The bare Faddeev-Popov (FP) Lagrangian for pure SU(N) Yang-Mills theory in a general covariant gauge is given by L=LYM,B+Lfix,B+LFP,B,(1)where LYM,B=-12Tr(FBμνFBμν),Lfix,B=-1ξBTr(∂μABμ∂νABν),LFP,B=∂μc¯BaDBμcBa.(2)Here, we have defined the bare gauge field ABμ as ABμ=ABaμTa,(3)where the Ta’s are SU(N) generators, chosen so that Tr(TaTb)=12δab.(4)ξB is the bare gauge parameter defining the covariant gauge, and FBμν is the bare field-strength tensor, FBaμν=∂μABaν-∂νABaμ+gBfbcaABbμABcν,(5)with [Ta,Tb]=ifabcTc.(6)The bare covariant derivative DBμ acting on the ghost and antighost fields cBa,c¯Ba reads (DBμ)ca=δca∂μ+gBfbcaABbμ.(7)L can be renormalized by introducing suitable renormalization factors ZA, Zc, and ZAc¯c for the gauge and ghost fields and for the coupling constant, respectively, and by defining new, renormalized gauge and ghost fields Aμa, ca, and c¯a, a renormalized coupling g and a renormalized gauge parameter ξ, according to ABμ=ZA1/2Aμ,ξB=ZAξ,cBa=Zc1/2ca,c¯Ba=Zc1/2c¯a,g2=gB2ZAZc2ZAc¯c2.(8)In terms of the renormalized fields, the Faddeev-Popov Lagrangian reads L=LYM+Lfix+LFP+Lc.t.,(9)where LYM=-12Tr(FμνFμν),Lfix=-1ξTr(∂μAμ∂νAν),LFP=∂μc¯aDμca,(10)and Lc.t. contains the renormalization counterterms. The renormalized field-strength tensor Fμνa and covariant derivative Dμ are defined as Fμνa=∂μAνa-∂νAμa+gfbcaAμbAνc,(Dμ)ca=δca∂μ+gfbcaAμb.(11)We note that Lc.t. does not contain a counterterm for the gauge-fixing term Lfix; indeed, the Slavnov-Taylor identities ensure that the bare gauge parameter ξB can be multiplicatively renormalized by the gauge field renormalization factor ZA alone.

Ordinary perturbation theory is defined by a split of the renormalized Lagrangian, L=L0+Lint+Lc.t.,(12)where L0=limg→0L is taken to be the noninteracting limit of L, L0=12Aμa[iΔ0abμν(p)-1]Aνb+c¯a[iG0ab(p2)-1]cb.(13)Here, the ordinary zero-order gluon and ghost propagators Δ0μνab and G0ab read Δ0μνab(p)=-iδabp2(tμν(p)+ξℓμν(p)),G0ab(p2)-1=iδabp2,(14)where tμν(p) and ℓμν(p) are the transverse and longitudinal projectors defined as tμν(p)=ημν-pμpμp2,ℓμν(p)=pμpνp2.(15)The interaction term Lint contains a three-gluon, four-gluon, and ghost-gluon interaction, Lint=L3g+L4g+Lc¯cg,(16)where L3g=-gfbca∂μAνaAbμAcν,L4g=-14gfbcafdeaAμbAνcAdμAeν,Lc¯cg=gfbca∂μc¯aAμbcc.(17)On the other hand, the term Lc.t. contains the field and coupling renormalization counterterms, Lc.t.=-12δAδabp2tμν(p)AμaAνb+δcδabp2c¯acb+⋯,(18)where δA=ZA-1 and δc=Zc-1. In particular, the gluon field renormalization counterterm is completely transverse.

At low energies, the ordinary perturbation theory of pure YM theory is known to be inconsistent due to the presence of an IR Landau pole in the running of the strong coupling constant. Moreover, constraints due to gauge invariance—when applied in the framework of ordinary perturbation theory–prevent the generation of an IR dynamical mass for the gluons, a phenomenon which by now has been well established mainly thanks to lattice calculations [4–15]. Addressing these issues is the main objective of the screened massive expansion.

The screened massive expansion of pure YM theory is defined by a shift of the expansion point of the Yang-Mills perturbative series, performed in such a way as to treat the transverse gluons as massive already at tree level [53,54]. Explicitly, a shifting term δL is added to the zero-order (kinetic) part of the gauge-fixed, renormalized Fadeev-Popov Lagrangian and subtracted back from its interaction part, L0′=L0+δL,Lint′=Lint-δL.(19)δL is chosen so that the zero-order transverse gluon propagator contained in L0′ is replaced by a massive one; in momentum space δL=12Aμa(p)i[iΔmab-1μν(p)-iΔ0ab-1μν(p)]Aνb(-p),(20)where Δmabμν(p)=δab{-itμν(p)p2-m2+ξ-iℓμν(p)p2}(21)is the new, massive zero-order gluon propagator. Since δL is added to and subtracted from the FP Lagrangian, the shift does not not modify the full action of Yang-Mills theory. Instead, it introduces a new free mass parameter m2 and changes the Feynman rules of YM theory in two respects. First of all, since the new zero-order Lagrangian L0′ reads L0′=12Aμa[iΔmab-1μν]Aνb+c¯a[iG0ab-1]cb,(22)the transverse gluons propagate with a massive propagator rather than with a massless one, see Eq. (21). Second of all, the interacting part of the Lagrangian, Lint′, contains a new two-point interaction, namely, -δL=-12Aμa(p)iδΓgabμν(p)Aνb(-p),(23)where the vertex δΓgabμν(p) is given by δΓgabμν(p)=-im2tμν(p)δab.(24)We refer to the latter as the gluon mass counterterm, not to be confused with the renormalization counterterms contained in Lc.t.. Neither the remaining interaction vertices—spelled out in Eq. (17)—nor the renormalization counterterms are affected by the shift.

The quantities of physical interest can be computed in the framework of the screened expansion using the Feynman rules described above. Since the vertex δΓg is not proportional to the coupling constant, diagrams with an arbitrary number of vertices—termed crossed diagrams if they contain at least one gluon mass counterterm—coexist at any given loop order. For this reason, the screened expansion is intrinsically nonperturbative.

The crossed diagrams can be computed as derivatives of noncrossed diagrams with respect to the gluon mass parameter. This easily follows from the equality [64] [Δm(p)·(δΓg(p)·Δm(p))n]abμν=-i(-m2)n(p2-m2)n+1tμν(p)δab=(-m2)nn!∂n∂(m2)nΔmabμν(p),(25)which is valid for every n≥1 and in any covariant gauge and carries over to the loop integrals.

Due to the massiveness of the zero-order gluon propagator in the screened expansion, new UV divergences arise in the loop integrals which are proportional to the gluon mass parameter m2. These divergences do not invalidate the renormalizability of the n-point functions of the theory, since they cancel as soon as crossed diagrams with a higher number of crossed vertices are taken into account [54,64]. The removal of mass divergences can (and indeed must) be adopted as a criterion for fixing the minimum number of crossed loops to be included when computing some quantity at a given loop order [54,64].

To one loop, the one-particle-irreducible (1PI) gluon polarization Πμνab(p) and ghost self-energy Σab(p2) were computed from the diagrams in Fig. 1. The crossed vertices in the figure represent the gluon mass counterterm δΓg. Diagrams (1c) and (2c) in the gluon polarization are required in order to eliminate the mass divergences that arise from diagrams (1b) and (2b), respectively; they have a total of three vertices. To one loop, there are two more diagrams with the same number of vertices, namely, diagram (1d) and the crossed diagram in the ghost self-energy (top right diagram in Fig. 1); these were also included in the one-loop calculation for consistency.

Since the shift that defines the screened expansion does not change the total action of pure YM theory, the full 1PI gluon polarization is known to be transverse by the Slavnov-Taylor identities. Therefore, we can write Πμνab(p)=Π(p2)tμν(p)δab,(26)where Π(p2) is the gluon’s scalar polarization. After the resummation of the 1PI diagrams, the transverse-gluon and ghost dressed propagators Δ(p2) and G(p2) can then be expressed as Δ(p2)=-i[p2-m2-Π(p2)]-1,G(p2)=i[p2-Σ(p2)]-1,(27)where Σ(p2) is the ghost self-energy. Diagram (1a) in Fig. 1 is easily shown to contribute to the gluon polarization with a constant term ΔΠ=-m2, Π(p2)=-m2+Π(loops)(p2),(28)where Π(loops)(p2) is the loop contribution to the polarization—diagrams (1b)–(2c) in Fig. 1. It is then easy to see that the tree-level mass term inherited from the shift cancels out with ΔΠ, so that the dressed propagator itself can be expressed as Δ(p2)=-i[p2-Π(loops)(p2)]-1.(29)From the above equation it is clear that in the screened expansion, rather than being a trivial effect of the shift of the expansion point, the gluon mass must come from the loops and is thus genuinely dynamical in nature; it does not coincide with the gluon mass parameter m2, which at this stage is just an undetermined dimensionful scale.

Quite interestingly, the existence of a finite mass-scale in YM theory has been derived in the Gaussian approximation from first principles [56,64], but, of course, the actual value of that scale can only arise from the phenomenology, since there is no energy scale in pure YM theory. The best variational Gaussian vacuum was shown to be the vacuum of a massive gluon, and the present screened expansion emerged has the perturbative loop expansion around that best massive vacuum [56]. While fermions have also been incorporated in the Gaussian formalism in the past [67], it is not clear if the screened expansion of full QCD, as is discussed in the present paper, can also be regarded as a loop expansion around a variational Gaussian vacuum which breaks the chiral symmetry.

In a general renormalization scheme and in the Landau gauge, the dressed gluon propagator Δ(p2) can be expressed as Δ(p2)=-iZΔp2(F(s)+F0),(30)where s=-p2/m2 and ZΔ and F0 are, respectively, a multiplicative and an additive renormalization constant.¹

The strong coupling constant αs was absorbed into the definition of ZΔ and F0 and makes no explicit appearance in what follows.

Together with the gluon mass parameter m2, ZΔ and F0 are the only free parameters determining the gluon propagator in the screened expansion. The multiplicative constant ZΔ can of course be fixed by renormalizing the propagator at some specified renormalization scale p2=-μ2, i.e., by requiring that Δ(-μ2)=-i-μ2.(33)The value of the additive renormalization constant F0, on the other hand, was optimized and fixed in Ref. [60] according to principles of gauge invariance. In more detail, it was shown that there exists a value of F0 in the Landau gauge, namely, F0=-0.876, which, when evolved to a general covariant gauge (ξ≠0), yields gauge-invariant poles p02 for the gluon propagator whose residues are also gauge invariant in phase to less than 0.3% [68–70].

In the same context (and in previous papers also, see, e.g., [55,58]), we found that the screened expansion’s gluon propagator has two complex-conjugate poles, whose adimensional positions z02=p02/m2 and z02¯ were determined in [60] from first principles. The existence of complex-conjugate poles has been related in the literature to the issue of the violation of positivity of the gluon spectral function and, more generally, to that of confinement [71,72]. The poles and phases of the residues of the gluon propagator, as computed in the (optimized) screened expansion, are reported in Table I.

Of particular relevance to this paper is the fact that the principal part of the gluon propagator, i.e., the term which contains its poles, well-approximates the full propagator itself [64], provided that the former is multiplied by a factor of 0.945. This is shown in Fig. 2.

With ZΔ and F0 fixed, the gluon mass parameter m2 is left as the only free parameter of the theory (at least as far as the gluon two-point function is concerned). m2 sets the energy units for the dimensionful quantities in the theory; as such, it cannot be determined from first principles and must be fixed from phenomenology. In this respect, the gluon mass parameter plays the same role as the QCD scale ΛQCD of ordinary perturbation theory.²

For a lengthy discussion on the conceptual similarities between the gluon mass parameter m2 and the QCD scale ΛQCD, see Ref. [63], where the issue was addressed in the context of the renormalization group improvement of the screened expansion.

The Lagrangian of full QCD with one flavor of quarks is given by LQCD=L+Lq,B,(34)where L is the Faddeev-Popov Lagrangian of pure Yang-Mills theory—Eq. (1)—and Lq,B is the quark Lagrangian expressed in terms of the bare fields, mass, and coupling, Lq,B=ψ¯B(iDB-MB)ψB.(35)Here, MB is the quark’s bare mass, while DB is the bare covariant derivative acting on the bare quark field ψB, DBμ=∂μ-igBABaμTa.(36)

In order to renormalize the quark Lagrangian, we introduce a quark field renormalization constant Zψ such that ψB=Zψ1/2ψ,(37)where ψ is the renormalized quark field. Then, Lq,B can be expressed as Lq,B=ψ¯(iD-MBZψ)ψ+Lq,c.t.,(38)where D is the renormalized covariant derivative acting on the renormalized quark field, Dμ=∂μ-igAμaTa,(39)with g and Aμa being the renormalized coupling and gluon field defined as in Sec. II A, while Lq,c.t. contains the quark’s field strength and quark-gluon vertex renormalization counterterms.

At this point, if the quark is not massless (i.e., MB≠0), one usually introduces a renormalized quark mass through a kinetic term of the form -MRψ¯ψ and includes the corresponding mass renormalization counterterm -δMψ¯ψ into Lq,c.t.. In ordinary perturbation theory, MR and MB are understood to be proportional and related to each other by radiative corrections which can be computed perturbatively at any given loop order. Due to dynamical mass generation, however, in the IR the light quarks acquire a mass which is much larger than their renormalized mass MR and nonproportional to it; indeed, the former would be nonzero (and of the order of the QCD scale ΛQCD) also in the case of chiral quarks (MB=0). Clearly, choosing MR as the mass of the zero-order propagator around which to expand the perturbative series is not optimal for the purpose of exploring the low-energy dynamics of the quark sector.

On the other hand, the situation could improve if an effective mass scale, mimicking the dynamically generated IR quark mass, was used in place of the renormalized mass MR. Our setup, therefore, employs the following scheme. As in ordinary perturbation theory, we add to the quark Lagrangian a mass term of the form -Mψ¯ψ. However, we do not interpret M as the renormalized counterpart of MB. Instead, we regard the former as being a nonperturbative mass scale arising from the low-energy dynamics of the theory and leave MB unrenormalized. Explicitly, we rewrite the quark Lagrangian as Lq,B=ψ¯(iD-M)ψ+ψ¯(M-MBZψ)ψ+Lq,c.t.(40)and treat M and MB as independent mass parameters; the difference MBZψ-M, which in ordinary perturbation theory would correspond to the mass renormalization counterterm δM, is not taken to be proportional to the coupling constant (i.e., small in the perturbative sense) nor is it regarded as fixed by the renormalization of the quark propagator. We anticipate that an appropriate choice of the diagrams to include in the one-loop quark propagator preserves the renormalizability of the theory also when using this nonstandard scheme.

The quark Lagrangian is now split into a kinetic (zero-order) term Lq,0, Lq,0=ψ¯(i∂-M)ψ,(41)in which M appears as the mass in the zero-order quark propagator; an interaction term Lq,int, Lq,int=ψ¯(gAaTa+M-MBZψ)ψ,(42)which contains the quark-gluon vertex and two new quadratic terms, proportional to M and MB; and a renormalization term Lq,c.t., Lq,c.t.=ψ¯(iδψ∂+gδgAaTa)ψ,(43)which contains the quark field strength renormalization counterterm δψ=Zψ-1 and a renormalization counterterm δg for the quark-gluon vertex.

The addition and subtraction of the mass term -Mψ¯ψ from the quark Lagrangian parallels what we did in the gluon sector of pure Yang-Mills theory. This is best seen in the chiral limit (MB→0), where the addition of a mass term of the form -MRψ¯ψ would be meaningless, since MR∝MB=0. As a nonperturbative mass parameter not directly related to MB, M has the same status of the gluon mass parameter m in the screened expansion of YM theory and is allowed to remain finite also in the chiral limit. For this reason, we refer to M as the chiral mass of the quark.

As in the screened expansion of YM theory, the shift of the quark Lagrangian changes the Feynman rules of the theory. First of all, the chiral mass M now figures as the tree-level mass in the zero-order quark propagator SM(p), SM(p)=ip-M.(44)Second of all, two new two-point vertices δΓq,1 and δΓq,2 arise in the interaction, δΓq,1(p)=iM,δΓq,2(p)=-iMBZψ.(45)We reiterate that in our framework these are treated as independent vertices. The quark-gluon interaction and renormalization vertices, on the other hand, are left unchanged, except for the quark mass renormalization counterterm, which must not be included in the calculation.

These Feynman rules must of course be supplied with those of the gluon sector, which were derived in Sec. II A in the context of pure YM theory. In particular, the transverse gluons propagate with a massive zero-order propagator—Eq. (21)—and a third two-point vertex, the gluon mass counterterm of Eq. (24), is included in the interaction.

As a consequence of the new Feynman rules, the screened expansion of full QCD is nonperturbative in nature. Like in pure YM theory, this is due to the two-point vertices δΓg, δΓq,1, and δΓq,2, which are proportional to the gluon and the quark mass parameters m2, M, and MB, and are not taken to be proportional to the strong coupling constant.

Let us now turn our attention to how to compute the quark propagator in the new framework. The dressed quark propagator S(p) can be expressed in terms of the 1PI quark self-energy Σ(p)³

Not to be confused with the ghost self-energy of Sec. II A.

From Eqs. (50) and (51), we see that in the chiral limit (MB→0), despite the absence of a tree-level mass for the quark propagator, the quark mass function M(p2) does not vanish; thanks to the finiteness of the nonperturbative scale M, one finds that ΣS(p2)≠0, which makes B(p2)≠0 and thus M(p2)≠0, also for vanishing MB. Since ΣS(p2) comes from the loops, the mass of the quark is genuinely dynamical, a feature that was already highlighted in Sec. II for the gluons in pure YM theory. For nonchiral quarks the situation is similar, the only difference being that B(p2) also contains one additional tree-level term which is proportional to the bare mass MB of the quark. As we will see in a moment, the fact that this term is not renormalized poses no issue of consistency.

To one loop, an infinite number of diagrams contributes to the 1PI quark self-energy. These have the structure of the ordinary one-loop diagram of standard perturbation theory—diagram (2a) in Fig. 4—with an arbitrarily large number of insertions of the gluon mass counterterm δΓg and of the quark mass counterterms δΓq,1 and δΓq,2. In order to chose a truncation scheme for this infinite series, let us have a look at the first few such diagrams.

The simplest one-loop self-energy diagram is the ordinary uncrossed loop—denoted by (2a) in Fig. 4. In a general covariant gauge, diagram (2a) has divergences in both its vector component and in its scalar component, Σ(2a)(p)=(c2aVp+c2aSM)2ε+⋯,(53)where c2aV and c2aS are O(g2) coefficients, ε=4-d is the regulator of dimensional regularization, and the dots denote finite self-energy terms. While the vector divergence can be straightforwardly absorbed into the renormalization constant Zψ—see the first of Eq. (50)—in order to remove the mass divergence c2aS, we would need to define a renormalized mass MR in terms of which MB=Zψ-1(MR-c2aSM2ε+scheme-dep.consts.),(54)see the second of Eq. (50). A relation like this mixes infrared entities (namely, the chiral mass M) to UV features (the divergence and the renormalization of the bare mass) with no apparent logic, aside from the mathematical convenience of it. Moreover, this type of renormalization cannot be employed in the chiral limit MB→0, when there is no bare mass in which to absorb the divergence. For these reasons, it must be rejected.

We note that, having been introduced through a term which is added and subtracted in the Lagrangian, the mass parameter M cancels in the total action; as a consequence, any divergence proportional to M must disappear when diagrams with a different number of mass counterterms are resummed at the same loop order.

In fact, diagram (2b) in Fig. 4 is easily shown to contain the same mass divergence of diagram (2a) with an opposite sign; since the crossed quark line in the diagram can be expressed as a derivative with respect to the quark’s chiral mass, ip-M(iM)ip-M=-M∂∂Mip-M.(55)The self-energy contribution from diagram (2b), Σ(2b)(p), can be obtained as a derivative of Σ(2a)(p), Σ(2b)(p)=-M∂∂MΣ(2a)(p).(56)It follows that Σ(2b)(p)=-c2aSM2ε+⋯,(57)that is, Σ(2a)(p) and Σ(2b)(p) have opposite mass divergences. As a consequence, the sum of diagrams (2a) and (2b) only contains a divergence in the vector component, coming entirely from Σ(2a)(p). This divergence can be shown to be the same as the one found in ordinary perturbation theory and is to be absorbed into the definition of Zψ, as we saw earlier.

Now, in the Landau gauge (ξ=0), the divergence contained in Σ(2a)(p) is known from ordinary perturbation theory to vanish. Therefore, not only does the sum Σ(2a)(p)+Σ(2b)(p) not contain mass divergences, but in the Landau gauge it is also fully finite. In particular, if we truncate the perturbative series to diagrams (2a) and (2b) in Fig. 4 and limit ourselves to the Landau gauge, then the term MBZψ that appears in the B(p2) function—see Eq. (50)—can be taken to be a finite constant. In other words, no renormalization of divergent constants or masses is required in the screened expansion of the Landau gauge quark propagator, provided that the latter is truncated to diagrams (2a) and (2b).

On the other hand, if ξ≠0, the vector divergence in Σ(2a)(p)+Σ(2b)(p) still needs to be absorbed into Zψ. For nonchiral quarks (MB≠0), if MB were taken to be finite, this would leave us with a divergent MBZψ term inside B(p2). Therefore, for ξ≠0 and MB≠0, a renormalized mass MR must still be introduced, even when truncating the quark self-energy to diagrams (2a) and (2b).

It is easy to see that a renormalized mass of the form MR=MBZψ would not have the ordinary behavior of a running mass under the renormalization group (RG). Indeed, if the RG equations were employed in the scheme, MR would run exclusively with the anomalous dimension of the quark field, rather than with the full anomalous dimension of the quark mass. This happens because we have left out one further divergent diagram from the calculation, namely, diagram (2c) in Fig. 4. The latter can be obtained from diagram (2a) by using the equality ip-M(-iMBZψ)ip-M=MBZψ∂∂Mip-M,(58)which can be exploited to write Σ(2c)(p)=MBZψ∂∂MΣ(2a)(p).(59)In particular, Σ(2c)(p)=c2aSMBZψ2ε+⋯.(60)As we can see, diagram (2c) has a scalar divergence proportional to MBZψ. When the latter is summed to the tree-level term in B(p2), one finds B(p2)=MBZψ(1+c2aS2ε)+⋯.(61)By simple dimensional arguments, it is easy to show that the remaining one-loop diagrams in the quark self-energy are finite. Therefore, the above expression spells out the complete divergent term of the scalar component of the one-loop self-energy, obtained by summing the divergences of diagrams (2a) to (2c) in Fig. 4. Such a term can indeed be equated, modulo finite constants, to a renormalized mass MR which would run like an ordinary quark mass if the RG equations were to be used, leaving us with B(p2)=MR+finite terms.(62)Beyond the Landau gauge, then, consistency with the renormalization group requires us to include diagram (2c) in the calculation. In the Landau gauge, on the other hand, diagram (2c) is not needed, in principle, since to one loop the sum of diagrams (2a) and (2b) already results in a finite quark 1PI self-energy.

Despite being necessary for theoretical consistency, if the renormalized quark mass MR is much smaller than the chiral mass M, the inclusion of diagram (2c) in the quark self-energy turns out not to be essential from a quantitative point of view. This is easily seen as follows. Let Σf(2a,2b,2c)(p) be the finite parts of the self-energy diagrams (2a), (2b), and (2c). Using Eqs. (56) and (59), Σf(2b)(p)+Σf(2c)(p)=-(M-MBZψ)∂∂MΣf(2a)(p).(63)Modulo higher-order corrections, we can set MB=MR, Zψ=1 in the above equation, so that Σf(2b)(p)+Σf(2c)(p)=-(M-MR)∂∂MΣf(2a)(p).(64)It is then clear that, as long as MR≪M, the contribution of diagram (2c) is completely negligible with respect to that of diagram (2b). In other words, for the light quarks, diagram (2c) can be taken to contribute only to the divergent part of the self-energy, i.e., to the renormalization of the bare mass.⁴

For the sake of completeness, we note that there is one catch in this argument; at high energies, the scalar part of the sum of diagrams (2a) and (2b) can be shown to vanish, see, e.g., Sec. IV A, so that, instead of being negligible, diagram (2c) actually makes up for the whole scalar self-energy. As long as we limit ourselves to low and moderate energies, this issue does not arise. At large energies, however, diagram (2c) and appropriate RG techniques are needed to account for the correct asymptotic behavior of the quark mass function.

To summarize, in every linear covariant gauge, diagram (2b) in Fig. 4 is needed in order to remove the mass divergence in diagram (2a). This mass divergence has no counterpart in ordinary perturbation theory, since it is proportional to the quark chiral mass M. Diagram (2c) is essential to renormalize the bare mass MB in compliance with the standard RG equations. Nonetheless, its finite part is completely negligible in the case of light quarks. Finally, in the Landau gauge the sum of diagrams (2a) and (2b) results in a finite self-energy. Since for the light quarks diagram (2c) is quantitatively negligible, in the Landau gauge one can simply exclude it from the self-energy and interpret the free parameters MB and Zψ as bare but finite quantities.

In the next section, we carry on with our analysis of the resummation of the one-loop quark propagator. Our main focus is on exploring different ways to treat the finite diagrams in Fig. 4.

Up to this point, we have discussed the self-energy diagrams which contribute to the divergent part of the one-loop quark propagator, namely, diagrams (2a) to (2c) in Fig. 4. Using simple dimensional arguments, it is easy to show that, to one loop, other insertions of the gluon and quark two-point mass counterterms indeed yield convergent diagrams. As an example, consider diagram (2d) in Fig. 4. This diagram has a superficial degree of divergence D D=d-1-2-2→-1<0,(65)where the -1 and the -2’s come from the internal quark and gluon lines, respectively, making diagram (2d) UV-finite in the limit d→4. Equivalently, observe that diagram (2d) can be expressed as a derivative of diagram (2a) with respect to the gluon mass parameter m2; using Eq. (25) with n=1, we find that Σ(2d)(p)=-m2∂∂m2Σ(2a)(p).(66)Since the divergent part of Σ(2a)(p) does not depend on m2, Σ(2d)(p) is again shown to be finite.

While divergent diagrams are included in the one-loop calculation based on principles of renormalizability, assessing which finite diagrams should be included as well is far more tricky. One option could be to adopt a minimalistic point of view and limit oneself to the one-loop diagrams needed for consistency, i.e., diagrams (2a) to (2b) or (2c) in Fig. 4. Yet another option could be to retain all the one-loop diagrams with a maximum of three vertices, as we did for the gluon propagator in Sec. II; in practice, this amounts to also including diagram (2d) in the self-energy. These two resummation schemes differ by how the internal gluon line is treated—explicitly, by whether the internal zero-order gluon propagator is corrected with its own mass counterterm or not. We refer to them as the minimalistic and the vertex-wise schemes, respectively. Schemes with a larger number of crossed diagrams (not shown in Fig. 4) are not considered in this paper.

In the next section, we fit and compare the results obtained in the minimalistic and vertex-wise schemes with the quenched lattice data of Ref. [73]. The reason for using quenched rather than unquenched lattice data is to exploit our previous results for pure YM theory and fix ab initio the value of the gluon mass parameter m2 that appears in the quark propagator—thus reducing the number of free parameters to be fitted. Indeed, observe that, to one loop, the quark self-energy diagrams for the quenched and unquenched theories coincide. Hence, in principle, our results could be used for comparisons with both quenched and unquenched data. However, in the framework of the screened massive expansion, the value of the gluon mass parameter m2 running in diagrams (2a)–(2d) (Fig. 4) can receive corrections from the quark loop in the gluon polarization (Fig. 5), which is only present in the unquenched theory. Thus, we expect the value of m2 to be different depending on which theory (quenched or unquenched) we are trying to fit. In order to reduce the freedom in the choice of free parameters, we decide not to make a new determination of the gluon mass parameter, but rather to use the quenched lattice data for our fits. The value m=0.6557  GeV was obtained in [60] by a fit of the lattice data of Ref. [15] for pure YM theory. With m fixed, the remaining free parameters of the quark propagator are the chiral mass M, the quark bare mass MB or renormalized mass MR, and the renormalization constants.

As we will see, the minimalistic and vertex-wise schemes are practically equivalent from the point of view of the fit, the only difference being in the values of the parameters needed to achieve the match with the lattice data. Both of them succeed in quantitatively reproducing the lattice mass function M(p2) with very good precision. On the other hand, in none of the two the Z function has the behavior displayed by the lattice data; Z(p2) is found to be a decreasing function of momentum, at variance with the lattice. To one loop, such a mismatch is not unseen, having been reported for another massive model, namely, the Curci-Ferrari model of Ref. [45].

One interesting question to ask is whether higher-order or nonperturbative corrections to the internal gluon line in the quark self-energy can sensibly change the behavior of the Z function. Indeed, as we noted in the Introduction, in the Landau gauge, to one loop and at sufficiently high energies, Z(p2)≈1, making the Z function sensitive to all kinds of contributions beyond the leading perturbative order. The near vanishing of the perturbative contribution makes the Z function a valid benchmark for investigating the role of condensates by the OPE. Indeed, the slightly increasing behavior which is observed on the lattice has been modeled by OPE [74–76] and shown to be consistent with the existence of a dimension-2 gluon condensate of the form ⟨A2⟩. In order to explore these issues, we introduce a third resummation scheme, which we term the complex-conjugate (CC) scheme for reasons that will become apparent in a moment.

In the CC scheme, instead of only summing the zero-order gluon propagator (minimalistic scheme) or its counterterm-corrected counterpart (vertex-wise scheme), we use the fully dressed gluon propagator as the internal gluon line of the one-loop quark self-energy (see Fig. 6). Switching to the dressed gluon propagator allows us to account for the full nonperturbative dynamics of the gluon, when computing the quark propagator.

While, in principle, using the dressed propagator would require us to resum and integrate an infinite number of higher-order diagrams, in practice we know that—in pure Yang-Mills theory—the principal part of the screened expansion’s one-loop gluon propagator provides a very good approximation to the dressed propagator, modulo a multiplicative factor (see Sec. II B, in particular, Figs. 2 and 3). Therefore, in the CC scheme, we use a zero-order gluon propagator which—in Euclidean space and in the Landau gauge—reads Δμν(c.c.)(p)={Rp2+p02+R¯p2+p02¯}tμν(p).(67)Here, p02 and p02¯ are the complex-conjugate poles of the dressed gluon propagator (hence, the name CC scheme) in the complexified Minkowski space, and R and R¯ are their normalized residues. The value of the modulus |R|—which depends both on the renormalization conventions for the dressed gluon propagator and on a multiplicative factor that converts between the full propagator and its principal part—does not actually affect the results for the quark propagator, provided that the free parameters are suitably redefined. Indeed, to one loop, the internal gluon line in the quark self-energy is multiplied by a factor of the strong coupling constant αs, so that |R| can be absorbed into the definition of the latter. Our convention for the definition of |R| (and thus also αs in the CC scheme) is discussed in Sec. IV C. As for p02 and the phase of R, we use the values reported in Table I (Sec. II B). These were obtained in pure Yang-Mills theory and are thus suitable for calculations in the quenched theory, in line with our discussion on the gluon mass parameter m2 in the minimalistic and vertex-wise schemes.

As we show in Appendix B, despite the poles p02 and p02¯ being complex, as long as the external momentum p2∈R, the loop integrals in the CC scheme can be computed by employing the usual machinery of Feynman parameter integrals and gamma functions. In particular, if we denote with Σm.(loops)(p) the loop contribution to quark self-energy computed in the minimalistic scheme—diagrams (2a) to (2c) in Fig. 4—then we can express the corresponding self-energy term Σc.c.(loops)(p) in the CC scheme as Σc.c.(loops)(p)=RΣm.(loops)(p)|m2=p02+R¯Σm.(loops)(p)|m2=p02¯(68)or equivalently Σc.c.(loops)(p)=2Re{RΣm.(loops)(p)|m2=p02}.(69)

As we will see, the Z function computed in the CC scheme indeed turns out to have a qualitatively different behavior than those computed in the minimalistic or vertex-wise scheme, closer to the one displayed by the quenched lattice data at moderately large momenta.

In the minimalistic resummation scheme, the loop diagrams included in the quark self-energy are those denoted by (2a), (2b), and, for the purpose of defining a renormalized mass MR, (2c) in Fig. 4. The quark mass function M(p2) can be expressed as M(p2)=3[k0+σS(m.)(p2)]h0-σV(m.)(p2),(81)where the analytic expressions for the scalar functions σS(m.)(p2) and σV(m.)(p2) are reported in Appendix A. By fixing m=655.7  MeV as discussed in Sec. III B and fitting the quenched lattice mass functions of Ref. [73] for the lattice masses Mlat=18, 36, 54, 72, 90 MeV, we obtained the values of h0 and k0 reported in Table II. In Table III, we list the corresponding values of αs, MB, and MR, computed by employing the definitions in Eqs. (73), (75), and (80).

As we can see from Fig. 7, the mass functions computed in the minimalistic scheme show a very good agreement with the lattice data. For all but one of the considered lattice masses—namely, Mlat=18  MeV, which we discuss separately in a moment—the fitted values of the chiral mass M are found to be in the range 320–337 MeV, while the bare masses MB are found to increase with Mlat, always keeping close to the latter.

The fact that MB≈Mlat can be easily explained by looking at the high-momentum limit of the functions σV(m)(p2) and σS(m)(p2). For p2≫m2,M2 we have σV(m)(p2)→-1-3m24p2+3m22p2lnp2m2→-1,σS(m)(p2)→2M2p2lnp2M2→0.(82)Therefore, in terms of MB and αs, M(p2)→MBZψZψ+αs3π≈MB(p2≫m2,M2),(83)where the approximation holds provided that the coupling is sufficiently small. The above equation shows that the scale of the high-momentum limit of the mass function is set by the bare mass MB; since on the lattice the same role is played by the lattice mass Mlat, we expect MB≈Mlat as long as our function fits well the lattice data.

In the limit of vanishing momenta, regardless of the lattice mass, the data saturate to a finite value of about 350–450 MeV.⁷

Note that this value is larger for the heavy quarks, as we show later on in Fig. 10.

In Table III, the value of the bare mass MB fitted for Mlat=18  MeV stands out for being negative (this is a direct consequence of k0<0 in Table II). Presumably, this physically meaningless result is an artifact of the fit caused by the highly oscillatory tail of the MB=18  MeV lattice mass function; the oscillations themselves are most likely due to discretization errors, as suggested by the large error bars in the original data (see Ref. [73]). A constrained fit forcing MB≥0 is not able to fix this issue, since, in the presence of the constraint, the fitting routine still tries to push MB to negative values, which implies that the lower boundary of the fitting interval, namely, MB=0, is inevitably hit. Thus, no meaningful result for MB is obtained by constraining the latter to be non-negative. Cutting the data at large momenta in order to avoid the oscillations (which begin at approximately 2.5–3 GeV), as well, would not improve the situation; since at low momenta the quark mass function is not very sensitive to the value of MB (provided, of course, that we assume MB≪M), employing a cut dataset would make it impossible to meaningfully establish the value of the bare mass by a fit. As an alternative, to test our results, we checked that fixing the value of MB by hand, instead of fitting it from the lattice data, still yields a mass function which—modulo oscillations—is in good agreement with the lattice. Some examples are shown in Fig. 8, where we plot the data for Mlat=18  MeV together with our minimalistic scheme mass function. Here, MB is set to 0,10,18 MeV, while the rest of the free parameters (reported in Table IV) are still obtained by fitting the data. Remarkably, as soon as the bare mass is fixed to small but positive values, the values of the parameters M and αs obtained from the constrained fit get closer to those found for Mlat=36–90  MeV (Table III), further evidence that MB>0 is a more consistent choice when compared to the raw result of the fit.

Being in possession of analytic expressions which give a good description of the quark mass function in the Euclidean space, we are in a position to extend the quark propagator to the complexified Minkowski space and look for its poles p02. These are defined as the solutions to the equation p02-M2(p02)=0,(84)where the argument p2 of the function M(p2) is a complexified Minkowski momentum squared, at variance with the convention used in this section, where we used the Euclidean momentum. For all the considered lattice masses, using the parameters in Tables II and III, we found that the quark propagator has a pair of complex-conjugate poles in the variable p2 (equivalently, two pairs in the variable p=p2); their positions p0 are reported in Table V. In the literature, the existence of complex-conjugate poles has been interpreted as proof of confinement, since the imaginary part of the poles has the effect of removing the particles from the asymptotic states of the theory [55,60,71]. In the minimalistic scheme, the real part of the poles was found to be between 388 and 424 MeV, while their imaginary part is roughly half these values, having been found in the range from 174 to 194 MeV. Fixing MB=10  MeV by hand for the lattice mass Mlat=18  MeV yields p0=±373.7±202.3i  MeV, a result which is more consistent with those of the other lattice masses, when compared with the one obtained from the raw fit. Indeed, we note that |Re(p0)| increases with Mlat, while |Im(p0)| decreases with it. We checked that using small but positive values of MB for Mlat=18  MeV yields similar poles to those reported above.

In Fig. 9, we show an example of the Z function computed in the minimalistic scheme using the parameters in Table III, compared with the lattice data for a quark with mass Mlat=54  MeV. As we can see, the behavior of Z(p2) is the complete opposite of that found on the lattice; while on the lattice the Z function increases with momentum, in the minimalistic scheme it decreases. This behavior is independent of the considered lattice mass, and we checked that it does not change if the parameters are fixed by fitting the Z function itself rather than the mass function. We believe that the mismatch with the lattice data may be due to the fact that, at least at sufficiently high energies, Z(p2)≈1, making the Z function very sensitive to higher-order and even nonperturbative corrections. This is supported by the results we obtained in the complex-conjugate resummation scheme, which show an improved agreement at large momenta (see Sec. IV C) and by recent findings reported in Ref. [65], where the Z function is computed in the context of the Curci-Ferrari model and shown to change its behavior at two loops.

While up to this point our main focus has been on the light quarks, it may be interesting to see what happens if we try to apply the screened expansion to heavier quarks. Therefore, to end this section, we compare the minimalistic scheme mass function with the lattice data for quarks of mass Mlat=126, 181, 271 MeV. The outcome is shown in Fig. 10; as in Fig. 7, the free parameters are fitted from the data themselves. It should be noted that when MB becomes of the same order as M, as is the case in these fits, the approximation that we employed throughout this paper, namely, to neglect the finite part of diagram (2c) in Fig. 4, becomes less justifiable, and the diagram should be fully included in the quark self-energy. Nevertheless, it appears that the mass functions in the minimalistic scheme still manage to fit well the lattice data. As for the light quarks, the Z functions computed in the minimalistic scheme for the heavier quark do not match the lattice data and are thus not reported.

In the vertex-wise resummation scheme, the loop diagrams included in the quark self-energy are those denoted by (2a), (2b), (2d), and, for defining a renormalized mass MR, (2c) in Fig. 4. The quark mass function M(p2) can be expressed as M(p2)=3[k0+σS(v.)(p2)]h0-σV(v.)(p2),(85)where the analytic expressions for the scalar functions σS(v.)(p2) and σV(v.)(p2) are reported in Appendix A. As in Sec. IV A, we fixed m=655.7  MeV and performed a fit to the quenched lattice mass functions of Ref. [73] for the lattice masses Mlat=18, 36, 54, 72, 90 MeV. The results of the fit are reported in Table VI, while in Table VII we list the corresponding values of αs, MB and MR.