The masses of gauge fields in higher spin field theory on AdS(4)

Higher spin field theory on AdS(4) is defined by lifting the minimal conformal sigma model in three dimensional flat space. This allows to calculate the masses from the anomalous dimensions of the currents in the sigma model. The Goldstone boson field can be identified


Introduction
We consider the higher spin field theory constructed on AdS(4)space by lifting of the O(N) vector minimal sigma model from R(3) [1]. It results a nonlocal interacting renormalized QFT denoted HS (4) and defined by its n-point functions which can each be computed by a 1/N expansion. Such AdS field theory has the unconventional property that mass renormalizations of the fundamental fields and composite fields can be calculated perturbatively. This is made possible by the fact that the corresponding flat conformal field theory permits to calculate the anomalous part η of the conformal dimension ∆ by 1/N expansion and that the mass m 2 of the AdS field and ∆ of the dual conformal field are related, e.g. for symmetric tensor fields of rank l by Obviously for conserved currents J (l) (x) with exact dimension we obtain a vanishing mass corresponding to the gauge fields h (l) on AdS space. Lifting of the flat conformal field theory is done with the bulk-to-boundary propagator (say, in the scalar case, see [2] for the general case) which satisfies the free field equation with m 2 as in (1) and D z a covariant second order differential operator. It is this equation which transforms dimension into mass. If the symmetric tracesless tensor current J (l) ( x) assumes an anomalous dimension η(l) (which it does for l ≥ 4) which can be 1/N expanded as we obtain masses for h (l) from (1) and (5) M. Porrati [3] has shown that a Higgs phenomenon is possible for a graviton to assume a mass. A massless symmetric tensor representation of the conformal group [∆, l] with ∆ = d − 2 + l| d=3 = l + 1 (8) which is massless, appears from an irreducible massive representation in the limit The second representation [l+2, l−1] is identified with the Goldstone field. L. Girardello et al.[4] propose to create the Goldstone field from tensoring the conserved current [l−1, l−2] corresponding to J (l−2) or h (l−2) with the scalar field α( x) on R(3) resp. σ(z) on AdS(4) of dimension two (all in the free field limit) which contains the Goldstone field representation for s = 1, n = 0. In the minimal sigma model the operator product of the current J (l) with the scalar field α contains by expansion the currents J (l,t) with dimension d − 2 + l + 2t, t ∈ N, (in the free field limit). Thus the Goldstone field of J (l) is J (l−1,1) . A Higgs mechanism for producing the masses of the gauge fields is possible therefore except for the graviton since in (10) l = 2 is excluded (the representation [1, 0] does not occur but is eliminated in favour of the dual representation [2,0] by the boundary condition of AdS (4)).
The gauge fields contain a traceless symmetric part h (l) and companion fields that are dynamically irrelevant. In HS(4) there exists a bilocal biscalar field B(z 1 , z 2 ) [1] which by operator product expansion decomposes into all h (l) , l ∈ 2N, and the scalar field σ(z) at leading order in N, and further operators at order 1/N and higher. The starting point for a calculation of the masses m 2 l is therefore the AdS four-point function The perturbative corrections to the bilocal fields Instead of the AdS Green function (11) we study the flat CFT Green function where b is defined from the Lorentz scalar O(N) vector fields ϕ( x) by the normal product This O(N) vector field is normalized to where δ is the conformal dimension of ϕ (we set d = 3 at the end) The first three terms in (16) are known [5].
Due to the contraction of O(N) vector indices the O(1) contribution to the Green function is from the graphs A 1 , A 3 with the result where the first (second) term describes the exchange of α ( ϕ 2 ), respectively, which are dual to each other in a representation theoretic sense. Expressions in (19) are The constants C 1,2 involve the coupling constant z squared between α and ϕ to leading order z 1 and are explicitly given as In CFT in flat space conformal invariance determines three-point functions up to a few normalizing constants. The same must be true then for complete exchange graphs. For the radiatively corrected (at the vertices) graph B com 2 of B 2 we obtain instead of (19),(21) where δ and κ are taken from (15)  can stay undetermined (see below).
In the subsequent section we shall decompose the O(1) contribution to the Green function (12) into "conformal partial waves", amplitudes for the exchange of irreducible conformal fields. The O(1/N) contribution to this Green function contains three different types of terms: 1. Power series in u and 1−v containing the same conformal partial waves as the order O(1). These imply a renormalization of the coupling constants in the exchange amplitudes; 2. Power series in u and 1 − v multiplied with logu. Consistency requires that conformal partial wave expansion of the power series yields the same exchange fields as the O(1) expansion. From the normalization we extract the combination where the field C is exchanged and the fields A, B, C form a vertex of the exchange graph. 3. Power series in u and 1 − v containing new conformal partial waves.
In the case of the Green function (12) the order O(1) gives the exchanged fields J (l) , l ≥ 2 even, and α. At O(1/N) there appear as new fields J (l,1) , which contain the Goldstone fields. Since we are interested here only in the anomalous dimensions of J (l) it suffices to extract the logu power series. These appear from the difference B com 2 − B 2 by expansion of the factor u κ and from four new graphs B 1 , B 3 , C 21 , C 22 , where the first two B 1,3 are obtained by crossing B 2 , C 21 is a box graph, and the last one, C 22 , is obtained by crossing C 21 The exchange graphs B 1 + B 3 are easily calculated and give ( [6], equ. (3.17), only the logu terms) From the box graph C 21 we get ( [7], only the logu terms) To obtain the box graph C 22 one can apply crossing explicitly by exchanging Inserting this into (34) we obtain the crossed graph contribution.

Deriving the anomalous dimensions
The Green function (12) can be submitted to a conformal partial wave decomposition by expressing it as a sum over exchange graphs (only the direct term) of α and J (l) at O(1), composites of two α and J (l,1) at O(1/N) and so on [8]. For example we consider exchange graphs of J (l,t) with conformal dimension ∆(l, t) Here we use an adhoc normalization This inversion formula was given in [8],(4.11), (4.12) in a more general version with two free parameters instead of only one (δ). At order O(1) we have the graphs A 1 + A 3 + B 2 . We skip the F α term and get for the remainder where a n,m = δ n,0 (δ m,0 We recognize immediately that a 0,0 = a 0,1 = 0 (44) which implies that a current J (l=0) which would be identical with ϕ 2 is not exchanged. The ansatz a n,m = ∞ l=2,even is easily solved first by setting n = 0 using the inversion formula (40) and then showing that it remains true for all n. The result is γ 2 l = 2((δ) l ) 2 (2δ − 1 + l) l for l ≥ 2, even = 0 for all other l (46) At order O(1/N) we obtain the relevant logu terms from

Conclusion
The result (58) for the mass squared of the gauge bosons at leading order of 1/N is surprisingly simple due to cancellations between the five graphs primarily. Contrary to the case of the four-point function of the scalar field α, the four-point function (12) does apparently not reduce to an elementary function at d = 3 (see (34)). The linear dependence of the mass squared on the spin l of the boson is of Regge trajectory type and suggests the existence of a string theory from which the higher spin field theory on AdS(4) can be derived.