IInstantaneous Dynamics of QCD
Patrick Cooper , a and Daniel Zwanziger , b , c Physics Department, Duquesne University, 600 Forbes Ave, Pittsburgh, PA 15282, USA Physics Department, New York University, 4 Washington Place, New York, NY 10003, USA
Abstract.
We start from the observation that, in the confining phase of QCD, the instan-taneous color-Coulomb potential in Coulomb gauge is confining. This suggests that, inthe confining phase, the dynamics, as expressed in the set of Schwinger-Dyson equations,may be dominated by the purely instantaneous terms. We develop a calculational schemethat expresses the instantaneous dynamics in the local formulation of QCD that includesa cut-o ff at the Gribov horizon. d In sections 1 through 9 we outline the properties and derivation of the instantaneous dynamics ex-pressed in the instantaneous Schwinger-Dyson equations (ISDE). In sect. 10, we write the ISDE indiagrammatic form.
Gribov’s insight into the mechanism of confinement [1] is substantiated by the theorem [2], V coul ( R ) ≥ V wilson ( R ) (1)for R → ∞ , where the color-Coulomb potential, V coul ( R ), is the instantaneous part of the zero-zerocomponent of the gluon propagator in Coulomb gauge, g D ( R , T ) = V coul ( R ) δ ( T ) + non − instantaneous , (2)and V wilson ( R ) is the gauge-invariant potential derived from a large rectangular Wilson loop. Accord-ingly, when V wilson ( R ) rises linearly, V coul ( R ) rises linearly or super-linearly. a e-mail: [email protected] b e-mail: [email protected] c Speaker, XIIth Quark Confinement and the Hadron Spectrum, 28 August to 4 September, 2016, Thessaloniki, Greece d This talk is dedicated to the memory of Martin Schaden. It is shown in [3] that V coul ( R ) and g A are renormalization-group invariants in Coulomb gauge. a r X i v : . [ h e p - t h ] D ec Motive and method of present approach
Motivated by this theorem, we conjecture that, in the Coulomb gauge, the instantaneous dynamicsis dominant. This will be expressed in a closed system of instantaneous Schwinger-Dyson equations(ISDE).We shall work within the framework of local quantum field theory, with a local action that en-codes the cut-o ff at the Gribov horizon [4], and moreover that is BRST-invariant, thus preserving thegeometric property of a gauge theory. This action is derived by the method of Maggiore-Schaden[5] in which BRST-symmetry is spontaneously broken [6]. A conjecture is o ff ered in [7, 8] for theidentification of physical operators and the physical subspace, but this important subject will not bediscussed here.We shall derive the ISDE by extending to the local action that enforces the cut-o ff at the Gribovhorizon the method which was previously applied to the Faddeev-Popov action [9]. We shall localize the non-local Euclidean action, S = S FP + γ H − γ (cid:90) d d xd ( N − , (3)where S FP is the (local) Faddeev-Popov action in Coulomb gauge, H is the non-local horizon functiongiven by H ≡ (cid:90) d d xd d y D ab µ ( x ) D ac µ ( y )( M − ) bc ( x , y ; A ) , (4)and γ / is the Gribov mass. H cuts o ff the functional integral at the Gribov horizon, as one sees fromthe eigenfunction expansion, ( M − ) bc ( x , y ; A ) = (cid:88) n ψ bn ( x ) ψ cn ( y ) λ n ( A ) , (5)where the λ n ( A ) are the eigenvalues of the Faddeev-Popov operator M ≡ −∇ · D ( A ) in Coulombgauge. The lowest non-trivial eigenvalue λ ( A ) approaches 0 as the Gribov horizon is approachedfrom within. The Gribov mass γ / is not an independent parameter, but is fixed by the horizon condition < H > = d ( N − (cid:90) d d x . (6)It is a remarkable fact that the horizon condition and the famous Kugo-Ojima confinement condition[10, 11] are the same statement − i (cid:90) d d x < ( D µ c ) a ( x )( D µ ¯ c ) a (0) > = d ( N − . (7)This may indicate that color confinement is assured in this theory, although the precise hypotheses ofthe Kugo-Ojima theorem are not satisfied in this approach. Horizon condition and dual Meissner effect
It is also a remarkable fact that the horizon condition is equivalent to the statement that the QCDvacuum is a perfect color-electric superconductor, which is the dual Meissner e ff ect [12], G ( (cid:126) k ) = d ( (cid:126) k ) (cid:126) k = (cid:15) ( (cid:126) k ) (cid:126) k (8) d − ( (cid:126) k = = ⇐⇒ (cid:15) ( (cid:126) k = , (9)where G ( (cid:126) k ) is the ghost propagator and (cid:15) ( (cid:126) k ) is the dielectric constant. Just as the Faddeev-Popov determinant is localized by introducing ghosts,det M = (cid:90) dcd ¯ c exp (cid:32) − (cid:90) d d x ¯ cMc (cid:33) , (10)likewise, the horizon function H may be localized by introducing “auxiliary" ghosts [4],exp( − γ H ) = (cid:90) d ϕ d ¯ ϕ d ω d ¯ ω exp (cid:32) − (cid:90) d d x (cid:104) ¯ ϕ M ϕ − ¯ ω M ω + γ / D · ( ϕ − ¯ ϕ ) (cid:105)(cid:33) . (11)For reviews of this approach, see [13, 14]. The Coulomb gauge is plagued by energy divergences. For example, the integrand of the ghost loopis independent of p , (cid:90) d pd p (cid:126) p (cid:126) p − (cid:126) k ) , (12)which leads to the horrible energy divergence (cid:90) d p = ∞ . (13)We get rid of such divergences by using the first-order formalism in which they cancel manifestly[3, 15]. This relies on the fact that the Coulomb gauge is a unitary gauge. The first-order action isobtained by writing exp (cid:32) − (cid:90) d x F i / (cid:33) = (cid:90) d π exp (cid:32)(cid:90) d x ( i π i F i − π / (cid:33) . (14)Here F i = ∂ A i − D i A , the Coulomb gauge condition ∂ i A i = π i is color-electric field. It is decomposed into its transverse and longitudinal parts, π i = τ i − ∂ i λ, (15)with ∂ i τ i =
0. As a result of the equality (23), given below, the c − ¯ c ghost loop cancels the λ − A loop, (cid:90) d d + p (cid:104) D A λ ( (cid:126) p ) D A λ ( (cid:126) k + (cid:126) p ) − D c ¯ c ( (cid:126) p ) D c ¯ c ( (cid:126) k + (cid:126) p ) (cid:105) = , (16)and likewise for the pairs of auxiliary ghosts. This cancellation eliminates the unwanted energy diver-gences. = (1) Figure 1.
Schematic form of the ISDE. The shaded circle represents an instantaneous dressed propagator and theempty circle a non-instantaneous dressed propagator. All three-point vertices are tree-level.
The local action is given by S = (cid:90) d d + x ( L + L + L ) (17) L = i τ i D A i + (1 / τ + ( ∂λ ) + B ] (18) L = i ∂ i λ D i A − ∂ i ¯ cD i c + ∂ i ϕ j D i ϕ j − ∂ i ϕ j D i ϕ j − ∂ i ω j D i ω j (19) L = γ / g f abc A bj ( ϕ − ϕ ) cja , (20)and B i is the color-magnetic field. Only the first term, i τ i ∂ A i , contains a time derivative. Because A i is identically transverse, ∂ i A i = τ i ), ∂ acts on the two would-be physical degrees of freedom.The remaining terms impose constraints in the local theory. The energy divergences cancel betweenpairs of fermi and bose ghost loops, including the first pair, i ∂ i λ D i A − ∂ i ¯ cD i c . The last term, L ,mixes bose-ghost and gluon fields.In the following it will be convenient to change variables from ϕ and ¯ ϕ to U and V defined by ϕ = ( U + iV ) / √ ϕ = ( U − iV ) / √ . (21)The V -field mixes with the gluon field A , whereas the U -field does not.The propagators of the fields λ and A are related to the ghost propagator D c ¯ c by (cid:32) D λλ D λ A D A λ D A A (cid:33) ab = δ ab − iD c ¯ c ( (cid:126) k ) − iD c ¯ c ( (cid:126) k ) Γ λλ ( (cid:126) k ) D c ¯ c ( (cid:126) k ) , (22)where Γ λλ is the 2-point vertex function. The equality of the Faddeev-Popov ghost propagator and thebose-ghost propagator, Γ − A λ = D λ A = − iD c ¯ c ( (cid:126) k ) . (23)assures that the corresponding fermi and bose loops yield Faddeev-Popov determinants that cancelexactly, D ( M ) D ( M ) = . (24)
10 Instantaneous Schwinger-Dyson equations (ISDE)
The truncation scheme which we use is represented schematically in Fig. 1, with detailed diagramsgiven in Figs. 2 and 3. Propagators such as D λ A and D AV represent mixing. For the action at finite temperature, see [16]. (0) V P V P ! D (0) AA ! D (0) ⌧ ⌧ ! D (0) A A ! D (0) AV ! D (0) A ! Figure 2.
Free propagators that appear in the ISDE. V P is the part of the bose-ghost propagator that mixes withthe gluon propagator. In Coulomb gauge, the propagators in general decompose into an instantaneous part and a non-instantaneous part, D ( (cid:126) x , x ) = D I ( (cid:126) x ) δ ( x ) + D N ( (cid:126) x , x ) (25) D ( (cid:126) p , p ) = D I ( (cid:126) p ) + D N ( (cid:126) p , p ) , (26)where lim p →∞ D N ( (cid:126) p , p ) =
0. The ISDE is obtained as follows.I. Cancel loops with two instantaneous bose-ghost propagators against similar fermi-ghostloops, to get rid of energy divergences, (cid:90) d d + p (cid:104) D I boson ( (cid:126) p ) D I boson ( (cid:126) k + (cid:126) p ) − D I fermion ( (cid:126) p ) D I fermion ( (cid:126) k + (cid:126) p ) (cid:105) energy divergence = . (27)II. Neglect loops with two non-instantaneous propagators, (cid:90) d d + p (cid:2) D N ( p ) D N ( k + p ) (cid:3) → . (28)III. Keep loops with one instantaneous propagator and one non-instantaneous propagator, (cid:90) d d + p (cid:104) D N ( (cid:126) p , p ) D I ( (cid:126) k + (cid:126) p ) (cid:105) . (29)Only the equal-time part of any non-instantaneous propagator contributes to the graphs we consider, (cid:90) d d + p (cid:104) D N ( (cid:126) p , p ) D I ( (cid:126) k + (cid:126) p ) (cid:105) = (cid:90) d d p (cid:104) D ET ( (cid:126) p ) D I ( (cid:126) k + (cid:126) p ) (cid:105) , (30)where the equal-time part of the propagator is defined by (cid:90) d p D N ( (cid:126) p , p ) = D ET ( (cid:126) p ) . (31)Thus, in all graphs that we consider, the non-instantaneous propagator gets replaced by its equal-timepart. The diagrams corresponding to the ISDE are given in Fig. 3.Calculations with the ISDE will be reported elsewhere [17]. The part of the bose-ghost loop, that is caused by mixing with the gluon propagator, survives. V = V - A AV VA V = A V - V AA V VVA A = A - V VA V AV - ⌧ ⌧A A AA ⌧ ⌧ = ⌧ - A A⌧ A ⌧A A = A - A A A A = - A A A A Figure 3.
Diagrams for the relevant ISDEs. The five di ff erent types of lines are explained in Fig. 2. As usual,shaded circles represent the instantaneous, full propagator and the empty circles are non-instantaneous. We have a local, renormalizable quantum field theory with the following interesting properties: • It provides a cut-o ff at the Gribov horizon. • The Kugo-Ojima color confinement condition is satisfied. • The vacuum is a perfect dielectric. • The Maggiore-Schaden shift provides a BRST-invariant Lagangian. • BRST-symmetry is spontaneously broken, but perhaps only in the unphysical sector. • In Coulomb gauge, the color-Coulomb potential rises linearly or super-linearly when the Wilsonpotential is linearly rising. • In the ISDE, all loops consist of one instantaneous propagator and the equal-time part of a non-instantaneous propagator.
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