pi-pi and pi-K scatterings in three-flavour resummed chiral perturbation theory
aa r X i v : . [ h e p - ph ] O c t ππ and πK scatterings in three-flavour resummedchiral perturbation theory S´ebastien Descotes-Genon
Laboratoire de Physique Th´eorique,CNRS/Univ. Paris-Sud 11 (UMR 8627), 91405 Orsay Cedex, FranceE-mail: [email protected]
Abstract.
The (light but not-so-light) strange quark may play a special role in the low-energydynamics of QCD. The presence of strange quark pairs in the sea may have a significant impactof the pattern of chiral symmetry breaking : in particular large differences can occur betweenthe chiral limits of two and three massless flavours (i.e., whether m s is kept at its physical valueor sent to zero). This may induce problems of convergence in three-flavour chiral expansions. Tocope with such difficulties, we introduce a new framework, called Resummed Chiral PerturbationTheory. We exploit it to analyse ππ and πK scatterings and match them with dispersive resultsin a frequentist framework. Constraints on three-flavour chiral order parameters are derived.
1. Two chiral limits of interest
Because of its intermediate mass, the strange quark has a special status in low-energy QCD. Itis light enough to allow for a combined expansion of observables in powers of m u , m d , m s aroundthe N f = 3 chiral limit (meaning 3 massless flavours): m u = m d = m s = 0. But it is sufficientlyheavy to induce significant changes from the N f = 3 limit to the N f = 2 limit: m u = m d = 0and m s physical. Each limit can engender its own version of Chiral Perturbation Theory ( χ PT).In the N f = 2 limit, the pions are the only degrees of freedom, whereas N f = 3 χ PT deals withpions, kaons and η . This second version of χ PT is richer, discusses more processes in a largerrange of energy, but contains more unknown low-energy constants (LECs) and may have a slowerconvergence. The details of the connection between the two theories remain under debate.Indeed, due to ¯ ss sea-pairs, order parameters such as the quark condensate and thepseudoscalar decay constant, Σ( N f ) = − lim N f h ¯ uu i and F ( N f ) = lim N f F π , can reachsignificantly different values in the two chiral limits (lim N f denoting the chiral limit with N f massless flavours) [1]. An illustation is provided by the quark condensate:Σ(2) = Σ(2; m s ) = Σ(2; 0) + m s ∂ Σ(2; m s ) ∂m s + O ( m s ) (1)= Σ(3) + m s lim m u ,m d → i Z d x h | ¯ uu ( x ) ¯ ss (0) | i + O ( m s ) (2)Here, ¯ ss -pairs are involved through the two-point correlator h (¯ uu )(¯ ss ) i , which violates the Zweigrule in the vacuum (scalar) channel. One expects [1] that this effect should suppress orderparameters when m s →
0: Σ(2) ≥ Σ(3) and F (2) ≥ F (3). Since the quark condensate(s) andthe pseudoscalar decay constant(s) are the leading-order LECs for the two versions of χ PT, astrong decrease from N f = 2 to 3 would have an impact on the structure of the two theories. . Two- and three-flavour chiral expansions A few years ago, the E865 collaboration provided new data on K ℓ decays [2]. Building uponthe dispersive analysis of ππ scattering [3], we extracted the two-flavour order parameters [4]: X (2) = ( m u + m d )Σ(2) / ( F π M π ) = 0 . ± . Z (2) = F (2) /F π = 0 . ± .
03 (3)A different analysis, with an additional theoretical input from the scalar radius of the pion, ledto an even larger value for X (2) [5]. The situation is somewhat modified by new data from theNA48 collaboration [6], which show some discrepancy with the E865 phase shifts in the higherend of the allowed phase space. The role of isospin breaking corrections is under discussioncurrrently. The preliminary values of the phase shifts [6] tend to increase the value of the I = J = 0 ππ scattering length, and to decrease the value of the two-flavour quark condensate,pushing X (2) down to 0.7. In any case, one would expect values closer to 1, since X (2) and Z (2) monitor the convergence of N f = 2 chiral expansions of F π M π and F π respectively. Suchexpansions in powers of m u and m d only should exhibit smaller NLO corrections (below 10%) [4].To include K - and η -mesons dynamically, one must use three-flavour χ PT around the N f = 3chiral limit. Strange sea-quark loops may affect chiral series by damping the leading-order(LO) term, which depends on F (3) and Σ(3), and by enhancing next-to-leading-order (NLO)corrections, in particular when violating the Zweig rule in the scalar sector. Take for instance: F π = F (3) + 16( m s + 2 m ) B ∆ L + 16 mB ∆ L + O ( m q ) (4)where B = − lim m u ,m d ,m s → h ¯ uu i /F π , and we have put together NLO low-energy constants andchiral logarithms ∆ L = L ( M ρ ) + 0 . · − , ∆ L = L ( M ρ ) + 0 . · − (enhanced by m s ).If we assume that the LO contribution is numerically dominant (i.e., F π = F (3) to a very goodapproximation), we can perform the following manipulations: F (3) F π = F (3) F (3) + O ( m q ) = 1 − M K + M π F π ∆ L − M π F π ∆ L + O ( m q ) = 1 − . − .
04 + O ( m q )(5)where we have used 1 / (1 + x ) = 1 − x and eq. (4) at the second step, and the second andthird terms of the last equality are obtained using L ( M ρ ) = 0 . · − and L ( M ρ ) = 1 . · − respectively [7]. This is clearly in contradiction with the orginal assumption F π ≃ F (3) . Theavailable dispersive estimates of L [9] yield a similar situation for F π M π citeresum. Therefore,a small positive value of L ( M ρ ) or L ( M ρ ) is enough to spoil the rapid convergence of N f = 3chiral series and to induce a numerical competition between formal LO and NLO contributions.
3. Constraints from ππ and πK scatterings The potentially “large” values of L and L lead to a numerical competition between formalLO and NLO contributions in chiral series. To deal with such a situation, we have introduceda framework, called Resummed Chiral Perturbation Theory (Re χ PT) [10], where we define theappropriate observables to consider and the treatment of their chiral expansion [10, 8]. It allowsfor a a resummation of the potentially large effect of the Zweig-rule violating couplings L and L [10, 8]. Since this framework copes with the possibility of a numerical competition between(formal) LO and NLO terms in chiral series, some usual O ( p ) results are not valid any longer:for instance r = m s /m is not fixed by M K /M π and may vary from 8 to 40.One can apply this framework to ππ and πK scatterings, which provide information on N f = 2and N f = 3 patterns of chiral symmetry breaking respectively, and in particular on r = m s /m ,the quark condensate X (3) = 2 m Σ(3) / ( F π M π ) and the decay constant Z (3) = F (3) /F π . Onecan exploit dispersive relations, such as Roy equations [3] and Roy-Steiner equations [7], toreconstruct the amplitudes from the phase shifts from threshold up to energies around 1 GeV. r C L pi pipi Kboth X(3) C L pi pipi Kboth Figure 1.
Profiles for the confidence levels of r = m s /m (left) and X (3) = 2 m Σ(3) / ( F π M π )(right). In each case, the results are obtained from ππ scattering only (dashed line), πK scattering only (dotted line), or both sources of information (solid line).Matching the dispersive and chiral representations of the amplitude in a frequentist frameworkprovides constraints (in terms of confidence levels) on the main parameters of interest for three-flavour χ PT [10]. Fig. 1 shows the situation for r = m s /m and X (3) = 2 m Σ(3) / ( F π M π ).The main impact of ππ scattering consists in constraining r = m s /m : indeed, ππ scatteringpins down the two-flavour condensate X (2) rather accurately, which can be related to r throughthe spectrum of pseudoscalar mesons [1, 8]. The combination of ππ and πK scatterings yields: r ≥ . , X (3) ≤ . , Y (3) ≤ . , . ≤ Z (3) ≤ . [68%CL] (6)
4. Conclusion
The presence of massive s ¯ s -pairs in the QCD vacuum may induce significant differences in thepattern of chiral symmetry breaking between the N f = 2 and N f = 3 chiral limits. This effect,related to the violation of the Zweig rule in the scalar sector, may spoil the convergence ofthree-flavour chiral expansions. We introduce Resummed Chiral Perturbation Theory to dealwith such a problem, and apply it to our experimental knowledge on ππ and πK scatterings.The outcome does not favour the usual picture of a large quark condensation independent ofthe number of massless flavours. Further experimental information is needed to constrain thepattern of chiral symmetry breaking efficiently and learn more about its variation with N f . Acknowledgments
Work supported in part by the EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”. [1] S. Descotes-Genon, L. Girlanda and J. Stern, JHEP 0001 (2000) 041.[2] S. Pislak et al. , Phys. Rev. D (2003) 072004.[3] B. Ananthanarayan, G. Colangelo, J. Gasser and H. Leutwyler, Phys. Rept. (2001) 207.[4] S. Descotes-Genon, N. Fuchs, L. Girlanda and J. Stern, Eur. Phys. J. C 24 (2002) 469.[5] G. Colangelo, J. Gasser and H. Leutwyler, Phys. Rev. Lett. 86 (2001) 5008.[6] G. Lamanna, ππ scattering from K e decaysdecays