The Color Antisymmetric Ghost Propagator and One-Loop Vertex Renormalization
aa r X i v : . [ h e p - ph ] J a n The Color Antisymmetric Ghost Propagatorand One-Loop Vertex Renormalization
Sadataka
Furui
School of Science and Engineering, Teikyo UniversityUtsunomiya 320-8551, Japan (Received September 19, 2007)The color matrix elements of the ghost triangle diagram that appears in the triple gluonvertex and the ghost-ghost-gluon triangle diagram that appears in the ghost-gluon-ghostvertex are calculated. The ghost-ghost-gluon triangle contains a loop consisting of two colordiagonal ghosts and one gluon and a loop consisting of two color antisymmetric ghosts andone gluon. Consequently, the pQCD argument in the infrared region based on the oneparticle irreducible diagram should be modified.Implications for the Kugo-Ojima color confinement and the QCD running coupling arediscussed. §
1. Introduction
In 1971, ’t Hooft showed that in massless Yang-Mills field theory, non-gaugeinvariant regulator fields can be incorporated, provided that in the limit of highregulator mass the gauge invariance can be restored by means of a finite numberof counter terms including ghost fields and longitudinally polarized gauge fields.After this work, Taylor pointed out that if the charge carried by the ghost andthe gauge field are the same, the Ward identity in QED, i.e. Z /Z = 1, where Z is the vertex renormalization factor and Z is the matter field wave functionrenormalization factor, can be extended to QCD as Z /Z = Z ¯ ψψA /Z ψ , where Z isthe gluon wave function renormalization factor, Z ψ is the matter field wave functionrenormalization factor, and Z ¯ ψψA is the vertex renormalization factor of the matterfield.In these arguments, vertices at tree level are considered, and the ghost propa-gators were assumed to be color diagonal. The ghost propagator is defined as theFourier transform (FT) of the matrix element of the inverse Faddeev-Popov operator, F T [ D abG ( x, y )] = F T h tr( Λ a { ( M ) − } xy Λ b ) i = δ ab D G ( q ) , where M = − ∂ µ D µ , and {} xy represents the matrix value. We define the ghostdressing function G ( q ) as q D G ( q ). In the Dyson-Schwinger (DS) approach, G ( q )at 0 momentum behaves as ∼ ( q ) − κ , and that of the gluon dressing function Z ( q )behaves as ∼ ( q ) κ . In lattice simulations, we calculated the overlap to obtain the color typeset using
PTP
TEX.cls h Ver.0.9 i Sadataka Furui diagonal ghost propagator, D G ( q ) = 1 N c − V tr D δ ab ( h Λ a cos q · x | f bc ( x ) i + h Λ a sin q · x | f bs ( x ) i ) E , and the color antisymmetric ghost propagator φ c ( q ) = 1 N V tr D f abc ( h Λ a cos q · x | f bs ( x ) i − h Λ a sin q · x | f bc ( x ) i ) E , where N = 2 for SU (2) and 6 for SU (3). Here, f cb ( x ) and f sb ( x ) represent thesolution f b ( x ) = M [ U ] − ρ b ( x ), where U is the lattice link variable, with ρ bc ( x ) =1 √ V Λ b cos q · x and ρ bs ( x ) = 1 √ V Λ b sin q · x , respectively. We normalize the SU (3)generator Λ a as Λ a = λ √ λ is that defined by Gell-Mann. In our notation,we have tr Λ a Λ b = δ ab .Lattice studies of the color antisymmetric ghost propagator in quenched SU (2) and unquenched SU (3) were motivated by analysis in the local composite op-erator (LCO) approach based on symmetry under the BRST (Becchi, Rouet, Storaand Tyutin) transformation. The presence of h f abc ¯ c b c c i condensates was ex-pected to manifest itself in the color anti-symmetric ghost propagator. The ghostcondensates, as the on-shell BRST partner of A condensates, were also discussed inRef. In the study of unquenched SU (3), the modulus of the color antisym-metric ghost propagator is found to be relatively large, and its infrared singularity,characterized by α ′ G ∼ .
9, is larger than that of the color diagonal ghost propagator α G ∼ . However, analysis of quenched SU (3) on a 56 lattice showed thatthe modulus of the color antisymmetric ghost propagator is small and its samplevariation is large.These data suggest that the ghost propagators of SU (2) and SU (3) are qualita-tively different, although the gluon propagators of the two are similar, and that thepresence of the dynamical quark affects the color antisymmetric ghost propagator.In the momentum subtraction scheme ( ^ M OM scheme), the running couplings inthe Coulomb gauge and in the Landau gauge are calculated as the product of colordiagonal ghost dressing function squared and the gluon propagator. The runningcoupling in the Landau gauge is α s ( q ) = q D G ( q ) D A ( q ) , where D A ( q ) is the gluon propagator in 4-dimensional space, and that in the Coulombgauge is α I ( q ) = q D G ( q ) D Atr ( q ) , where D Atr ( q ) is the 3-dimensional transverse gluon propagator.In the Coulomb gauge, the color-Coulomb potential defines another runningcoupling α Coul ( q ) = N c − N f N c q V Coul ( q ). Because we do not fix A , the variation of α Coul ( q ) in the infrared region is large. he color antisymmetric ghost propagator α s ( q ), has apeak around 2.3, near q = 0 . α I ( q ) exhibits freezing to a constant around 3. This qualitative difference is believed to be due to the difference between the ghostpropagators: That in the Coulomb gauge is 3 dimensional and instantaneous, whilethat in the Landau gauge is 4 dimensional and propagating in the 4th direction.In usual perturbative QCD (pQCD), a ghost is assumed to be color diagonal,which is valid at high energy. In a study of the Gribov horizon, Zwanziger introduceda real Bose (ghost) field φ abµ ( x ) and the additional action S = S cl + γS + S S cl = 14 X µ,ν,a Z d x ( F aµν ( x )) S = − (2 C ) − Z Z d xd y trA µ ( x ) M [ A ] − A ν ( y ) S = Z d x (cid:18) φ abµ M bd [ A ] φ adµ + iγ / C − / f abd φ adµ A µb (cid:19) , (1.1)in order to restrict the gauge configuration in the fundamental modular region. Here, C is the value of the SU (3) Casimir operator of the fundamental representation.In the Gribov-Zwanziger Lagrangian, the equation of motion for φ adµ becomes ( D µ φ ν ) ab = ∂ µ φ ν ab f acd A cµ φ ν db , (1.2)and its solution is φ µab = − γ √ f abc M [ A ] − A µc . (1.3)When A µc is replaced by q µ c c , where c c is the ghost field and f deh is multiplied, weobtain f deh φ abµ = − γ √ δ ae δ bh − δ be δ ah ) M [ A ] − q µ c c . (1.4)The color antisymmetric ghost, φ c ( q ), multiplied by the momentum q µ and f deh is similar to Zwanziger’s ghost, φ abµ . In an analytical one-loop calculation, basedon the Gribov-Zwanziger Lagrangian, including two additional ghost field denotedby { φ abµ , ¯ φ abµ } and { ω abµ , ¯ ω abµ } , infrared freezing of the running coupling α s ( q ) wasdemonstrated. Although Zwanziger’s ghost, φ abµ , was introduced in order to re-strict the gauge configuration into the fundamental modular region, its relation tothe A condensates is discussed in Ref. Although the color antisymmetric ghostmultiplied by the momentum q µ and f deh and Zwanziger’s ghost are different, weexpect similar corrections to the infrared exponents of the gluon propagator and theghost propagator through the color antisymmetric ghost propagator in the Landaugauge, when it is incorporated properly.The organization of this paper is as follows. In § §
3, the contribution of the
Sadataka Furui ghost loop in the gluon propagator is investigated, and in §
4, the contribution of thegluon-ghost-ghost loop in the ghost propagator is studied. Discussion of the quark-gluon vertex, the Kugo-Ojima confinement parameter and QCD running coupling isgiven in §§ § §
2. The triple gluon vertex and the ghost-gluon-ghost vertex
In this section we study the contribution of the color antisymmetric ghost prop-agator in the triple gluon vertex and the ghost-gluon-ghost vertex.The triple gluon vertex defined in Fig. 1 in the pQCD is given by the tree-leveldiagram (Fig. 2), ghost loop diagram (Fig. 3) and the gluon loop diagram (Fig. 4) a bc
Fig. 1. The triple gluon vertex. a bc
Fig. 2. The bare triple gluon vertex. a bc
Fig. 3. The dressed triple gluon vertex. Thedashed line represents a ghost and the thinline a gluon. a bc
Fig. 4. The triple gluon vertex. The thin linesrepresent gluons.
The color indices of the ghost loop in the triple gluon vertex are assigned as inFig. 5. We express the ghost propagator, which is assigned as d , as a combination he color antisymmetric ghost propagator δ a ′′ c ′′ D G ( k ) + 2 f a ′′ c ′′ d φ d ( k ) . (2.1)The same combination is assumed for the ghost assigned as h . The color factor f a ′′ a ′ a f b ′′ b ′ b f c ′′ c ′ c (2.2)is multiplied at the three edges of the triangle. a ´ bc a’a’’c’’c’ b’’e b’d h Fig. 5. The triple gluon vertex. The dashed line represents a ghost and the thin line is a gluon.
The ghost-gluon-ghost vertex is expressed as in Fig. 6 and its tree-level diagramis given in Fig. 7. At the one-loop level, the ghost-ghost-gluon loop shown in Fig. 8contributes. The assignment of the color indices is shown in Fig. 9. a bc
Fig. 6. The gluon-ghost-ghost vertex. Thedashed line represents a ghost and the thinline a gluon. a bc
Fig. 7. The gluon-ghost-ghost vertex. Thedashed line represents a ghost and the thinline a gluon.
In the calculation of the color matrix element of the one loop vertex diagramin Figs. 5 and 9, we fix a, b, c and carry out a summation over the color indices a ′ , a ′′ , b ′ , b ′′ , c ′ , c ′′ , d, e, h . The matrix elements of the quark gluon vertices are pro-portional to f abc , which are given by Sadataka Furui a bc
Fig. 8. The gluon-ghost-ghost vertex. Thedashed line represents a ghost and the thinline a gluon. a ´ bc a’a’’c’’c’ b’’e b’d h Fig. 9. The gluon-ghost-ghost vertex. Thedashed line represents a ghost and the thinline a gluon. • f = 1, • f = f = f = f = f = f = 12 , • f = f = √
32 .Using the definitions D c ′ b ′′ ≡ D e , D a ′ b ′ ≡ D h and D a ′′ c ′′ ≡ D d , we obtain inthe case of SU (2) the coefficients given in Table I. In this table, abc and deh arethe SU (2) color indices, D h D d D e represents all color diagonal D h φ d φ e , and D e φ h φ d represent one propagator assigned as D h or D e are color diagonal but the rest arecolor antisymmetric.When deh are not elements of the Cartan subalgebra, there appear terms pro-portional to φφφ , but the coefficient is purely imaginary, and both plus sign termsand minus sign terms appear, and they cancel. abc deh D h D d D e D h φ d φ e D d φ e φ h D e φ h φ d
123 333 -1 -4 0 0Table I. The SU (2) color matrix elements of the ghost triangle diagram. In the case of SU (3), we obtain the coefficients which are to multiply f abc , as inTable II. The gluon that has the color index in the Cartan subalgebra couples witha pair of color antisymmetric ghost propagators and the contribution has the samesign as the contribution of the gluon that couples with a pair of color diagonal ghostpropagators.The qualitative difference between the color matrix elements of SU (2) and SU (3)comes from the anomaly cancelling equation of the triangle diagramtr {{ Λ a , Λ b } Λ c } = 0 , (2.3)where {} represents the anti-commutator, is satisfied for SU (2) but not for SU (3). he color antisymmetric ghost propagator abc deh D h D d D e D h φ d φ e D d φ e φ h D e φ h φ d
123 888 -1.5 -1.5 1.5 1.5333 -1.5 -4.5 0 0833 -1.5 0 -0.5 0383 -1.5 0 0 -0.5338 -1.5 -4.5 0 0838 -1.5 0 0 1.5883 -1.5 -1.5 0 0388 -1.5 0 1.5 0147 888 -1.5 0 0 1.5333 -1.5 -1 1 -0.5338 -1.5 -1 0 −√ / −√ −√ √ / −√ √ / −√ √ / −√ / −√ −√ / −√ −√ / −√ −√ −√ / √ √ − √ √ − √ − √ √ − √ √ √ − √ − √ √ / √ √ / √ √ / √ √ √ / SU (3) color matrix elements of the ghost triangle. The coefficients of the
Dφφ terms of f and f differ only in their signs.When the color indices are deh = 333 and abc = 123, the same structure as in the SU (2) appears. But in the case of deh = 888 and abc = 147 since f and f arethe only coefficient that do not vanish when coupled to the color source 8, the colorantisymmetric pair of d and e appear in the link.In the case of the unquenched SU (3) configuration with the Kogut-Susskind Sadataka Furui b c
Fig. 10. The gluon propagator dressed by theghost propagator. b c
Fig. 11. The gluon propagator dressed bygluon propagator. ab cdeh ´ Fig. 12. The ghost loop contribution in thegluon propagator. The dashed line repre-sents a ghost. The × indicates the dressingof a gluon whose color index is a . ab cdh e Fig. 13. The ghost loop contribution in thegluon propagator. The dashed line repre-sents a ghost and the thick line is a quark. fermion MILC f , the ratio of the dressing function q φ ( q ) /G ( q ) for q ∼ .
18 GeV isabout 0.2 and for q ∼ . while in the case of quenched SU (2), theratio for q ∼ . and for q ∼ . . − . §
3. The ghost loop in the gluon propagator
At one-loop order, the vacuum polarization tensor that contributes to the gluonself-energy consists of a) a quark loop, b) a ghost loop (Fig. 10), c) a gluon tadpoleand d) a gluon loop (Fig. 11). In b) there is a ghost-gluon-ghost vertex, and in d)there is a triple gluon vertex at two-loop order, in which the color antisymmetricghost could contribute. In this section, we study the effect of the ghost loop on thegluon propagator.The product of the color antisymmetric ghost propagator contributes to theghost-loop diagram as shown in Fig. 12. There the cross indicates a coupling to thequark loop, as shown in Fig. 13, or to gluon loops. The quark loop or the gluon loopcan couple to other gluons, and among the ghost propagator specified by d, e and h ,one is color diagonal and the other two are color antisymmetric. The color matrixelements we calculated in the triple gluon vertex imply that if the ghost propagatorthat does not couple with the gluon of color index a is color diagonal and the restare color antisymmetric, the color index a should belong to the Cartan subalgebra,in order that the matrix element becomes real.But because other gluons can couple to the quark loop, the color index a isnot necessarily in the Cartan subalgebra. The propagator is proportional to f abc ,and since the signs of the color matrix element given in TableII are random, itscontribution is expected to be small, and thus the color mixing of gluons should besmall. Lattice simulations also indicate that the gluon propagator is diagonal in the he color antisymmetric ghost propagator §
4. The ghost-ghost-gluon loop in the ghost propagator
At one-loop order, the ghost self-energy is given by the gluon-ghost loop shownin Fig. 14.At two-loop order, the ghost-ghost-gluon loop contributes in such a mannerthat a pair of color antisymmetric ghost propagator and the gluon propagator areincorporated. We consider the production of the color antisymmetric ghost pair froma gluon of color index a . The propagator shown in Fig. 15 is proportional to f abc ,and its trace vanishes. However, the product of these propagators shown in Fig. 16does not vanish, and they contribute to the dressing of the external ghosts of theghost-gluon-ghost vertex. b c Fig. 14. The ghost propagator. ab cdeh ´ Fig. 15. The dressing of the ghost propagatorby the gluon. The dashed line is a ghostthe thin line is a gluon. The × indicatesdressing of a gluon in Cartan subalgebra. When the gluon with color index a can be treated as a background field thatcouples to a quark loop, as shown in Fig. 17 the color antisymmetric ghost propagatorcontributes. In contrast to the gluon propagator, the color mixing of the ghost isnot ruled out from the lattice simulations. ab cdeh a’b’ c’d’e’h’ ´ Fig. 16. The two-ghost propagator. The × in-dicates a gluon of color indices a = a ′ inthe Cartan subalgebra. ab cdh e. . Fig. 17. The dressing of the ghost propagatorby the gluon. The dashed line represents aghost the thin line a gluon, and the thickline a quark. Sadataka Furui
The dressing of the ghost propagator by the gluon is depicted in Fig. 14. Theintegral is given by d ( b, c ) = Z d q (2 π ) δ bc (cid:18) δ µν q µ q ν ( p − q ) q κ ) − q µ q ν ( p − q ) µ ( p − q ) ν ( p − q ) q κ ) (cid:19) Z (( p − q ) ) . (4.1)When Z (( p − q ) ) is taken as a constant, the formulae in Ref. give d ( b, c ) ∝ K (1 + κ, , p ) p − ( K (1 + κ, , p ) p − L (1 + κ, , p ) p + M (1 + κ, , p ) p ) , (4.2)where Z d q q µ q ν q a ( P − q ) b = K ( a, b, p ) p µ p ν + K ( a, b, p ) p δ µν , Z d q q µ q ν q ρ q a ( p − q ) b = L ( a, b, p ) p µ p ν p ρ + L ( a, b, p ) p ( p µ δ νρ + p ν δ ρµ + p ρ δ µν ) , Z d q q µ q ν q ρ q σ q a ( p − q ) b = M ( a, b, p ) p µ p ν p ρ p σ + M ( a, b, p ) p ( δ µν p ρ p σ + δ µρ p ν p σ + δ µσ p ρ p µ + δ νρ p µ p σ + δ νσ p ρ p µ + δ ρσ p µ p ν )+ M ( a, b, p ) p ( δ µν δ ρσ + δ µρ δ νσ + δ µσ δ ρν ) . (4.3)According to the ansatz of the DS equation, the exponent of the ghost dressingfunction is given by κ ∼ α G and it is related to the exponent of the gluon dressingfunction α D as − κ ∼ α D . When κ = 0 .
5, corresponding to an infrared finitegluon propagator, the integral becomes δ bc d ( b, c ) = − . p ) . − ( 6 . p ) . − . p ( p ) . − . p ( p ) . ) (4.4)Analysis of the exponent of the gluon dressing function of lattices with a long timeaxis is given in Ref. In reality, κ in the infrared and ultraviolet regions could bedifferent, and the above numerical values should be regarded as simple estimations.The loop integral of the ghost-ghost-gluon triangle is given by I ( p, a, b, c ) = Z d q (2 π ) (cid:18) δ µν − ( p − q ) µ ( p − q ) ν ( p − q ) (cid:19) Z (( p − q ) )( p − q ) × δ a ′′ c ′′ q κ ) + 2 if a ′′ c ′′ d φ d ( q ) ! ( − gf a ′′ aa ′ q µ ) × δ b ′ a ′ q κ ) + 2 if b ′ a ′ h φ h ( q ) ! ( − gf b ′′ bb ′ q ν ) × X x (loop term)( t a ) xx . (4.5)where the quark loop contribution is expressed as the loop term. he color antisymmetric ghost propagator abc dh D h D d D Ae D Ae φ h φ d
321 88 1.5 -1.533 1.5 -4.583 1.5 038 1.5 0854 88 1.5 -4.533 1.5 -1.583 1.5 √ /
238 1.5 √ / −√ /
238 1.5 −√ / SU (3) color matrix elements of the ghost-ghost-gluon triangle. The color matrix elements are given in Table IV, where D Ae indicates the colordiagonal gluon propagator. The coherent contribution, D e φφ from dh =88 and 33suggests that color mixing occurs in the infrared region. The contribution of thecolor antisymmetric ghost propagator would be enhanced in an unquenched latticesimulation, in which case the quark loop becomes a source of the color antisymmetricpair.The exponent of the color diagonal ghost dressing function α G is ∼ . and that of the color antisymmetric ghost dressing function defined at q ∼ . α ′ G ∼ . By using the formulae in Ref. and assuming Z (( p − q ) ) is constantin the relevant integration region, we obtain I ( p, a, b, c ) ∝ f abc ( c K (2(1 + κ ) , , p ) p + c K (2(1 + κ ′ ) , , p ) p ) − (cid:0) c ( K (2(1 + κ ) , , p ) p − L (2(1 + κ ) , , p ) p + M (2(1 + κ ) , , p ) p (cid:1) − (cid:0) c ( K (2(1 + κ ′ ) , , p ) p − L (2(1 + κ ′ ) , , p ) p + M (2(1 + κ ′ , , p ) p (cid:1) (4.6)where c is the color matrix element of D Ae D h D d , and c is the color matrix elementof D Ae φ h φ d . Using κ = 0 . D h , D d and κ ′ = 0 . φ h , φ d , we obtain I ( p, a, b, c ) ∝ f abc ( c ( − ∞ p ) + c ( − . p . ) − (cid:18) c ( − ∞ p − π p + π p ) (cid:19) − (cid:18) c ( − . p . − . p . + 2 . p . ) (cid:19) . (4.7)We remark that the divergent terms cancel.The dressed ghost propagator can be incorporated into the ghost loop in thegluon propagator, as shown in Fig. 18, and thereby modify the infrared behavior ofthe gluon propagator in the lattice Landau gauge.The loop of quarks in the fundamental representation yields tr( t c ′ t c ′′ ) = δ c ′ ,c ′′ .2 Sadataka Furui c’c’’ ´ a’a’’ b’b’’a bh’ d’h’’d’’e’e’’ Fig. 18. The gluon propagator with a ghostloop. The dashed line represents a ghost,the thin line a gluon. The × indicatesdressing of the gluon in Cartan subalgebra. c’c’’a’a’’ b’b’’a be’h’ d’e’’h’’d’’ Fig. 19. The gluon propagator with a ghostloop and quark loop contributions. Thedashed line represents a ghost, the thin linea gluon, and the thick line a quark.c—ab 11 22 33 44 55 66 77 883 -0.25 -0.25 2.25 0.5 0.5 0.5 0.5 0.758 0.75 0.75 0.75 0 0 0 0 2.25Table IV. Color matrix elements of the gluon propagator.
The sum of the color indices becomes2 g ( a, b, c ′ , c ′′ ) = X a ′ ,a ′′ ,b ′ ,b ′′ δ c ′ ,c ′′ f aa ′ a ′′ f bb ′ b ′′ f c ′ a ′ b ′ f c ′′ a ′′ b ′′ . (4.8)This gives 4 . δ ab . The D Ae φ h φ d loop contribution, denoted by c , exists only when c ′ , c ′′ = 3 ,
8. There is no restriction to the D Ae D h D d loop contribution denoted by c , but the infrared singularity of the c term is stronger than that of the c term. §
5. The quark-gluon vertex
The quark-gluon vertex is calculated from the longitudinal photon quark cou-pling q µ Γ µ ( p, q ) = G ( q )[(1 − H ( q, p + q )) S − ( p ) − S − (1 − H ( q, p + q ))] , (5.1)where G ( q ) is the ghost dressing function and H ( q, p + q ) is the ghost-quark scat-tering kernel. In the continuum theory, the ghost-quark scattering kernel in thequark-gluon vertex is studied in Ref. Its role in the DS approach is examined inRef. and it is studied by lattice simulation in Ref.
In the quark-gluon vertex, a ghost-triangle couples with an external gluon, andtwo internal gluons couple to the quark, as shown in Fig.20.The color of the external gluon is specified by a , and the two internal gluons arespecified by b and c . When the ghost-quark scattering kernel is approximated by adressed gluon exchange, the amplitude of the quark-gluon vertex becomes M ( p, s, a, x, y ) = Z d k (2 π ) Z d q (2 π ) he color antisymmetric ghost propagator a bc a’a’’c’’c’ b’’b’d he k k + pq - kpq p + q Fig. 20. The ghost triangle. The color indices of the external gluons and internal ghosts and themomentum assignments. × δ a ′′ c ′′ k κ ) + 2 if a ′′ c ′′ d φ d ( k ) ! ( − gf a ′′ aa ′ k ) × δ c ′ b ′′ ( q − k ) κ ) + 2 if c ′ b ′′ e φ e ( q − k ) ! (cid:16) − gf c ′′ cc ′ ( q − k ) (cid:17) × δ b ′ a ′ ( k + p ) κ ) + 2 if b ′ a ′ h φ h ( k + p ) ! (cid:16) − gf b ′′ bb ′ ( k + p ) (cid:17) × ( igt c ) xz ( igt b ) zy Z (( s − q ) ) − i (/ s − / q ) + M ( s − q ) + M × ( δ µν − q µ q ν q ) Z ( q ) q × ( δ ηδ − ( q + p ) η ( q + p ) δ ( q + p ) ) Z (( q + p ) )( q + p ) . (5.2)It is important to note that the color antisymmetric ghost does not appear asan external line, but, rather, it appears in the vertex as an internal line. The gluondressing indicated by the cross can be contracted as δ c ′ c ′′ and becomes a propagatorin the Cartan subalgebra. This could effectively introduce the ghost condensateseffect, i.e. a choice of a specific color direction. §
6. Effects on the Kugo-Ojima color confinement parameter and therunning coupling
Kugo and Ojima constructed a two-point function u ab ( q ) from the Lagrangianin the Landau gauge that satisfies BRST symmetry. They claimed that if its valueat momentum zero is −
1, it is evidence of color confinement:4
Sadataka Furui ( δ µν − q µ q ν q ) u ab ( q ) = 1 V X x,y e − ip ( x − y ) h tr (cid:16) Λ a † D µ M − [ A ν , Λ b ] (cid:17) xy i ,u ab (0) = − δ ab c. (6.1)In this argument and in the proof of the Slavnov-Taylor identity regarding the ratioof the vertex renormalization factor and the wave function renormalization factor, Z Z = ˜ Z ˜ Z = Z ¯ ψψA Z ψ , the ghost propagator is assumed to be color diagonal.The replacement h c a ( x )¯ c b ( y ) i = D G ( x − y ) δ ab + f abc φ c ( x − y ) (6.2)does not affect the argument of the tree level, since the expectation value of φ ( x − y )is 0. However, when there is a ghost color mixing of the type depicted in Fig. 15,we have h c ( A ν × ¯ c ) i P I = iq ρ h ( A ρ × c )( A ν × ¯ c ) i P I . (6.3)Thus, when we define h c ¯ c i ≡ − q G ( q ) , h ( A µ × c )¯ c i P I ≡ − iq µ F ( q )and h D µ c ¯ c i = h ∂ µ c ¯ c i + h ( A µ × c )¯ c i ≡ iq µ (1 + F ( q )) 1 q G ( q ) , (6.4)we obtain h D µ c ( A µ × ¯ c ) i = h ∂ µ c ¯ c ih c ( A µ × ¯ c ) i P I + h ( A µ × c )( A ν × ¯ c ) i P I = ( δ ρµ − q µ q ρ q ) h c ( A µ × ¯ c ) i P I . (6.5)In other words, although the lattice simulation confirms G (0) = 0, there shouldbe a contribution of the ghost propagator of the type appearing in Fig. 15 between c and ¯ c that is not proportional to δ ab , and 1 + u (0) = 1 + F (0) in eq.(6.4) is notnecessarily equal to G (0) = 0.In the S -matrix theory, the ghost intermediate state cancels the intermediatestates with a non-physically polarized gluon. The longitudinal gluon with polariza-tion vector − ip µ couples with a set of diagrams of a given order in g and a givennumber of transverse gluons on mass shell, which are expressed as circles in the t’Hooft definition, and it was shown that all the contributions cancel, yielding 0, whenthe ghosts are color diagonal.Our analysis suggests that if the longitudinal gluon forms a pair of color anti-symmetric ghosts and the loop intersects with the circle, the contribution remains he color antisymmetric ghost propagator Λ in the t’ Hooft definition cancelsthe color-diagonal ghost contribution. This results in an artificial suppression of theghost propagator and the QCD running coupling.No infrared suppression occurs in α I ( q ), since the longitudinal gluon polarizedin the 4th direction does not contribute to the three-dimensional calculation.The color indices of the color antisymmetric ghost pair could be ordered if thereis a long-range background gluon field that couples to the pair. In this case, Eq.(6.5)would be satisfied and the Kugo-Ojima parameter c would become close to 1.There is no direct proof that the quark loop introduces ordering, but the dif-ference between the Binder cumulants of the color antisymmetric ghost propagatorof the quenched simulation and the unquenched simulations strongly suggeststhis effect. The temperature dependence of the parameter c could also be explainedby this mechanism. §
7. Discussion and outlook
We investigated whether the deviation of the Kugo-Ojima parameter c from 1and the infrared suppression of the effective coupling α s ( q ) in the Landau gaugecould be due to the color antisymmetric ghost propagator.In the case of the Coulomb gauge, the ghost dressing function is three dimen-sional, no dissipation of the flow occurs and the running coupling freezes.The lattice Landau gauge results show some discrepancies from the results of theDS equation with regard to the infrared exponents of the ghost propagator and thegluon propagator. We investigated the possibility that this discrepancy comes fromthe color-mixing of the ghost propagator induced by the ghost-gluon-ghost trianglediagram. A study employing the DSE is left as a future project.The color antisymmetric ghost pair couples to a gluon whose color is in theCartan subalgebra. The lattice data suggest that the gluon in the Cartan subalgebracouples with a dynamical quark loop, and this introduces a difference between thecolor antisymmetric ghost propagator of the quenched configurations and that of theunquenched configurations.We also studied the contribution of ghost triangle diagram in the quark-gluonvertex. For the estimation of the latter effect, it is necessary to perform a two-loopcalculation, and this is left to a future. Acknowledgements
The main part of this work was done at the Department of Theoretical Physics ofUniversity of Graz in August 2007. The author thanks Kai Schwenzer and ReinhardAlkofer for helpful and enlightening discussions in Graz and Hideo Nakajima for thecollaboration in the lattice simulation cited in this paper. Thanks are also due to theAustrian Academic Exchange Service for their support during the author’s stay inGraz and the Japan Society for the Promotion of Science (JSPS) for support throughthe Scientist Exchange Program that enabled the collaboration.6
Sadataka Furui
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