A determination ofand the non-perturbative vacuum energy of Yang-Mills theory in the Landau gauge

We discuss the 2-point-particle-irreducible (2PPI) expansion, which sums bubble graphs to all orders, in the context of SU(N) Yang-Mills theory in the Landau gauge. Using the method we investigate the possible existence of a gluon condensate of mass dimension two,, and the corresponding non-zero vacuum energy. This condensate gives rise to a dynamically generated mass for the gluon.


Introduction.
Recently there has been growing evidence for the existence of a condensate of mass dimension two in SU (N ) Yang-Mills theory with N colours. An obvious candidate for such a condensate is A a µ A a µ . The phenomenological background of this type of condensate can be found in [1,2,3]. However, if one first considers simpler models such as massless λφ 4 theory or the Gross-Neveu model [4] and the role played by their quartic interaction in the formation of a (local) composite quadratic condensate and the consequent dynamical mass generation for the originally massless fields [4,5,6], it is clear that the possibility exists that the quartic gluon interaction gives rise to a quadratic composite operator condensate in Yang Mills theory and hence a dynamical mass for the gluons. The formation of such a dynamical mass is strongly correlated to a lower value of the vacuum energy. In other words the causal perturbative Yang Mills vacuum is unstable. From this viewpoint, mass generation in connection with gluon pairing has already been discussed a long time ago in, for example, [7,8,9,10]. There the analogy with the BCS superconductor and its gap equation was examined. It was shown that the zero vacuum is tachyonic in nature and the gluons achieve a mass due to a nontrivial solution of the gap equation. Moreover, recent work using lattice regularized Yang Mills theory has indicated the existence of a non-zero condensate, A a µ A a µ , [11]. There the authors invoked the operator product expansion, (OPE), on the gluon propagator as well as on the effective coupling α s in the Landau gauge. Their work was based on the perception that, even in the relatively high energy region (∼ 10GeV), a discrepancy existed between the expected perturbative behaviour and the lattice results. It was shown that, within the momentum range accessible to the OPE, that this discrepancy could be solved with a 1/q 2 power correction * . They concluded that a non-vanishing dimension two condensate must exist. Further, the results of [12] give some evidence that instantons might be the mechanism behind the low-momentum contribution to condensate. As has been argued in [3], only the low-momentum content of the squared vector potential is accessible with the OPE. Moreover, they argue that there are also short-distance non-perturbative contributions to A 2 µ . It is no coincidence the Landau gauge is used for the search for a dimension two condensate. Naively, the operator A 2 µ is not gauge-invariant. Although this does not prevent the condensate A 2 µ showing up in gauge variant quantities like the gluon propagator, we should instead consider the gauge-invariant operator , where V T is the space-time volume and U is an arbitrary gauge transformation in order to assign some physical meaning to the operator. Clearly from its structure this operator is non-local and thus is difficult to handle. However, when we impose the Landau gauge, it reduces † to the local operator A 2 µ . Moreover, it has been shown that A 2 µ is (on-shell) BRST invariant [14,15,16,17]. Another motivation for studying A 2 µ is the perceived connection between the gluon propagator and confinement. (See [18] and references therein.) More precisely, the gluon propagator exhibits an infrared suppression, as has been reported in many lattice simulations, [19,20,21] and using the Schwinger-Dyson approach, [22,23,24]. A dynamical gluon mass might serve as an indication for such a suppression. An attempt has already been made to explain confinement by a dual Ginzburg-Landau model or an effective string theory, in the Landau gauge, with the help of A 2 µ [25]. The fact that A 2 µ might be central to confinement, is supported by the observation that it undergoes a phase transition due to the monopole condensation in three dimensional compact QED [2,3].
From these various analyses the importance of A 2 µ must have become clear. Therefore, the aim of this article is to provide some analytical evidence that gluons do condense. To our knowledge, [26] is the only paper which effectively calculates A 2 µ , without referring * The 1 q 4 power correction due to the G 2 µν condensate is too weak at such energies to be the cause of the discrepancy.
† Although this equality is somewhat disturbed by Gribov ambiguities [13]. In this paper Gribov copies are neglected since we will work in the perturbative Landau gauge and sum a certain class of bubble diagrams in this particular gauge. It is a pleasant feature of the Landau gauge that A 2 µ can be given a gauge invariant meaning. In another gauge, the bubbles will no longer correspond to A 2 µ and the correspondence with A 2 µ min is more of academic interest.
to lattice regularization. In [26] the standard way of calculating the effective potential for a particular quantity was followed, and all the problems concerning the fact that the considered quantity was a composite operator were elegantly solved.
In a previous paper [27], we have discussed the 2P P I expansion for the Gross-Neveu model and found results close to the exact values for the Gross-Neveu mass gap and the vacuum energy. The 2P P I expansion does not rely on the effective action formalism of [26]. Instead it is directly based on the path integral and the topology of its Feynman diagrammatical expansion. In this paper we will discuss how to apply it to SU (N ) Yang-Mills theories in the Landau gauge. Of course, it is not our aim to provide a complete picture of A 2 µ but rather give further evidence for its existence since it lowers the vacuum energy.
2 The 2P P I expansion.
The SU (N ) Yang Mills Lagrangian in d-dimensional Euclidean space time is given by where G a µν is the gluon field strength, 1 ≤ a ≤ N 2 − 1, L gauge+F.P. implements the Landau gauge and its corresponding Faddeev-Popov part and ξ and ξ denote the ghosts and antighost fields respectively. Issues concerning the counterterm part of (2.1) will be discussed later. First, we consider the diagrammatical expansion for the vacuum energy which we denote by E. As is well known, this is a series consisting of one particle irreducible, (1P I), diagrams. These 1P I diagrams can be divided into two disjoint classes: • those diagrams not falling apart into two separate pieces when two lines meeting at the same point x are cut, which we call 2-point-particle-irreducible, (2P P I); (an example is given in Fig. 1) • those diagrams falling apart into two separate pieces when two lines meeting at the same point x are cut which we call 2-point-particle-reducible, (2P P R), while x is called the 2P P R insertion point; (an example is given in Fig. 2).
We may now resum this series of 2P P R and 2P P I graphs, where the propagators are the usual massless ones, by retaining only the 2P P I graphs, whereby the 2P P R insertions, or bubbles, are resummed into an effective (mass) 2 m 2 2P P I ≡ m 2 . The bubble graph gluon polarization is then given by where f abc are the SU (N ) structure constants. We define the vacuum expectation value of The global SU (N ) symmetry can then be used to show that which results in an effective mass, m, running in the 2P P I propagators, given by If we let E 2P P I be the sum of the 2P P I vacuum bubbles, calculated with the effective 2P P I propagator, then this E 2P P I is not equal to the vacuum energy E, because simply removing all 2P P R insertions is too naive. For instance, there is a double counting problem which is already visible in the 2P P R diagram of Fig. 2 where each bubble can be seen as a 2P P R insertion on the other one. However, we can resolve this ambiguity. A dimensional argument results in where c = 0 will accomodate the double counting. To determine the appropriate value of c, we use the path integral which gives (2.8) The first two terms contribute unambiguously to the 2P P I part. For the last term, we rewrite Using (2.4) and the properties of the structure constants f abc , we obtain Then a simple diagrammatical argument gives which is a local gap equation, summing the bubble graphs into m 2 . Using this together with (2.12) finally gives 14) It is easy to show that the following equivalence holds To summarize, we have summed the bubble insertions into an effective (mass) 2 , m 2 . The vacuum energy is expressed by We stress the fact that (2.16) is only meaningful if the gap equation (2.15) is satisfied. This means we cannot consider m or ∆ as a real variable on which E depends. It is a quantity which has to obey its gap equation, otherwise the 2P P I expansion loses its validity. This also means that E(m), or equivalently E(∆), is not a function depending on m (∆), in contrast ‡ with the usual concept of an effective potential V (ϕ) which is a function of the constant field ϕ.
In order to ensure the usefulness of the 2P P I formalism for actual calculations, we should prove it can be fully renormalized with the counterterms available from the original (bare) Lagrangian, (2.1). However, it is sufficient to say that all our derived formulae remain valid and are finite when the counterterms are included. This also implies the 2P P I mass m is ‡ V (ϕ) also makes sense if dV dϕ = 0.
renormalized and gives rise to a finite, physical mass § , m phys . Furthermore, no new counterterms are needed to remove the vacuum energy divergences. The renormalizability of the 2P P I expansion has been discussed in detail in [27] in the case of the Gross-Neveu model. Since the arguments for Yang Mills theory are completely analogous, we refer to [27] for technical details concerning the renormalization. Another point worth emphasising here, is the renormalization group equation, (RGE), for E. The first diagram of E 2P P I is given by the O-bubble. Using the MS renormalization scheme in dimensional regularization in d dimensions, we find Since E is a physical quantity, it should not depend on the subtraction scale µ. This is expressed by the RGE where β(g 2 ) governs the scaling behaviour of the coupling constant and κ(g 2 ) is the anomalous dimension of m 2 The coefficients can be found in [28,29] for β and in [26,29,30] for γ A 2 µ ,

(2.23)
When we combine all this information and determine µ dE dµ up to lowest order in g 2 , we find Apparently, it seems that E does not obey its RGE. However, this is not a contradiction because of the demand that the gap equation (2.15) must be satisfied. The gap equation implies that ln m 2 µ 2 ∝ 1 g 2 + constants. The consequence is that all leading logarithms contain terms of order unity. Hence, we cannot show that the RGE for E is obeyed order by order. The same phenomenon extends to higher orders. In other words, knowledge of µ dE dµ up to a certain order n, would require knowledge of all leading and subleading log terms to order n, to show explicitly that µ dE dµ = 0. We must therefore be careful not to interpret the non-vanishing of the RGE as a reason to introduce a "non-perturbative" β-function, as is sometimes done, [31]. § m phys is the pole of the gluon propagator. Results.
Up to 2-loop order in the 2P P I expansion (see Fig. 3), we find in the MS scheme where ζ(n) is the Riemann zeta function, and Cℓ 2 (x) is the Clausen function. We have computed the relevant two loop vacuum bubble diagrams to the finite part using the massive gluon and massless ghost propagators which are respectively in the Landau gauge. The expressions for the general massive and massless two loop bubble integrals were derived from the results of [32] and implemented in the symbolic manipulation language Form, [33]. It is easy to check that solving the gap equation ∂E ∂∆ = 0, with µ 2 set equal to 3 4 g 2 N N 2 −1 ∆ to kill potentially large logarithms, gives no solution at 1-loop or 2-loop, apart from the trivial one ∆ = 0. This does not imply A 2 µ does not exist but that the MS scheme might not be the best choice for the 2P P I expansion. To address this we first remove the freedom existing in how the mass parameter ∆ is renormalized by replacing ∆ by a renormalization scheme and scale independent quantity ∆. This can be accommodated by ¶ (3.5) ¶ Barred quantities refer to the MS scheme, otherwise any other massless renormalization scheme is meant.
A change in massless renormalization scheme corresponds to relations of the form With these, it is easily checked that (3.4) is renormalization scheme and scale independent. The explicit solution of (3.5) reads Since the gap equation is still a series expansion in g 2 N 16π 2 and we hope to find (at least qualitatively) acceptable results, g 2 N 16π 2 should be small. We will therefore choose to renormalize the coupling constant in such a scheme so that E is of the form for E is not obeyed order by order, see also [27]. Once a solution ∆ * of the gap equation is found, then we will always have If the constructed value for g 2 N 16π 2 is small enough, then we can trust, at least qualitatively, the results we will obtain. Now we are ready to rewrite (3.1) in terms of ∆. After a little algebra, one finds Next, we determine b 0 and b 1 so that (3.13) reduces to (3.20) We do not list the value for b 1 since it is no longer required. From the β-function we find the two loop expression for the coupling constant is where Λ is the scale parameter of the corresponding massless renormalization scheme. We will express everything in terms of the MS scale parameter Λ MS . In [34], it was shown that We will also derive a value for the αs π G 2 µν condensate from the trace anomaly (3.23) This anomaly allows us to deduce for N = 3 the following relation between the vacuum energy and the gluon condensate At 1-loop order, the results for N = 3 are while the scale parameter µ 2 ≈ (184MeV) 2 . We have used Λ MS ≈ 233MeV which was the value reported in [11]. We see that the 1-loop expansion parameter is quite large and we conclude that we should go to the next order where the situation is improved. We find Although there is a sizeable difference between 1-loop and 2-loop results, the relative smallness of the 2-loop expansion parameter, indicates that the 2-loop values are qualitatively trustworthy. It is well known that in order to find reliable perturbative results, one must go beyond 1-loop, and even beyond 2-loop approximations. Therefore, one should not attach a firm quantitative meaning to the numerical values. Let us compare our results with what was found elsewhere with different methods. A combined lattice fit resulted in A 2 µ ≈ 1.64GeV, [11]. We cannot really compare this with our result for ∆, since the lattice value was obtained with the OPE at a scale µ = 10GeV in a specific renormalization scheme (MOM). However, it is satisfactory that (3.30) is at least of the same order of magnitude. More interesting is the comparison with what was found in [26] with the local composite operator formalism for A 2 µ . In the MS scheme at 2-loop order, it was found that g 2 N 16π 2 ≈ 0.141247 while E vac ≈ − 0.789Λ 4

MS
which is in quite good agreement with our results. An estimate of the tree level gluon mass of ∼ 500MeV was also given in [26] which compares well with the lattice value of ∼ 600MeV of [35,36]. With the 2P P I method, one does not really have the concept of a tree level mass. Instead, the determination of m phys would need the calculation of the highly non-trivial 2-loop 2P P I mass renormalization graphs which is beyond the scope of this article.
In conclusion we note that the perturbative Yang-Mills vacuum is unstable and lowers its value through a non-perturbative mass dimension two gluon condensate A 2 µ . We have omitted quark contributions in our analysis but it is straighforward to extend the 2P P I expansion to QCD with quarks included. Indeed an idea of the effect they have could be gained by an extension of [26].