Non-perturbative QCD effective charges
aa r X i v : . [ h e p - ph ] O c t Non-perturbative QCD effective charges
Arlene C. Aguilar a and Joannis Papavassiliou ba Federal University of ABC, CCNH,Rua Santa Ad´elia 166, CEP 09210-170, Santo Andr´e, Brazil b Department of Theoretical Physics and IFIC,University of Valencia-CSIC, E-46100, Valencia, Spain
Using gluon and ghost propagators obtained from Schwinger-Dyson equations (SDEs), we construct the non-perturbative effective charge of QCD. We employ two different definitions, which, despite their distinct field-theoretic origin, give rise to qualitative comparable results, by virtue of a crucial non-perturbative identity.Most importantly, the QCD charge obtained with either definition freezes in the deep infrared, in agreement withtheoretical and phenomenological expectations. The various theoretical ingredients necessary for this constructionare reviewed in detail, and some conceptual subtleties are briefly discussed.
1. Introduction
One of the challenges of the QCD is the self-consistent and physically meaningful definitionof an effective charge. This quantity providesa continuous interpolation between two physi-cally distinct regimes: the deep ultraviolet (UV),where perturbation theory works well, and thedeep infrared (IR), where non-perturbative tech-niques, such as lattice or SDEs, must be em-ployed. The effective charge depends strongly onthe detailed dynamics of some of the most fun-damental Green’s functions of QCD, such as thegluon and ghost propagators [1].In this talk we will focus on two characteris-tic definitions of the effective charge, frequentlyemployed in the literature. Specifically we willconsider (i) the effective charge of the pinch tech-nique (PT) [2,3] and (ii) the one obtained fromthe ghost-gluon vertex [4].
2. Two non-perturbative effective charges
We first introduce the notation and the basicquantities entering into our study. In the covari-ant renormalizable ( R ξ ) gauges, the gluon prop-agator ∆ µν ( q ) has the form∆ µν ( q ) = − i (cid:20) P µν ( q )∆( q ) + ξ q µ q ν q (cid:21) , (1) where ξ denotes the gauge-fixing parame-ter, P µν ( q ) = g µν − q µ q ν /q is the usual trans-verse projector, and ∆ − ( q ) = q + i Π( q ), withΠ µν ( q ) = P µν ( q )Π( q ) the gluon self-energy. Inaddition, the full ghost propagator D ( q ) andits dressing function F ( q ) are related by iF ( q ) = q D ( q ). The all-order ghost vertex willbe denoted by IΓ µ ( k, q ), with k representing themomentum of the gluon and q the one of the anti-ghost; at tree-level IΓ (0) µ ( k, q ) = − q µ .An important ingredient for what follows is thetwo-point function Λ µν ( q ) defined by [1,5] i Λ µν ( q ) = λ Z k H (0) µρ D ( k + q )∆ ρσ ( k ) H σν ( k, q ) , = g µν G ( q ) + q µ q ν q L ( q ) , (2)where λ = g C A , C A is the Casimir eigen-value of the adjoint representation, and R k ≡ µ ε (2 π ) − d R d d k , with d the dimension ofspace-time. H µν ( k, q ) is represented in Fig. 1, andits tree-level, H (0) µν = ig µν . In addition, H µν ( k, q )is related to IΓ µ ( k, q ) by q ν H µν ( k, q ) = − i IΓ µ ( k, q ) . (3) The PT definition of the effective charge relieson the construction of an universal ( i.e. , process-1 H σν ( k, q ) = H (0) σν + k, σk + q qν Figure 1. Diagrammatic representation of H .independent) effective gluon propagator, whichcaptures the running of the QCD β function, ex-actly as happens with the vacuum polarization inthe case of QED [2,6] (See Fig. 2). One impor-tant point, explained in detail in the literature,is the (all-order) correspondence between the PTand the Feynman gauge of the BFM [2,7]. In fact,one can generalize the PT construction [2] in sucha way as to reach diagrammatically any value ofthe gauge fixing parameter of the BFM, and inparticular the Landau gauge. In what follows wewill implicitly assume the aforementioned gener-alization of the PT, given that the main identitywe will use to relate the two effective charges isvalid only in the Landau gauge.To fix the ideas, the PT one-loop gluon self-energy reads b ∆ − ( q ) = q (cid:20) bg ln (cid:18) q µ (cid:19)(cid:21) , (4)where b = 11 C A / π is the first coefficient of theQCD β -function. Due to the Abelian WIs sat-isfied by the PT effective Green’s functions, thenew propagator-like quantity b ∆ − ( q ) absorbs allthe RG-logs, exactly as happens in QED withthe photon self-energy. Then, the renormaliza-tion constants of the gauge-coupling and of thePT gluon self-energy, defined as g ( µ ) = Z − g ( µ ) g , b ∆( q , µ ) = b Z − A ( µ ) b ∆ ( q ) , (5)where the “0” subscript indicates bare quantities,satisfy the QED-like relation Z g = b Z − / A . There-fore, the product b d ( q ) = g b ∆ ( q ) = g ( µ ) b ∆( q , µ ) = b d ( q ) , (6)forms a RG-invariant ( µ -independent) quan-tity [2]. For asymptotically large momenta one c ∆ c ∆ c ∆( a ) ( b ) ( c ) g gg g g g Figure 2. The universal PT coupling. may extract from b d ( q ) a dimensionless quantityby writing, b d ( q ) = g ( q ) q , (7)where g ( q ) is the RG-invariant effective chargeof QCD; at one-loop g ( q ) = g bg ln ( q /µ ) = 1 b ln (cid:0) q / Λ QCD (cid:1) . (8)where Λ QCD denotes an RG-invariant mass scaleof a few hundred MeV.Eq. (6) is a non-perturbative relation; there-fore it can serve unaltered as the starting pointfor extracting a non-perturbative effective charge,provided that one has information on the IR be-havior of the PT-BFM gluon propagator b ∆( q ).Interestingly enough, non-perturbative informa-tion on the conventional gluon propagator ∆( q )may also be used, by virtue of a general rela-tion connecting ∆( q ) and b ∆( q ). Specifically, aformal all-order relation known as “background-quantum” identity [8] states that∆( q ) = (cid:2) G ( q ) (cid:3) b ∆( q ) . (9)Note that, due to its BRST origin, the above re-lation must be preserved after renormalization.Specifically, denoting by Z Λ the renormalizationconstant relating the bare and renormalized func-tions, Λ µν and Λ µν , through g µν + Λ µν ( q, µ ) = Z Λ ( µ )[ g µν + Λ µν ( q )] , (10)then from (9) and Z g = b Z − / A follows the addi-tional relation Z − g = Z / A Z Λ , (11)which is useful for the comparison with the cou-pling discussed in the following subsection.It is now easy to verify, at lowest order, thatthe 1 + G ( q ) obtained from Eq. (2) restores the β function coefficient in front of UV logarithm.In that limit [5]1 + G ( q ) = 1 + 94 C A g π ln (cid:18) q µ (cid:19) , ∆ − ( q ) = q (cid:20) C A g π ln (cid:18) q µ (cid:19)(cid:21) . (12)Using Eq. (9) we therefore recover the b ∆ − ( q ) ofEq. (4), as we should.Then, non-perturbatively, one substitutes intoEq. (9) the 1 + G ( q ) and ∆( q ) obtained fromeither the lattice or SD analysis, to obtain b ∆( q ).This latter quantity is the non-perturbative gen-eralization of Eq. (4); for the same reasons ex-plained above, the combination b d ( q ) = g ∆( q )[1 + G ( q )] , (13)is an RG-invariant quantity. In principle, a definition for the QCD effectivecharge can be obtained starting from the vari-ous QCD vertices; however, in general, such aconstruction involves more than one momentumscales, and further assumptions about their valuesneed be introduced, in order to express the chargeas a function of a single variable. The ghost-gluonvertex has been particularly popular in this con-text, especially in conjunction with Taylor’s non-renormalization theorem and the correspondingkinematics [4].We next define the following renormalizationconstants∆( q ) = Z − A ∆ ( q ) , F ( q ) = Z − c F ( q ) , IΓ ν ( k, q ) = Z IΓ ν ( k, q ) , g ′ = Z − g ′ g . (14)Notice that a priori Z g ′ defined as Z g ′ = Z Z − / A Z − c , does not have to coincidewith the Z g introduced in (5); however, as wewill see shortly, they do coincide by virtue of thebasic identity we will derive in next section.It turns out that for the so-called Taylor kine-matics (vanishing incoming ghost momentum, k µ → − q µ ), one may impose the additional con-dition Z = Z g ′ Z / A Z c = 1 ⇒ Z − g ′ = Z / A Z c . (15)Thus, the combination b r ( q ) = g ′ ∆( q ) F ( q ) , (16)is a RG-invariant ( µ -independent) quantity.Therefore, for asymptotically large q , in anal- ( ) − = ( ) − + kq q q k + q Figure 3. The SDE for the ghost propagator.ogy to Eq. (7) one can define an alternative QCDrunning coupling as b r ( q ) = g ( q ) q . (17)It is easy to verify that g ( q ) and g ( q ) displaysthe same one-loop behavior, since, perturbatively,the function 1 + G ( q ) is the inverse of the ghostdressing function F ( q ) (this is due to the generalidentity of Eq. 20).
3. An important identity
In this section, we discuss a non-trivial identity,valid only in the Landau gauge, relating the F ( q )with the G ( q ) and L ( q ) of (2).The derivation proceeds as follows. First, con-sider the standard SD equation for the ghostpropagator, represented in Fig. 3, and written as iD − ( q ) = q + iλ Z k Γ µ ∆ µν ( k )IΓ ν ( k, q ) D ( p ) , (18)where p = k + q . Then, contract both sides of thedefining equation (2) by the combination q µ q ν toget[ G ( q )+ L ( q )] q = λ Z k q ρ ∆ ρσ ( k ) q ν H σν ( k, q ) D ( p ) . (19)Using the Eq. (3) and the transversality of thefull gluon propagator, we can see that the rhs ofEq. (19) is precisely the integral appearing in theghost SDE (18). Therefore, in terms of the ghostdressing function F ( q ),1 + G ( q ) + L ( q ) = F − ( q ) . (20)Eq. (20), derived here from the SDEs, has beenfirst obtained in [9], as a direct consequence ofthe BRST symmetry.Let us study the functions G ( q ) and L ( q )more closely. From Eq. (2) we have that (in d dimensions) G ( q ) = 1( d − q (cid:0) q Λ µµ − q µ q ν Λ µν (cid:1) ,L ( q ) = 1( d − q (cid:0) dq µ q ν Λ µν − q Λ µµ (cid:1) . (21)In order to study the relevant equations fur-ther, we will approximate the two vertices, H µν and IΓ µ , by their tree-level values, Then, setting f ( k, q ) ≡ ( k · q ) /k q , one may show that [1] F − ( q ) = 1 + λ Z k [1 − f ( k, q )]∆( k ) D ( k + q ) , ( d − G ( q ) = λ Z k [( d − f ( k, q )]∆( k ) D ( q + k ) , ( d − L ( q ) = λ Z k [1 − d f ( k, q )]∆( k ) D ( q + k ) , (22)It turns out that if F and ∆ are both IR finite,Eq. (22) yields the important result L (0) = 0 [1].Of course, all quantities appearing in Eq. (22)are unrenormalized. It is easy to recognize, forexample, by substituting in the corresponding in-tegrals tree-level expressions, that F − ( q ) and G ( q ) have exactly the same logarithmic depen-dence on the UV cutoff, while L ( q ) is finite atleading order.Since the origin of (20) is the BRST symme-try, it should not be deformed after renormaliza-tion. To that end, using the definition of (10), inorder to preserve the relation (20) we must im-pose that Z Λ = Z c . In addition, by virtue of (3),and for the same reason, we have that, in theLandau gauge, IΓ ν ( k, q ) and H σν ( k, q ) must berenormalized by the same renormalization con-stant, namely Z [ viz. Eq. (14)]; for the Taylorkinematics, we have that Z = 1 [see Eq. (15)](for some additional subtleties see [1]).Returning to the effective charges, first of all,comparing Eq. (6) and Eq. (16), it is clear that g ( µ ) = g ′ ( µ ), by virtue of Z Λ = Z c . Therefore,using Eq. (9), one can get a relation betweenthe two RG-invariant quantities, b r ( q ) and b d ( q ),namely b r ( q ) = [1 + G ( q )] F ( q ) b d ( q ) . (23)From this last equality follows that α PT ( q ) and α gh ( q ) are related by α gh ( q ) = [1 + G ( q )] F ( q ) α PT ( q ) . (24) Gluon Propagator ( ) = 0.21 and = 4.3 GeV ( ) = 0.16 and = 10 GeV ( ) = 0.13 and = 22 GeV ( q ) [ G e V - ] q [GeV ] Figure 4. Numerical solutions for the gluon prop-agator.After using Eq. (20), we have that α PT ( q ) = α gh ( q ) (cid:20) L ( q )1 + G ( q ) (cid:21) . (25)Evidently, the two couplings can only coincide attwo points: (i) at q = 0, where, due to the factthat L (0) = 0, we have that α gh (0) = α PT (0),and (ii) in the deep UV, where L ( q ) approachesa constant.
4. The nonperturbative analysis
We next turn to the dynamical information re-quired for the various ingredients entering intothe effective charges defined above. To that end,we solve numerically the system of SDEs for∆( q ) F ( q ), G ( q ) and L ( q ) obtained in [5]In Figs. (4) and (5) we show the resultsfor ∆( q ) and F ( q ) renormalized at threedifferent points, µ = { . , , } GeV with α ( µ ) = { . , . , . } respectively. On theright panel we plot the corresponding F ( q )renormalized at the same points. Notice that thesolutions obtained are in qualitative agreementwith recent results from large-volume lattices [10]where the both quantities, ∆( q ) and F q ), reachfinite (non-vanishing) values in the deep IR.The results for 1 + G ( q ) and L ( q ), renor-malized at the same points, are presented inFig. 6. As we can see, the function 1 + G ( q )is also IR finite exhibiting a plateau for values Ghost dressing ( ) = 0.21 and = 4.3 GeV ( ) = 0.16 and = 10 GeV ( ) = 0.13 and = 22 GeV F ( q ) q [GeV ] Figure 5. Numerical solutions for the ghost dress-ing function.of q < . . In the UV region, we insteadrecover the perturbative behavior (12). On theother hand, L ( q ), Fig. 7, shows a maximum inthe intermediate momentum region, while, as ex-pected, L (0) = 0.With all ingredients defined, the first thing onecan check is that indeed Eq. (13) gives rise to aRG-invariant combination. Using the latter def-inition, we can combine the different data setsfor ∆( q ) and [1 + G ( q )] at different renormal-ization points, to arrive at the curves shown inFig. 8. Indeed, we see that all curves, for differ-ent values of µ , merge one into the other provingthat the combination b d ( q ) is independent of therenormalization point chosen. + G ( q ) q [GeV ] ) ( ) = 0.21 and = 4.3 GeV ( ) = 0.16 and = 10 GeV ( ) = 0.13 and = 22 GeV Figure 6. 1 + G ( q ) determined from Eq. (22). L(q ) ( ) = 0.21 and = 4.3 GeV ( ) = 0.16 and = 10 GeV ( ) = 0.13 and = 22 GeV L ( q ) q [GeV ] Figure 7. L ( q ) determined from Eq. (22) d ( q ) [ G e V - ] q [GeV ] RGI product d(q )=g (q ) ( )=0.21 and =4.3 GeV ( )=0.16 and =10 GeV ( )=0.13 and =22 GeV ^ ^ ^ Figure 8. The b d ( q ) obtained by combining ∆( q )and [1 + G ( q )] according to (13).From the dimensionful b d ( q ) we must now ex-tract a dimensionless factor, g ( q ), correspond-ing to the running coupling (effective charge).Given that ∆( q ) is IR finite (no more “scal-ing”!), the physically meaningful procedure isto factor out from b d ( q ) a massive propagator[ q + m ( q )] − , b d ( q ) = g ( q ) q + m ( q ) , (26)where for the mass we will assume “power-law”running [11], m ( q ) = m / ( q + m ).Thus, it follows from Eq. (26), that the effectivecharge α PT = g ( q ) / π is identified as being4 πα PT ( q ) = [ q + m ( q )] b d ( q ) , (27) q [GeV] PT (q ) gh (q ) Figure 9. α gh ( q ) vs α PT ( q ), for m = 500 MeV.Finally we compare numerically the two effec-tive charges, α PT ( q ) and α gh ( q ) in Fig. 9. First,we determine α PT ( q ) obtained using (27), thenwe obtain α gh ( q ) with help of (25) and the re-sults for 1 + G ( q ) and L ( q ), Fig. 6 and Fig. 7.As we can clearly see, both couplings freeze atthe same finite value, exhibiting a plateau for val-ues of q < .
02 GeV , while in the UV both showthe expected perturbative behavior. They differonly slightly in the intermediate region where thevalues of L ( q ) are appreciable.
5. Conclusions
In this talk we have presented a comparisonbetween two QCD effective charge, α PT ( q ) and α gh ( q ), obtained from the PT and the ghost-gluon vertex, respectively.Despite their distinct theoretical origin, due toa fundamental identity relating the various ingre-dients entering into their definitions, the two ef-fective charges are almost identical in the entirerange of physical momenta. In fact, the coincideexactly in the deep infrared, where they freeze ata common finite value, signaling the appearanceof IR fixed point in QCD [12], also required froma variety of phenomenological studies [13]. Acknowledgments:
The authors thank the or-ganizers of LC09 for their hospitality. Theresearch of JP is supported by the EuropeanFEDER and Spanish MICINN under grantFPA2008-02878, and the “Fundaci´on General” ofthe University of Valencia.
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