ar X iv :h ep -p h/ 05 11 14 7v 1 1 1 N ov 2 00 5 IPPP/05/67 DCPT/05/134 Studying unquenching effects in QCD with Dyson-Schwinger equations ∗ Christian S. Fischera, Reinhard Alkoferb, Wolfgang Cassingc, Felipe Llanes-Estradad, Peter Watsone, aIPPP, University of Durham, Durham DH1 3LE, U.K. bInstitute of Physics, Graz University, Universitätsplatz 5, A-8010 Graz, Austria cInstitute for Theoretical Physics, Univ. of Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany dFisica Teorica I, Univ. Complutense, Madrid 28040, Spain eInstitute for Theoretical Physics, University of Tübingen, D-72076 Tübingen, Germany We summarise recent results on the properties of gluons, quarks and light mesons from the Green’s functions approach to QCD. We discuss a self-consistent, infrared power law solution for the Schwinger-Dyson equations of the 1PI-Greens functions of Yang-Mills theory. The corresponding running coupling has a universal fixed point at zero momentum. Based on these analytical results a truncation scheme for the coupled system of Schwinger-Dyson equations for the propagators of QCD and the Bethe-Salpeter equation for light mesons has been formulated. We compare numerical results for charge eigenstate vector and pseudoscalar meson observables with corresponding lattice data. The effects of unquenching the system are found to be small but not negligible. One of the most fascinating problems of QCD is to find a low energy description of colourless bound states (hadrons) in terms of their non- perturbative, coloured constituents (quarks and gluons). Lattice simulations are not entirely sat- isfactory in this respect. They provide values for the global properties of hadrons (masses, de- cay widths etc.,), but they may not be capable to provide enough information on their internal structure to understand all dynamical aspects of low energy QCD. An alternative field theoretical and relativistic method which is well suited to de- liver this information is the Schwinger-Dyson and Bethe-Salpeter formalism [1,2]. In principle this approach allows one to derive meson properties directly from the fundamental building blocks of the field theory, the Green’s functions. Lattice simulations and the Green’s functions approach are complementary to each other in several re- spects. Lattice simulations are ab initio whereas the Green’s functions approach has to rely on an ∗Invited talk given by C. S. F. at the ’Workshop on com- putational hadron physics’, Sept. 13 - 17, Nikosia, Cyprus. (educated) approximation scheme. The Green’s functions approach is continuum based. It allows for analytical investigations in the infrared and all aspects of chiral symmetry and its breaking pattern are respected. In this talk we summarise recent results in the Green’s functions framework that dwell on these advantages. We discuss an- alytical results on the infrared exponents of the 1PI-Green’s functions of SU(Nc)-Yang-Mills the- ory in Landau gauge. We report on numerical re- sults for the ghost, gluon and quark propagators as well as light meson observables in a truncation scheme that is based on the analytical findings. Our focus in particular is on unquenching effects due to light quark loops in the gluon polarisation. 1. Infrared exponents for the Green’s func- tions of Yang-Mills theory The infrared behaviour of the Green’s functions of SU(Nc)-Yang-Mills theory is related to confine- ment in several ways. A particularly interesting example is the Kugo-Ojima confinement criterion of a well-defined global colour charge. The crite- 1 http://arXiv.org/abs/hep-ph/0511147v1 2 rion is satisfied in Landau gauge if the dressing function of the ghost propagator is singular and the gluon propagator is finite or vanishes in the infrared. Provided BRST-symmetry is conserved nonperturbatively, the cohomology of the BRST- operator then defines a physical state space with colourless states only [3]. A convenient starting point to investigate the infrared behaviour of one-particle-irreducible (1PI) Green’s functions is the Schwinger-Dyson equation (SDE) for the ghost-gluon vertex, shown diagrammatically fig. 1. The dressed ghost-gluon vertex Γabc µ = Γµ(p, q)fabc can be represented by the two tensor structures Γµ(p, q) = pµA(p2, q2) + kµB(p2, q2), (1) where pµ and qµ are the momenta of the out- going and incoming ghost and kµ = −pµ − qµ is the gluon momentum. In Landau gauge, the momentum qµ of the incoming ghost factorises from the vertex dressing, as can be seen from fig. 1: Since the gluon propagator Dµν is trans- verse in Landau gauge, its contraction with the bare ghost-gluon vertex lµ in the loop of the SDE gives lµDµν(l − q) = qµDµν(l − q). Let us assume for the moment that (I) the loop-integral is finite in the infrared. (We come back to this assumption in the para- graph below eq. (6).) We then observe that the dressing of the full ghost-gluon vertex vanishes if qµ goes to zero, Γµ(p, q) = pµ(A − B) − qµB qµ→0−→ pµ, (2) and thus neither A(p2, q2) nor B(p2, q2) can be singular in this limit [4]. Since factorisation of the other ghost momentum is trivial, the same happens for pµ → 0. One thus concludes that the dressing of the ghost-gluon vertex is finite in the infrared and may be well approximated at small momenta by the bare vertex. This has been confirmed by lattice and SDE-studies [5,6,7]. A finite ghost-gluon vertex at small momenta admits the following power law solution for the ghost-SDE in the infrared: Writing the ghost and gluon propagators as DG(p2) = −G(p2) p2 , = + q − l l q Figure 1. Schwinger-Dyson equation for the ghost-gluon vertex. Figure 2. Schwinger-Dyson equation for the ghost propagator. Dµν(p2) = ( δµν − pµpν p2 ) Z(p2) p2 , (3) one finds power laws for the ghost and gluon dressing functions with interrelated exponents given by G(p2) ∼ (p2)−κ, Z(p2) ∼ (p2)2κ . (4) [NB: This can be checked easily by just counting anomalous dimensions on both sides of the equa- tion. Note that the loop-integral is dominated by momenta of the same magnitude as the external momentum. Thus, for small external momenta one can replace the propagators in the loop by their infrared approximation, eq. (4).] In this no- tation the Kugo-Ojima criterion translates to the condition κ ≥ 0 for the ghost dressing function and κ ≥ 0.5 for the gluon dressing function. On general grounds, the exponent κ is known to be positive [4], independent of any truncation of the SDEs. For a bare ghost-gluon vertex in the in- frared one obtains κ = (93 − √ 1201)/98 ≈ 0.595 [8,9], which satisfies both criteria. This specific value of κ is found to vary only slightly for a large class of possible dressings of the ghost-gluon- vertex [9]. Similar values have been determined 3 Figure 3. Exact Schwinger-Dyson equation for the three-gluon vertex and lowest order in a skele- ton expansion of the four- and five-point func- tions. All internal propagators in the diagrams are to be understood as fully dressed. from exact renormalisation group equations [10]. The basic idea to determine the infrared be- haviour of the other (1PI) Green’s functions is to investigate their Schwinger-Dyson equations or- der by order in a skeleton expansion (i.e. a loop expansion using full propagators and vertices). This program has been carried out in ref. [11]. It turns out that in this expansion the Green’s func- tions can only be infrared singular, if all external scales go to zero. Thus to determine the degree of possible singularities it is sufficient to investi- gate the SDEs in the presence of only one external scale p2 ≪ Λ2 QCD. As an example we summarise the treatment of the SDE for the three-gluon ver- tex. In fig. 3 we show the full equation as well as the lowest order in a skeleton expansion of the four-and five-point functions. In the presence of one (small) external scale the approximated SDE has a selfconsistent power law solution given by Γ3g(p2) ∼ (p2)−3κ. (5) [NB: Again this can be seen easily by counting anomalous dimensions on both sides of the equa- tions. The leading diagram on the right hand side is the one involving ghosts, diagram (a), the oth- ers are less singular (recall κ > 0). The loops are again dominated by momenta of the same magni- tude as the external scale.] One can see by induc- tion that this solution is also present if terms to arbitrary high order in the skeleton expansion are taken into account. Thus the skeleton expansion is stable wrt. the infrared solution of the SDEs. This technique can also be applied to any other SDE. A self-consistent solution of the whole tower of SDEs is then given by [11] Γn,m(p2) ∼ (p2)(n−m)κ. (6) Here Γn,m(p2) denotes the infrared leading dress- ing function of the 1PI-Green’s function with 2n external ghost legs and m external gluon legs. By counting anomalous dimensions it can be checked easily that the expression eq. (6) indeed solves the full three-gluon vertex SDE in fig. 3 selfconsis- tently. Furthermore, inserting Γ1,2(p2) ∼ (p2)−κ together with the power laws (4) into the SDE for the ghost-gluon vertex, fig. 1, one can verify the assumption (I) that the loop-integral of the ver- tex dressing is indeed finite in the infrared. Thus eq. (6) is a truly selfconsistent infrared solution of the tower of SDEs. [NB: It is worth mentioning that the solution (6) also has the correct scal- ing behaviour such that the Slavnov-Taylor iden- tities of the renormalisation constants are sat- isfied. Since the theory is multiplicative renor- malisable these functions scale with the renor- malisation point µ2 in the same way as the 1PI- functions with the external scale p2. E.g. the relation Z1/Z3 = Z̃1/Z̃3 between the three-gluon vertex, gluon propagator, ghost-gluon vertex and ghost propagator renormalisation constant leads to Z1(µ 2) = (µ2)−3κ, which agrees with eq. (5).] Certainly, selfconsistency is not enough to es- tablish eq. (6) as the ’true’ solution of Yang-Mills theory in the infrared, since there may be other selfconsistent solutions of the SDEs. However, one may argue that the solution (6) has an in- teresting property that qualifies it as a promising candidate: it leads to qualitative universality of the running coupling in the infrared. Renormal- 4 isation group invariant couplings can be defined from either of the primitively divergent vertices of Yang-Mills-theory, i.e. from the ghost-gluon vertex (gh), the three-gluon vertex (3g) or the four-gluon vertex (4g) via αgh(p2) = g2 4π G2(p2)Z(p2) p2 →0∼ c1 Nc , (7) α3g(p2) = g2 4π [Γ0,3(p2)]2 Z3(p2) p2 →0∼ c2 Nc , (8) α4g(p2) = g2 4π [Γ0,4(p2)]2 Z4(p2) p2 →0∼ c3 Nc . (9) Using the SDE-solution (6) it is easy to see that all three couplings approach a fixed point in the infrared. The constants ci may be different for each coupling and depends on the respective choice of the tensor component used to extract the vertex dressing functions Γ (this ambiguity is well know in the literature [12]). For the coupling (9) of the ghost-gluon vertex this fixed point can be explicitly calculated using propagator dressing functions alone. Employing a bare ghost-gluon vertex one obtains αgh(0) ≈ 8.92/Nc [9]. Re- cently it has been shown that this value together with the infrared exponent κ ≈ 0.595 are invari- ant in a class of transverse gauges that interpolate between Landau and Coulomb gauge [13]. 2. Unquenching effects in propagators Dynamical chiral symmetry breaking is, be- sides confinement, the most important low energy property of QCD. It is a truly nonperturbative effect, since there is no dynamical mass genera- tion at every order in perturbation theory. On the quark level, the Schwinger-Dyson formalism is well suited to investigate the chiral symmetry breaking pattern also in the chiral limit of van- ishing current quark masses. It is therefore com- plementary to lattice simulations, which provide reliable results for large quark masses and vol- umes, but are yet severely restricted close to the chiral limit. In the SDE-framework the effects of dynamical chiral symmetry breaking in quenched and (partially) unquenched QCD have been in- vestigated in refs. [14,15]. Based on the analytical results summarised in the last section, ansätze for the vertices have been constructed such that the −1 = −1 - + + −1 = −1 - −1 = −1 - π, K... = π, K... 1 Figure 4. A diagrammatical representation of the coupled system of ghost, gluon and quark Schwinger-Dyson equations and the meson Bethe- Salpeter equation. Filled blobs denote dressed propagators and empty circles denote dressed ver- tex functions. system of SDEs for the ghost, gluon and quark propagators are closed and can be solved numer- ically (for details of the truncation scheme see refs. [14,16]). Here we focus in particular on un- quenching effects in these propagators, which are generated by quarks in the gluon SDE, cf. fig. 4. Numerical solutions for the ghost and gluon propagators can be seen in fig. 5. In the infrared, the numerical SDE-results reproduce the analyti- cal power laws, eqs.(4). (This can be seen explic- itly on a log-log-plot, displayed e.g. in Ref. [16].) In the ultraviolet they reproduce the correct one- loop running from resummed perturbation the- ory. Compared to the results of recent lattice cal- culations [17,18,19] (see also [20]) we find good agreement for large and small momenta. Small deviations for the value of the infrared expo- nent κ between continuum-SDE and lattice re- sults on a finite volume may have methodical reasons. The resulting running coupling on the lattice does not reproduce the fixed point be- haviour in the continuum but vanishes in the in- 5 0 1 2 3 4 5 6 7 8 9 10 p [GeV] 0 1 2 3 4 Z (p 2 ) N f =0, Sternbeck et al (2005) N f =0, Bowman et al (2004) N f =2+1, Bowman et al (2004) N f =0, SDE N f =2+1, SDE 0 1 2 3 4 5 6 p [GeV] 1 2 3 4 G (p 2 ) N f =0, Langfeld et al. (2004) N f =0, Sternbeck et al. (2005) N f =0, SDE N f =2+1, SDE Figure 5. Comparison of the quenched and unquenched ghost and gluon dressing functions with recent lattice data [17,18,19]. The sea-quark masses are mu/d ≃ 16 MeV, ms ≃ 79 MeV in the lattice simulations and mu/d ≃ 3.9 MeV, ms ≃ 84MeV in the SDE-approach. frared [17,21,22]. These effects are also seen when one solves Schwinger-Dyson equations on a torus and are discussed in detail elsewhere [23] (see also [24]). In the intermediate momentum region one clearly sees unquenching effects in the gluon dressing function due to the formation of quark- antiquark pairs from the vacuum. The screening effect from these pairs decreases the bump in the gluon dressing function considerably. The over- all difference in the size of the bump between the 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 p 2 [GeV 2 ] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 M ( p2 ) [G eV ] strange, N f =2+1 strange, N f =0 up/down, N f =2+1 up/down, N f =0 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 p 2 [GeV 2 ] 0.6 0.8 1.0 Z f ( p2 ) strange, N f =2+1 strange, N f =0 up/down, N f =2+1 up/down, N f =0 Figure 6. Comparison of the quenched and un- quenched quark mass (upper diagram) and wave- functions (lower diagram). lattice and the SDE-results is a measure of the influence of the (neglected) gluon-two-loop dia- grams in the gluon-SDE. The up/down and strange quark propagator functions, S(p) = Zf (p2)/(−ip/ + M(p2)) (10) are plotted in fig. 6. We clearly observe a large amount of dynamical mass generated in the in- frared. This mass is reduced by roughly 10 per- cent once quark-loops are taken into account. For large momenta the numerical solutions re- produce the logarithmic running known from re- 6 summed perturbation theory. There are notice- able unquenching effects in the intermediate mo- mentum region, which are, however, much smaller in size than those observed in the gluon propaga- tor. By explicitly solving the quark-SDE in the complex plane one finds a pair of complex conju- gate poles at (0.47 ± 0.29ı)GeV (quenched) and (0.45±0.27ı)GeV (unquenched). Thus unquench- ing hardly has any effect on the position of these singularities (see however ref. [25] for a discussion of the possible influence of scalar tensor pieces in the quark-gluon vertex on the analytical structure of the propagator). 3. Unquenching effects in light meson ob- servables The results for the gluon, ghost and quark propagators, discussed in the last section, serve as input for a calculation of light meson observables employing a Bethe-Salpeter equation (BSE), cf. fig. 4. The crucial link between the bound states and their quark and gluon constituents is pro- vided by the axialvector Ward-Takahashi iden- tity. It relates the quark self energy with the quark-quark interaction kernel in the BSE and thereby guarantees the Goldstone nature of the pions and kaons [26]. For details of the imple- mentation of this identity within the truncation scheme discussed here see ref. [14]. Results for the masses of the pseudoscalar and vector meson (in the isospin symmetric limit) as a function of the current quark mass are shown in fig. 7. For mq → 0 (the chiral limit) we observe a mass- less pion as expected. As for unquenching effects both the pseudoscalar and vector masses with larger quark masses are increased ∼ 30MeV when quarks loops are taken into account. The values for the current quark masses at the physical point together with the resulting pseu- doscalar meson masses, leptonic decay constants and vector meson masses are given in Table 1. When fitted to the experimental pion and kaon masses the resulting up/down and strange-quark masses are lowered when quark loop effects are taken into account. This has also been observed in corresponding lattice simulations [27,28]. We furthermore see that the results for fK and mρ 0 0.05 0.1 m q (GeV) 0 0.2 0.4 0.6 0.8 1 1.2 M ps ( G eV ) N f =0, full SDE N f =3, full SDE 0 0.05 0.1 m q (GeV) 0.6 0.8 1 1.2 1.4 1.6 M ve ( G eV ) N f =0, full SDE N f =3, full SDE Figure 7. Pseudoscalar (upper diagram) and vec- tor (lower diagram) meson masses as functions of the quark mass parameter. We compare results for the quenched and unquenched theory using three degenerate sea quarks. are quite insensitive to whether or not the sys- tem is unquenched (in the restricted sense of fig. 4). This leads to the conclusion that once the in- teraction has been fitted to the pseudoscalar ob- servables, the vector meson mass is largely fixed. The likely explanation for this is that the ground state pseudoscalar and vector mesons are both states with the lowest orbital angular momentum (L = 0) in the sense of the naive (quantum me- chanical) quark model – meaning that they are determined largely by the lowest spin contribu- 7 Nf 0 3 2+1 PDG [29] mu 4.17 4.06 4.06 3-5 ms 88.2 86.0 80-130 mπ 139.7 139.7 140.0 139.6 fπ 130.9 131.1 131.0 130.7 (−〈q̄q〉)1/3 266 271 271 mK 494.5 493.3 493.7 fK 165.6 169.5 160.0 mρ 708.0 690.0 695.2 770.0 Table 1 Parameter sets and results for mπ, fπ, mK , fK and mρ for the quenched case (Nf = 0), the un- quenched case with three degenerate ’sea’-quarks (Nf = 3) and the physical quark configuration case (Nf = 2 + 1) with two up/down quarks and one strange quark. The quark masses and the condensate have been determined using a large renormalisation point and subsequently evolved down to the scale µ = 2 GeV according to their one-loop running. All units are MeV. tions of the kernel in the Bethe-Salpeter equa- tion given by the ladder approximation (which is used here). The interaction then plays the same role in both channels, hence the similarity in re- sults. Note that the ρ meson calculated within the framework of the truncated Bethe-Salpeter equation here refers to a pure quark-antiquark meson with no allowed decay channel. A first step towards including the non-trivial decay width of the physical ρ meson in this formalism has been made in ref. [30] (see also ref [31] for a calcula- tion of gπρρ using quenched Bethe-Salpeter am- plitudes). Here the decay of ρ meson will lead to an additional shift of the ρ meson mass peak as evident from dispersion relations. As a con- sequence our present results should not directly compared to experiment. Although the ρ mass is slightly low the dispersive corrections via its two pion decay (and the inclusion of quark-gluon ver- tex corrections) might yield a satisfactory answer. A suitable quantity to compare results from the Green’s functions framework with lattice data is the vector meson mass as a function of the pseu- doscalar meson mass. Since both quantities are 0 0.2 0.4 0.6 0.8 1 1.2 M ps [GeV] 0.6 0.8 1 1.2 1.4 1.6 M ve [ G eV ] N f =3, full SDE N f =0, full SDE N f =3, phen. model N f =0, phen. model N f =2, JLQCD(2003) N f =0, JLQCD (2003) N f =2, CP-Pacs (2002/4) Figure 8. Vector meson masses as a function of pseudoscalar meson masses. We compare the quenched and degenerate unquenched cases. The lattice results are taken from refs. [27,28]. physical no scheme ambiguities arise. The results are shown in fig. 8. From the SDE/BSE-approach we compare results from the full SDE-setup, de- scribed above, with those obtained employing a phenomenological model for the quark-gluon in- teraction (the details of the model are described in ref. [14]). The model interaction and the effec- tive interaction of the full SDE-setup are comple- mentary to each other in the sense that the model interaction is confined to a quite narrow momen- tum region, whereas the interaction of the full SDE-setup has considerable strength in the in- frared and extends into the ultraviolet according to the correct one-loop scaling known from per- turbation theory. Together, both setups represent a measure for the theoretical error of our calcu- lation. This error is obviously of the same size as the combined systematic error of the different lat- tice simulations. In general, the results are in nice agreement with the lattice data. For pion masses below 240 MeV, where no lattice data are avail- able, the results show a nonlinear dependence of the vector meson mass on the pseudoscalar one. The effect of unquenching – when viewed as a function of the pseudoscalar meson mass – be- comes the same for both schemes: the vector me- 8 son mass is slightly increased when quark loops are taken into account. This trend is also seen in the lattice simulations [27,28], where the ef- fect is even more pronounced. However, these un- quenching effects are small compared to the dif- ferences between both, the truncation schemes we employed and the systematic errors of the lattice results. Acknowledgements It is a pleasure to thank the organisers of the Workshop on computational hadron physics for all their efforts which made this highly interesting conference possible. C. S. Fischer and P. Watson thank M. Pennington for inspiring discussions. The work summarised here has been supported by a grant from the Ministry of Science, Re- search and the Arts of Baden-Württemberg (Az: 24-7532.23-19-18/1 and 24-7532.23-19-18/2), the Deutsche Forschungsgemeinschaft (DFG) under contract Fi 970/2-1, the Virtual Institute for Dense Hadronic Matter and QCD Phase Tran- sitions and the Spanish FPA2005-02327. REFERENCES 1. R. Alkofer and L. von Smekal, Phys. Rept. 353, 281 (2001). 2. P. Maris and C. D. Roberts, Int. J. Mod. Phys. E 12, 297 (2003). 3. N. Nakanishi and I. Ojima, World Sci. Lect. Notes Phys. 27, 1 (1990). 4. P. Watson and R. Alkofer, Phys. Rev. Lett. 86 (2001) 5239. 5. A. Cucchieri, T. Mendes and A. Mihara, JHEP 0412 (2004) 012. 6. A. Sternbeck et al., Proc. Sci. LAT2005, 333 (2005); these proceedings. 7. W. Schleifenbaum, A. Maas, J. Wambach and R. Alkofer, Phys. Rev. D 72, 014017 (2005). 8. D. Zwanziger, Phys. Rev. D 65, 094039 (2002). 9. C. Lerche and L. von Smekal, Phys. Rev. D 65, 125006 (2002). 10. J. M. Pawlowski, D. F. Litim, S. Nedelko and L. von Smekal, Phys. Rev. Lett. 93 (2004) 152002 ; C. S. Fischer and H. Gies, JHEP 0410 (2004) 048, 11. R. Alkofer, C. S. Fischer and F. J. Llanes- Estrada, Phys. Lett. B 611, 279 (2005). 12. P. Pascual and R. Tarrach, Nucl. Phys. B 174 (1980) 123 [Erratum-ibid. B 181 (1981) 546]. 13. C. S. Fischer and D. Zwanziger, Phys. Rev. D 72 (2005) 054005. 14. C. S. Fischer. P. Watson and W. Cassing, ac- cepted for PRD, arXiv:hep-ph/0509213. 15. C. S. Fischer and R. Alkofer, Phys. Rev. D 67 (2003) 094020. 16. C. S. Fischer and R. Alkofer, Phys. Lett. B 536, 177 (2002); C. S. Fischer, R. Alkofer and H. Reinhardt, Phys. Rev. D 65, 094008 (2002). 17. A. Sternbeck, E. M. Ilgenfritz, M. Mueller- Preussker and A. Schiller, Phys. Rev. D 72 (2005) 014507. 18. P. O. Bowman et al., Phys. Rev. D 70 (2004) 034509. 19. J. Gattnar, K. Langfeld and H. Reinhardt, Phys. Rev. Lett. 93, 061601 (2004). 20. O. Oliveira and P. J. Silva, AIP Conf. Proc. 756, 290 (2005). 21. B. Alles et al., Nucl. Phys. B 502 (1997) 325. 22. P. Boucaud et al., JHEP 0304, 005 (2003); P. Boucaud et al., arXiv:hep-ph/0505150. 23. C. S. Fischer, B. Gruter and R. Alkofer, arXiv:hep-ph/0506053; C. S. Fischer, arXiv:hep-lat/0509031. 24. J. M. Pawlowski, D. F. Litim, S. Nedelko and L. von Smekal, AIP Conf. Proc. 756 (2005) 278. 25. R. Alkofer, W. Detmold, C. S. Fischer and P. Maris, Phys. Rev. D 70 (2004) 014014. 26. P. Maris, C. D. Roberts and P. C. Tandy, Phys. Lett. B 420 (1998) 267. 27. Y. Namekawa et al. [CP-PACS Collabora- tion], Phys. Rev. D 70 (2004) 074503. A. Ali Khan et al. [CP-PACS Collaboration], Phys. Rev. D 65 (2002) 054505 [Erratum-ibid. D 67 (2003) 059901]. 28. S. Aoki et al. [JLQCD Collaboration], Phys. Rev. D 68 (2003) 054502. 29. S. Eidelman et al. [Particle Data Group], Phys. Lett. B 592 (2004) 1. 30. P. Watson and W. Cassing, Few Body Syst. 35 (2004) 99. 31. D. Jarecke, P. Maris and P. C. Tandy, Phys. Rev. C 67 (2003) 035202. http://arXiv.org/abs/hep-ph/0509213 http://arXiv.org/abs/hep-ph/0505150 http://arXiv.org/abs/hep-ph/0506053 http://arXiv.org/abs/hep-lat/0509031 Infrared exponents for the Green's functions of Yang-Mills theory Unquenching effects in propagators Unquenching effects in light meson observables