From Running Gluon Mass to Chiral Symmetry Breaking
Orlando Oliveira, P. Bicudo, D. Dudal, T. Frederico, W. de Paula, N. Vandersickel
aa r X i v : . [ h e p - ph ] S e p From Running Gluon Mass to Chiral SymmetryBreaking
Orlando Oliveira , , P. Bicudo , D. Dudal , T. Frederico , W. de Paula , N.Vandersickel Departamento de F´ısica, Universidade de Coimbra, 3004-516 Coimbra, Portugal Departamento de F´ısica, Instituto Tecnol´ogico de Aeron´autica, 12228-900 S˜aoJos´e dos Campos, SP, Brazil Departamento de F´ısica, I.S.T., Av Rovisco Pais, 1049-001 Lisboa, Portugal Ghent University, Department of Physics and Astronomy, Krijgslaan 281-S9,B-9000 Gent, Belgium
In recent years the non-perturbative computation of the two point correlation func-tions of pure Yang-Mills theory have attracted a lot of attention. For what concernsthe Landau gauge gluon propagator, the subject of this communication, lattice QCDsimulations, Schwinger-Dyson equations and non-perturbative quantization of Yang-Mills theories provide essentially the same results. Recent reviews can be found in[1]. In this sense, we can claim to have a fair description of this two point functionover the entire range of momentum. Here we review the large volume lattice simu-lations performed by one of the authors, its interpretation in terms of modeling thegluon propagator and we describe how the infrared propagator can be incorporatedinto an effective field theory model which approximates QCD at low energies. Thiseffective theory connects gluon confinement with a gluon mass m g with chiral sym-metry breaking. Further, the model predicts a particular simple relation between m g and the light quark masses m q , i.e. that m q /m g is constant. This relation is testedusing the solutions of the Schwinger-Dyson equations and found to be valid in thelow energy regime below the 10% accuracy level.From the point of view of lattice simulations there are still some open questions.On the lattice, one numerically checks whether the selected Landau gauge configura-tions belong to the so-called Gribov region, which is the set of all transverse gaugeconnections with positive Faddeev-Popov operator. Within this set, one still findsGribov copies, and it is not 100% well established if/how these additional copiesinfluence the propagator in the (very deep) infrared region. However, previous simu-lations suggest that by choosing a different Gribov copy, the accompanying error liestypically within the statistical error of the propagator - see, for example, [2].1 D ( p ) [ G e V - ] β = 6.4 -- 80 , 4.32 fm β = 6.2 -- 80 , 5.84 fm β = 6.0 -- 64 , 6.50 fm β = 6.0 -- 80 , 8.16 fm Renormalized Gluon Propagator -- µ = 3 GeV D ( p ) [ G e V - ] β = 6.4 -- 80 , 4.32 fm β = 6.2 -- 80 , 5.84 fm β = 6.0 -- 64 , 6.50 fm β = 6.0 -- 80 , 8.16 fm Renormalized Gluon Propagator -- µ = 3 GeV Figure 1: D ( p ) for the largest volumes computed for each of the lattice spacings.The plot on the right shows D ( p ) for the infrared region defined as p < In this section we report on the Landau gauge quenched lattice gluon propagator D abµν ( p ) = δ ab δ µν − p µ p ν p ! D ( p ) (1)computed for large volumes L a > . L a ≈
17 fm [4] andfor SU(2) up to
L a ≈
27 fm, but using a ∼ . ∼ a = 0 .
102 fm ( β = 6 . a = 0 . β = 6 .
2) and a = 0 . β = 6 .
4) and for various volumes up to
L a ≈ . p = 3 GeV anda zoom of the infrared region. For momenta above 1 GeV all data sets are, withinerrors, compatible. In the ultraviolet region, i.e. for p above ∼ . D ( p ) = Z h ln p Λ i − γ p , (2)where γ = 13 /
22 is the gluon anomalous dimension. Indeed, for each set, the fits to (2)2 M g l u e [ M e V ] Linear ExtrapolationM glue = 634(40) D ( p ) [ G e V - ] β = 6.0 -- 80 Renormalized Gluon Propagator - µ = 3 GeV Figure 2: Gluon mass from the infrared propagator (left) computed assuming a simplepole. The point on the right is the extrapolated mass to infinite volume. The rightplot illustrates a typical fit. Note that the simple pole overestimates D (0). β L a D (0)(fm) (GeV − )6.0 6.50 9.22(21)6.0 8.16 8.96(45)6.2 5.84 8.58(43)6.4 4.32 9.24(37)Table 1: D (0) for the various simulations shown in Figure 1.where used to renormalize the gluon propagator according to the MOM prescription D R ( p ) (cid:12)(cid:12)(cid:12) p = µ = 1 µ . (3)The renormalization procedure is described in detail in [2, 5]. Note, however, that inthe present work, to renormalize, only the data for momenta above p ∼ . χ /d.o.f. ∼ β = 6 .
2, althoughhaving a smaller physical volume (5.84 fm) , is below all the remaining data sets. For p = 0, the large statistical error hides the differences between the various D (0). Forthe various simulations reported here, the corresponding D (0) are given in Table 1.This numbers should be compared with the large volume, β = 5 .
7, Wilson action SU(3)simulations performed by the Berlin-Moscow-Adelaide (BMA) group. Using the samedefinitions and setting the scale in the same way, i.e. from the string tension, the3MA data reads D (0) = 8.68(37), 8.09(36), 7.59(56), 7.17(31) and 7.53(19) GeV − for L a = 8.09, 11.76, 13.23, 14.70 and 16.17, respectively, given in fm.Qualitatively, the propagators computed with the different values of β are similar.The D ( p ) from the β = 5 . D (0)and extrapolates its parameters to the infinite volume - see also [6, 7].Let us assume that the infrared propagator is described by a simple mass pole D ( p ) = Zp + M . (4)This functional form can only describe the propagator within a limited range ofmomenta. For example, a propagator described by (4) does not violate positivity.Violation of positivity is a well established property of the non-perturbative gluonpropagator. Anyway, at minimum, a fit to (4) defines an interval range where onecan approximate D ( p ) by such a mass pole. The outcome of the fits as a function ofthe maximum range of momenta p max are β L a p max Z M χ /d.of. (fm) (MeV) (MeV)6.0 6.50 504 4.12(10) 657(11) 1.36.0 8.16 505 3.95(12) 633(14) 1.26.2 5.84 522 4.35(30) 694(33) 1.26.4 4.32 493 3.82(20) 634(23) 0.7It follows that the lattice propagator can be described by a pole mass up to p ∼ M ( V ) = M ∞ + M /L and fit the lattice data, then the infinite volume extrapolated mass is M ∞ = 634(40) MeV. The fit has a χ /d.o.f. = 1 .
4. The data for the different M andthe extrapolation can be seen in Figure 2. Note that the simple pole (4) overestimates D (0), see right plot in Figure 2, and that is the reason why we do not provide thedetails of the extrapolation of Z and D (0).The lattice data can be described by a propagator of type (4) if Z and M arefunctions of momentum, i.e. for D ( p ) = Z ( p ) p + M ( p ) . (5)A momentum dependent gluon mass together with a Z ( p ) were investigated in [8],where the same functional forms were used to fit the decoupling solutions of the4 D ( p ) [ G e V - ] p D ( p ) Figure 3: Gluon propagator and the fit to (5) using the definitions (6). Note that thefunctional form used here also overestimates D (0). β L a χ d.o.f. M M + m λ p max La is given in fm andthe mass scales and higher momenta in the fitting p max are given in power of GeV.Schwinger-Dyson equations. According to this work the lattice data can be describedby Z ( p ) = z h ln p + r m Λ i γ and M ( p ) = m p + m (6)up to momenta p max = 4 . z = 1 . , Λ = 1 . , r = 7 . m = 671(9) MeV . The gluon data and the fit to (5) are reported in Figure 3. Note that, as in the caseof simple pole mass, the fit overestimates D (0).5 D ( p ) [ G e V - ] β = 6.0 - 8.13 fm β = 6.2 - 5.81 fm Renormalized Gluon Propagator - µ = 3 GeV Figure 4: Gluon propagator and the fits to (7) for the largest lattice volumes. Notethat the RGZ reproduces well all the lattice data points, including D (0).The refined Gribov-Zwanziger (RGZ) action is an improvement over the usualFaddeev-Popov quantization procedure for Yang-Mills theories, in the sense that itprovides a better way to handle the problem of the Gribov copies by restrictingthe functional integration space to the so-called Gribov region. The RGZ actionis renormalizable, in the perturbative sense, and introduces new auxiliary bosonicand fermionic fields. In what concerns the gluon propagator, the RGZ tree levelpropagator is given by D ( p ) = p + M p + ( M + m ) p + 2 g N γ + M m , (7)where M is a mass scale related to the new auxiliary fields, m is another mass scalerelated with the h A i condensate and γ is the Gribov parameter. γ is not a freeparameter but is fixed by the so-called horizon condition [9]. In the following we shallintroduce the shorthand λ = 2 g N γ + M m . The RGZ being a non-perturbativequantization for the Yang-Mills theories, one hopes that its tree level predictionsprovide a good description for the infrared. The propagator (7) can rewritten as D ( p ) = 1 p + M ( p ) with M ( p ) = m + 2 g N γ p + M . (8)In this sense, the RGZ action predicts a momentum dependent effective gluon masswhich is essentially the functional form analyzed previously, i.e. M ( p ) given byequation (6). The tree level expression for D ( p ) does not include the observedlogarithmic corrections at high energies and, therefore, one expects (7) to deviate6 D ( p ) [ G e V - ] β = 5.7 -- 96 (17.6 fm) β = 5.7 -- 88 (16.2 fm) β = 6.0 -- 80 ( 8.2 fm) β = 6.2 -- 80 ( 5.8 fm) Renormalized Gluon Propagator - µ = 3 GeV Figure 5: Gluon propagator and the fit to (7) using the linear extrapolated param-eters. The red line was computed using the extrapolated β = 6 . β = 6 . with La = 4 .
88 fm, 64 with La = 6 .
50 fm and 80 with La = 8 .
13 fm for β = 6 .
0; (ii) 48 with La = 3 .
49 fm, 64 with La = 4 .
65 fm and80 with La = 5 .
81 fm for β = 6 .
2. The fits are summarized in Table 2. The latticedata and the fits for the largest physical volumes are reported in Figure 4. Note thatthe RGZ propagator reproduces well all the lattice data, including D (0).The infrared propagator can be extrapolated to the infinite volume if one assumesa linear dependence on 1 / ( La ). The extrapolations give β χ d.o.f. M χ d.o.f. M + m χ d.o.f. λ D (0) = . − from the β = 6 . , . − from the β = 6 . . (9)In the computation of D (0), given the poor quality of the linear extrapolation for λ , we have used instead the fitted value from the largest physical volume. Figure 5shows the lattice gluon propagator for the largest physical volumes computed usingdifferent lattice spacings and the extrapolated fits to (7) as described above.7n the RGZ propagator, the parameter m is related to the h A i . If one uses thefigures from the extrapolations it follows h g A i = . from the β = 6 . , . from the β = 6 . . (10)or h g A i
10 GeV = . from the β = 6 . , . from the β = 6 . . (11)The values are slightly below those reported in [5]. A gluon mass term in the QCD action is forbidden by gauge invariance and, therefore,a gluon mass m g has to be generated dynamically. A non-vanishing mass means thatthe gluon field is short ranged. We should be careful not to attribute a physical mean-ing to this “massive” gluon, given the already mentioned positivity violation, whichis a indication of the unphysical (confined) nature of the gluon. Besides providingthe screening of the gluon, one may ask if there are additional implications of having m g = 0. In this section, we show that, within an effective field theory for low energyQCD, a gluon mass is connected with chiral symmetry breaking, i.e. the theory eitherhas m g = 0 and chiral symmetry is broken or chiral symmetry is restored and thegluon is a long range field. This section is based in the work [10].In QCD the fundamental fields are associated with quarks and gluons. However,to describe the low energy regime of QCD other fields can be included to define aneffective theory. Let us assume that the non-perturbative physics is mainly associatedwith the gluon sector. Pure Yang-Mills theory has multi-gluon configurations asbound states. The simplest of these bound states is a two gluon state. Given that thegluon belongs to the adjoint representation, the two gluon state can be decomposedaccording to 8 ⊗ ⊕ ⊕ ⊕ ⊕ ⊕ . The lowest dimensional irrep is a singletand can be identified with glueball states. The lightest glueball state has J P C = 0 ++ and a predicted mass of ∼ . ∼ φ a ∝ d abc F bµν F c µν , (12)8here F aµν is the non-abelian Maxwell tensor. Of course, one can add to the abovedefinition a quark contribution given by, for example, q t a q , where t a are the generatorsof the fundamental representation. Adding the two terms enables to estimate thecontribution of quarks and gluons to the effective field, φ a ≈ h F i Λ + h q q i Λ , (13)where Λ ∼ Λ QCD is a non-perturbative mass scale. Plugging into this the gluoncondensate α s h F i = 0 .
04 GeV and the light quark condensate h q q i = ( −
270 MeV) ,it follows that the ratio gluon to quark content of φ a is around 7.Let us consider an effective theory which includes the gluon field A µ , the quarkfields q f , where f is a flavor index, and an effective scalar field φ a that belongs tothe adjoint representation of the SU(3) color group. In the following we will assumethat the non-perturbative physics is contained in φ a . Furthermore, being an effectivefield theory, it should describe hadronic physics only in the low energy regime and itdoes not need to be renormalizable. The effective Lagrangian reads L = − F aµν F a µν + X f q g { iγ µ D µ − m f } q f + 12 D µ φ a D µ φ a − V oct ( φ a φ a ) + L GF + L ghost − G X f h q f t a q i φ a − G X f h q f q i φ a φ a − F X f h q f q i d abc φ b φ c − F X f h q f t a γ µ q i D µ φ a − F X f h q f t a γ µ D µ q i φ a + h.c. (14)where D µ = ∂ µ + igT a A µ is the covariant derivative, T a the SU(3) generators, m f thecurrent quark mass associated with flavor f , V oct the effective potential associatedwith the scalar field. L GF is the gauge fixing part of the Lagrangian and L ghost contains the ghost terms. The Lagrangian is gauge invariant, excepts for the L GF term. The effective gauge coupling constant g parameterizes residual interactions andit should be a small number, i.e. one expects the theory can be treated perturbatively.The new interactions with the scalar field, the terms proportional to G , G , F , F and F , where written assuming flavor independence of strong interactions. L includes the QCD Lagrangian and verifies the usual soft-pion theorems of chiralsymmetry at low energy. The new interactions introduce new vertices, not presentin the original QCD Lagrangian, which contribute to quark processes. Note that theonly new quark color singlet operator mimics the P model describing OZI-allowedmesonic strong decays. 9he φ a kinetic term couples to a quadratic gluon term through the operator12 g φ c ( T a T b ) cd φ d A aµ A b µ . (15)If the scalar fields acquires a vacuum expectation value without breaking color sym-metry, i.e. h φ a i = 0 and h φ a φ b i = v δ ab , (16)given that for the adjoint representation tr ( T a T b ) = N c δ ab , the gluon mass reads m g = N c g v , (17)where N c = 3. From the definition it follows that h φ a φ b i , i.e. v , and thereforethe gluon mass is gauge invariant. The proof of gauge invariance follows from thetransformation properties of φ a .In the same way, the operator G [ q q ] φ a φ a shifts the quark masses giving rise toa constituent quark mass M f = m f − ( N c − G v = m f − N c − N c G g m g . (18)For light quarks, the constituent quark mass is given by the quark self energy which,in the model, is linked with the gluon mass. Note, for our definitions, that G M f ∝ m g in the effectivemodel links chiral symmetry with a finite effective gluon mass.The quark condensate h q q i , an order parameter for chiral symmetry breaking, canbe computed in the model as a function of the constituent quark mass, the gluon massand the theory cut-off - see [10] for details. Then, if one identifies the gluon mass withthe mass measured from the lattice using a simple pole propagator, m g = 634 MeV,together with M f = 330 MeV and h q q i = ( −
270 MeV) , one is able to estimate someof the theory parameters: ω = 879 MeV , gv = 366 MeV and G g = − .
31 GeV − , where ω is the theory’s cut-off. The effective model relates the constituent quark mass M and the gluon mass m g through equation (18). For a vanishing current mass, equation (18) predicts a constantvalue for the ratio M/m g , at least at tree level. This result can be tested looking10 p [GeV ] (BC) [GeV]M (CP) [GeV]m [GeV ] p [GeV ]00.20.40.60.811.21.41.61.82 M Q / M [ G e V - ] M (BC) / M M (CP) / M Figure 6: On the left hand side, the plot shows the quark masses from solving thefermionic SDE gap equation, using different ans¨atze for the quark-gluon vertex, andthe squared gluon mass computed from quenched lattice simulations. Note that M ( p )depends slightly on the definition of the quark-gluon vertex. On the right hand side,the plot shows the ratio M/m g .at the solutions of the Schwinger-Dyson equations. In the following we will use theresults published in [12]. For the gluon and ghost propagators, the authors usedthe results of lattice QCD simulations and solved the gap equation for a masslessfermion. The calculation does not take into account fermion loops and can be viewedas a quenched approximation.In [12], the fermionic gap equation was solved for two different ans¨atze for thequark-gluon vertex, a non-Abelian improved version of the Ball-Chiu vertex and animproved version of the Curtis-Pennington vertex. The choice of vertex leads toslightly different quark mass. In order to distinguish, the results of the Ball-Chiuvertex will be referred as BC, while the results from using the Curtis-Penningtonvertex will be referred as CP. Figure 6 shows M computed from the Schwinger-Dysonequations for the different vertex ans¨atze, together with m g , as a function of p and, onthe right hand side, the ratio M/m g . The plots shows that M/m g increases slightly.If one looks at the maximal momentum range where the lattice gluon propagator canbe fitted by a simple pole, i.e. if one compares the ratios up to momenta p ∼ . M/m g changes by less than 8%, relative to its zero momentum value, whenusing the BC quark-gluon vertex and less than 10% when using the CP vertex. We have currently a fair description of the gluon propagator over all momentumranges. To extract the various parameters modeling the propagator, it would be11esirable to perform a high statistic and large volume simulation.The results of lattice simulations and Schwinger-Dyson equations show that thegluon propagator behaves as a dynamically massive gauge boson in the infrared region,see also the discussion in [13] and references therein, and, the effective model sketchedhere, shows a connection between the gluon mass and chiral symmetry breaking.Comparing the tree level mass ratio prediction with the solutions of the Schwinger-Dyson equations we found good agreement in the low energy regime.The authors acknowledge financial support from the Brazilian agencies FAPESP(Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo), CNPq (Conselho Na-cional de Desenvolvimento Cient´ıfico e Tecnol´ogico) and Research-Foundation Flan-ders (FWO Vlaanderen). OO acknowledges financial support from FCT under con-tract PTDC/FIS/100968/2008.
References [1] A. Maas, arXiv:1106.3942; Ph. Boucaud, J. P. Leroy, A. Le Yaouanc, J. Micheli,O. P`ene J. Rodr´ıguez-Quintero, arXiv:1109.1936.[2] P. J. Silva, O. Oliveira, Nucl. Phys.
B690 , 177 (2004).[3] A. Cucchieri, T. Mendes, PoS (
LAT2007 ), 297 (2007) [arXiv:0710.0412].[4] I. L. Bogolubsky, E.-M. Ilgenfritz, M. M¨uller-Preussker, A. Sternbeck, Phys.Lett.
B676 , 69 (2009).[5] D. Dudal, O. Oliveira, N. Vandersickel, Phys. Rev.
D81 , 074505 (2010).[6] O. Oliveira, P. J. Silva PoS (
LAT2009 ), 226 (2009) [arXiv:0910.2897].[7] O. Oliveira, P. J. Silva PoS (
QCD-TNT09 ), 033 (2009) [arXiv:0911.1643].[8] O. Oliveira, P. Bicudo, J. Phys.
G38 , 045003 (2011).[9] D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel, H. Verschelde, Phys. Rev.
D78 , 065047 (2008).[10] O. Oliveira, W. de Paula, T. Frederico, arXiv:1105.4899.[11] Y. Chen, A. Alexandru, S.J. Dong, T. Draper, I. Horvath, F.X. Lee, K.F. Liu, N.Mathur, C. Morningstar, M. Peardon, S. Tamhankar, B.L. Young, J.B. Zhang,Phys. Rev. D73 (2006) 014516.[12] A. C. Aguilar, J. Papavassiliou, Phys. Rev.