On the Yang-Mills two-loop effective action with wordline methods

We derive the two-loop effective action for covariantly constant field strength of pure Yang-Mills theory in the presence of an infrared scale. The computation is done in the framework of the worldline formalism, based on a generalization procedure of constructing multiloop effective actions in terms of the bosonic worldline path integral. The two-loop beta-function is correctly reproduced. This is the first derivation in the worldline formulation, and serves as a nontrivial check on the consistency of the multiloop generalization procedure in the worldline formalism.


I. INTRODUCTION
The physics of strong but slowly varying chromomagnetic and electric fields may provide some insight to the non-trivial vacuum structure of QCD. For example, the Euler-Heisenberg Lagrangian in QED exhibits truly nonperturbative effects; it allows the investigation of the nonlinear regime of QED, and has also been studied beyond one loop, see e.g. [1,2,3]. In QCD non-linearities already are present on the classical level due to its non-Abelian nature. It is yet unknown how the full effective action changes beyond one loop. The computation of multiloop terms in the effective action is cumbersome, in particular for a non-Abelian gauge group. These computations simplify if background field methods are employed and the multiloop terms are evaluated for specifically chosen background configurations such as covariantly constant or selfdual configurations. This gives access to beta functions and parts of the effective action beyond one loop, see e.g. [4,5,6]. Wordline methods [7,8,9,10,11,12,13,14,15,16,17,18,19,20] have been shown to lead to a striking simplification of certain computations. A summation of Feynman diagrams is already implemented without loop momentum integrals and Dirac traces [10,11].
In the present work we provide for the first time a numerically accessible expression for the two loop effective action of Yang-Mills theory for covariantly constant fields in the presence of a physical infrared cut-off (at finite correlation length). This computation serves many purposes: firstly, it completes the construction of multiloop worldline methods for Yang-Mills theories as initiated in [13,14]. The two-loop beta function serves as a nontrivial consistency check. Secondly, it is the necessary input for an RG-improved non-perturbative computation of the effective action within a Wilsonian framework. Finally one can study the stabilisation of the Savvidy vacuum beyond one loop.

II. WORDLINE REPRESENTATION BEYOND ONE LOOP
We briefly recapitulate the analysis of [13,14]. The starting point is the generating functional of pure Yang-Mills theory in the presence of a background field configuration A, where the gauge-fixed Yang-Mills action is given by with tr t a t b = −δ ab /2 in the fundamental representation, where Z n comprises the n-loop contribution to the generating functional. The generating functional (3) is gauge invariant under the gauge transformation A → A + Dω, and its logarithm is the gauge invariant Wilsonian effective action of pure Yang-Mills. The one loop contribution Z 1 [A] for a subset of field configurations, e.g. covariantly constant fieldstrength F , has been computed in [12], for related results within standard methods see [21]. So far, a full computation of the two loop contribution Z 2 [A] is lacking. After some algebra, (1) can be turned into a more convenient representation for Z 2 [A], see [13]. However, in all representations the formal expression (1) suffers both from UV and IR divergences.
In the present work we regularise and renormalise these divergences separately: for the UV divergences we employ dimensional regularisation for analytic purposes and construct a gauge-invariant proper-time cutoff interesting for numerical work. The divergences are then cured by appropriate counter terms. Additionally we introduce a physical IR cut-off. IR divergences are absent, if putting the theory into a box of size L. Effectively this can be implemented by introducing gauge invariant masses m ∼ 1/L to the propagating degrees of freedom of the theory. The latter also has the advantage of implementing a physical mass gap on the level of the Green functions. This offers a path towards a self-consistent investigation of QCD in the confining regime in an effective field theory approach that is quite close to the fundamental theory. We emphasise that the approach leads to a fully gauge invariant effective action. However, the choice of the gauge-fixing parameter is directly related to the choice of different physical boundary conditions on the surface of the box.
Within this framework the renormalised two loop contribution Z 2 [A] is provided by [13] with antisymmetric tensor field α µν = −α νµ , and We emphasise again that m 2 serves a twofold though related purpose. Firstly it accounts for a possible nonperturbative mass-gap, secondly it mimics the implementation of a finite volume. The space-time integration over y 1 , y 2 can be used for regularising the generating functional (4) by means of the dimensional regularisation In Feynman gauge ξ = 1, the expression simplifies as the last term in (6) drops out, and the non-trivial tensor structure disappears. In (4) we have not specified the counter terms indicated by c.t., that shall be discussed later. Within the representation introduced above the effective action derived from the generating functional (4) reads with the contribution of purely gluonic loops, Γ gluon , has been computed in [14], and the result is summarised in Appendix B. We complete the analysis of [14] by computing the ghost contribution Γ ghost as well as the renormalisation insertions. To that end we turn (8) into Euclidean wordline integrals, and arrive at [13] with the abbreviations and Eq. (9) stands for the fully renormalised ghost contribution to the two loop effective action. We have employed an additional ultraviolet regularisation in the propertime integrations which entails a gauge invariant momentum cut-off. Such a cut-off scheme is amiable to numerical computation, whereas the dimensional regularisation facilitates analytic computations. In the following we shall conveniently project onto either regularisation by simply switching off either the regularisation parameter τ → 0 or ǫ → 0. We have not specified the counter terms indicated by c.t., which in general depend both on the proper-time regularisation via τ and on the dimensional regularisation via ǫ. The computations of these counter terms will be discussed in the next section. For explicit computations we employ pseudo-Abelian su(2) with constant field strength, where n a is a constant unit vector in color space with n a n a = 1. The gauge fields (11) satisfy the Fock-Schwinger gauge x µ A µ = 0. With (11) we rewrite the Lorentz matrices M and N as with T − = n a λ a . The computation of Γ ghost is straightforward but tedious [22]. It results in The integrand I ghost of the proper-time integral is given by .
The expression (13) with (14) is numerically accessible, after the counter terms in (13) are specified.

III. RENORMALISATION
Now we discuss the UV subtraction terms hidden in the counter terms that render Z 2 , Γ 2 finite, and in particular, (13) finite. Apart from applying a standard dimensional regularisation convenient for analytic considerations, we have introduced a gauge invariant UV regularisation by cutting off the proper-time integrals in (13) at a finite proper time τ , T i ≥ τ . This translates via a Laplace transform into a gauge invariant momentum cut-off if the effective action is formulated in terms of momentum loops. Such a scheme makes numerical computations accessible where the dimensional regularisation only can be employed in exceptional cases. Indeed, a fully non-perturbative wordline formulation of quantum field theories would provide a tool for devicing gauge-invariant momentum cut-off schemes on the non-perturbative level which would be highly interesting. The ghost action (9) is then written as Γ ghost = Γ ghost,reg + c.t. with The regularised expression Γ ghost,reg diverges if the regularisation parameters τ, ǫ are removed. Here we first concentrate on the proper-time regularisation with ǫ = 0. Then the regularised expression Γ ghost,reg in (16) diverges with powers of 1/τ , more precisely with 1/τ n (ln τ ) m . Moreover, since we are dealing with the two-loop effective action, the divergent terms are not necessarily polynomial, and the counter terms cannot be determined in a polynomial expansion. The non-polynomial terms can be attributed to the divergence of one loop sub-diagrams which can be used to construct the related counter terms. However, here we want to set-up a procedure with which these counter terms can be derived systematically from Γ ghost,reg by means of derivatives. Such a procedure mimics the standard BPHZ-type subtraction schemes in momentum space. The divergences in τ are extracted by appropriate τ -derivatives with the help of the identity Applying the above τ -derivative to Γ ghost,reg , the physically finite term drops out. Hence appropriate subtractions render the action Γ finite. Note that such a procedure in principle only properly provides the renormalised effective action by a careful discussion of the finite renormalisation that originates in the subtractions in (18). In the present two loop case it suffices to only take one τderivative, e.g.
This reduces the number of proper-time integrations and makes the divergence structure analytically accessible. This procedure deserves further studies. We still have to compute the traces over the field strength. Following [25], we work in the Lorentz frame in which the electric and magnetic fields are parallel, and thereby the field strength takes on a simple form with only two non-zero symplectic block elements, i.e., it can be written as with and where ǫ and η are the magnitudes of the magnetic and electric fields, respectively. We close with a remark on the explicit computation of the proper-time integrals. In particular for numerical purposes it is advantageous to convert the integrals into less divergent expressions. Indeed, I ghost as well as the corresponding integrand I gluon can be integrated analytically over T 3 and hence can be written as a total T 3 -derivative. The computations are deferred to Appendix A and Appendix B respectively and the results read whereÎ ghost is given in (A.4), and whereÎ gluon is given in (B.11). The counter terms in (23), (24) are τ -dependent and can be computed from the derivative procedure outlined in (17), (18).

IV. TWO LOOP β FUNCTION
In the remainder of this work we concentrate on the the question of full two-loop consistency of the wordline formalism suggested in [13,14], the construction of which we have completed here. To that end we discuss the running of the coupling at two loop which is universal in mass-independent renormalisation schemes. As this concerns an analytic computation we employ a dimensional regularisation with τ = 0. Moreover, we use a minimal subtraction scheme that renders the renormalisation constants mass-independent, and hence projects onto the the universal result for the two loop β-function. How such a mass-independent scheme is fixed in the presence of general IR cutoffs has been discussed in detail in [23], and we can straightforwardly use the related arguments for the mass terms for gluon and ghosts employed in the present work. The β-function can be read-off from the running of the wave function renormalisation Z A of the gauge field. Expanding the two loop contribution to the effective action Γ 2,reg in powers of F we are led to The F 2 -coefficient of the ghost effective action Γ ghost,reg reads where the coefficients C ′ i are given by [14] with Now we are in the position to perform the UV renormalisation for the ghost term. Note that by power counting, the UV divergence can appear at most in the quadratic term in the expansion, we write where and C ′ finite is its finite part. This introduces the minimal subtraction scheme in the ghost part. The integral in (13) is changed by the additional integrands proportional to C ′ ghost F 2 rendering a finite expression. We remark that it can be explicitly checked that the renormalisation constants are mass-independent at two loop. This constitutes a mass-independent RG-scheme and hence β 2 is universal.
The two loop β-function is provided by The background field formalism allows us to directly extract the two loop β-function from Z A : the effective action Γ[A] is gauge invariant and consequently the combination gA is RG-invariant, leading to Z g = Z −1/2 A , and hence With (32) and (33) we conclude that and we directly read off the two loop β-function from the subtraction terms computed in the last section. Using (28) in (26) we arrive at the ghost contribution β 2,ghost to the two loop coefficient β 2 , The gluon loop contribution has been computed in [14] as β 2,gluon = −11/2. It is left to compute the contributions of the one loop counter terms. They arise from the insertion of the one loop RG constants of coupling and propagating field at one loop. Note that the propagating field is the fluctuation field a with one loop wave function renormalisation The worldline counter terms reduce to the standard one loop graphs. The corresponding diagrams shown in Fig.1 and 2 result from the one-loop renormalisation of the fluctuation field a and its vertices, while those in Fig.3 and 4 arise from the one-loop renormalisation of ghost field and its vertices. The computation of the counter term in Fig.1 requires a gluon mass renormalisation with m 2 → Z a Z m m 2 . In Feynman gauge it is given by [24] Z a Z m = 1 − 1 ǫ This counter term has been considered in [14], which gives a contribution of 10/3ǫ to the two loop coefficient β 2 . We also remark that the gauge-fixing term does not renormalise, ξ → Z a ξ.
Note that the ghost mass term is not renormalised. The counter terms from Fig.1-Fig.4 give rise to pole contributions and we are led to β 2 = − 11 2 + 11 6 + 35 6 + 10 3 which agrees with the well-known result, e.g. [4,5]. As we are only interested in diagrams with external background fields we could have rescaled the fluctuation field a and the ghost field with the renormalisation factors. With these rescaled fields, the diagrams above reduce to terms proportional to the renormalisation of the gauge-fixing term introduced by this rescaling and the renormalisation of the mass terms which also changes by this rescaling (e.g. the ghost mass renormalises with these rescaled fields). This has been used in [5]. Of course, this does not change the result. For comparison we list the different contributions  In the middle column the different contributions from a direct computation (right), and from one with rescaled fluctuation fields (left) are listed.
We can use the above results on the consistent renormalisation in the presence of an infrared mass-scale to define the renormalised two loop contribution Γ 2 [A] by means of appropriate subtractions instead of the dimensional regularisation used above. This allows us to numerically compute the full two loop effective action Γ = Γ 1 + Γ 2 as a function of F .

V. OUTLOOK
In the present work we have completed the wordline construction of the two loop effective action initiated in [13,14]. In particular we have provided a crucial consistency check of the construction by computing the universal two loop β-function within the wordline formalism.
We also have set-up a practical ultraviolet BPHZ-type renormalisation scheme in the proper-time which makes numerical computations accessible. For example, this can be used to numerically compute the two loop effective action for covariantly constant fields.
The inclusion of fermions in the present approach is straightforward, and is, in our opinion, the physically most interesting extension of the present work.
Acknowledgements We thank G. Dunne, H. Gies and C. Schubert for useful discussions.

A. GHOST CONTRIBUTION
One of the T -integrations in (13) can be done analytically. This is achieved by integrating I ghost over T 3 . Performing all the traces in the integrand of (14) and summing over them gives with a 1 = coth(aT 1 ) + coth(aT 2 ). The square-rooted determinant term reads where C 1 = a 2 b 2 csch(aT 1 ) csch(bT 1 ) csch(aT 2 ) csch(bT 2 ). We definê 3) after analytically performing the integration over T 3 one finds (b 1 = a 1 (a ↔ b)) The gluon loop contribution to the two loop effective action reads [14] ×tr cos 2F (T 1 + T 2 ) + δ( Following the same procedure as we extracted Eq. (26) from (13), we obtain the renormalisation part of the effective action above at the second order of F