Scheme invariants in phi^4 theory in four dimensions

We provide an analysis of the structure of renormalisation scheme invariants for the case of $\phi^4$ theory, relevant in four dimensions. We give a complete discussion of the invariants up to four loops and include some partial results at five loops, showing that there are considerably more invariants than one might naively have expected. We also show that one-vertex reducible contributions may consistently be omitted in a well-defined class of schemes which of course includes MSbar.


Introduction
Beyond leading order it is well-known that the values of β-function coefficients are schemedependent. On the other hand one would expect that statements with physical meaning should be expressible in a scheme-independent way. A notable recent example is the issue of the existence of an a-function; i.e. a function which generates the β-functions through a gradient-flow equation. For this to be feasible, the β-function coefficients must satisfy a set of consistency conditions, which must clearly be scheme-invariant; as has been verified for various field theories in three [1], four [2] and six [3] dimensions. One can count the expected number of scheme independent combinations at each loop order as the difference of the number of β-function coefficients and the number of independent variations of coefficients; however the number of independent invariants actually found is considerably larger. This may be understood in a pragmatic way in terms of the structure of the expressions for the scheme changes of the coefficients; however a possibly deeper insight is afforded by Hopf algebra considerations. A general discussion of scheme dependence with a particular focus on one-particle reducible (1PR) structures was recently given in [5], and here the study of scheme invariant combinations was initiated with reference to the N = 1 scalar-fermion theory. The present paper is to be seen as a companion to a forthcoming article [6] where the ideas of scheme invariance and the relation to Hopf algebra will be explored in general and also exemplified for the case of φ 3 theory in six dimensions; our purpose here is to extend the discussion to φ 4 theory in four dimensions. We shall summarise results of Ref. [6] where necesary to render the present discussions self-contained. An additional complication in φ 4 theory is due to the existence of one-vertex reducible (1VR) graphs. These are one-particle irreducible (1PI) graphs which may be separated into two distinct portions by severing a vertex. They have no simple poles when using minimal subtraction and dimensional regularisation, and hence a vanishing β-function coefficient in this scheme. It would be convenient to be able to omit these coefficients from our considerations. Indeed we shall show that although we may if desired include such coefficients, we may also consistently confine our attention to a well-defined subset of schemes in whch these coefficients are absent.

One, two and three loop calculations
In this section we establish our notation and obtain the invariants up to three loop order (the first non-trivial case for φ 4 theory). We consider the action in d dimensions The anomalous dimension γ ij may be expressed as a series of two-point 1PI diagrams with 4-point vertices representing the contractions of couplings. Up to three loops we have where here and elsewhere we suppress indices as far as possible. We are neglecting here and elsewhere contributions from "snail" diagrams in which a bubble is attached to a propagator. Such contributions do not arise in minimal subtraction and will not be generated by redefinitions if the redefinitions themselves do not include such diagrams. The β-function β ijkl may then be decomposed into 1PI pieces together with one-particle reducible pieces determined by the anomalous dimension, in the form: withβ denoting the 1PI contributions and S 4 the sum over the four terms where γ is attached to each external line. Up to three loops the contributions toβ are given bỹ For later convenience we introduce the notation that g λ 3a is the graph corresponding to c 3a , and g γ 2 is the graph corresponding to d 2 , etc. We note that in Eq. (2.4) the graph g λ 3f is primitive in that it has no divergent subgraph.
Changes of scheme may be parametrised as g ′ijkl = (g + f (g)) mnpq C mi C nj C pk C ql (2.5) where C ij = δ ij + c ij .
(2. 6) f (g) and c(g) may be represented by similar diagrammatic series to those for the β-function and anomalous dimension respectively, but where we introduce the notation c X →δ X and d X →ǫ X . After a scheme change the β-function and anomalous dimension are also represented by a similar diagrammatic series, but with As explained in the Appendix (which in turn is a summary of the discussion in Ref. [6]), it is useful to parametrise the scheme change by v defined implicitly by Eq. (A.4). We assume that v is also represented by a similar diagrammatic series, but with c X → δ X and d X → ǫ X . At one and two loops we have At three loops we find using Eqs. (A.10), (A.11) Note that c 3e and c 3f are individually invariant-which in the case of c 3f follows immediately from the fact that it corresponds to a primitive graph. In deriving invariant combinations of coefficients it is important to note that We now start the search for these invariant combinations of coefficients at lowest (threeloop) order. A priori since at this order there are nine three-loop coefficients and five variationsδ 1 ,δ 2 1 ,δ 2 ,ǫ 2 ,δ 2R , one's naive expectation would be 9−5 = 4 invariants. However, the variations on the right-hand side of Eq. (2.9) are expressed in terms of only three independent quantities, X λλ 1,2 , X γλ 2,1 and X λλ 1,2R , and so in fact we should have 9 − 3 = 6 independent invariant combinations of three-loop coefficients. Indeed, we easily find from Eqs. (2.9) that are four independent invariant combinations (making a total of six invariants with the individually invariant c 3e and c 3f ).

The four and five loop calculations
The full list of four loop diagrams was presented in Ref. [4]. The anomalous dimension is given at this order by while the 1PI part of the β-function will be parametrised as In Eq. (3.2) the graph g λ 4s is the only primitive one. We find (again using Eqs. (A.10), (A.11)) variations of the four-loop cofficients given by for the one-vertex irreducible coefficients, for the 1VR coefficients and δd 4a =0, for the anomalous dimension coefficients. TheX λλ quantities are defined bŷ in other words as for X λλ in Eq. (2.10) but with the β-function quantities c X,Y replaced by hatted quantitiesĉ X,Y . Similar definitions apply to X λγ , etc, but with d X,Y replaced by hatted quantitiesd X,Y where relevant. Hereĉ 1 = c 1 ,ĉ 2 = c 2 ,d = d 2 , while the quantitieŝ c 3a etc are defined byĉ with δc 3a as defined as in Eq. (2.9), and similar expressions forĉ 3b etc, and alsod 3 . The additional terms in the hatted quantities derive from the first Lie derivative term on the right-hand side of Eq. (A.10).
The relations between theδ,ǫ and the δ, ǫ are given bỹ (3.8) Now again we look for invariants at this order. Note that c 4m , c 4n , c 4s , d 4a are individually invariant-which again in the case of c 4s follows immediately from the fact that it corresponds to a primitive graph. There are thirty four-loop coefficients whose variations are given in Eqs. (3.3), (3.4), (3.5); and eighteen variations up to the three-loop level, namelỹ δ 3a−3f,3aR,3bR ,ǫ 3 ,δ 3 1 ,δ 1δ2 ,δ 1δ2R ,δ 1ǫ2 ,δ 2 ,ǫ 2 ,δ 2R ,δ 2 1 ,δ 1 . We would therefore naively expect 30−18 = 12 invariants. However, the variations on the right-hand sides of Eqs. (3.9) We call these 18 invariants "linear". We also find three "quadratic" invariants which are a consequence of the relations respectively. Altogether we have found twenty-one invariants, considerably more than (in fact almost double) the twelve which might naively have been expected.
We note that one may derive a fourth identity which leads to an invariant is a linear combination of invariants already found in Eqs. (3.9), (3.10).
We now proceed to a very partial five-loop calculation. The number of diagrams at five loops is dauntingly high, so we have not undertaken a complete calculation of all the invariants. A natural place to start is with the five-loop anomalous dimension which has only eleven terms: (3.14) We find from Eqs. (A.10), (A.11) that their variations are given by The hatted X-type terms are defined in a similar manner to Eq. (3.6), i.e. by replacing β-function quantities c X,Y and d X,Y , in Eq. (2.10) by hatted quantitiesĉ X,Y , andd X,Y . The hatted coefficients are in turn defined in terms of the corresponding unhatted quantities in a manner similar to Eq. (3.7). However, in the case of four-loop anomalous dimension coefficients, we need to defined where δ ′ d 4a is defined as in Eq. (3.5), but with hatted replaced by unhatted quantities. This is simply a consequence of Eqs. (A.10), (A.11) and has not arisen until now simply because the three-loop variations in Eq. (2.9) are already expressed in terms of unhatted quantities.
However it proves impossible to construct an invariant combination purely of anomalous dimension coefficients and in fact we need to include some 1VR four-point contributions, depicted below: The variations of the corresponding coefficients are given by However we also find several quadratic invariants, namely where J denotes the frequently occurring combination defined by These owe their existence to relations like together with similar relations for 3b-3f , 3aR, 3bR; together with The number of invariants is as expected, since the eleven relations of the form Eqs. In the absence of a complete calculation, one may estimate the total number of invariants which will be found at five loops. Ref. [7] lists 124 1PI 5-loop 4-point diagrams, making 135 coefficients after including the 11 5-loop anomalous dimension coefficients. There are 67 independent variations at 5 loops, implying a naive expectation of 135-67=68 linear invariants. On the other hand there are 57 5-loop X-type terms (some of which of course appear in Eq. (3.15)), which following the argument explained at four loops implies an actual total of 135-57=78 linear invariants. But furthermore there are altogether 27 identities of the form Eqs. (3.22), (3.23), constructed from the one one-loop quantity, the three two-loop quantities and the nine three-loop quantities. This implies an additional 27 quadratic invariants making 105 invariants in total. As at four loops, there are considerably more invariants than might have been expected. One may also speculate on the possible existence of higher-order invariants based on higher-order Jacobi-style identities.

One-vertex reducible graphs
In this section we briefly discuss the issue of β-function contributions from one-particle redicible (1VR) graphs. It is well-known that no such contributions arise using minimal subtraction within dimensional regularisation (MS), as may easily be established by consideration of the diagram-by-diagram subtraction process. It would be convenient if when considering scheme redefinitions one could restrict attention to schemes which have the same feature. In fact, if we start from a scheme such as MS in which the β-function coefficients corresponding to 1VR graphs G R are zero, i.e. c G R = 0, it is clear from Eqs. (A.10), (A.11) that the simple conditions δ G R = 0 (4.1) will ensure that the redefined coefficients will also satisfy c ′ G R = 0. This relies on the fact that for L, L ′ loop graphs G, G ′ , with L + L ′ ≥ 3, if (in the notation of the appendix) [L G , G ′ ] contains 1VR graphs, then at least one of G or G ′ must itself be 1VR. We therefore have a simple all-orders prescription given by Eq. (4.1) for defining schemes with no 1VR contributions.
It is interesting to see what implications the conditions Eq. (4.1) have for the new couplings g ′ defined by Eq. (A.4). At one loop we simply findδ 1 = δ 1 . At two loops we findδ so that the condition for 1VI graphs is At three loops the relations betweenδ 3a and δ 3a have already been given in Eq. (3.8). It is easy to confirm using Eq. (4.2) that δ 2R = δ 3aR = δ 3bR = 0 corresponds tõ The emerging pattern is clear; the value forδ G R is the product of the δs for its 1VI subgraphs. At four loops we find Using Eqs. (3.8), (4.2) we find that δ G R = 0 up to this level corresponds to taking so that each four-loop 1VR δ is the product of the δs for its 1VI subgraphs, as expected.
It seems highly likely that this simple pattern persists to all orders, but we have not been able to construct a proof.
When considering the scheme invariants, we can therefore restrict ourselves to those schemes with c G R = 0. The counting of invariants is then slightly different. Upon setting c 3aR = c 3bR = 0 in Eq. (2.13), there are then just three invariant combinations, namely I . We have lost two coefficients (c 3aR and c 3bR ) and one independent variation (X λλ 1,2R ) and so we expect to lose 2 − 1 = 1 invariants. The pattern is similar at four loops; if we impose Eq. (4.4), then we have δc 4aR−4gR = 0 and so we can can consistently set c 4aR−4gR = 0 in Eq. (3.9). We now have 23 coefficients and the 14 variationsδ 3a−3f ,ǫ 3 ,δ 3 1 ,δ 1δ2 ,δ 1ǫ2 ,δ 2 ,ǫ 2 ,δ 2 1 ,δ 1 , leading to a naive expectation of 23-14=9 invariants. On the other hand, out of the original eighteen linear invariants in Eq. (3.9) we are left with eleven invariant linear combinations, plus the four individual invariants, making 15. Again this is as anticipated, since we have lost the seven coefficients c 4aR−4gR and the four independent variations X λλ 1,3aR , X λλ 1,3bR , X λλ 2,2R and X λγ 2R,2 so we lose 7 − 4 = 3 invariant linear combinations. Furthermore it is clear that in the 1VI case only one of the identities in Eq. (3.11) remains, and consequently only one of the quadratic invariants in Eq. (3.10) survives. The total number of invariants is therefore 16; once again, almost double the naively expected number.
Finally we can consistently set c 5aR = c 5bR = c 5cR = 0 in Eq. (3.19), to obtain a invariant constructed solely from anomalous dimension coefficients (4.7)

Relation with Hopf algebra
Scheme invariants may be described graphically by adopting and extending rules described by Panzer [8] using the Hopf algebra coproduct ∆ : G → G ⊗ G, where G is the vector space spanned by the set of conected 1PI superficially divergent graphs and the disconnected products of such graphs. The action of the reduced coproduct ∆ on a Feynman graph g ∈ G is defined by (5.1) Here g/g i denotes the graph obtained from g by contracting each connected 1PI graph in the subgraph to a single vertex, or a single line if the connected 1PI graph has two external lines. Further details and a general discussion will be presented in Ref. [6], but this brief overview is sufficient for our present purposes. The invariants of Eqs. (3.9), (3.10) and (3.19) correspond to combinations of graphs with a symmetric coproduct, following the general results of Ref. [6]. We readily derive the following useful results: At three loops and at four loops we have for the 4-point graphs and for the 2-point graphs At five loops, the basic co-products are At three loops, the coproducts for g λ 3e and g λ 3f are cocommutative and zero respectively, corresponding to the individual invariance of c 3e , c 3f . Corresponding to the invariants in Eq. (2.13) we have the cocommutative combinations where The combinations of coefficients in the invariants of Eq. (2.13) maybe obtained [6] by substituting g λ i → c i , g γ j → d j and multiplying by S i , S ′ j respectively, where S i are the symmetry factors for the 4-point graphs,and S ′ i those for the 2-point graphs. The relevant symmetry factors are given by At four loops, the coproducts for g λ 4m , g λ 4n and g γ 4a are cocommutative and that for g λ 4s is zero, corresponding to the individual invariance of c 4m , c 4n , c 4s and d 4a . Corresponding to the invariants in Eq. (3.9) we have the cocommutative combinations Here, rather than give explicit expressions on the right-hand side, we use C (l)L i ∈ G ⊗ s G to denote l-loop cocommutative coproducts corresponding to linear invariants. Since their exact form is not especially significant, we relegate the full expressions to Appendix B. The noteworthy new feature here is the necessity sometimes to add quadratic terms, not of course present in the original invariants of Eq. (3.9), in order to obtain co-commutative results. The need for this is not currently understood in detail, but is also observed in φ 3 theory [6].
Corresponding to the quadratic invariants in Eq. (3.10) we have The relevant graph combination corresponding to the additional invariant in Eq. (3.13) may be derived from those already given and hence is not displayed here. Here we use C (l)Q i ∈ G ⊗ s G to denote l-loop cocommutative coproducts corresponding to quadratic invariants. The coefficients of the linear invariants in Eq. (3.9) may be obtained from the linear terms on the left-hand side of Eq. (5.9) by substitutions similar to those described at three loops after Eq. (5.6). Likewise, the coefficients of the quadratic invariants in Eq. (3.10) may be obtained from the quadratic terms on the left-hand side of Eq. (5.10) by similar substitutions. Here the relevant symmetry factors are given by together with those in Eq. (5.8). We also find corresponding to Eq. (3.19) Corresponding to the quadratic invariants in Eq. (3.20), we find

a-function considerations
A good deal of effort has been invested in recent years [9][10][11][12] on the search for an a-theorem, a generalisation of Zamolodchikov's two-dimensional c-theorem [13] to four dimensions (or indeed to other dimensions higher than two [14,[16][17][18][19]). From our point of view the crucial development is the demonstration that the β-functions in theories in four and six dimensions obey a gradient flow equation similar to one which plays a critical role in the derivation of the c-theorem [20][21][22][23]. These gradient flow equations often place constraints relating the β-function coefficients, as has been shown for four-dimensional gauge theories [25] and six-dimensional φ 3 theories [26] (similar gradient flows have been demonstrated in three dimensions [27][28][29] though here the theoretical underpinning has not yet been provided). Our purpose in this section is to apply the same considerations to our four-dimensional φ 4 theory where we are able to confirm our results using the results available to a high loop order. We start by presenting the basic results in general notation in the interests of clarity and brevity. For a theory with couplings g I , the corresponding β-functions are defined by The results of Ref. [22] then imply the existence of a function A such that where ∂ I ≡ ∂ ∂g I and where g IJ is an arbitrary symmetric matrix. At lowest order we have an a-function given by and Eq. (6.2) simply implies 3A (4) (the factor of 3 on the right-hand side derives from the multiplicity factor of S 3 for the corresponding term in the β-function). At the next order we have and now Eq. (6.2) entails Here T (4) represents the coefficient of the single fourth-order metric term. The figure below shows the contraction of this with a dg (represented by a cross) and a β (1) (represented by a diamond).
In Eq. (6.8) there are two equations and three unknowns resulting in one residual free parameter. This corresponds to the invariance under reflecting the freedom described by Eq. (6.4) at lowest order (with g IJ = g (3)δ IJ ). The six-loop a-function is given by The values of the coefficients may be extracted from Ref. [4] and are given at one and two loops by and at three loops by The solution of Eq. (6.13) is then Here we have nine equations for ten unknowns, again resulting in one free parameter. This corresponds to the invariance under 3 + 4g (4) , A 4 → A 4 + g (4) , A 5 → A 5 + 4g (4) , T 3 + 8g (4) , (6.17) reflecting the freedom under Finally, the seven-loop a-function is parametrised as These seven-loop vacuum diagrams were given in Fig. 6 of Ref. [31] and we have retained their ordering (similarly, the five and six loop vacuum diagrams were depicted in their Figs. 4 and 5 respectively). Since there are 24 6-loop metric contributions, we have introduced a compact notation to avoid depicting them all individually. Eq. (6.20) shows the sixloop vacuum diagrams; seen already in Eq. (6.11), but now with some vertices labelled. We introduce the notation T  The number of T -type contributions is the number of distinct ways of selecting an ordered pair of vertices from the diagrams shown in (6.20), namely 24.

Differential operators for scheme changes
Following the general considerations of Ref. [6] we may define differential operators for all λ, r.
In the case of φ 4 theory we find at lowest order and at next-to-leading order Note that here we suppress the label r in the case of the one-loop β-function and the two-loop γ-function where there is only one coefficient.
The Y λlr and Y γlr defined according to Eq. (7.2) satisfy the commutation relations with, correspondingly, the commutation relations

Conclusions
We have shown how scheme changes in φ 4 theory may be analysed within a compact and efficient framework. In particular we have derived the full set of scheme invariants up to four loop order and shown that their number is consistent with general expectations, though considerably higher than might be expected from a naive counting. In particular we have identified the existence of quadratic invariants which would be missed in a naive counting. Furthermore, we have shown that in the context of the Hopf algebra approach to renormalisation, each invariant is associated with a cocommutative combination of graphs.
We have also considered the construction of the a-function generating the β-functions up to four-loop order via a gradient flow equation. In particular we have analysed the consistent conditions which guarantee this construction, again showing that their number is as expected and furthermore that, as expected, they may be expressed in terms of linear combinations of the scheme invariants. Finally we have considered one-vertex reducible diagrams and shown that there is a natural family of schemes in which these do not contribute to the β-function.
Future work might explore the Hopf algebra connection further. Although the 1PI diagrams in the cosymmetric combination corresponding to linear invariants are determined by the general theory, this does not apply to the additional disconnected products of graphs which are also required. Moreover, the general theory still requires extension to encompass the quadratic invariants. Finally, at higher orders than we have yet considered there might be the possibility of cubic and higher order invariants.
We choose to parametrise the redefined coupling as g ′ = e vg g. (A.4) We then find using the easily proved result For our purposes it is useful to use this result in the form A useful identity for products of three graphs G 1 , G 2 , G 3 each of which only has a single subgraph is where for instance g 1 ⊂ G 1 and G 1 /g 1 =ḡ 1 , similarly for G 2 and G 3 . It wll be noticed that the expression simplifies when for instanceḡ 1 = g 1 . Also the result for the case when G 3 (for instance) has no subgraphs may be obtained simply by removing terms involving g 3 andḡ 3 .