Variational and Dyson--Schwinger Equations of Hamiltonian Quantum Chromodynamics

The variational Hamiltonian approach to Quantum Chromodynamics in Coulomb gauge is investigated within the framework of the canonical recursive Dyson--Schwinger equations. The dressing of the quark propagator arising from the variationally determined non-perturbative kernels is expanded and renormalized at one-loop order, yielding a chiral condensate compatible with the observations.


II. QCD IN COULOMB GAUGE
In Coulomb gauge the QCD Hamiltonian reads [32] where Π a i = −iδ/δA a i is the canonical momentum, B a i is the chromomagnetic field, ψ and ψ † are the fermion field operators, α i and β are the usual Dirac matrices, m is the bare current quark mass, and A = A a t a are the (transverse) gauge fields with t a being the hermitian generators of the su(N c ) algebra. The last term in Eq. (1) is the so-called Coulomb term: it describes the interaction of the colour charge density through the Coulomb kernel where is the Faddeev-Popov operator of Coulomb gauge with f acb being the structure constants of the su(N c ) algebra. Finally, J A = Det G −1 A is the Faddeev-Popov determinant of Coulomb gauge.
In coordinate space we employ a notation where a numerical index stands collectively for the spatial coordinate as well as for the colour and Lorentz indices. A repeated numerical index like in Eq. (4) implies integration over the spatial coordinate and summation over the discrete indices. Matrix elements of an operator O between physical states Φ and Ψ are given by the functional integral where µ = ξ † (1) S 0 (1, 2) ξ(2) = d 3 p (2π) 3 ξ † (p) S 0 (p) ξ(p), is the integration measure of the fermionic coherent states, which involves the bare quark propagator For the vacuum wave functional of QCD we take the ansatz where S A and S f define, respectively, the wave functionals of pure Yang-Mills theory and of the quarks interacting with the gluons. We choose the latter in the form where K 0 and K are variational kernels, whose form will be specified in more detail later.
Once the functional derivatives in Eq. (7) are taken, expectation values of operators boil down to quantum averages of functionals of the fields with an "action" This equivalence between expectation values in the Hamiltonian approach and quantum averages in the functional integral formulation of a Euclidean field theory in d = 3 dimensions is the basis for the Dyson-Schwinger approach [30,31] employed in this work. With the help of familiar Dyson-Schwinger techniques the various one-particle irreducible equal-time Green functions of the Hamiltonian approach can be related to the kernels occurring in the "action" Eq. (11), i.e. in the vacuum wave functional Eq. (9), by means of an infinite tower of integral equations. These are named canonical recursive Dyson-Schwinger equations (CRDSEs) in order to make clear that, while they look like standard DSEs, their physical content is somewhat different. (The bare n-point vertices are not given by the action of the theory but by variational kernels of the vacuum wave functionals.) Notice that the variational kernels K 0 and K i in Eq. (10) enter the action Eq. (11) (and therefore the CRDSEs) only in the combinationsγ andΓ In the following we will refer toγ as the biquark kernel, and toΓ 0 as the bare quark-gluon vertex. 1 The choice of the variational kernels in Eq. (10) is subject to a restriction: the vacuum wave functional must be invariant under global colour rotations. These are generated by the total charge operator i.e. the spatial integral of the colour charge density Eq. (2). Invariance under global colour rotations generated by Q a implies that the wave functional Eq. (9) (or, equivalently, the quantities S A and S f occurring in its exponent) must be annihilated by Q a , which leads to the condition In order to satisfy this condition the variational kernels should obey the colour structure K 0 ∼ 1 and K a i ∼ t a . Furthermore, since the Λ ± are orthogonal projectors [see Eq. (5)] it is obvious from Eq. (12) that the variational kernel K 0 must possess non-trivial Dirac structures.
In principle, K 0 could have the general form with complex scalar functions s 1 , s 2 , s 3 . The resulting biquark kernel Eq. (12) becomes in momentum spacē As one observes, the complex kernels s 1 , s 2 , and s 3 enter the biquark kernelγ (and therefore all vacuum expectation values of observables as well as the CRDSEs) never individually but only in the combination It is hence sufficient to consider only one of them; moreover, in the chiral limit m = 0 the scalar kernel s 2 drops out. Therefore, the general ansatz Eq. (14) is not necessary. Instead the relevant physics can be captured by the much simpler choice which leads to the biquark kernel [Eq. (12)] For the vector kernel we choose the ansatz [10] K mn,a where V , and W are variational kernels: For simplicity we write only their dependence on the quark-anti-quark momenta, as momentum conservation implicitly fixes the gluon momentum. Note that the vectorial character of the quark-gluon coupling is entirely given by the Dirac matrix α i , i.e. the variational kernels V and W are scalar functions which may depend only on p 2 , q 2 , and p · q, implying e.g. V (−p, −q) = V (p, q). The bare quark-gluon vertex Eq. (13) becomes with Eq. (17) in the chiral limit When both vector kernels are omitted, V (p, q) = 0 = W (p, q), the wave functional Eq. (9) reduces to the BCS-type wave functional used in Refs. [22,25,27], while keeping only V corresponds to the choice of Refs. [28]. The above ansatz for the fermionic wave functional defined by Eqs. (9), (10), (15), and (17) was also chosen in Refs. [10,29], where the QCD variational principle was formulated in the ordinary operator language of second quantization, avoiding the introduction of Grassmann fields. As shown in Ref. [29] this ansatz has the advantage that all UV divergences cancel in the quark gap equation.

B. Quark Propagator and Quark-Gluon Vertex CRDSEs
As shown in Refs. [30,31] the formal equivalence between expectation values in the Hamiltonian approach and quantum averages of a Euclidean field theory can be used to write down DSE-like equations, referred to as CRDSEs to express the n-point functions by the variational kernels of the vacuum wave functional. The CRDSE for the fermion propagator where is the bare fermion propagator, is the gluon propagator, andΓ is the full quark-gluon vertex defined by The latter also obeys a CRDSE, which is represented diagrammatically together with Eq. (19) in Fig. 1. The explicit form is not relevant for the present work but the first term on the right-hand side is given indeed byΓ 0 [Eq. (18)], thus justifying its interpretation as bare quark-gluon vertex. Equation (19) may be conveniently written in momentum space: with the explicit form Eq. (16) of the biquark kernel we obtain For the inverse quark propagator, which we assume to be colour diagonal, we must consider in principle the following Dirac structure which can be inverted to give From the CRDSE (22) we obtain the following system of coupled equations for the dressing functions Eq. (23) of the quark propagator where is the gluon propagator Eq. (20), conveniently parametrized in terms of the quasi-gluon energy Ω(p). At this point it should be mentioned that the fermion propagator Q is not the physical quark propagator, which in the Hamiltonian approach is defined by The commutator arises from the equal-time limit of the time-ordered operator product in the full time-dependent theory. The quark propagator S and the propagator Q are related by with S 0 (p) being the free quark propagator, Eq. (8). As long as no confusion is possible we will keep referring indiscriminately to both S(p) and Q(p) as quark propagator.

C. The QCD Vacuum Energy Density
The vacuum expectation value of the QCD Hamiltonian has been evaluated in Ref. [31], to which we refer the reader for the details of the calculation; here we will merely quote the relevant contributions to the energy density e ≡ H /(V · N c ) in momentum space. The Dirac Hamiltonian [second line in Eq. (1)] yields where is the quadratic Casimir invariant of the fundamental representation of the su(N c ) algebra. The fermionic contribution to the kinetic energy of the gluons [first term on the right-hand side of Eq. (1)] is given by For simplicity, in Eqs. (27) and (28) we have omitted the dependence of the vertex functions on the gluon momentum, which follows from the fermionic momenta kept in the above equations by momentum conservation. Furthermore, we have assumed that the propagators are colour diagonal and that the colour structure of the full quark-gluon vertex is given by the generator t a as for the bare vertex. Finally, the Coulomb interaction of the fermionic charges reads Here, F (p) is the expectation value of the Coulomb kernel Eq. (3), which in the following calculations will be approximated by the simple form [9] with σ C being the Coulomb string tension. Since the expectation value e (0) D of the single-particle Hamiltonian [first term in Eq. (27)] and the Coulomb interaction Eq. (29) do not depend on the full quark-gluon vertex, the Dirac traces can be worked out explicitly, yielding respectively and where we have introduced the abbreviation for the denominator of the quark propagator Eq. (24).

III. INFRARED BEHAVIOUR OF THE DRESSING FUNCTIONS
Before we proceed to derive the equations of motion of our variational approach by minimizing the energy density with respect to the variational kernels, we discuss here which conditions the dressing functions A(p), . . . , D(p) of the quark propagator Eq. (23) must satisfy in order to guarantee confinement and chiral symmetry breaking. For given variational kernels of the wave functional these dressing functions are determined by the quark propagator CRDSE (25), while the variational kernels themselves are determined by minimizing the energy density.
For simplicity we assume that the vector kernels V (p, q) and W (p, q) are real and symmetric, and that the scalar kernel s(p) is real (we can always restrict our variational ansatz to these class of kernels): then, consistent solutions with D(p) = 0 and C(p) = 0 exist, see Eqs. (B1) below. We will furthermore restrict our considerations to chiral quarks, m = 0.
As we have shown in Sec. II B, the physical quark propagator S is related to the propagator Q of the Grassmann fields by Eq. (26) and can be expressed through the dressing functions A and B as In order to prevent the notation from becoming excessively cluttered we have expressed the momentum dependence of the dressing functions through a subscript. Inspired by the form of the bare quark propagator [Eq. (8)] we define the running mass M p and the and the quark dressing function Z p by From Eqs. (34) and (35) we obtain where p = |p|. These equations can be inverted to express the dressing functions A p and B p in terms of M p and Z p as Note that the approximation A p = 1 is equivalent to Z p = 1. We will now exploit these relations to investigate the IR behaviour of the dressing functions A p and B p . An IR finite mass function M (p = 0) ≡ M 0 = 0 is an indicator of chiral symmetry breaking. Therefore we investigate now which conditions the functions A p and B p must fulfil at vanishing momentum so that M 0 = 0. From Eq. (36) follows immediately that an IR diverging B p and an IR finite A p would give rise to a vanishing (negative!) mass function. The dressing function B p must therefore have an finite IR limit B 0 . Furthermore, from the first equation in (36) follows that the dressing function A p must also have an finite IR limit A 0 satisfying the condition Hence for real B 0 and A 0 we find that A 0 ∈ [0, 2]. From the second expression in Eq. (36) we find in the limit of vanishing momentum assuming that Eq. (38) holds Like Eq. (38), the right-hand side of Eq. (39) is well defined only for A 0 ∈ [0, 2]. An infrared suppressed propagator Z 0 < 1 requires A 0 > 1, and an IR vanishing quark propagator requires A 0 = 2, which in view of Eq. (38) implies B 0 = 0. For the mass function to be still non-vanishing in the IR, the dressing function A should have zero slope at vanishing momentum, as it can be seen by Taylor expanding Eq. (36). From this IR analysis there emerges a possible Gribov-Zwanziger-like scenario which includes both confinement and chiral symmetry breaking: an IR vanishing dressing function B p and a dressing function A p satisfying A(0) = 2 and A (0) = 0 yield an IR finite running mass (i.e. spontaneous breaking of chiral symmetry) and an IR vanishing (i.e. confined) quark propagator. The same conclusions follow of course from Eq. (37) taken at zero momentum For an infrared vanishing quark propagator, Z 0 = 0, we find immediately A 0 = 2 and B 0 = 0.
The above results are based in the analysis of the unrenormalized CRDSEs and may hence change after renormalization. However, the renormalization affects mostly the UV behaviour.

IV. MASSIVE ADLER-DAVIS MODEL
To make contact with previous work and for the sake of illustration, in the present section let us neglect the quarkgluon coupling in the QCD Hamiltonian and consider the quark sector only. The remaining contributions to the energy density are therefore Eqs. (31) and (32). If we neglect the coupling of the quarks to the transverse (spatial) gluons in the vacuum wave functional Eq. (9), (10), i.e. V = 0 = W , the bare quark-gluon vertex Eq. (18) vanishes, Γ 0 = 0. Furthermore, if the scalar kernel s p is real both dressing functions C p and D p vanish identically. Then the energy density reduces to while the dressing functions Eq. (25) of the quark propagator become Inserting these expressions into Eq. (40) yields .
where we have introduced the abbreviation Variation of e AD with respect to s p (or, equivalently, with respect to w p ) yields the gap equation with Putting m = 0 in Eqs. (43) and (44) and approximating the Coulomb potential F (p) [Eq. (30)] by its infrared part 8πσ C /p 4 yields precisely the gap equation obtained by Adler and Davis [25]. Equations (43) and (44) give the extension of their model to finite current quark masses. The integral on the right-hand side of Eq. (43) appears also in Ref. [27], where a slightly extended phenomenological model for the quark-quark interaction was considered. From the dressing functions Eq. (41) we can calculate the quark propagator Q [Eq. (24)] , and after elementary but somewhat lengthy algebra the true quark propagator S [see Eq. (26)] can be cast into the form where the mass function M p is related to the variational kernel s p through Equation (45) gives a quasi-particle approximation to the full quark propagator: It has the same form as the freefermion propagator S 0 [Eq. (8)] except that the current quark mass m is replaced by a running mass M p . Note also that in this case the quark dressing function becomes Z p = 1. Equation (46) can be used to trade the kernel s p in the gap equation Eq. (43) for the running mass M p yielding The same equation has been derived in Ref. [33] from a truncated system of DSEs in the so-called first order formalism.

V. THE BARE-VERTEX APPROXIMATION
Let us now return to the full equations of motion of Sec. II with the quark-gluon vertex included. We are interested here mainly in recovering the results of Refs. [10,29] within the present CRDSE approach. For this purpose we replace in the following the full quark-gluon vertexΓ [Eq. (21)] by the bare oneΓ 0 [Eq. (13)]. We are aware that this approximation might not yet be entirely sufficient to provide a realistic description of the mechanism of spontaneous breaking of chiral symmetry, i.e. to yield realistic values for quark condensate in agreement with low-energy meson phenomenology. Nevertheless, it is certainly worthwhile to investigate first the bare-vertex approximation in order to get a better understanding of the structure of the equations of motion of the present approach. In addition, the use of a bare quark-gluon vertex is sufficient to carry out the renormalization of these equations, since the leading UV behaviour of the dressed vertex agrees with that of the bare one, due to asymptotic freedom.

A. The Quark CRDSE
After replacing the full vertices in the CRDSE (25) by bare ones, the Dirac traces can be worked out and the coupled equations (25) for the dressing functions of the quark propagator reduce in the chiral limit m = 0 to the set of equations (B1) given in Appendix B. Equations (B1b) and (B1c) for the dressing functions B p and C p can be collected into a single equation for the complex quantity H = B + iC where we have introduced the abbreviations while ∆ q is given by Eq. (33). Similarly, the equations (B1a) and (B1d) for A p and D p can be added and subtracted, yielding If the variational vector kernels have the symmetry |V (p, q)| = |V (q, p)|, |W (p, q)| = |W (q, p)| then the Eqs. (B1a) and (B1d) for the form factors A p and D p decouple and there exists always the trivial solution D p = 0. Finally, notice that for vanishing vector kernels V = 0 = W these equations reduce to The quark propagator is then entirely determined by the scalar variational kernel s p , which corresponds to the BCStype model considered in Refs. [22,25,27], see Sect. IV.

B. Determination of the Variational Kernels
From both continuum [33] as well as lattice [34,35] studies there exists no indication that the quark propagator in Coulomb gauge contains a term proportional to the Dirac matrix βα i [see Eq. (24)]. Furthermore, when the energy variable of the full propagator is integrated out to yield the equal-time propagator, the term in the quark propagator proportional to the unit matrix vanishes too. Therefore, we expect the physical quark propagator Eq. (24) to be characterized by C p = D p = 0. It is not difficult to see that the quark CRDSEs (B1) allow for consistent solutions with C p = D p = 0 when the variational kernels s, V and W are real, and the vector kernels V and W are symmetric in the quark momenta. Under these assumptions the quark propagator CRDSEs (B1) reduce to while the contributions to the energy density [see Eqs. (27), (28), and (32)] become with ∆ [Eq. (33)] reducing to The energy density contributions (50) contain the scalar kernel s p only implicitly through the dressing functions A p and B p , while the vector kernels V and W enter both explicitly and implicitly. From Eq. (49a) we find the derivatives of the dressing function A p with respect to the vector kernels and similarly the derivatives of B p δB k δV (p, q) = g 2 C F 2 The ellipsis on the right-hand side of these equations stand for the one-loop terms, which we will usually neglect since they would give rise to more than one loop in the equations of motion of the vector kernels.
In the same way we can evaluate the functional derivatives of the dressing functions A p and B p with respect to the scalar kernel s p At one-loop order the previous equations reduce to In a diagrammatic language, differentiating with respect to the vector kernel implies removing one quark-gluon vertex from the diagram. Since the energy contributions contain at most two loops, the variational equations for V and W are free of loops. To this order, we can ignore the Coulomb energy Eq. (50c) and include only the explicit dependence on V and W in the second term of Eq. (50a) and in Eq. (50b), yielding δ e (1) as well as δ e (1) In the first term of Eq. (50a), however, we must take into account also the dependence of the dressing functions A p and B p on the kernels V and W . This yields and by using Eqs. (52) and (54) we find Requiring that the sum of Eqs. (57) and (58) vanishes fixes the vector kernel V to To simplify this and the following expressions we introduce the ratio and cast Eq. (59) into the form At leading order we find from Eq. (49) A p = 1 and b p = s p , and Eq. (61) reduces to the kernel found in Ref. [10]. Furthermore, at large momenta we recover the leading-order perturbative result [36]. The variation of the energy with respect to W is carried out in an analogous way by using Eqs. (53) and (55). This yields the equation of motion Also this kernel reduces to the one found in Ref. [10] at leading order. Both kernels V and W turn out to be real and negative, as we might have expected from e The variation of the energy density with respect to the scalar kernel s p is slightly more involved than the variational derivative with respect to the vector kernels. For the second term in the single-particle energy density Eq. (50a), as well as for the contributions of the gluonic kinetic term Eq. (50b) and of the Coulomb interaction Eq. (50c) it is sufficient to keep only the leading order of Eq. (56), while for the first term in Eq. (50a) we need also the one-loop contributions. Then the variation with respect to s p yields where we have expressed the resulting equations in terms of b p Eq. (60) instead of s p . In order to reproduce the loop expansion of Ref.
[10] 2 on the right-hand side of Eq. (63) it is sufficient to replace b p → s p and A p → 1, while on the left-hand side the factor With this expression we can define a renormalized propagator The scale in our calculations is fixed by the Coulomb string tension σ C occurring in the colour Coulomb potential Eq. (30). Lattice and continuum calculations [39][40][41] 62)] are not the same. We believe that this is an artefact of the one-loop expansion. The mass function Eq. (70) stays however constant over almost three orders of magnitude before slowly bending over (see Fig. 2). Furthermore, the integral I B (p) is rather small in comparison to the (renormalized) integral I A . While the latter has an important effect on the chiral condensate, the mass function Eq. (70) is, apart from the deep IR, almost indistinguishable from the mass function of Ref. [29] extracted from the unrenormalized quark propagator Eq. (64) as shown in Fig. 2. While our mass function vanishes in the deep infrared, the plateau value reads which, due to the uncertainty in the Coulomb string tension, is in the range between 135 and 170 MeV.

VII. MASS FUNCTION IN THE FULL AND STATIC PROPAGATOR
As mentioned before, in Ref. [29] the renormalization of the propagator was ignored and the value of the quarkgluon coupling constant was chosen to reproduce the phenomenological value of the chiral condensate. The mass function, however, was not significantly enhanced in comparison to the Adler-Davis model [25] (see Sec. IV), showing an infrared value of 135 MeV (for σ C = 2.5σ). Similar results have been obtained also in the previous section: although our rough one-loop calculation is capable of reproducing the correct value of the chiral condensate, the mass function is not significantly influenced by the coupling to the transverse gluons. This seems at odds with the common lore that the infrared value of the mass function should be around the value of the constituent quark mass, i.e. roughly 300 MeV. Here we show that this apparent contradiction might result from comparing the mass functions of the full and equal-time propagators. Before discussing this issue in Coulomb gauge we address the question in Landau gauge, for which we have solutions of the Dyson-Schwinger equations at our disposal.
Suppressing colour indices, the quark propagator in Landau gauge is usually written as For symmetry reasons the contribution proportional to γ 4 p 4 vanishes and we are left with S 3 (p) = iγ · p dp 4 2π Analogously to the definition of the quark mass function M we can introduce the equal-time mass function M 3 (p 2 ) as ratio of the coefficients of the 1 and γ i terms of the equal-time propagator, yielding Numerical solutions for the mass function always show a monotonically decreasing function of the four-momentum. Therefore, since M (p 2 ) ≤ M (0) we see from Eq. (74) that M 3 (0) < M (0). For typical results for the Landau gauge quark propagator we find that M 3 (0) lies between 50% and 60% of M (0), see Fig. 3a. Furthermore, the equal-time quark propagator Eq. (73) can be brought into the form (69) .  The situation might be similar in Coulomb gauge. Being non-covariant, the propagator depends separately on p 4 and p and has therefore four Dirac components instead of two The mixed structure γ 4 γ i does not arise at one-loop level in perturbation theory [37] and is not found in lattice calculations [34,35] either; therefore we will set A d = 0 in the following. The propagator in Coulomb gauge takes therefore the form .
As for the quark propagator in Landau gauge we expect also in Coulomb gauge that the effective quark mass extracted from the static propagator is considerably smaller than the one extracted from the four-dimensional propagator.

VIII. CONCLUSIONS
The gap equation of Ref. [29] has been rederived within the framework of the canonical recursive Dyson-Schwinger equations. We have shown that the additional Dirac structure in the bare quark-gluon vertex of the vacuum wave functional not only eliminates the UV divergences from the gap equation (as shown already in Refs. [10,29]) but is also crucial to ensure multiplicative renormalizability of the quark propagator. We have performed a quenched semi-perturbative calculation assuming a bare quark-gluon vertex. Unlike the covariant functional approaches in Landau gauge, where the dressing of the (four-dimensional) quark-gluon vertex is crucial for obtaining spontaneous breaking of chiral symmetry, in the present Hamiltonian approach the bare quark-gluon vertex in the vacuum wave functional is sufficient to reproduce the phenomenological value of the quark condensate. In the present approach the dominant IR contribution, which triggers the spontaneous breaking of chiral symmetry, comes from the confining Coulomb potential. We have also shown that, depending on the details of the momentum dependence, the effective quark mass obtained in the Hamiltonian approach cannot be compared with the (constituent) mass extracted from the corresponding four-dimensional propagator and is expected to be considerably smaller than the latter. The results obtained in the present paper are quite encouraging for a fully self-consistent solution of the coupled variational and CRDSEs. and to introduce the Grassmann-valued Dirac spinor fields ξ + (p) := 1 √ 2E p s u(p, s) ξ + (p, s), which satisfy Λ ± (p) ξ ± (p) = ξ ± (p).
From Eqs. (A2) follow the inverse relations to Eq. (A3) For simplicity we will simply write |ξ instead of |ξ + , ξ * − . With these definitions we find Furthermore, the coherent-state representation of a Fock state |Φ of the Dirac fermions is given by In the following it will be also convenient to assemble the independent fields ξ + and ξ − in a single Grassmann-valued spinor ξ(p) = ξ + (p) + ξ − (p), ξ ± (p) = Λ ± (p) ξ(p).