Condition for confinement in non-Abelian gauge theories
aa r X i v : . [ h e p - t h ] A p r Condition for confinement in non-Abeliangauge theories
Masud Chaichian and Marco Frasca Department of Physics, University of HelsinkiP.O. Box 64, 00014 Helsinki, Finland Via Erasmo Gattamelata, 300176 Rome, Italy
Abstract
We show that a criterion for confinement, based on the BRST invariance, holds infour dimensions, by solving a non-Abelian gauge theory with a set of exact solutions.The confinement condition we consider was obtained by Kugo and Ojima somedecades ago. The current understanding of gauge theories permits us to applythe techniques straightforwardly for checking the validity of this criterion. In thisway, we are able to show that the non-Abelian gauge theory is confining and thatconfinement is rooted in the BRST invariance and asymptotic freedom.
The question of why quarks are never seen as single particles is central to a deeper under-standing of the Standard Model, especially to the QCD sector, which describes the strongforce ([1] and refs. therein). In the course of years, several mechanisms have been proposedbut nobody has been able to derive this property directly from the theory. Rather, somecriteria have been devised that can grant confinement in the four dimensional theory. Forexample, Kugo and Ojima proposed a condition from BRST invariance based on chargeannihilation [2, 3]. On a similar ground, Nishijima and collaborators [4–8] obtained aconstraint on the amplitudes of unphysical states signaling confinement. These authorsshowed that colour confinement arises as a consequence of BRST invariance and asymp-totic freedom. Indeed, these approaches are deeply linked. In supersymmetric models,confinement is proven in certain conditions as a condensation of monopoles, similar toType II superconductors [9, 10]. For a comparison of different confinement theories andtheir overlapping regions, see [11]. On the other hand, the study of the propagators in theLandau gauge, initiated by Gribov [12] and further extended by Zwanziger [13], seemedto point to a confining theory with the gluon propagator running to zero as momenta goto zero and an enhanced ghost propagator running to infinity faster than the free case inthe same limit of momenta.Studies of the gluon and ghost propagators on the lattice, mostly in the Landau gauge,[14–16] and the spectrum [17, 18] proved that a mass gap appears in a non-Abelian gaugetheory without fermions, in evident contrast with the scenario devised by Gribov andZwanziger. Theoretical support for these results was presented in [19–24] providing aclosed form formula for the gluon propagator. A closed analytical formula for the gluonpropagator is pivotal to obtain the low-energy behavior of QCD in a manageable effective masud.chaichian@helsinki.fi [email protected] u function of Kugo and Ojima [3].We point out that our first aim is to consider QCD without quarks, namely to provethat a non-Abelian gauge theory with no fermions is confining in four dimensions. Inprinciple, it provides a rigorous proof that the theory is confining, besides having a massgap coming from the derived correlation functions. At this stage, one can state thatconfinement is due to the BRST invariance and the asymptotic freedom of the theory, aswell the existence of a mass gap.The paper is structured as follows: in Sec. 2 we introduce the condition for confinementthat is obtained from BRST invariance. In Sec. 3 we present the correlation functions ofa non-Abelian gauge theory without fermions, quantized by using a set of exact solutions.In Sec. 4 we show that the confinement condition is satisfied in this case. In Sec. 5 wepresent the exact β function. Finally, in Sec. 6 the conclusions are given. In this section we present the approach to confinement proposed in [4–8] and show howthis reduces to the Kugo–Ojima criterion [3]. We emphasize that our proof is for thetheory without fermions.The Lagrangian of QCD is given by L = L inv + L gf + L F P , (1)where L inv denotes the classical gauge-invariant part, L gf the gauge-fixing terms and L F P the Faddeev–Popov (FP) ghost term characteristic of non-Abelian gauge theories: L inv = − F µν · F µν + ¯ ψ ( γ µ D µ − m ) ψ , L gf = ∂ µ B · A µ + 12 αB · B , L F P = i∂ µ ¯ c · D µ c , (2)in the usual notation, with the convention (1 , − , − , −
1) for the metric signature. Wedenote by α the gauge parameter and D µ represents the covariant derivative whose explicitforms are given by D µ ψ = ( ∂ µ − igT · A µ ) ψ , µ c = ∂ µ c + gA µ × c . (3)In general, the BRST transformations of a generic field φ are given in terms of theBRST charges Q B and ¯ Q B by [2] δ φ = i [ Q B , φ ] ∓ , ¯ δ φ = i [ ¯ Q B , φ ] ∓ , (4) Q B = ¯ Q B = Q B ¯ Q B + ¯ Q B Q B = 0 . (5)We choose the − (+) sign in (5) when φ is even (odd) in the ghost fields c and ¯ c , whichare anticommuting scalar fields.The BRST transformations of the gauge field A µ and the quark field ψ are defined byreplacing the infinitesimal gauge function by the FP ghost field c or ¯ c , in their respectiveinfinitesimal gauge transformations: δA µ = D µ c , δψ = ig ( c · T ) ψ , ¯ δA µ = D µ ¯ c , ¯ δψ = ig (¯ c · T ) ψ . (6)Requiring to have for the auxiliary fields B , c and ¯ cδ L = ¯ δ L = 0 , (7)we find δ B = 0 , δ ¯ c = iB , δ c = − g ( c × c ) , ¯ δ ¯ B = 0 , ¯ δ c = i ¯ B , ¯ δ ¯ c = − g (¯ c × ¯ c ) , (8)where ¯ B is defined by B + ¯ B − ig ( c × ¯ c ) = 0 . (9)On the other hand, the conserved current, from Noether theorem (up to a total diver-gence) is defined as j µ = X { Φ } ∂ L∂ ( ∂ µ Φ) δ Φ = B a ( D µ c ) a − ∂ µ B a c a + i
12 g f abc ∂ µ ¯ c a c b c c , (10)with { Φ } the set of all fields present in the Lagrangian, from which we get the correspond-ing charge Q B : Q B = Z d x (cid:18) B a ( D c ) a − ˙ B a c a + i
12 g f abc ˙¯ c a c b c c (cid:19) . (11)So, the Lagrangian with the gauge-fixing term is then L gf + L F P = δ ( − i∂ µ ¯ c · A µ − i α ¯ c · B ) (12)and evidently we have δ L inv = 0 . (13)Namely, L inv is closed and L gf + L F P is exact, and δ L = 0 . (14)3his Lagrangian yields the equations of motion D µab F bµν + j bν = iδ ¯ δA bν , (15)where the contribution on the right-hand side comes from the auxiliary fields in the La-grangian. At the tree level, these represent massless particles. Besides, the B field doesnot propagate. This means that the current due to these fields should not yield contri-butions to the physical spectrum of the theory. Also, since ∂ ν ( iδ ¯ δA ν ) = 0, this current isconserved. In order to evaluate it, we need to study the behavior of the amplitude h iδ ¯ δA aµ ( x ) , A bν ( y ) i . (16)Then, according to current conservation, the most general form of its Fourier transformcan be expressed as − δ ab ( δ µν − p µ p ν p + iǫ ) R dm σ ( m ) p − m + iǫ + Cδ ab p µ p ν p + iǫ . (17)As we see, we cannot exclude massless excitations from the spectrum at this stage. Thiswill imply no confinement, as we would get massless gluons. But if the theory is confining,massless states cannot be physical states. Then, ∂ µ h iδ ¯ δA aµ ( x ) , A bν ( y ) i = iδ ab C∂ ν δ ( x − y ) , (18)that can be cast into the form of an equal-time commutator: δ ( x − y ) h | (cid:2) iδ ¯ δA a ( x ) , A bj ( y ) (cid:3) | i = iδ ab C∂ j δ ( x − y ) , ( j = 1 , , . (19)Based on the preceding considerations, we have seen that the confinement condition isrealized with no massless excitations appearing in the physical spectrum and the currentarising from the auxiliary fields has no effect on the amplitudes of the processes.We can link this conclusion with the Kugo–Ojima criterion, which is also a no-masslesspole criterion. This can be seen in the following way. Using the Kugo–Ojima formalism,one has, δ ¯ δA aµ = −{ Q B , { ¯ Q B , A aµ }} . (20)Using the fact that h | Q B = Q B | i = ¯ Q B | i = h | ¯ Q B = 0, it is clear that h iδ ¯ δA aµ ( x ) , A bν ( y ) i = h i ¯ δA aµ ( x ) , δA bν ( y ) i = i h D µ ¯ c a ( x ) , D ν c b ( y ) i . (21)For this correlator, Kugo and Ojima showed [3] that Z d d xe ipx h D µ ¯ c a ( x ) , D ν c b ( y ) i = δ ab (cid:18) δ µν − p µ p ν p − iǫ (cid:19) u ( p ) − δ ab p µ p ν p − iǫ , (22)and the no-pole condition yields here1 + u ( p = 0) = 0 , (23)which is the Kugo–Ojima condition for confinement.Thus, our aim will be to derive the u ( p ) function and evaluate it for p = 0.4t this stage we note that a possible mapping exists between the Nishijima conditionand the Kugo–Ojima condition when the infrared limit p → − δ ab (cid:18) δ µν − p µ p ν p + iǫ (cid:19) Z dm σ ( m ) p − m + iǫ + Cδ ab p µ p ν p + iǫ p → → δ ab (cid:18) δ µν − p µ p ν p + iǫ (cid:19) Z dm σ ( m ) m + Cδ ab p µ p ν p + iǫ . (24)On the other hand, the no-massless pole condition must be taken into account as C − Z dm σ ( m ) m = 0 . (25)This is analogous to the Kugo–Ojima condition 1 + u ( p = 0) = 0 in the infrared limit. The correlation functions for a pure non-Abelian gauge theory, without matter fields,have been computed in [24], where the Dyson–Schwinger equations were solved with theapproach devised in [37]. In these computations, the Dyson–Schwinger equations aresolved with no truncation involved but computations are performed to obtain at least thetwo-point function exactly. For the sake of completeness, we give a summary of them inthe appendix. Below, we present the solutions.We note that G a µ ( x ) can be written as in (49) G a µ ( x ) = η aµ φ ( x ) , (26)where φ ( x ) = µ (cid:16) Ng (cid:17) · sn( px, − η aµ constants and p = µ p N g /
2. Thus, thegiven set of Dyson–Schwinger equations can be solved exactly. For the two-point functionin the Landau gauge we can write G abµν ( x ) = δ ab (cid:18) g µν − p µ p ν p (cid:19) ∆( x − y ) , (27)provided that ∂ ∆( x − y ) + 3 N g φ ( x )∆( x − y ) = δ ( x − y ) ,P a ( x ) = 0 ,∂ P am ( x − y ) = δ am δ ( x − y ) ,K am κ ( x − y ) = 0 (28)and G ac νρ (0) = 0, G bcm µνκ (0 , x − y ) = 0, G µbdem µνκ (0 , , x − y ), K bcm κ (0 , x − y ) = 0, a behavior ofthe 3- and 4-point functions in agreement with lattice results [38, 39]. This shows that theset of Schwinger–Dyson equations for Yang–Mills theory can be exactly solved, at leastto the level of two-point functions.The propagator is given by [24]∆( p ) = π K ( − ∞ X n =0 e − ( n + ) π e − (2 n +1) π (2 n + 1) p − m n + iǫ , (29)5ith K ( −
1) being an elliptic integral that yields the numerical constant 1 . . . . and given the mass spectrum m n = (2 n + 1) π K ( − (cid:18) N g (cid:19) σ , (30)that is indeed the spectrum of the theory. Here σ is an integration constant having thedimension of mass. It is easy to see how this propagator recovers asymptotic freedom[4–8]. In the high-energy limit, we make the momenta run to infinity. This yields∆( p ) p →∞ = π K ( − ∞ X n =0 e − ( n + ) π e − (2 n +1) π (2 n + 1) p − = p − , (31)as the sum adds to 1. We just note that this propagator is a leading order approximationwhen one can neglect the corrections due to mass renormalization to the spectrum of thetheory.The theory has no massless excitation and thus, already at this stage, we can concludethat the approach devised in [2–8] appears sound. We will complete the proof in the nextsection. Now, we are in a position to evaluate the confinement condition by computing the u ( p )function and evaluating it at 0. For the sake of simplicity we limit our analysis to SU ( N )and the numerical analysis to SU (3). This extends the analysis, performed on the lattice,presented in [40, 41]. We note that, from eq. (22), Z d xe ipx h D µ ¯ c a ( x ) , D ν c b (0) i = Z d xe ipx h (cid:0) ∂ µ − igT c A cµ ( x ) (cid:1) ¯ c a ( x ) , (cid:0) ∂ ν − igT d A dν (0) (cid:1) c b (0) i . (32)This yields Z d xe ipx h D µ ¯ c a ( x ) , D ν c b (0) i = − δ ab p µ p ν p − g Z d xe ipx h T c A cµ ( x )¯ c a ( x ) , T d A dν (0) c b (0) i , (33)where it has been taken into account that h A aµ ( x ) i = 0 and we used the free ghost propa-gator. Now, as shown in the preceding section, the ghost field decouples from the gluonfield and the above equation can be written as follows: Z d xe ipx h D µ ¯ c a ( x ) , D ν c b ( y ) i = − δ ab p µ p ν k (34) − ( N − N g δ ab (cid:18) δ µν − p µ p ν p (cid:19) Z d p ′ (2 π ) | p − p ′ | ∆( p ′ ) , where we identify u ( p ) = − ( N − N g Z d p ′ (2 π ) | p − p ′ | ∆( p ′ ) . (35)Then, we have to evaluate the integral u (0) = − ( N − N g Z d p (2 π ) p ∞ X n =0 B n p + m n − ( N − N g ∞ X n =0 B n m n Z d p (2 π ) (cid:18) p − p + m n (cid:19) , (36)with B n = π K ( − e − ( n + 12 ) π e − (2 n +1) π (2 n + 1) . This integral is divergent and needs to be renor-malized. We can evaluate it by dimensional regularization. We use I d = − Z d d p (2 π ) d (cid:18) p − p + m n (cid:19) = ( m n ) d/ − (4 π ) d Γ(1 − d/ , (37)then set ǫ = 4 − d and expand. This yields I ǫ = m n (4 π ) (cid:18) πµ m n (cid:19) ǫ Γ (cid:16) ǫ − (cid:17) = m n (4 π ) (cid:20) − ǫ − γ + ln (cid:18) m n πµ (cid:19) + O ( ǫ ) (cid:21) , (38)where we have reintroduced the scale factor µ arising by going to d dimensions and γ isthe Euler–Mascheroni constant. From this we can extract the finite part, that is I ′ = m n (4 π ) (cid:20) − γ + ln (cid:18) m n πµ (cid:19)(cid:21) , (39)which is explicitly dependent on the cut-off µ . Then, u (0) = ( N − N α s π " − γ + ∞ X n =0 B n ln (cid:18) m n πµ (cid:19) , (40)where use has been made of the identity P ∞ n =0 B n = 1 and α s = g / π .One can see that, if for the Kugo–Ojima function holds u (0) = − α s ( µ ) given by the following equation( N − N α s ( µ )4 π " − γ + ∞ X n =0 B n ln (cid:18) m n πµ (cid:19) = − . (41)This equation, consistently with our approach, is exact. Indeed, in the high-energy limit,we get the asymptotic freedom limit for SU (3) as α s ( µ ) = 3 π (cid:16) µ σ (cid:17) , (42)where use has been made of eq. (30) for the spectrum of the theory and we have introducedthe string tension σ = (0 .
44 MeV) obtained from experimental data that we keep herefixed. The square root of the string tension represents the gap into the spectrum of thetheory and, when one accounts for quarks, characterizes the glueball spectrum. Thisresult should compare with the asymptotic freedom limit given by [42] α s ( µ ) = 12 π (33 − n f ) ln (cid:16) µ σ (cid:17) , (43)with n f being the number of flavours of quarks that here we take to be 0 and we assume σ also here for the integration constant coming from the equation of the renormalizationgroup. This is just for reasons of numerical comparison but we note that it is physically7eaningful anyway. In this way, one gets the ratio between eqs.(43) and (42) equal to96 / ≈ .
97, very near 1, but we should remember that the former is a perturbativeresult in an asymptotic series.We can also compare with the experimental value of α s at M Z , the mass of the Zvector boson. From [42] one has the world average value α s ( M Z ) = 0 . ± . α s ( M Z ) = 0 . ± . √ σ . The agreement is within an error of about 7%. We have not accounted contributionof quarks in this computation. We just note that the analogous limit from perturbativeQCD has a higher error. Also, the perturbative result is very near to this value beingabout 0 . ± . β function So far, we have evaluated the running coupling, given by eq. (41), fixing the gap in thespectrum of the theory, given by eq. (30). This requires solving the eq. (41) by iteration.Notwithstanding, this yields excellent results for asymptotic freedom; we need to see ifthis agreement will extend for all the energy range. This can be done by deriving the β function from eq. (41) without any approximation. We do it by noting that the spectrumdepends on α s and, normally, we set for the string tension (the gap in the spectrum) σ = σ p πN α s . (44)The idea is to use σ as an energy scale for the ultraviolet cut-off µ we introduced in thepreceding section after renormalization of the u function. Given this, we can derive the β function from eq. (41) in a straightforward manner. This gives the renormalization groupequation dα s dl = − β α s − β α s , (45)with β = ( N − / πN . We have set l = ln( µ /σ ) as an independent variable. Thisresult should compare with the exact β function obtained for SUSY Yang–Mills theory[43, 44] dα s dl = − N π α s − π N α s , (46)and the Ryttov and Sannino hypothesis for Yang–Mills theory [45] dα s dl = − N π α s − π N α s . (47)It should be pointed out that the Ryttov–Sannino hypothesis, also being inspired by theSUSY result of eq. (46), is founded on the perturbative results of asymptotic freedom asgiven in [42].It is interesting to note that, in the formal limit α s → ∞ , SUSY Yang–Mills theorygives for the β function 3 α s / α s in the same limit. However,Ryttov and Sannino would get about 0 . α s in the same limit. Using the approaches developed in [2, 3] and [4–8], we were able to give a rigorous proofof confinement for non-Abelian gauge theories in four dimensions as a consequence of8he BRST invariance and the asymptotic freedom. Our results are based on the exactsolutions obtained in [24] for the correlation functions. These are obtained by solvingthe set of Schwinger–Dyson equations exactly, without truncation, to obtain the exacttwo-point function. As a by-product, we get an exact equation for the running couplingof the theory.We hope to extend this proof to the case of QCD with fermions in a future communi-cation.
Acknowledgements
We are deeply indebted to Taichiro Kugo for several enlightening discussions and com-ments on the manuscript, which have improved the results and the conclusions of the workto a significant degree. Our thanks also go to Carl Bender for enlightening discussions andto David Dudal and Silvio Sorella for pointing to us several originally weak points, wherethe exact solutions were confronted with the lattice simulations and as well for usefulsuggestions. Last but not least, we are grateful to Marco Ruggieri for useful discussionsabout the various questions discussed in the work.
Appendix: Dyson–Schwinger equations
The correlation functions are obtained when a given exact solution is known for the one-point function i.e., one has to solve exactly the equations ∂ µ ∂ µ A aν − α ∂ ν ( ∂ µ A aµ )+ gf abc A bµ ( ∂ µ A cν − ∂ ν A cµ )+ gf abc ∂ µ ( A bµ A cν )+ g f abc f cde A bµ A dµ A eν = 0 . (48)In the Landau gauge ( α → A aν ( x ) = η aν (cid:18) N g (cid:19) µ · sn( px, − , (49)with sn( px, −
1) the Jacobi snoidal elliptic function and η aµ being a set of constants tobe determined depending on the problem at hand (e.g., for SU (2) one can take η = η = η = 1, all other components being zero) and µ an integration constant with thedimension of an energy. This holds provided the following dispersion relation holds p = r N g µ . (50)Solutions given in eq.(49) appear as massive solution, due to the dispersion relation (50),even if we started from a massless theory.Then, if we use these solutions as one-point function of the set of Schwinger–Dysonequations for a non-Abelian gauge theory without fermions, given by [24], we are able tocompute the two-point functions exactly, without any approximation or truncation. Weuse the approach devised in [37]. Indeed, to get the Schwinger–Dyson equations one hasto start from the quantum equations of motion that have the form ∂ µ ∂ µ A aν + gf abc A bµ ( ∂ µ A cν − ∂ ν A cµ ) + gf abc ∂ µ ( A bµ A cν ) + g f abc f cde A bµ A dµ A eν = gf abc ∂ ν (¯ c b c c ) + j aν ,∂ µ ∂ µ c a + gf abc ∂ µ ( A bµ c c ) = ε a . (51)9e fix the gauge to the Landau gauge, α →
0, and c, ¯ c are the ghost fields. Averagingon the vacuum state and dividing by the partition function Z Y M [ j, ¯ ε, ε ], one has ∂ G ( j ) a ν ( x ) + gf abc ( h A bµ ∂ µ A cν i − h A bµ ∂ ν A cµ i ) Z − Y M [ j, ¯ ε, ǫ ] + gf abc ∂ µ h A bµ A cν i Z − Y M [ j, ¯ ε, ε ]+ g f abc f cde h A bµ A dµ A eν i Z − Y M [ j, ¯ ε, ε ] = gf abc h ∂ ν (¯ c b c c ) i Z − Y M [ j, ¯ ε, ε ] + j aν ,∂ P ( ε ) a ( x ) + gf abc ∂ µ h A bµ c c i Z − Y M [ j, ¯ ε, ε ] = ε a . (52)The one-point function is given by G ( j ) a ν ( x ) Z Y M [ j, ¯ ε, ǫ ] = h A aν ( x ) i ,P ( ε ) a ( x ) Z Y M [ j, ¯ ε, ǫ ] = h c a ( x ) i . (53)Deriving once with respect to currents, at the same point because of the averages on thevacuum (see [37]), one has G ( j ) ab νκ ( x, x ) Z Y M [ j, ¯ ε, ǫ ] + G ( j ) a ν ( x ) G ( j ) b κ ( x ) Z Y M [ j, ¯ ε, ǫ ] = h A aν ( x ) A bκ ( x ) i ,P ( ε ) ab ( x, x ) Z Y M [ j, ¯ ε, ǫ ] + ¯ P ( ε ) a ( x ) P ( ε ) b ( x ) Z Y M [ j, ¯ ε, ǫ ] = h ¯ c b ( x ) c a ( x ) i ,∂ µ G ( j ) ab νκ ( x, x ) Z Y M [ j, ¯ ε, ǫ ] + ∂ µ G ( j ) a ν ( x ) G ( j ) b κ ( x ) Z Y M [ j, ¯ ε, ǫ ] = h ∂ µ A aν ( x ) A bκ ( x ) i ,K ( ε,j ) ab ν ( x, x ) Z Y M [ j, ¯ ε, ǫ ] + P ( ε ) a ( x ) G ( j ) b ν ( x ) Z Y M [ j, ¯ ε, ǫ ] = h c a ( x ) A bν ( x ) i , (54)and twice G ( j ) abc νκρ ( x, x, x ) Z Y M [ j, ¯ ε, ǫ ] + G ( j ) ab νκ ( x, x ) G ( j ) c ρ ( x ) Z Y M [ j, ¯ ε, ǫ ] + G ( j ) ac νρ ( x, x ) G ( j ) b κ ( x ) Z Y M [ j, ¯ ε, ǫ ] + G ( j ) a ν ( x ) G ( j ) bc κρ ( x, x ) Z Y M [ j, ¯ ε, ǫ ] + G ( j ) a ν ( x ) G ( j ) b κ ( x ) G ( j ) c ρ ( x ) Z Y M [ j, ¯ ε, ǫ ] = h A aν ( x ) A bκ ( x ) A cρ ( x ) i . (55)These give us the first set of Schwinger–Dyson equations as ∂ G ( j ) a ν ( x ) + gf abc ( ∂ µ G ( j ) bc µν ( x, x ) + ∂ µ G ( j ) b µ ( x ) G ( j ) c ν ( x ) − ∂ ν G ( j ) µbc µ ( x, x ) − ∂ ν G ( j ) b µ ( x ) G ( j ) µc ( x ))+ gf abc ∂ µ G ( j ) bc µν ( x, x ) + gf abc ∂ µ ( G ( j ) b µ ( x ) G ( j ) c ν ( x ))+ g f abc f cde ( G ( j ) µbde µν ( x, x, x ) + G ( j ) bd µν ( x, x ) G ( j ) µe ( x )+ G ( j ) eb νρ ( x, x ) G ( j ) ρd ( x ) + G ( j ) de µν ( x, x ) G ( j ) µb ( x ) + G ( j ) µb ( x ) G ( j ) d µ ( x ) G ( j ) e ν ( x )) = gf abc ( ∂ ν P ( ε ) bc ( x, x ) + ∂ ν ( ¯ P ( ε ) b ( x ) P ( ε ) c ( x ))) + j aν ,∂ P ( ε ) a ( x ) + gf abc ∂ µ ( K ( ε,j ) bc µ ( x, x ) + P ( ε ) b ( x ) G ( j ) c µ ( x )) = ε a . (56)By setting the currents to zero and noticing that, by translation invariance, one has G ( x, x ) = G ( x − x ) = G (0), G ( x, x, x ) = G (0 ,
0) and K ( x, x ) = K (0), we get ∂ G a ν ( x ) + gf abc ( ∂ µ G bc µν (0) + ∂ µ G b µ ( x ) G c ν ( x ) − ∂ ν G νbc µ (0) − ∂ ν G b µ ( x ) G µc ( x ))+ gf abc ∂ µ G bc µν (0) + gf abc ∂ µ ( G b µ ( x ) G c ν ( x ))+ g f abc f cde ( G µbde µν (0 ,
0) + G bd µν (0) G µe ( x )+ G eb νρ (0) G ρd ( x ) + G de µν (0) G µb ( x ) + G µb ( x ) G d µ ( x ) G e ν ( x )) = gf abc ( ∂ ν P bc (0) + ∂ ν ( ¯ P b ( x ) P c ( x ))) ,∂ P a ( x ) + gf abc ∂ µ ( K bc µ (0) + P b ( x ) G c µ ( x )) = 0 . (57)10his set of Schwinger–Dyson equations can be solved exactly in the Landau gauge withthe aforementioned exact solutions. This is so by noting that the contributions comingfrom G ab µν (0), P ab (0), G µbde µν (0 ,
0) and K bc µ (0) are zero in this case due to the fact thatthey give a symmetric group contribution against the antisymmetric structure constantsof the group itself. Then, one gets that the ghost one-point function decouples and canbe assumed to be a constant and does not contribute to the gluon one-point function.The Schwinger–Dyson equation for the two-point functions can be obtained by furtherderiving eq. (56). One has ∂ G ( j ) am νκ ( x − y ) + gf abc ( ∂ µ G ( j ) bcm µνκ ( x, x, y ) + ∂ µ G ( j ) bm µκ ( x − y ) G ( j ) c ν ( x ) + ∂ µ G ( j ) b µ ( x ) G ( j ) cm νκ ( x − y ) − ∂ ν G ( j ) µbcm µκ ( x, x, y ) − ∂ ν G ( j ) bm µκ ( x − y ) G ( j ) µc ( x )) − ∂ ν G ( j ) b µ ( x ) G ( j ) µcm κ ( x − y ))+ gf abc ∂ µ G ( j ) bcm µνκ ( x, x, y ) + gf abc ∂ µ ( G ( j ) bm µκ ( x − y ) G ( j ) c ν ( x )) + gf abc ∂ µ ( G ( j ) b µ ( x ) G ( j ) cm νκ ( x − y ))+ g f abc f cde ( G ( j ) µbdem µνκ ( x, x, x, y ) + G ( j ) bdm µνκ ( x, x, y ) G ( j ) µe ( x ) + G ( j ) bd µν ( x, x ) G ( j ) µem κ ( x − y )+ G ( j ) acm νρκ ( x, x, y ) G ( j ) ρb ( x ) + G ( j ) eb νρ ( x, x ) G ( j ) ρdm κ ( x − y )+ G ( j ) de νρ ( x, x ) G ( j ) ρbm κ ( x − y ) + G ( j ) µb ( x ) G ( j ) dem µνκ ( x, x, y ) + G ( j ) µbm κ ( x − y ) G ( j ) d µ ( x ) G ( j ) e ν ( x ) + G ( j ) µb ( x ) G ( j ) dm µκ ( x − y ) G ( j ) e ν ( x ) + G ( j ) µb ( x ) G ( j ) d µ ( x ) G ( j ) em νκ ( x − y ))= gf abc ( ∂ ν K ( jε ) bcm κ ( x, x, y ) + ∂ ν ( ¯ P ( ε ) b ( x ) K ( jε ) cm κ ( x, y )))+ ∂ ν ( ¯ K ( jε ) bm κ ( x, y ) P ( ε ) c ( x ))) + δ am g νκ δ ( x − y ) ,∂ P ( ε ) am ( x − y ) + gf abc ∂ µ ( K ( ε,j ) bcm µ ( x, x, y ) + P ( ε ) bm ( x − y ) G ( j ) c µ ( x ) + P ( ε ) b ( x ) K ( jε ) cm µ ( x − y )) = δ am δ ( x − y ) ,∂ K ( jε ) am κ ( x − y ) + gf abc ∂ µ ( L ( ε,j ) bcm µκ ( x, x, y ) + K ( jε ) bm κ ( x − y ) G ( j ) c µ ( x ) + P ( ε ) b ( x ) G ( j ) cm µκ ( x − y )) = 0 . (58)By setting currents to zero and using translation invariance, the above mentioned relationsyield ∂ G am νκ ( x − y ) + gf abc ( ∂ µ G bcm µνκ (0 , x − y ) + ∂ µ G bm µκ ( x − y ) G c ν ( x ) + ∂ µ G b µ ( x ) G cm νκ ( x − y ) − ∂ ν G µbcm µκ (0 , x − y ) − ∂ ν G bm µκ ( x − y ) G µc ( x )) − ∂ ν G b µ ( x ) G µcm κ ( x − y ))+ gf abc ∂ µ G bcm µνκ (0 , x − y ) + gf abc ∂ µ ( G bm µκ ( x − y ) G c ν ( x )) + gf abc ∂ µ ( G b µ ( x ) G cm νκ ( x − y ))+ g f abc f cde ( G µbdem µνκ (0 , , x − y ) + G bdm µνκ (0 , x − y ) G µe ( x ) + G bd µν (0) G µem κ ( x − y )+ G acm νρκ (0 , x − y ) G ρb ( x ) + G eb νρ (0) G ρdm κ ( x − y ) + G de νρ (0) G ρbm κ ( x − y ) + G µb ( x ) G dem µνκ (0 , x − y ) + G µbm κ ( x − y ) G d µ ( x ) G e ν ( x ) + G µb ( x ) G dm µκ ( x − y ) G e ν ( x ) + G µb ( x ) G d µ ( x ) G em νκ ( x − y ))= gf abc ( ∂ ν K bcm κ (0 , x − y ) + ∂ ν ( ¯ P b ( x ) K cm κ ( x − y ))) + ∂ ν ( ¯ K bm κ ( x − y ) P c ( x ))) + δ am g νκ δ ( x − y ) ∂ P am ( x − y ) + gf abc ∂ µ ( K bcm µ (0 , x − y ) + P bm ( x − y ) G c µ ( x ) + P b ( x ) K cm µ ( x − y )) = δ am δ ( x − y ) ,∂ K am κ ( x − y ) + gf abc ∂ µ ( L bcm µκ (0 , x − y ) + K bm κ ( x − y ) G c µ ( x ) + P b ( x ) G cm µκ ( x − y )) = 0 . 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