Gluon and ghost propagators in the Landau gauge: Deriving lattice results from Schwinger-Dyson equations
A. C. Aguilar,1 D. Binosi,2 and J. Papavassiliou1
de F?sica Te?rica and IFIC, ? o
Centro Mixto, Universidad de Valencia-CSIC, E-46100, Burjassot, Valencia, Spain
arXiv:0802.1870v2 [hep-ph] 19 Jun 2008
European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), Villa Tambosi, Strada delle Tabarelle 286, I-38050 Villazzano (TN), Italy
We show that the application of a novel gauge invariant truncation scheme to the SchwingerDyson equations of QCD leads, in the Landau gauge, to an infrared ?nite gluon propagator and a divergent ghost propagator, in qualitative agreement with recent lattice data.
PACS numbers: 12.38.Lg, 12.38.Aw, 12.38.Gc
Introduction – The infrared sector of Quantum Chromodynamics (QCD)  remains largely unexplored, mainly due to the fact that, unlike the electroweak sector of the Standard Model, it does not yield to a perturbative treatment. The basic building blocks of QCD are the Green’s (correlation) functions of the fundamental physical degrees of freedom, gluons and quarks, and of the unphysical ghosts. Even though it is well-known that these quantities are not physical, since they depend on the gauge-?xing scheme and parameters used to quantize the theory, it is widely believed that reliable information on their non-perturbative structure is essential for unraveling the infrared dynamics of QCD . The two basic non-perturbative tools for accomplishing this task are (i) the lattice, where space-time is discretized and the quantities of interest are evaluated numerically, and (ii) the in?nite set of coupled non-linear integral equations governing the dynamics of the QCD Green’s functions, known as Schwinger-Dyson equations (SDE) [3–5]. Even though these equations are derived by an expansion about the free-?eld vacuum, they ?nally make no reference to it, or to perturbation theory, and can be used to address problems related to chiral symmetry breaking, dynamical mass generation, formation of bound states, and other non-perturbative e?ects . While the lattice calculations are limited by the lattice size used and the corresponding extrapolation of the numerical results to the continuous limit, the fundamental conceptual di?culty in treating the SDE resides in the need for a self-consistent truncation scheme, i.e., one that does not compromise crucial properties of the quantities studied. It it generally accepted by now that the lattice yields in the Landau gauge (LG) an infrared ?nite gluon propagator and an infrared divergent ghost propagator. This rather characteristic behavior has been ?rmly established recently using large-volume lattices, for pure Yang-Mills (no quarks included), for both SU (2)  and SU (3) . To be sure, lattice simulations of gauge-dependent quantities are known to su?er from the problem of the Gribov copies, especially in the infrared regime, but it is generally believed that the e?ects are quantitative rather than qualitative. The e?ects of the Gribov ambiguity on the ghost propagator become more pronounced in the infrared, while their impact on the gluon propagator usually stay within the statistical error of the simulation . In what follows we will assume that in the lattice results we use the Gribov problem is under control. In this article we show that the SDEs obtained within a new gauge-invariant truncation scheme furnish results (in the LG) which are in qualitative agreement with the lattice 2
data. As has been ?rst explained in , obtaining an infrared ?nite result for the gluon self-energy from SDEs, without violating the underlying local gauge symmetry, is far from trivial, and hinges crucially on one’s ability to devise a self-consistent truncation scheme that would select a tractable and, at the same time, physically meaningful subset of these equations. To accomplish this, in the present work we will employ the new gauge-invariant truncation scheme derived in , which is based on the pinch technique [9, 11] and its correspondence  with the background ?eld method (BFM) . SDEs in the gauge-invariant truncation scheme – The gluon propagator ??ν (q) in the covariant gauges assumes the form ??ν (q) = ?i P?ν (q)?(q 2 ) + ξ q ? qν , q4 (1)
where ξ denotes the gauge-?xing parameter, P?ν (q) = g?ν ? q? qν /q 2 is the usual transverse projector, and, ?nally, ??1 (q 2 ) = q 2 + iΠ(q 2 ), with Π?ν (q) = P?ν (q) Π(q 2 ) the gluon selfenergy. In addition, the full ghost propagator D(p2 ) and its self-energy L(p2 ) are related by iD?1 (p2 ) = p2 ? iL(p2 ). In the case of pure (quarkless) QCD, the new SD series  for the gluon and ghost propagators reads (see also Fig. 1) ??1 (q 2 )P?ν (q) = q 2 P?ν (q) + i 4 (ai )?ν i=1 , [1 + G(q 2 )]2 Γ? ??ν (k)Γν (p, k)D(p + k) ,
iD?1 (p2 ) = p2 + iλ iΛ?ν (q) = λ
(0) H?ρ D(k + q)?ρσ (k) Hσν (k, q) ,
where λ = g 2 CA , with CA the Casimir eigenvalue of the adjoint representation [CA = N for SU (N )], and
≡ ?2ε (2π)?d dd k, with d = 4 ?
the dimension of space-time. Γ? is
the standard (asymmetric) gluon-ghost vertex at tree-level, and Γν the fully-dressed one. G(q 2 ) is the g?ν component of the auxiliary two-point function Λ?ν (q), and the function Hσν is de?ned diagrammatically in Fig. 1. Hσν is in fact a familiar object : it appears in the all-order Slavnov-Taylor identity (STI) satis?ed by the standard three-gluon vertex, and is related to the full gluon-ghost vertex by q σ Hσν (p, r, q) = ?iΓν (p, r, q); at tree-level, Hσν = igσν . When evaluating the diagrams (ai ) one should use the BFM Feynman rules ; notice in particular that (i) the bare three- and four-gluon vertices depend explicitly on 1/ξ, (ii) the coupling of the ghost to a background gluon is symmetric in the ghost momenta, (iii) 3
q (a1 ) k (a2 ) k (a3 ) (a4 )
k, σ Hσν (k, q) = Hσν +
)?1 = (
q, ν k+q
FIG. 1: The new SDE for the gluon-ghost system. Wavy lines with white blobs are full gluon propagators, dashed lines with white blobs are full-ghost propagators, black blobs are full vertices, and the grey blob denotes the scattering kernel. The circles attached to the external gluons denote that, from the point of view of Feynman rules, they are treated as background ?elds.
there is a four-?eld coupling between two background gluons and two ghosts. Thus, for the gluonic contributions we ?nd (a1 )?ν = λ 2 Γ?αβ ?αρ (k)Γνρσ ?βσ (k + q) ,
(a2 )?ν = λg?ν
?ρ (k) + λ ρ
1 ?1 ξ
with Γ?αβ(q, p1 , p2 ) = Γ?αβ(q, p1 , p2 )+(pβ g ?α?pα g ?β )ξ ?1, Γ?αβ the standard QCD three-gluon 2 1 vertex, and Γ?αβ is the fully-dressed version of Γ?αβ . For the ghost contributions, we have instead (a3 )?ν = ?λ
Γ? D(k)D(k + q)Γν , D(k) ,
(a4 )?ν = 2λg?ν
with Γ? (q, p1 , p2 ) = (p2 ? p1 )? , and Γ? its fully-dressed counterpart. Due to the Abelian all-order Ward Identities (WIs) that these two full vertices satisfy (for all ξ), namely q ? Γ?αβ = i??1 (k + q) ? i??1 (k) and q ? Γ? = iD?1 (k + q) ? iD?1 (k), one can demonstrate αβ αβ that q ? [(a1 ) + (a2 )]?ν = 0 and q ? [(a3 ) + (a4 )]?ν = 0 . For the rest of the article we will study the system of coupled SDEs (2) in the LG (ξ = 0), in order to make contact with the recent lattice results of [6, 7]. This is a subtle exercise, because one cannot set directly ξ = 0 in the integrals on the rhs of (3), due to the terms proportional to 1/ξ. Instead, one has to use the expressions for general ξ, carry out explicitly 4
the set of cancellations produced when the terms proportional to ξ generated by the identity k ? ??ν (k) = ?iξkν /k 2 are used to cancel 1/ξ terms, and set ξ = 0 only at the very end. It is relatively easy to establish that only the bare part Γναβ of the full vertex contains terms that diverge as ξ → 0. Writing Γναβ = Γναβ + Kναβ , we thus have that Kναβ is regular in that limit, and we will denote by Kναβ its value at ξ = 0. Introducing ?t (q) = P?ν (q)?(q 2 ), we ?ν get
(ai )?ν = λ
9 Γαβ ?t (k)?t (k + q)Lρσ ? g?ν ? αρ βσ ν 4 k (k + q)β [Γ + L]αβ + ν (k + q)2
?t (k) α?
k? (k + q)ν , k 2 (k + q)2
where L?αβ = Γ?αβ +K?αβ satis?es the WI q ? L?αβ = Pαβ (k + q)??1 (k + q) ? Pαβ (k)??1 (k). Contracting the lhs of (5) by q ? one can then verify that it vanishes, as announced. Next, following standard techniques, we express L?αβ and Γ? as a function of the gluon and ghost self-energy, respectively, in such a way as to automatically satisfy the corresponding WIs. Of course, this method leaves the transverse (i.e., identically conserved) part of the vertex undetermined. The Ansatz we will use is L?αβ = Γ?αβ + i Γ? = Γ? ? i q? [Παβ (k + q) ? Παβ (k)] , q2 (6)
q? [L(k + q) ? L(k)] , q2
whose essential feature is the presence of massless pole terms, 1/q 2 . Longitudinally coupled bound-state poles are known to be instrumental for obtaining ??1 (0) = 0 ; on the other hand, due to current conservation, they do not contribute to the S-matrix. For the conventional ghost-gluon vertex Γν , appearing in the second SDE of (2) we will use its tree-level expression, i.e., Γν → Γν = ?pν . Note that, unlike Γν , the conventional Γν satis?es a STI of rather limited usefulness; the ability to employ such a di?erent treatment for Γν and Γν without compromising gauge-invariance is indicative of the versatility of the new SD formalism used here. Finally, for Hσν we use its tree-level value, Hσν . With these approximations, the last two equations of (2), together with (4) and (5), give (in Euclidean space)
[1 + G(q 2 )]2 ??1 (q 2 ) = q 2 ?
?(k)?(k + q)f1 +
q2 k 2 (k + q)2
(k · q)2 D(k)D(k + q) ? 2 D(k) , (7) q2 k k (q 2 )2 20 10 q2 2(k + q)2 f1 = 20q 2 + 18k 2 ? 6(k + q)2 + , ? (k · q)2 2 + 2 + 2 + (k + q)2 k q k (k + q)2 q2k2 k2 27 q2 (k · q)2 (k · q)2 f2 = ? ? 8 +8 +4 2 ?4 2 , 2 (k + q)2 (k + q)2 k (k + q)2 q (k + q)2 +λ 4 3 k2 ? (p · k)2 ?(k) D(p + k) , k2 k (k · q)2 2 + 2 2 ?(k)D(k + q) . k q p2 ?
D?1 (p2 ) = p2 ? λ G(q 2 ) = ? λ 3
Since [(a1 ) + (a2 )]?ν and [(a3 ) + (a4 )]?ν are transverse, in arriving at (7) we have used [(a1 )+ (a2 )]?ν = Tr[(a1 ) + (a2 )]P?ν (q) and [(a3 ) + (a4 )]?ν = Tr[(a3 ) + (a4 )]P?ν (q), substituted into (2), and then equated the scalar co-factors of both sides. Thus, the transversality of the answer cannot be possibly compromised by the ensuing numerical treatment (e.g. hard ultraviolet cuto?s), which may only a?ect the value of the co-factor. Numerical results – Before solving numerically the above system of integral equations, one must introduce renormalization constants to make them ?nite. The values of these constants will be ?xed by the conditions ??1 (?2 ) = ?2 , D?1 (?2 ) = ?2 , and G(?2 ) = 0,
2 with the renormalization point ?2 of the order of MZ . It is relatively straightforward
to verify that the perturbative expansion of (7) and (8) furnishes the correct one-loop results. Speci?cally, keeping only leading logs, we have 1 + G(q 2 ) = 1 + while D?1 (p2 ) = p2 [1 + αs = g 2 /4π. The crux of the matter, however, is the behavior of (7) as q 2 → 0, where the “freezing” of the gluon propagator is observed. In this limit, Eq.(7) yields λ (Tg + Tc ) , [1 + G(0)]2 15 3 Tg = ?(k) ? k 2 ?2 (k), 4 k 2 k ??1 (0) = Tc = ?2
k 3CA αs 16π 3CA αs 16π
ln(q 2 /?2 ),
ln(p2 /?2 )] and ??1 (q 2 ) = q 2 [1 +
13CA αs 24π
ln(q 2 /?2 )], where
(9) (10) (11)
k 2 D2 (k).
Perturbatively the rhs of Eq.(9) vanishes by virtue of the dimensional regularization result
k lnn k2 k2
= 0 n = 0, 1, 2, . . . which ensures the masslessness of the gluon to all orders. How-
ever, non-perturbatively ??1 (0) does not have to vanish, provided that the quadratically 6
divergent integrals de?ning it can be properly regulated and made ?nite, without introducing counterterms of the form m2 (Λ2 )A2 , which are forbidden by the local gauge invariance of 0 UV ? the fundamental QCD Lagrangian. It turns out that this is indeed possible: the divergent integrals can be regulated by subtracting appropriate combinations of “dimensional regularization zeros”. Speci?cally, as we have veri?ed explicitly and as can be clearly seen in Fig.2 (left panel), for large enough k 2 the ?(k 2 ) goes over to its perturbative expression, to be denoted by ?pert (k 2 ); it has the form ?pert (k 2 ) =
an lnk2k , where the coe?cient an are
known from the perturbative expansion. For the case at hand, measuring k 2 in GeV2 , using ? ≈ 100 GeV and αs (?) = 0.1, after inverting and re-expanding the ??1 (k 2 ) given below Eq.(8), we ?nd a0 ≈ 1.7, a1 ≈ ?0.1, a2 ≈ 2.5 × 10?3 . Then, subtracting
reg from both sides of Eq.(10), we obtain the regularized Tg given by (k 2 = y) k
?pert (k 2 ) = 0
15 s dy y [?(y) ? ?pert (y)] = 4 0 3 s ? dy y 2 ?2 (y) ? ?2 (y) . pert 2 0
A similar procedure can be followed for Tc (see below). The obvious ambiguity of the regularization described above is the choice of the point s, past which the two curves, ?(y) and ?pert (y), are assumed to coincide. Ideally, one should then: (i) solve the system of integral equations under the boundary condition ?(0) = C, where C is an arbitrary positive parameter; (ii) substitute the solutions for ?(q) and D(q) in the (regularized) integrals on the rhs of (9), together with the obtained value for G(0), and denote the result by ??1 (0); (iii) check that the self-consistency reg requirement ??1 (0) = C ?1 is satis?ed; if not, (iv) a new C must be chosen and the procereg dure repeated. In practice, due to the aforementioned ambiguity, we cannot pin down ?(0) completely, and we will restrict ourselves to providing a reasonable range for its value. We have solved the system for a variety of initial values for C, ranging between 1 ? 50 GeV?2 , and obtained from (12) the corresponding ??1 (0). On physical grounds reg one does not expect the perturbative expression ?pert (k 2 ) to hold below 5 ? 10 GeV2 , and therefore, when computing ??1 (0), s should be chosen around that value. For values of C reg between 10 ? 25 GeV?2 the corresponding ??1 (0) can be made equal to C ?1 by choosing reg values for s within that (physically reasonable) range. For example, for C = 14.7 GeV?2 , the value of the lattice data at the origin, we must choose s ≈ 10 GeV2 . The solutions for ?(q), D(p), and 1 + G(q) obtained for that special choice, C = 14.7 GeV?2 , are shown in Fig.2. 7
FIG. 2: Left panel: The gluon propagator obtained from the solution of the SDE system (blue continuous line) compared to the lattice data of ; the red dashed line represents the perturbative behavior. In the inset we show the function 1 + G(q 2 ) (blue continuous line) and its perturbative behavior (red dashed line). Right panel: The ghost propagator obtained from the SDE system (blue continuous line), the one-loop perturbative result (red dashed line), and the corresponding lattice data of . In the inset we show the function p2 D(p2 ) from the SDE.
In order to enforce the equality ??1 (0) = C ?1 for higher values of C one must assume the reg validity of perturbation theory uncomfortably deep into the infrared region; for example, for C = 50 GeV?2 one must choose s below 1 GeV2 . We emphasize that the non-perturbative transverse gluon propagator, being ?nite in the IR, is automatically less singular than a simple pole, thus satisfying the corresponding Kugo-Ojima (KO) con?nement criterion , essential for ensuring an unbroken color charge in QCD . Note that for q 2 ≤ 10 GeV2 both gluon propagators (lattice and SDE) shown in Fig.2 may be ?tted very accurately using a unique functional form, given by ??1 (q 2 ) = a + b (q 2 )c?1 . Speci?cally, [measuring q 2 in GeV2 and the χ2 per degrees of freedom], the lattice data are ?tted by a = 0.07, b = 0.15, and c = 2.54 (χ2 ? 10?2 ), while our SDE solution is described setting a = 0.07, b = 0.77, and c = 2.01 (χ2 ? 10?4 ). Let us now consider the ghosts. The D(p2 ) obtained from the ghost SDE diverges at the origin, in qualitative agreement with the lattice data. From the SDE point of view, this divergent behavior is due to the fact that we are working in the LG and the vertex Γν employed contains no 1/p2 poles, as suggested by previous lattice studies . The rate of divergence of our solution is particularly interesting, because it is related to the KO con?nement criterion for the ghost , according to which the non-perturbative ghost 8
propagator (in the LG) should be more singular in the infrared than a simple pole. Motivated by this, we proceed to ?t the function p2 D(p2 ) [see inset in right panel of Fig.2]. First we use a ?tting function of the form p2 D(p2 ) = c1 (p2 )?γ (p2 in GeV2 ); a positive γ would indicate that the SDE solution satis?es the KO criterion. Our best ?t, valid for p2 ≤ 10, gives the values γ = 0.02 and c1 = 1.30, which lead to a χ2 ? 10?3 . Interestingly enough, an even better ?t may be obtained using a qualitatively di?erent, physically motivated functional form, namely p2 D(p2 ) = κ1 ? κ2 ln(p2 + κ3 ) (with κ3 acting as a gluon “mass”). Our best ?t, valid for the same range, gives κ1 = 1.3, κ2 = 0.05, and κ3 = 0.05, with χ2 ? 10?6 . This second ?t suggests that p2 D(p2 ) reaches a ?nite (positive) value as p2 → 0. Even though not conclusive, our ?tting analysis seems to favor a ghost propagator displaying no powerlaw enhancement, in agreement with recent results presented in ; clearly, this question deserves further study. Turning to the tadpole contributions Tc of (11), the subtraction of 0 = we get (s ≈ 1 GeV2 )
s s k
k ?2 regularizes
Tc , yielding a rather suppressed ?nite value for Tcreg . For example, using the ?rst ghost ?t,
dy [yD(y) ? 1] +
dy y 2 D2 (y) ? 1 (13)
? ?2γ s ln s , which is numerically negligible.
Discussion – The present work has focused on the derivation of an infrared ?nite gluon propagator from a gauge-invariant set of SDEs for pure QCD in the LG, and its comparison with recent lattice data. Following the classic works of , the ?niteness of the gluon propagator is obtained by introducing massless poles in the corresponding three-gluon vertex. The actual value of ??1 (0) has been treated as a free parameter, and was chosen to coincide reg with the lattice point at the origin. The curves shown in Fig.2 were then obtained dynamically, from the solution of the SDE system, for the entire range of momenta. Comparing the solution for the gluon propagator with the lattice data we see that, whereas their asymptotic behavior coincides (perturbative limits), there is a discrepancy of about a factor of 2-2.5 in the intermediate region of momenta, especially around the fundamental QCD mass-scale [re?ected also in the di?erent values of the two sets of ?tting parameters (a, b, c)]. In the case of the ghost propagator the relative di?erence increases as one approaches the deep infrared, given that both curves diverge at a di?erent rate. These discrepancies may be ac9
counted for by extending the gluon SDE to include the “two-loop dressed” graphs, omitted (gauge-invariantly) from the present analysis, and/or by supplying the relevant transverse parts of the vertex given in (6). We hope to be able to make progress in this direction in the near future. In our opinion, the analysis presented here, in conjunction with the recent lattice data, fully corroborates Cornwall’s early description of QCD in terms of a dynamically generated, momentum-dependent gluon mass . In this picture the low-energy e?ective theory of QCD is a non-linear sigma model, known as massive gauge-invariant Yang-Mills, obtained from the generalization of St¨ckelberg’s construction to non-Abelian theories . This model u admits vortex solutions, with a long-range pure gauge term in their potentials, which endows them with a topological quantum number corresponding to the center of the gauge group [ZN for SU (N )], and is, in turn, responsible for quark con?nement and gluon screening [21, 22]. Speci?cally, center vortices of thickness ? m?1 , where m is the induced mass of the gluon, form a condensate because their entropy (per unit size) is larger than their action. This condensation furnishes an area law to the fundamental representation Wilson loop, thus con?ning quarks. On the other hand, the adjoint potential shows a roughly linear regime followed by string breaking when the potential energy is about 2m, corresponding to gluon screening . Acknowledgments: Work supported by the Spanish MEC grants FPA 2005-01678 and FPA 2005-00711, and the Fundaci?n General of the UV. We thank Professor J.M.Cornwall o for several useful comments.
 W. J. Marciano and H. Pagels, Phys. Rept. 36, 137 (1978).  See, for example, J. Greensite, Prog. Part. Nucl. Phys. 51, 1 (2003), and references therein.  F. J. Dyson, Phys. Rev. 75, 1736 (1949).  J. S. Schwinger, Proc. Nat. Acad. Sci. 37, 452 (1951); Proc. Nat. Acad. Sci. 37, 455 (1951).  J. D. Bjorken and S. D. Drell, “Relativistic Quantum Field Theory”, chapter 19; C. Itzykson and J. B. Zuber, “Quantum Field Theory,” chapter 10.  A. Cucchieri and T. Mendes, PoS LATTICE, 297 (2007); P. O. Bowman et al., in full Phys. Rev. D 76, 094505 (2007).
 I. L. Bogolubsky et al., PoS LATTICE, 290 (2007).  A. G. Williams, Prog. Theor. Phys. Suppl. 151, 154 (2003); A. Sternbeck et al., AIP Conf. Proc. 756, 284 (2005); P. J. Silva and O. Oliveira, Nucl. Phys. B 690, 177 (2004)  J. M. Cornwall, Phys. Rev. D 26, 1453 (1982).  D. Binosi and J. Papavassiliou, Phys. Rev. D 77, 061702 (2008).  J. M. Cornwall and J. Papavassiliou, Phys. Rev. D 40, 3474 (1989).  D. Binosi and J. Papavassiliou, Phys. Rev. D 66, 025024 (2002); Phys. Rev. D 66, 111901 (2002); J. Phys. G 30, 203 (2004); A. Pilaftsis, Nucl. Phys. B 487, 467 (1997).  L. F. Abbott, Nucl. Phys. B 185, 189 (1981).  A. C. Aguilar and J. Papavassiliou, JHEP 0612, 012 (2006).  R. Jackiw and K. Johnson, Phys. Rev. D 8, 2386 (1973); J. M. Cornwall and R. E. Norton, Phys. Rev. D 8 (1973) 3338; E. Eichten and F. Feinberg, Phys. Rev. D 10, 3254 (1974).  T. Kugo and I. Ojima, Prog. Theor. Phys. Suppl. 66, 1 (1979).  See, e.g., C. S. Fischer, J. Phys. G 32, R253 (2006), and references therein.  A. Cucchieri, T. Mendes and A. Mihara, JHEP 0412, 012 (2004).  Ph. Boucaud et al., arXiv:0803.2161 [hep-ph].  J. M. Cornwall, Phys. Rev. D 10, 500 (1974).  J. M. Cornwall, Nucl. Phys. B 157, 392 (1979).  C. W. Bernard, Nucl. Phys. B 219, 341 (1983).  J. M. Cornwall, Phys. Rev. D 57, 7589 (1998)