Effective gluon mass and infrared fixed point in QCD
aa r X i v : . [ h e p - ph ] S e p Effective gluon mass andinfrared fixed point in QCD
Arlene C. Aguilar and Joannis Papavassiliou
Departamento de F´ısica Te´orica and IFIC, Centro Mixto,Universidad de Valencia – CSICE-46100, Burjassot, Valencia, Spain
Abstract.
We report on a special type of solutions for the gluon propagator of pure QCD,obtained from the corresponding non-linear Schwinger-Dyson equation formulated in theFeynman gauge of the background field method. These solutions reach a finite value in thedeep infrared and may be fitted using a massive propagator, with the crucial characteristicthat the effective “mass” employed depends on the momentum transfer. Specifically, thegluon mass falls off as the inverse square of the momentum, as expected from the operator-product expansion. In addition, one may define a dimensionless quantity, which constitutesthe generalization in a non-Abelian context of the universal QED effective charge. Thisstrong effective charge displays asymptotic freedom in the ultraviolet whereas in the low-energy regime it freezes at a finite value, giving rise to an infrared fixed point for QCD.
Keywords:
Schwinger-Dyson equations, pinch technique, gluon propagator.
PACS:
A plethora of theoretical and phenomenological studies spanning more than twodecades have corroborated the possibility of describing the infrared (IR) sector ofQCD in terms of an effective gluon mass (for an extended list of references see [1]).According to this picture, even though the gluon is massless at the level of thefundamental Lagrangian, and remains massless to all order in perturbation theory,the non-perturbative QCD dynamics generate an effective, momentum-dependentmass, without affecting the local SU (3) c invariance, which remains intact [2].The most standard way for studying such a non-perturbative effect in thecontinuum is the (appropriately truncated) Schwinger-Dyson equation (SDE) forthe gluon propagator ∆ µν ( q ), defined (in the Feynman gauge) as∆ µν ( q ) = − i (cid:20) P µν ( q )∆( q ) + q µ q ν q (cid:21) , P µν ( q ) = g µν − q µ q ν q . (1)Specifically, one looks for solutions having ∆( q ) reaching finite (non-vanishing)values in the deep infrared, that may be fitted by “massive” propagators of theform ∆ − ( q ) = q + m ( q ). The crucial characteristic is that m ( q ) is not “hard”,but depends non-trivially on the momentum transfer q . When the renormalization-group logarithms are properly taken into account, one obtains in addition the non-perturbative generalization of g ( q ), the QCD running coupling (effective charge).The presence of m ( q ) in the argument of g ( q ) tames the Landau singularityassociated with the perturbative β function, and the resulting effective charge isasymptotically free in the ultraviolet (UV), “freezing” at a finite value in the IR.he running of m ( q ) is of central importance for the self-consistency of thisapproach, mainly because the value of ∆ − (0) is determined by integrals involving∆( q ), m ( q ), and g ( q ) over the entire range of (Euclidean) momenta. The UVconvergence of these integrals depends crucially on how m ( q ) behaves as q → ∞ .If m ( q ) drops off asymptotically faster than a logarithm, then ∆ − (0) is finite.This, in turn, is crucial because the finiteness of ∆ − (0) guarantees essentially therenormalizability of QCD.In earlier studies of linear SDE [2, 1] the m ( q ) obtained drops in the deepUV as an inverse power of a logarithm. The main result reported in this talk isthe existence of a new type of solutions for m ( q ) that drop asymptotically as an inverse power of momentum (multiplied by logarithms) [3].These solutions are found in the study of nonlinear SDE, in the frameworkdefined from the combination of the Pinch Technique (PT) [2, 4, 5] and theFeynman gauge of the Background Field Method (BFM) [6, 7], known as PT-BFM truncation scheme [1]. One of the most powerful features of the PT-BFMformalism is that, by virtue of the Abelian Ward identities satisfied by the variousvertices, gluonic and ghost contributions are separately transverse, within each order in the “dressed-loop” expansion [1]. This, in turn, allows one to truncate theseries meaningfully, by considering only the diagrams ( a ) and ( a ) shown in Fig.1,(no ghosts included), without compromising the transversality of the answer.In order to reduce the algebraic complexity of the problem, we perform oneadditional approximation, dropping the longitudinal terms from the gluon propa-gators inside the integrals, i.e. we set ∆ αβ → − ig αβ ∆. Omitting these terms doesnot interfere with the transversality of the resulting propagator, provided that onedrops, at the same time, the longitudinal pieces in the WI of Eq.(4) [1, 3].After these steps, the scalar function, ∆ − ( q ) = q + i Π( q ), (where Π( q ) is thegluon-self energy given by the diagrams ( a ) and ( a ) in Fig.1 ) can be written as i P µν ( q )∆ − ( q ) = i P µν ( q ) q − C A g Z [ dk ] e Γ αβµ ∆( k ) e IΓ ναβ ∆( k + q )+ 4 C A g g µν Z [ dk ] ∆( k ) , (2)where the tree-level vertex e Γ µαβ appearing in (2) is given by e Γ µαβ ( q, p , p ) = ( p − p ) µ g αβ + 2 q β g µα − q α g µβ , (3)and e IΓ να ′ β ′ represents the full three-gluon vertex. FIGURE 1.
The gluonic “one-loop dressed” contributions to the SDE. s a next step we will employ the “gauge technique” [8], expressing e IΓ as afunctional of ∆, in such a way as to satisfy (by construction) the all-order Wardidentity q µ e IΓ µαβ ( q, p , p ) = i [∆ − αβ ( p ) − ∆ − αβ ( p )] , (4)characteristic of the PT-BFM. Specifically, we propose the following form for thevertex [3] e IΓ µαβ = L µαβ + T µαβ + T µαβ , (5)with L µαβ ( q, p , p ) = e Γ µαβ ( q, p , p ) + ig αβ q µ q [Π( p ) − Π( p )] ,T µαβ ( q, p , p ) = − i c q (cid:0) q β g µα − q α g µβ (cid:1) [Π( p ) + Π( p )] ,T µαβ ( q, p , p ) = − ic (cid:0) q β g µα − q α g µβ (cid:1) (cid:20) Π( p ) p + Π( p ) p (cid:21) . (6)Then, substituting Eqs.(3) and (5) into (2), introducing q ≡ x , k ≡ y , anddefining the renormalization-group invariant quantity d ( q ) = g ∆( q ) , we arriveat d − ( x ) = K ′ x + ˜ b X i =1 b A i ( x ) + d − (0) , (7)with b A ( x ) = − (cid:18) c (cid:19) x Z ∞ x dy y L ( y ) d ( y ) , b A ( x ) = 6 c x Z ∞ x dy L ( y ) d ( y ) , b A ( x ) = − (cid:18) c (cid:19) x L ( x ) d ( x ) Z x dy y L ( y ) d ( y ) , b A ( x ) = (cid:18) − − c c (cid:19) Z x dy y L ( y ) d ( y ) , b A ( x ) = − (cid:18) c (cid:19) L ( x ) d ( x ) Z x dy y L ( y ) d ( y ) , b A ( x ) = 6 c Z x dy y L ( y ) d ( y ) , b A ( x ) = 25 L ( x ) d ( x ) x Z x dy y L ( y ) d ( y ) , b A ( x ) = 15 x Z x dy y L ( y ) d ( y ) . (8)The renormalization constant K ′ is fixed by the condition d − ( µ ) = µ /g , (with µ ≫ Λ ), and L ( q ) ≡ ˜ b ln ( q / Λ ), where Λ is QCD mass scale. Due to the polesontained in the Ansatz for e IΓ, d − (0) does not vanish, and is given by the(divergent) expression d − (0) = 3˜ b π " c ) Z d k L ( k ) d ( k ) − (1 + 2 c ) Z d k k L ( k ) d ( k ) , (9)which can be made finite using dimensional regularization, and assuming that m ( q ) drops sufficiently fast in the UV [1].In order to determine the asymptotic behavior that Eq.(7) predicts for m ( x ) atlarge x , we perform the following replacements in the r.h.s. of (8) x L ( x ) d ( x ) → , L ( x ) d ( x ) → /x, L ( y ) d ( y ) = ˜∆( y ) , ˜∆( y ) = 1 y + m ( y ) . (10)Next, use the identity y ˜∆( y ) = 1 − m ( y ) ˜∆( y ) in all b A i ( x ), keeping only termslinear in m . Then separate all contributions that go like x from those that golike m on both sides, and match them up [9]. This gives rise to two independentequations, one for the “kinetic” term, which simply reproduces the asymptoticbehavior x ln x on both sides, and an equation for the terms with m ( x ), given by m ( x ) ln x = C − a Z ∞ x dy m ( y ) ˜∆( y ) + a x Z x dy ym ( y ) ˜∆( y )+ a x Z x dy y m ( y ) ˜∆( y ) + a x Z ∞ x dy m ( y ) ˜∆ ( y ) , (11)with a = 65 (1 + c − c ) , a = 45 + 6 c , a = − , a = 1 + 6 c , (12)and C ≡ ˜ b − d − (0) + a Z ∞ dy m ( y ) ˜∆( y ) . (13)Now, the important point to appreciate is that, in order for (11) to have solutionsvanishing in the UV, it is necessary to be sure that the constant term on ther.h.s. vanishes, i.e. C = 0. Since we know that d − (0) and the integral appearingin the r.h.s. of Eq.(13) are manifestly positive quantities, it follows immediatelythat the C will be zero if and only if a <
0. Notice that Eq.(13) restricts the rangeof allowed values of the parameters c and c through Eq.(12). In addition, andmore importantly, it constrains the momentum dependence of m ( x ) in the IR andintermediate regimes to be such that both terms on the r.h.s of Eq.(13) cancelagainst each other.Assuming that Eq.(13) is satisfied, it can be shown that Eq.(11) admits thefollowing asymptotic solutions for m ( x ) [3], m ( x ) = λ (ln x ) − (1+ γ ) , m ( x ) = λ x (ln x ) γ − , (14)here λ and λ are two mass-scales, and γ = − a , γ = a .The first type of solutions, m ( x ), are familiar from studying linearized versionsof Eq.(2), see for example [2, 1]. The second type of solutions, m ( x ), displayingpower-law running, are particularly interesting because they are derived for the firsttime in the context of SDE. The possibility of an effective gluon mass dropping inthe UV as an inverse power of the momentum was first conjectured in [2], and wassubsequently obtained in the context of the operator-product expansion [10]; therethe resulting gluon self-energy was identified as the effective gluon mass, leadingto the relation m ( x ) ∼ h G i /x , where h G i is the dimension four gauge-invariantgluon condensate.Which of the two types of solution will be actually realized depends on the detailsof the three-gluon vertex, e IΓ, and more specifically on the values of the parameters c and c . Our numerical analysis reveals that the sets of values for c and c givingrise to logarithmic running belong to an interval that is disjoint and well-separatedfrom that producing power-law running. In what follows we will focus our attentionon the latter type of solutions. In Fig.2 we present typical solutions for the d ( q ), m ( q ) and the effective charge α ( q ) = g ( q ) / π . FIGURE 2.
Upper panel: the numerical solution for d ( q ). Lower panels: Left: dynam-ical mass with power-law running, for m = 0 . and ρ = 1 .
046 in Eq.(17). Right:the running charge, α ( q ) = g ( q ) / π , fitted by Eqs.(15) and (16). he way to extract from d ( q ) the corresponding m ( q ) and g ( q ) is by castingthe numerical solutions shown in Fig.2 into the form d ( q ) = g ( q ) q + m ( q ) , g ( q ) = (cid:20) ˜ b ln (cid:18) q + f ( q , m ( q ))Λ (cid:19) (cid:21) − , (15)with f ( q , m ( q )) = ρ m ( q ) + ρ m ( q ) q + m ( q ) + ρ m ( q )[ q + m ( q )] , (16)where ρ , ρ , and ρ are fitting parameters.The functional form used for the running mass is m ( q ) = m q + m " ln (cid:18) q + ρ m Λ (cid:19) . ln (cid:18) ρ m Λ (cid:19) γ − . (17)In the deep UV Eq.(17) goes over to m ( q ), whereas at q = 0 it reaches the finitevalue m (0) = m . Note that f ( q , m ( q )) is such that f (0 , m (0)) >
0; as a resultthe perturbative Landau pole in the running coupling is tamed, and g ( q ) reachesa finite positive value at q = 0, leading to an infrared fixed point [2, 11, 12].To summarize our results, from a gauge-invariant SDE for the gluon propagatorwe have derived an integral equation that describes the running of the effectivegluon mass in the UV, and have demonstrated that, depending on the values oftwo basic parameters appearing in the three-gluon vertex, one finds solutions thatdrop as inverse powers of a logarithm of q , or much faster, as an inverse power of q . Moreover, we have extracted an asymptotically free effective (running) charge,that freezes in the low-momentum region, implying the existence of a IR fixed pointfor QCD. ACKNOWLEDGMENTS
This work was supported by the Spanish MEC under the grants FPA 2005-01678and FPA 2005-00711. The research of JP is funded by the Fundaci´on General ofthe UV.
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