ) II = / S ( s ) I . Hence a pole on the second sheet occurs if andonly if S ( s ) has a zero on the first sheet. Accordingly, we have an exact equation, whichallows us to study the behaviour of the amplitude in the vicinity of the pole and find outwhether – in the limited region of the first sheet where the Roy equations are valid –there are any resonances with the quantum numbers of the vacuum: S ( s ) = . (5)As emphasized above, the representation of the S-matrix element that follows fromthe Roy equations exclusively involves observable quantities and can be evaluated forcomplex values of s just as well as for real values – for the above formula, an analyticcontinuation is not needed.Inserting our central representation for the scattering amplitude in (5), we find that,in the region where the Roy equations are valid, the function S ( s ) has two zeros in thelower half of the first sheet: one at √ s = − i
272 MeV, the other in the vicinity of 1GeV [1]. While the first corresponds to the state f ( ) , commonly referred to as the s ,the second zero represents the well-established resonance f ( ) . Our analysis shedslittle light on the properties of the latter, because the location of the zero is sensitive tothe input used for the elasticity h ( s ) – the shape of the dip in h ( s ) and the position ofthe zero represent two sides of the same coin. For this reason, I only discuss the s . DISCUSSION
We are by no means the first to find a resonance in the vicinity of the above position. Inthe list of papers quoted by the Particle Data Group [25], the earliest one with a pole inthis ball park appeared more than 20 years ago [26]. What is new is that we can performa controlled error calculation, because our method is free of the systematic theoreticalerrors inherent in models and parametrizations. For this purpose, it is convenient to splitthe right hand side of the Roy equation for t ( s ) into three parts:1. Subtraction terms2. Contribution from Im t ( s ) below K ¯ K threshold3. Contributions from higher energies and other partial waves ad 1. As discussed above, the subtraction terms are determined by the S-wave scatter-ing lengths. The prediction of [11] reads a = . ± . , a = − . ± . a , we find that an increase by 0.005 shifts thepole position by ( − . + i . ) MeV, while the response to an increase in a by 0.0010 isa shift of ( . − i . ) MeV [1]. These numbers show that the error in the pole positiondue to the uncertainties in the subtraction constants are small. ad 2.
Below K ¯ K threshold, the S-waves are elastic to a very good approximation. Thefunction Im t ( s ) shows a broad bump, nearly hits the unitarity limit somewhere between Model independent determination of the s pole October 26, 2018 7 .3 0.4 0.5 0.6 0.7 0.8 0.9 1 GeV d Roy solutions with 78.3 o < d A < 92.3 o Bugg 2006Achasov & Kiselev 2007Albaladejo & Oller 2008
GeV d Roy solutions with 78.3 o < d A < 92.3 o KPY III 2008HPR 2008IAM with physical values of l , l FIGURE 4.
Behaviour of d below K ¯ K threshold
800 and 900 MeV and then rapidly drops, because the phase steeply rises, reaching 180 ◦ in the vicinity of 2 M K . Hence there is a pronounced dip in Im t ( s ) near K ¯ K threshold:the behaviour of the imaginary part is controlled almost entirely by the phase shift d ( s ) .Fig. 4 shows several recent representations for this phase. Replacing the integral over ourcentral representation for Im t ( s ) from 4 M p to 4 M K by the one of Bugg [27] and leavingeverything else as it is, the pole moves to 444 - i 267 MeV. Repeating the exercisewith the representations of Achasov and Kiselev [28], Albaladejo and Oller [29] andKami´nski, Peláez and Ynduráin [30], the pole is shifted to 438 - i 274 MeV, 451 - i 257MeV and 458 - i 253 MeV, respectively. As mentioned above, the solution of the Royequation for t ( s ) depends on the value of d A ≡ d (
800 MeV). The shaded band in thefigure shows the behaviour of the Roy solutions for 78 . ◦ < d A < . ◦ , the range usedin [1]. If the imaginary part of t ( s ) is evaluated with the lower edge of this band, thepole occurs at 435 - i 276 MeV, while the upper edge corresponds to 456 - i 262 MeV.The representation used by Ynduráin and collaborators underwent a sequence ofgradual improvements. The most recent edition is definitely better than the earlierversions, but an important difference to our representation remains: as can be seen on theright panel of Fig. 4, the phase d ( s ) of KPY III [30] still contains a kink (discontinuityin the first derivative) at 932 MeV, as well as a hump below that energy. The kinkarises because (i) two different parametrizations are used below and above 932 MeVand (ii) the correlations between the two regions imposed by causality are ignored. Thedeficiency is discussed in detail in [31], where it is shown that both the hump and the kinkare artefacts, produced by the use of a parametrization that is not flexible enough. Theanalysis in [24] fully confirms this conclusion: among the 42 conformal parametrizationsconstructed there, all of those with an acceptable behaviour above 900 MeV run withinthe band of Roy solutions specified above.The hump is also responsible for the disagreement between [11] and [30] regardingthe scattering lengths of the partial waves with angular momentum ℓ = , , ℓ = , , ,
3: the contributions from Im t ( s ) to the sum rules for thesequantities are not the same (the sum rules are listed in Eqs. (14.1) and (14.3) of [23]).Within errors, the difference between these contributions reproduces the difference inthe quoted results, without exception. Model independent determination of the s pole October 26, 2018 8 add a remark concerning the model of Hanhart, Peláez and Ríos [32], who applythe inverse amplitude method to improve the one loop approximation to the chiralperturbation series of SU(2) × SU(2). In the original formulation of the model, the chiralexpansion t ( s ) = t ( s ) + t ( s ) + . . . is unitarized with t ( s ) = t ( s ) / { − t ( s ) / t ( s ) } , butthis recipe fails in the vicinity of the Adler zero, because the term t ( s ) does not vanishthere. The deficiency is readily cured. It suffices to replace the IAM formula with t ( s ) = ˜ t ( s ) − ˜ t ( s ) / ˜ t ( s ) , ˜ t ( s ) = t ( s ) − t ( s A ) , ˜ t ( s ) = t ( s ) + t ( s A ) , (6)where s A is the position of the Adler zero in one loop approximation. Since t ( s A ) represents a term of O ( p ) , the chiral expansion of (6) reproduces the one loop approx-imation of c PT, also in the vicinity of the Adler zero. A similar recipe is used in [32].The model exclusively involves the coupling constants F p , ℓ , . . . , ℓ of the effectiveLagrangian. As discussed above, ℓ and ℓ are known quite well; ℓ and ℓ can bedetermined on phenomenological grounds [11]. The result for the phase shift obtainedby inserting the numerical values in the above formula is indicated on the right panel ofFig. 4. This shows that the model yields a decent approximation only below 500 MeV.The parametrization used by Hanhart eta al. [32] is better, because these authors treat thecoupling constants ℓ and ℓ as free parameters. This extends the range of energies wherethe IAM parametrization makes sense, but since the model does not account for the sharpincrease in the phase towards K ¯ K threshold, it can at best give a semi-quantitative pictureof the s . For the parameter values adopted in [32], the zero of the denominator in (6)occurs at 444(6) - i 218(10) MeV: the mass is OK, but the width is too low by 100 MeV.Inserting the observed values of ℓ and ℓ , the zero moves to 413(12) - i 269(12) MeV:now the width is OK, but the mass is too low. ad 3. Finally, I turn to the contributions of the third category: higher energies andother partial waves. Among these, the one from the P-wave, for example, is by no meansnegligible, but, as mentioned above, this wave is known very well. In fact, in the vicinityof the zero of S ( s ) , the sum of the contributions of this category can be worked outquite accurately. In [1], we estimated the net uncertainty in the pole position from thissource at ± ± i 6 MeV. As a check, we can simply replace our central representationfor the contributions of category 3 by the one in [30], retaining our own representationonly for the remainder. The operation shifts the pole position by - 0.6 - i 1.2 MeV, wellwithin the estimated range. CONCLUSION
Adding the errors up in square, the result for the pole position becomes [1] √ s s = + − − i + − . MeV . (7)The error bars account for all sources of uncertainty and are an order of magnitudesmaller than for the crude estimate √ s s = (400 - 1200) - i (250 - 500) MeV quoted bythe Particle Data Group [25]. The dispersive representation of the S-matrix element also Model independent determination of the s pole October 26, 2018 9 llows us to calculate the residue of the pole occurring on the second sheet, t ( s ) II = r s s − s s + . . . (8)Our preliminary result for the magnitude of the residue is | r s | = . + . − . GeV .I thank Irinel Caprini and Gilberto Colangelo for a very pleasant collaboration andChris Sachrajda, Daisuke Kadoh, Yoshinobu Kuramashi, David Bugg, Kolia Achasov,Lesha Kiselev, Miguel Albaladejo and José Antonio Oller for informative discussionsand correspondence. Also, it is a pleasure for me to thank George Rupp for the invitationand for warm hospitality during my stay at Lisbon. REFERENCES
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