Anomalous dimensions for $\phi^n$ in scale invariant $d=3$ theory

Recently it was shown that the scaling dimension of the operator $\phi^n$ in scale-invariant $d=3$ theory may be computed semiclassically, and this was verified to leading order (two loops) in perturbation theory at leading and subleading $n$. Here we extend this verification to six loops, once again at leading and subleading $n$. We then perform a similar exercise for a theory with a multiplet of real scalars and an $O(N)$ invariant hexic interaction. We also investigate the strong-coupling regime for this example.


Introduction
Renormalizable theories with scale invariant scalar self-interactions exist in four (φ 4 ), six (φ 3 ) and three (φ 6 ) dimensions. There has been considerable recent interest in the latter, in particular in theories involving complex scalar fields and a U(1) invariance, with (φ * φ) 3type interactions 3 . The anomalous dimension of the operator φ n , γ φ n , was calculated at the two loop level (in usual perturbation theory in powers of coupling constant λ) in Ref. [5] for the U(1) invariant pure scalar theory, and the result compared with a semiclassical calculation valid to all orders in the product λn. Of particular interest in this context is large n, because large charge operators are of peculiar relevance in conformal field theory. Also, amplitudes corresponding to many external lines are increasingly relevant in particle physics phenomenology, as collider energies increase, so insights gained by the study of them in simpler theories is worthwhile in itself. Agreement was found in . Ref [5] between perturbative and semiclassical results at the level of the leading and sub-leading terms in an expansion in powers of 1/n. Here we extend the straightforward perturbative calculation to the six loop level and once again obtain agreement with the semiclassical calculation for the two leading terms in the same expansion.
In Ref. [8], the calculations of Ref. [5] were extended from the U(1) to the O(N) case. Accordingly, we also perform our perturbative checks up to six loops for the semiclassical O(N) result as well. Furthermore, the U(1) semiclassical result was compared with an effective field theory valid at large λn. Accordingly we also examine the semiclassical O(N) results at large charge and find that we can obtain exact results for the N-dependent part of the coefficients in a large-charge expansion.
The paper is organised as follows: In Section 2 we describe the semiclassical calculation in the U(1) case, following Ref. [5], and then compare with the perturbative(i.e. small λn) results at four and six loops. In Section 3 we discuss the extension to the O(N) case as in Ref. [8], and perform a similar perturbative comparison. In Section 4 we describe the large charge limit and show how to compute the N-dependent parts of the coefficients in the large charge expansion. We offer some concluding remarks in Section 5. Finally in the Appendix we give a pedagogical description of the various methods used in our computation of the four-loop and six-loop Feynman diagrams involved in our perturbative check.

The U (1) case
The lagrangian of the theory is We shall be using dimensional regularisation with d = 3 − ǫ. The agreement between the semiclassical and perturbative results is expected to hold at the conformally invariant fixed point. However, because the β-function starts at two-loop order in d = 3, the theory is conformally invariant up to O(λ), and this is already sufficient for the agreement of the leading and subleading terms in n. The scaling dimension ∆ φ n is expanded as (returning to general d for the present, in order to facilitate the later discussion of convergence issues) For the leading and subleading terms in n, knowledge of ∆ −1 and ∆ 0 is sufficient. The semiclassical computation is performed by mapping the theory via a Weyl transformation to a cylinder R × S d−1 , where S d−1 is a sphere of radius R; where the Rφ * φ term (R being the Ricci curvature) generates an effective m 2 φ * φ mass term with m = d−2 2R . It was shown in Ref. [5] that stationary configurations of the action are characterised by a chemical potential µ, It was further shown in Ref. [5] that ∆ −1 may be written (2.5) (For convenience we give the results for d = 3 in Eqs. (2.3), (2.5).) Expanding to quadratic order around stationary configurations results in an action with two modes ω ± given by where is the eigenvalue of the Laplacian on the sphere. The dispersion relation for ω + describes a "gapped" mode, while that for ω − describes a "Type I" (relativistic) Goldstone boson [6]. The one-loop correction ∆ 0 is then determined by the fluctuation determinant corresponding to this quadratic action, which is given by where is the free theory dispersion relation, is the multiplicity of the laplacian on the d-dimensional sphere, and where ω ± are defined in (2.6). It was shown in Ref. [5] that after analytic continuation to negative d, we may obtain a regularised form for ∆ 0 convergent for d → 3, and we obtain in d = 3 where µ, ω ± are given by Eqs. (2.3), (2.6) (with now m = 1 2R ) and where is defined by subtracting positive and zero powers of l in the large-l expansion of Eq. (2.8) so as to give a convergent sum. With a slight abuse of notation, we use the same notation ∆ 0 (λn) for both the unregularised and regularised forms of the fluctuation operator.
The results for the various diagrams in terms of a basic set of Feynman integrals is shown  Table 1 (a factor of κ 2 is also implicitly assumed for each graph). The notation for these integrals conforms to Ref. [7]. Using the results for these integrals as listed in that paper, and also recapitulated in the Appendix, the total for the four loop anomalous dimension is thus (remembering to multiply the simple pole contribution by a loop factor of four)   [5].
We now turn to the six-loop calculation. At this loop order we focus from the outset on the contributions leading and subleading in n. The leading order six-loop contributions come solely from the diagrams depicted in Fig. 2 (of course these also produce contributions of lower order in n). Once again, the extraction of the poles in ǫ from these diagrams is described in some detail in the Appendix; the small black circles at the vertices will be explained in that context. The next-to-leading contributions six-loop contributions come The resulting simple poles for each diagram are tabulated in Table 2, together with the The contribution to the six-loop anomalous dimension from the diagrams in Figs. 2, 3 is then obtained by adding the products of corresponding symmetry factors and simple poles in Table 2 and multiplying by the usual loop factor of six and a factor κ 3 . The contribution at leading and next-to-leading n is given by κ 3 n 7 + −13 + 10 9 π 2 + 3 32 π 4 n 6 + . . . , (2.19) in agreement with Eq. (2.16).

The O(N ) case
In the four-dimensional case the U(1) computation of Ref. [5] was extended to O(N) in Ref. [8]. A similar agreement between the semiclassical and perturbative calculations was found, up to three-loop order in perturbation theory. It seems natural to perform a similar extension to O(N) in the case at hand, especially as the group theory and other results developed in Ref. [8] may straightforwardly be adapted to d = 3. Of course this represents a generalisation of the U(1) calculation, since the latter may be recovered as the special case N = 2; but now we may also wish to consider the limit of large N, for example.
In the O(N) case we have a multiplet of fields φ i , i = 1 . . . N, and the Lagrangian is now As shown in Ref. [8], the fixed-charge operator of charge Q may be taken to be where T i 1 i 2 ...i Q is symmetric, and traceless on any pair of indices. The scaling dimension ∆ T Q is expanded in a similar fashion to Eq. (2.2) as As in the U(1) case, we initially work in general d. The semiclassical computation of ∆ −1 and ∆ 0 proceeds in a similar manner to the U(1) case, but now the chemical potential µ is related to the cylinder radius R by The computation of the leading contribution is entirely analogous to the U(1) case and is given by where F −1 is as defined in Eq. (2.5). As in the U(1) case, for simplicity we give in Eq. (3.4) the result for d = 3. The non-leading corrections ∆ 0 are once more given by the determinant of small fluctuations. There are two modes corresponding to those in the abelian case, with the dispersion relation in Eq. (2.6). In addition there are N 2 − 1 "Type II" (nonrelativistic) [6] Goldstone modes and N 2 − 1 massive states with dispersion relation with J l as defined in Eq. (2.7). We then find that ∆ 0 is given by Here n l defined in Eq. (2.10) is again the multiplicity of the laplacian on the d-dimensional sphere, and ω ± are defined in (2.6) but with R, µ now related by Eq. (3.4). As before, with a slight abuse of notation, after analytic continuation we replace ∆ where C 2,l , C 3,l were defined in Eq. (2.14), and Performing the summations, and combining Eqs.  We note that the U(1) result in the previous section may be obtained by setting N = 2 and making the substitution g 2 = 1 6 λ 2 . The contributions from individual diagrams for this case are shown in Tables 3 and 4. Factors ofκ 2 at four loops andκ 3 at six loops are implicit. As mentioned before, the U(1) results may be recovered by setting N = 2 and making the substitution g 2 = 1 6 λ 2 . Once again, after adding the diagrammatic contributions and including a loop factor of 4 and 6 respectively, the leading and subleading four and six loop contributions agree with the semiclassical result in Eq. (3.13). It is noteworthy that the N dependence in Eq. (3.13) involves purely powers of π 2 ; and this feature in fact appears to persist to higher orders. It would be interesting to be able to associate this with a generic topological property of the relevant Feynman diagrams.

Large gQ
In the U(1) case, the result for the anomalous dimension may be expanded for large λn and compared with the effective theory for the gapless Goldstone mode corresponding to ω − . In the O(N) case, we can do an analogous expansion for large gQ. Following Ref. [5],

Symmetry Factor
Simple Pole 2(a)     and (4.4), derive from a numerical fit to ∆ 0 as given by Eq. (3.8), following the procedure explained in Ref. [5] and in more detail in Ref. [13].
The values in Eqs. (4.3) were essentially given already in Ref. [5], after making allowance for the change from λn to gQ. The numerical coefficients c i in Eq. (4.4) are therefore the only new features of the O(N) case at large gQ. We note the intriguing fact that c 3/2 = 5c 1/2 . This fact and indeed the values of the remaining c i may be explained quite simply. It is convenient to consider an expansion of ∆ 0 in powers of v = Rµ, rather than gQ; of course in view of Eq. (3.4), large gQ implies large Rµ. We find from redoing the numerical matching with and with J 2 l as in Eq. (2.7), but with d = 3. We would now be able to reproduce Eq. (4.5) with Eqs. (4.7), (4.8), if we could perform the summations over l. However, it turns out that we can only make progress on this in the case of ∆ where B n are the Bernoulli numbers. In the second sum in Eq. (4.13), we have accounted for the fact that the series in Eq. (4.10), Eq. (4.12), start at l = 0, l = 1, respectively; of course this makes no difference in the first sum. We obtain the following expressions for the coefficients: recalling that B n = 0 for n odd, except for n = 1. Comparing with Eq. (4.8), we find surprisingly good agreement. Turning now to a 0 and a 2 , the cancellation of leading powers of l in Eq. (3.9) appears to guarantee the vanishing of these coefficients as observed in Eq. (4.8), even though the ζ-function sums defined by Eq. (4.13) do not give vanishing results for the v 2 and v 0 terms in Eq. (4.10). We have checked that for other functions sharing the crucial properties mentioned above, we similarly obtain a i = 0 for i ≤ 2 and even; and the ζ-function sums correctly give a i for i ≤ 1 and odd. However, a 3 remains a problem. There is no v 3 term in Eq.

Conclusions
Neutron stars, and high density quark matter can both be described in terms of a superfluid effective field theory for a Goldstone boson field [14] [15]. As explained in Ref. [5], relevant issues may also be addressed in terms of the relativistic theory of a complex scalar field φ with λ This theory has a conformal fixed In this paper we have extended the calculation of the anomalous dimension of the operator φ n embarked upon in Ref. [5] from two loops to four and six loops. We continue to find agreement between the straightforward perturbative (in λ 2 ) calculation and the results of a semiclassical calculation, along the lines explained in Ref. [13]. This agreement interpolates between large and small λn.
We performed similar calculations for an O(N) theory with (φ i φ i ) 3 interactions, which includes the U(1) case described above as the special case N = 2. Here both semiclassical and perturbative approaches were pursued in Ref [8], for (φ i φ i ) 2 theory in d = 4 − ǫ, which similarly has a conformal fixed point with a coupling constant of O(ǫ). It turns out to be quite straightforward to adapt these calculations to the d = 3 − ǫ case, and once again we find that the two approaches interpolate seamlessly into one another. Finally in the O(N) case we have shown how to compute exactly the N-dependent parts of the coefficients in the large charge expansion. Figure 4: Two-and four-loop momentum integrals using "infra-red (IR) rearrangement". 4 This involves judiciously setting selected external momenta to zero, leaving a single momentum entering at one vertex and leaving at another, in order to obtain a more tractable integral. It will be useful to focus on G 4 for our pedagogical introduction. For convenience we have labelled the vertices of G 4 in Fig. 4 by A, B, C. We first consider the case where a momentum enters at A and leaves at B. The basic momentum integral is given by There is also a divergent two-loop subgraph, with a divergence I 2 , which needs to be subtracted to obtain a local result. We obtain The process of correctly subtracting the subdivergences is here denoted R. For more details see Ref. [11] where the procedure is well explained (with reference to the four-dimensional case). In general there may be several distinct ways of implementing the IR rearrangement. Any IR rearrangement which avoids the introduction of spurious IR divergences will give the same result for the final counterterm, after making the appropriate subtractions. In the case of I 4 , for instance, we may also consider the case where a momentum enters at C and leaves at B. The basic momentum integral is then given by and we now have As emphasised earlier, the same result is obtained for the counterterm I 4 . In general, in the process of IR rearrangement, the same entry and exit points must be used for the subtracted diagrams as for the original. For a different IR rearrangement of a given diagram, the pole terms for the original diagram and the subtracted diagrams will typically be individually different, but will combine to give the same total counterterm. In the current case, the subtractions are the same for the two IR rearrangements. The same overall result is nevertheless obtained for the pole term since both G 4 and G ′ 4 also have the same poles, though of course differ in their finite parts.
There is a final possible IR rearrangement, where the momentum enters at A and leaves at C. This requires a more careful treatment. In four dimensions one is familiar with the basic IR divergence from a double propagator; in the current case of three dimensions, the basic IR divergence is a double propagator followed by a single one, as shown in Fig. 5. This structure leads to an effective propagator Figure 5: Basic IR-divergent structure where the IR divergence in three dimensions is clearly revealed. It appears in this third IR rearrangement which consequently leads to a spurious IR divergence. We may avoid this spurious divergence by using the R * procedure, which augments the R procedure with a subtraction for the IR divergences [12]. We start by considering the basic two-loop IR-divergent diagram in Fig. 6. This is given by the divergence coming from Γ(d−3) where the positive sign for d signals the infra-red nature of the divergence. The IR subtraction for this simple, single IR divergence is consequently to replace Eq. (A.8) by The IR divergence is cancelled in Fig. 6 when (A.10) is used to replace (A.8). Just as we saw earlier for the case of IR rearrangement combined with the standard R procedure, the Figure 6: Two-loop IR-divergent integral same process must be applied to the subtracted diagrams. Expressed diagrammatically, denoting the IR-subtracted propagator of Eq. (A.10) by a line with a box, we have Here we denote the momentum entrance and exit points by a small black circle. A diagram with a single black circle has coincident momentum entrance and exit points and vanishes in dimensional regularisation. Once again, the same result is obtained for I 4 .
The corresponding pole terms for the remaining four-loop diagrams in Fig. 4 are given by where G 22 and G 4bbb are implicitly defined in terms of G-functions.
We now turn to the six-loop computation, for which the diagrams are shown in Figs. 2 and 3. These are again logarithmically divergent. Once again we use IR rearrangement, so that we retain just a single incoming and outgoing momentum; in all our examples, this momentum may be thought of as entering at the φ n vertex (i.e. the lozenge) since it turns out that this ensures wherever possible that the result may readily be expressed in terms of G-functions. The momentum leaves at the vertex marked by the small black circle. As observed earlier in the case of I 4 , the choice of momentum entrance and exit is not unique; but once made, must also be used for the subtracted diagrams. We have made the choice of momentum exit so as (for simplicity) where possible to avoid introducing infra-red divergences, either in the six-loop diagram itself or in its subtractions ; even though such IR divergences may be accommodated using the R * procedure. In the case of diagrams with a structure such as Fig. 5, the potential IR divergence may be avoided by choosing the central vertex in Fig. 5 as the exit for the momentum. It will be observed that this has been done in Fig. 2 Fig. 3(b) has been made to avoid an IR divergence in the two-loop subtracted diagram. The process of infra-red rearrangement also reduces the number of independent integrals; for instance, with the choice of momentum exit indicated, Figs. 3(e) and 2(c) correspond to the same integral, despite the structure Fig. 5 being reversed in the latter diagram. Figs. 3(d) and 2(b) look different, since the lower single loop is on different sides of the momentum exit point. However, either of the alternative IR rearrangements using one of the other (φφ * ) 3 vertices as exit would make the pair of diagrams look identical, and therefore would demonstrate that Figs. 3(d) and 2(b) produce the same counterterm after subtractions. Of course we could have used one of those alternative IR rearrangements, but at the expense of being obliged to use the (slightly more complicated) R * procedure.
In all the cases mentioned so far, IR rearrangement leads to graphs which may easily be evaluated in terms of G-functions. For those where the simple R procedure is sufficient, we find Here we use I 2a (for instance) to represent the counterterm resulting (after subtraction of subdivergences) from the Feynman integral G 2a corresponding to Fig. 2(a), just as I 4 results from G 4 . We shall give a complete list of explicit expressions for the pole terms later, after discussing the range of general procedures required for the different classes of diagram. We emphasise that, as explained earlier, the expression for I 3d in terms of diagram plus subtractions for the IR rearrangement shown in Fig. 3(d) would be different from that shown explicitly for I 2b , but the final total would be the same. In just one instance, Fig. 2(c) (or equivalently Fig. 3(e)), the process of IR rearrangement inevitably introduces infra-red divergences due to the presence of two IR-divergent structures of the form Fig. 5. One of these must then be dealt with using the R * operation [12] explained earlier in the context of I 4 . We find Once again, diagrams with a single black circle have coincident momentum entrances and exits and vanish in dimensional regularisation.
In a couple of cases, namely Figs. 3(b) and 3(g), there is no IR rearrangement which leads simply to an expression in terms of G-functions, and we need to use an identity derived using the "integration by parts" trick, which enables us to simplify integrals of the form shown in Fig. 7 which occur as substructures in these graphs. In this diagram, α i , i = 1 . . . 6, represent the weights of the corresponding propagators. This identity is given here in diagrammatic form.