The a-function for gauge theories

The a-function is a proposed quantity defined for quantum field theories which has a monotonic behaviour along renormalisation group flows, being related to the beta-functions via a gradient flow equation involving a positive definite metric. We construct the a-function at four loop order for a general gauge theory with fermions and scalars, using only one and two loop beta-functions; we are then able to provide a stringent consistency check on the general three-loop gauge beta-function. In the case of an N=1 supersymmetric gauge theory, we present a general condition on the chiral field anomalous dimension which guarantees an exact all-orders expression for the a-function; and we verify this up to fifth order (corresponding to the three-loop anomalous dimension).


Introduction
It is natural to regard quantum field theories as points on a manifold with the couplings {g I } as co-ordinates, and with a natural flow determined by the β-functions β I (g). At fixed points the quantum field theory is scale-invariant and is expected to become a conformal field theory. It was suggested by Cardy [1] that there might be a four-dimensional generalisation of Zamolodchikov's c-theorem [2] in two dimensions, such that there is a function a(g) which has monotonic behaviour under renormalisation-group (RG) flow (the strong a-theorem) or which is defined at fixed points such that a UV − a IR > 0 (the weak a-theorem). It soon became clear that the coefficient (which we shall denote 1 4 A) of the Gauss-Bonnet term in the trace of the energy-momentum tensor is the only natural candidate for the a-function. A proof of the weak a-theorem has been presented by Komargodski and Schwimmer [3] and further analysed and extended in Refs. [4,5].
In other work, a perturbative version of the strong a-theorem has been derived [6] from Wess-Zumino consistency conditions for the response of the theory defined on curved spacetime, and with x-dependent couplings g I (x), to a Weyl rescaling of the metric [7]. This approach has been extended to other dimensions in Refs. [8,9]. The essential result is that we can define a functionÃ byÃ where A is defined above and W I is well-defined as an RG quantity on the theory extended as described above, such thatÃ satisfies the crucial equation Here G IJ = G JI ,ρ I and Q J may all be computed perturbatively within the theory extended to curved spacetime and x-dependent g I ; for weak couplings G IJ can be shown to be positive definite in four dimensions (in six dimensions, G IJ has recently been computed to be negative definite at leading order [10]). Eq. (1.2) implies thus verifying the strong a-theorem so long as G IJ is positive. Crucially Eq. (1.2) also imposes integrability conditions which constrain the form of the β-functions and are the focus of this paper. These conditions relate contributions to β-functions at different loop orders.
We should mention here that for theories with a global symmetry, β I in these equations should be replaced by a B I which is defined, for instance, in Ref. [6]; however it was shown in Ref. [11,12] that the two quantities only begin to differ at three loops; and in Ref. [13] The (hermitian) gauge generators for the scalar and fermion fields are denoted respectively t ϕ A = −t ϕ A T and t ψ A , A = 1 . . . n V , where n V = dim G, and obey and gauge invariance requires Y a t ψ A + t ψ A T Y a = t ϕ A ab Y b , t ϕ A ae λ ebcd ϕ a ϕ b ϕ c ϕ d = 0. In order to simplify the form of our results, it is convenient to assemble the Yukawa couplings into a matrix and This corresponds to using the Majorana spinor Ψ = ψ i −C −1ψiT .
We should mention here that in our present calculations we have ignored potential parity violating counterterms (i.e. containing ǫ-tensors). The analysis of Ref. [6] was recently extended [28] to the case of theories with chiral anomalies, including the possibility of parity violating anomalies. It would be interesting to carry out the detailed computations necessary to exemplify the general conclusions of Ref. [28].
The one-and two-loop gauge β-functions are given by with T A as defined in Eq. (2.4). We follow Ref. [15] in removing factors of 1/16π 2 which arise at each loop order by redefining The one-loop Yukawa β-function is given by 8) and the one-loop scalar β-function is given by The leading terms in the metric G IJ in Eq. (1.2) may be written as [6] where σ is given (using dimensional regularisation, DREG) by [6,20] We emphasise here that y andŷ are not independent; and furthermore, the result of a trace is unchanged by interchanging y andŷ.
The lowest-order contributions toÃ are given implicitly in Ref. [20] as To proceed to the next order, we shall need the two-loop Yukawa β-function in addition to the one-loop scalar β-function in Eq. (2.9). The two-loop β-function is given in general by Refs. [26,27] in the form + tr[c 24 y aŷb y cŷc + c 25 y aŷc y bŷc + c 26 g 2Ĉ ψ y aŷb ] The contributions G y α are depicted in Table (1);Ĝ y α is the transpose of G y α . A solid or open box represents g 2 C ψ or g 2 C ϕ respectively. A box with a letter "A" represents the gauge generator gT A . Note that for each of G y α , there is an alternation between "hatted" and "unhatted" y matrices, as can be seen in Eq. (2.13) for G y α , α = 20, . . . 28. To give a couple of examples, G y 4 represents (G y 4 ) a = λ abcd y bŷc y d , (2.14) and G y 19 corresponds to  We present here the results for the coefficients evaluated using standard dimensional regularisation, DREG [26,27]: There are 33 coefficients altogether (counting c 20 and c 28 as three each). We do however have the freedom to redefine the couplings, corresponding to a change in renormalisation scheme; at this order we may consider δy a = µ 1 y bŷa y b + µ 2 (y bŷb y a + y aŷb y b ) + µ 3 tr[y aŷb ]y b This results in a change in the β-function We observe that the redefinitions corresponding to µ 1−4 are not all independent; for instance we may remove µ 4 by redefining This is a general consequence of the form of the redefinition given by Eq. (2.18), which implies that a redefinition δy a = β (1) y a , δg = β (1) g (2.21) has no effect on β (2) y a ; however µ 5 yields an independent redefinition due to the fact that there happens to be no corresponding C φ ab y b term in β y a . It then follows that µ 1−5 and ν 1−3 yield only 7 independent redefinitions; we therefore have 33 − 7 = 26 independent coefficients in the two-loop β-function. Under the change Eq. (2.17) which corresponds to taking δσ = 4 ν 1 C G + ν 2 R ψ + ν 3 R φ in Eq. (2.10).
Applying Eq. (1.2), we requireÃ (4) to satisfy d yÃ (4) = dy · T (3) yy · β (1) The contributions to dy · T yy · d ′ y at this order are depicted in Table (2). Here a diamond represents d ′ y and a cross dy. As an example, G T 1 represents yy is symmetric up to the order at which we are working. The β-functions β There are no "off-diagonal" fermion-scalar contributions to this order. We parameteriseÃ (4) as where the different contributions G A α are depicted in Table (3), with a similar notation to Table (2). We have included G A 28 as a reflection of the general freedom to redefinẽ Table 3: Contributions to A (4) in the non-supersymmetric case together with a related redefinition of T IJ ; see Ref. [15] for further details. The purely gdependent contributions toÃ (4) of course cannot be determined from Eq. (2.23). Eq. (2.23) entails the system of equations where the c α are given in Eq.
together with conditions on the β-function coefficients The conditions on T 1−6 in Eq. (2.29) were already derived in Ref. [15]. Reassuringly, the conditions Eq. (2.30) are satisfied by the coefficients in Eq. (2.16), and also by the redefinitions in Eq. (2.19). These six constraints in principle leave only 19 of the 25 independent coefficients in the two-loop β-function to be determined by perturbative computation.
It turns out that Eq. (2.23) is sufficient to determine the Yukawa or λ-dependent part ofÃ (4) up to three free parameters; here are the results for the case of dimensional regularisation: where β 0 is given in Eq. (2.5). Since A 6 only appears in Eq. (2.28) in the combination 4A 6 + 2A 28 , we have set A 6 = 0 in line with Ref. [15]. We note that under the redefinitions in Eq. (2.17), Moreover the effect of these redefinitions on the metric coefficients in Eq. (2.23) (as parametrised in Eq. (2.24)) may easily be computed using Eq. (2.10) as Using Eq. (2.19)), these results are easily seen to agree with Eq. (2.29).
It is remarkable that no knowledge of the "metric" coefficients T α is required to determine the A α in this fashion; of course the t i in Eq. (2.31), which define the "off-diagonal" fermion-gauge metric in Eq. (2.23), could be determined by a perturbative calculation if required, as was accomplished for the fermion-scalar case in Ref. [15]. The results in Eq. (2.31) will be used in Sect. 3 in a check of the three-loop β g .
In Ref. [15] the extension to three loops was accomplished by first inferring the threeloop Yukawa β-function for a chiral fermion-scalar theory, using the three-loop results derived in Ref. [30] for the standard model, combined with the results for the supersymmetric Wess-Zumino model. Such an approach will not work in the gauged case, unfortunately; the results of Ref. [30] are only for the SU(3) colour gauge group, which of course is not sufficient to determine how the three-loop Yukawa β-function depends on a general gauge coupling.

The three-loop gauge β-function
The three-loop gauge β-function was computed in Ref. [25] for a general gauge theory coupled to fermions and scalars. In this section we shall show that our result forÃ (4) is compatible with this result via Eq. (1.2). In fact, our result forÃ (4) determines the 16 terms in β (3) g with Yukawa couplings up to 4 (see later) unknown parameters. It is rather striking that the two-loop calculation of β λ ) have thereby provided so much information on a three-loop RG quantity. This is an example of the "3 − 2 − 1" phenomenon noted in Refs. [23,31]; namely that the gauge-gauge, fermion-fermion and scalar-scalar contributions to the metric G IJ start at successive loop orders.
In our notation, β where the G A α are implicitly defined in Table (3). The purely g-dependent terms are not determined in this analysis. It is then easy to show, using Eqs. (2.26), (2.31), that we can in the form, where β (1) g , β (2) g are given in Eq. (2.5). We notice that T (2) gg agrees with the result for σ in Eq. (2.11). T (3) gg takes the form Unfortunately we have no means of disentangling the separate purely g-dependent contributions inÃ 4 and in T gg β g , without a three-loop calculation; but all the Yukawa or λ dependent contributions match exactly. If then we would have T IJ symmetric at this order; but as demonstrated in Ref. [15], at three loops T IJ is not symmetric even for a pure fermion-scalar theory for a general renormalisation scheme.
Had we not known β g then it would have been determined by Eq. (3.2) up to the four parameters consisting of the two coefficients in T (2) gy and the two coefficients in T

The supersymmetric case
Here the analysis is extended to a general N = 1 supersymmetric gauge theory, which may in principle be obtained from the general non-supersymmetric theory discussed in Sect. 2 by an appropriate choice of fields and couplings. Such a theory can of course be rewritten in terms of n C chiral and corresponding conjugate anti-chiral superfields, and indeed perturbative computations are enormously simplified through the use of this formalism; moreover, in the light of the non-renormalisation theorem and the NSVZ formula [32,33] for the exact gauge β-function, the renormalisation of the theory is essentially entirely determined by the chiral superfield anomalous dimension γ (at least in a suitable renormalisation scheme). In this section we shall therefore start anew using results derived using superfield methods. Nevertheless, in Sect 5 we show that (at least up two loops) the results obtained using the two approaches match, as indeed they must.
The crucial new feature in the supersymmetric context is the existence of a proposed exact formula for the a-function [17][18][19]. This exact form was verified up to two loops in Ref. [20] for a general supersymmetric gauge theory, and up to three loops [15] in the case of the Wess-Zumino model. Moreover in Ref. [15] a sufficient condition on γ to guarantee the validity of this exact result was found and shown to be satisfied up to three loops; related considerations appear in Refs. [18,19], see later for a discussion. In this section we shall generalise this condition to the gauged case and check that it is satisfied up to three loops, using the results of Ref. [22].
The couplings g I are now given by g I = {g, Y ijk ,Ȳ ijk } withȲ ijk = (Y ijk ) * . The supersymmetric Yukawa β-functions are expressible in terms of the anomalous dimension matrix γ i j in the form where for arbitrary ω i j we define We also introduce a scalar product for Yukawa couplings 3 and it is further useful to define The gauge β-function is assumed to have the form where, with R A the gauge group generators, 6) and n V is the dimension of the gauge group. For gauge invariance we must have Under a change g → g ′ (g) = g + O(g 3 ) then in Eq. (4.5) assuming g ′ is independent of Y,Ȳ . For an infinitesimal change δf = f ∂ g δg − δg ∂ g f and δγ = −δg ∂ g γ. The NSVZ form for the β-function is obtained if (4.9) The resulting expression for β g originally appeared (for the special case of no chiral superfields) in Ref. [29], and was subsequently generalised, using instanton calculus, in Ref. [32].
(See also Ref. [33].) We note here that this result (called the NSVZ form of β g ) is only valid in a specific renormalisation scheme, which we correspondingly term the NSVZ scheme. The exact expression generalises one and two-loop results obtained in Refs. [34][35][36]. These results were computed using the dimensional reduction (DRED) scheme; though in any case, the DRED and NSVZ schemes only part company at three loops [39].
The one and two-loop results for γ are given by [37,38] γ (1) = P , where P and S 1 are defined by We use here the notation and conventions of Ref. [22].
In the supersymmetric theory Eq. (1.2) is assumed to now take the form (with a similar equation for dȲÃ). We have written the RHS in terms ofβ g , effectively absorbing the factor f (g) in Eq. (4.5) into T Y g and T gg . We omit potential β Y terms in the first of Eqs (4.12) since are not necessary to the order we shall consider. For N = 1 supersymmetric theories there is, at critical points with vanishing β-functions, an exact expression for a [17] in terms of the anomalous dimension matrix γ or alternatively the R-charge R = 2 3 (1 + γ). Introducing terms linear in β-functions there is a corresponding expression which is valid away from critical points and this can then be shown to satisfy many of the properties associated with the a-theorem [18], [19]. For the theory considered here, with n C chiral scalar multiplets, these results take the form whereβ g is given by Eq. (4.5) and we require (4.14) For the remainder of this section we omit for simplicity the term involving H in Eq. (4.13); but return to it in Sect. 5. In Refs. [18] and [19] Λ, λ are Lagrange multipliers enforcing constraints on the R-charges. At lowest order the result for Λ and also the metric G obtained in Ref. [18] are equivalent, up to matters of definition and normalisation, with those obtained here. The general form forÃ given by Eq. (4.13) was verified up to twoloop order (for the anomalous dimension) in Ref. [20]. Λ may be constrained by imposing Eq. (4.12). Then We also have Hence if Λ, λ are required to obey where making the indices explicit Θ • dȲ → Θ i j,klm dȲ klm and θ → θ i j , Eq. (4.13) then satisfies Eq. (4.12) if we take Here T gY = 0. However from Eq. (4.14) which may be used to write Eq. (4.18) in equivalent forms with non-zero T gY .
A related result to Eq. (4.17), with effectively Θ, θ = 0, is contained in Ref. [18] and also discussed in Ref. [19]. For supersymmetric theories, satisfying Eq. (4.17) is consequently essentially equivalent to requiring Eq. (4.12), although terms involving Θ are necessary at higher orders. However, the work of Refs. [18,19] implies that at least in the pure gauge case, there may be renormalisation schemes in which θ may be set to zero. It is striking that only minor modifications to the condition proposed in Ref. [15] are required for the extension to the gauged case.
The condition (4.17) does not fully determine λ, θ since we have the freedom for arbitrary µ. There is also a similar freedom in Λ, Θ.
At lowest order Θ, θ do not contribute so that (4.17) becomes and we may simply take from Eqs. (4.10), (4.11) At the next order we require sinceβ g (1) = Q, with Q as defined in Eq. (4.6). We may parameterise Λ (2) and Θ (1) by (4.11). We then find Eq. (4.24) requires, since Hence  T gg (2) = 4λ n V g 3 Q . (4.30) As a consequence of (4.20)λ is arbitrary. The computation in Ref. [20] (specialising the DRED version of Eq. (2.11) to the supersymmetric case; and adjusting for the differing definition of the "gg" metric) for T gg (2) fixes 31) in this scheme.
At third order we require now in order to satisfy Eq. (4.17) where we write  (4). Here a "blob" represents an insertion of the one-loop anomalous dimension. The 3-point vertices alternate between Y andȲ . As an example, G Λ 6 represents a contribution and Eq. (4.4) then implies a contribution to (Ȳ Λ (3) ) of the form Here P, S 1 are given in Eq. (4.11) and S 2 is defined by Similarly we write where the G Θ α are shown diagrammatically in Table ( 5). A term in S 1 is apparently possible in θ (2) but is excluded since there is no contribution to γ (3) involving g 2 QS 1 . As a consequence of (4.20) the resulting equations depend only on 2λ 3 +θ 3 , 2λ 4 + θ 4 .
We expand the three-loop anomalous dimension as with and with Q, P , S 1,2 as defined in Eqs. (4.6), (4.11), (4.36). The remainder of the distinct tensor contributions are depicted in diagrammatic form in Table ( 6). The basis for γ (3) is restricted by the absence of one particle reducible contributions such as P 3 , P 2 C R , S 1,2 P , P S 1,2 .
Using Eqs. (4.10), (4.25) in Eq. (4.32) leads to a large number of consistency equations which constrain γ (3) . If g = 0 they reduce to which requires These results were obtained in Ref. [15]. The other special case is for Y,Ȳ = 0 when In this case applying Eq. (4.32) with Λ, Θ → 0 it is necessary to require the conditions as well as The relations in Eq. (4.41) were obtained in Refs. [18,19].
We are therefore obliged for consistency to use the result for the anomalous dimension corresponding to this NSVZ scheme. The required transformation was presented in Ref. [39] and its effect on γ (3) given in Ref. [40]. In fact it is only γ 17 and γ 22 which are affected.
In the Wess-Zumino case considered in Ref. [15] the existence of an a-function satisfying Eq. (1.2) implied that γ 1 − 2γ 2 − γ 3 was an invariant (in a sense described in Ref. [15]) but did not impose a specific value; thus showing that Eq. (4.17) is sufficient but not necessary. We might expect similar remarks to apply to the other conditions in Eq. (1.2). It is all the more striking that these conditions are in fact satisfied by the anomalous dimension as computed.
We may count the independent parameters in the anomalous dimension as we did in Section 2 for the Yukawa β-function. The essential Eqs. and also assuming δβ g is given in terms of δγ in accord Eq. (4.5), for  In the g = 0 case, the only coefficient in γ (3) with a κ-dependence, γ 4 , corresponds to a non-planar graph. In the general case there is no such obvious association between non-planar Feynman graphs and coefficients in γ (3) with κ-dependence (evaluated using DRED). However, an intriguing observation is that a redefinition given by choosing where are the contributions corresponding to the Feynman diagrams shown in Table (7). The implication is that there is a scheme in which the κ-dependent terms in γ (3) are generated solely by non-planar diagrams.

Reduction of non-supersymmetric results to supersymmetric case
In this section we shall check that the a-function obtained using the methods of Section 2 for a general theory is compatible, upon specialisation to the supersymmetric case, with the a-function presented in Section 4 (at least up to two loops). The reduction of the non-supersymmetric theory presented in Section 2 to the supersymmetric case (with n ψ = n V + n C , n ϕ = 2n C ) may be accomplished by writing and with y a ϕ a = y i φ i +ȳ iφ i , where λ is the gaugino field.ŷ i andŷ i may be obtained fromȳ i and y i by interchanging the upper left and lower right 2 × 2 blocks of the 4 × 4 matrices. We also have and consequently, from Eq. (2.6), The scalar potential is now given by In making the reduction from the general theory to the supersymmetric case, we must start from two-loop β-functions corresponding to DRED, since the RG functions used in Section 3 were evaluated using this scheme; as we mentioned earlier, the DRED and NSVZ schemes coincide up to the two-loop order we are considering in this Section. We use the results given in Ref. [42], which may be obtained from the DREG results by a coupling redefinition as in Eq. (2.17) given by These changes are a consequence of making the transformations Eq. (5.6) and also upon A (3) and A (2) respectively in Eq. (2.12). Presumably the transformation in Eq. (5.9) represents a part of the two-loop transformation from DREG to DRED (namely the Yukawa dependent contribution to the transformation of g). To the best of our knowledge this has not been computed in full, though results have been given for the pure gauge case in These coefficients correspond to a three-loop calculation (see Eq. (2.23)) and, in view of Eq. (4.47), depend on the value of γ 17 , which has a different value for the NSVZ scheme than for DRED. It is beyond the scope of this article to consider how Eq. (5.14) would be modified within DRED or indeed within DREG. Since our whole approach is predicated on the NSVZ scheme, it would probably be naive to assume that the DRED form of Eq. (5.14) would be obtained simply by using the DRED result for γ 17 .
Eq. (5.11) extends the result of Eq. (7.30) in Ref. [15] (with a = 3α − 1 12 ) to the gauge case-once again, modulo pure gauge terms which are not captured by the methods used in Section 2. We see again the ambiguity in the form ofÃ expressed in general by Eq. (2.27).
Of course this check is guaranteed to work but nevertheless given the indirect manner in which we have obtainedÃ and the possibility of subtleties regarding scheme dependence, it is satisfying to "close the loop" in this fashion.
Finally, we remark that although the form forÃ presented in Eq. (5.11) is appealingly simple (arguably even more so than Eq. (4.13)), the obvious extension to higher loops does not appear to be viable.

Conclusions
In this article we have extended the results of Ref. [15] to the case of general gauge theories. In the non-supersymmetric case we have constructed the terms in the four-loop a-function containing Yukawa or scalar contributions, using the two-loop Yukawa β-function and oneloop scalar β-function. Our main result here is Eq. (2.26) with Eq. (2.31). This enabled a comparison with similar terms in the three-loop gauge β-function. In general, as a consequence of the properties of the coupling-constant metric, one can obtain information on the (n + 1)-loop gauge β-function from the n-(and lower-) loop Yukawa β-function and the (n − 1) (and lower) loop scalar β-function. This is reminiscent of the way in which the (n + 1)-loop gauge β-function is determined by the lower order anomalous dimensions in a supersymmetric theory, via the NSVZ formula.
In the supersymmetric case we have given a general sufficient condition for the exact a-function of Refs. [17][18][19], given in Eq. (4.13), to be valid, and shown that it is satisfied by the three-loop anomalous dimension. This condition is displayed in Eq. (4.17) and is our main result for the supersymmetric case.
One feature of interest is that Eq. (4.17) imposes extra conditions on the anomalous dimension beyond the mere requirements of integrability from Eq. (1.2); but which are nevertheless satisfied by the explicit results as computed. Indeed we remark here (without giving further details since it is beyond our remit in this article on the gauged case) that we have observed similar features in the Wess-Zumino model at four loops, using the results of Ref. [41].
These properties certainly hint that there might be some underlying reason why Eq. (4.17) must be satisfied; it would be interesting to explore this further. If this were indeed the case, one could imagine exploiting Eq. (4.17) to expedite higher-order calculations of the anomalous dimension such as the full gauged case at four loops; possibly combined with additional information such as the necessary vanishing of γ in the N = 2 case. Unfortunately, a preliminary check indicates that these constraints are far from sufficient to determine γ completely, even at three loops; and therefore a considerable quantity of perturbative calculation would still be unavoidable.
Finally, in Ref. [15] we explored in some detail the freedoms to redefine the various quantities we have considered, and it would be interesting to extend these discussions to the current gauged case. In particular it would be useful to extend Eq. (4.17), which in its current form is predicated upon the NSVZ renormalisation scheme, to a form valid for any scheme.