ЯДЕРНАЯ ФИЗИКА, 2008, том 71, № 6, с. 1136-1142
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
THE GOLDSTONE FIELDS OF INTERACTING HIGHER SPIN FIELD THEORY IN AdS(4)
© 2008 W. Ruhl*
Department of Physics, Kaiserslautern University of Technology, Germany
Received August 20, 2007
A higher spin field theory on AdS(4) possesses a conformal theory on the boundary R(3) which can be identified with the critical O(N) sigma model of O(N) invariant fields only. The notions of quasiprimary and secondary fields can be carried over to the AdS theory. If de Donder's gauge is applied, the traceless part of the higher spin field on AdS(4) is quasiprimary and the Goldstone fields are quasiprimary fields to leading order, too. Those fields corresponding to the Goldstone fields in the critical O(N) sigma model are odd-rank symmetric tensor currents which vanish in the free-field limit.
PACS: 75.10.Jm
1. INTRODUCTION
If we move from the free-field limit N = to of the critical O(N) sigma model to the interacting conformal field theory CFT(3) by a renormalization group transformation [1, 2], gauge symmetry of the higher spin field theory HS(4) on the AdS(4) side is broken and a Goldstone field arises. Such Goldstone modes are proper dynamical degrees of freedom that must be represented also in the boundary conformal field theory. Its quantum numbers being known, we must try to find such field in the list of fields for CFT(3).
To derive such list one looks first for all quasiprimary fields in the free-field limit N = to. A simple algorithm tells us [3] how to choose the quasiprimary fields from all composites of the derivatives of the free massless fields and select the O(N) invariant ones from them. Symmetric traceless tensor fields are then always of even rank (odd-rank tensors belong to nontrivial O(N) representations and are eliminated). Then one perturbs these composites by switching on the interaction. They obtain this way anomalous dimensions. Several quasiprimary fields in the free theory have degenerate quantum numbers (tensor type and dimension). Certain linear combinations of them form eigenvectors with respect to the anomalous dimensions as eigenvalues. These are the interacting quasiprimary fields.
Among these quasiprimary fields in the free-field theory are exceptional elementary representations which belong to conserved currents, their divergence is zero. If we switch on the interaction, these currents
E-mail: wue_ruehl@t-online.de
obtain not only a (positive) anomalous dimension, their divergence is nonzero and gives a new quasipri-mary field. Their two-point function is proportional to their anomalous dimension. In this fashion we can obtain symmetric traceless tensor fields with odd tensor rank. The Goldstone degree of freedom on the boundary of AdS(4) is represented by these field operators (in de Donder's gauge). Thus the renor-malization group which looks continuous, produces a discontinuous jump in the field-theory Hilbert space when we leave the free-field point.
In this article we summarize the relevant properties of the O(N) invariant critical sigma model (Section 2). In particular, this section is devoted to the currents in this field theory. Section 3 deals with the Goldstone field in higher spin field theory HS(4). In this section we introduce the concept of quasiprimary and secondary fields in HS(4). Using this we can identify the Goldstone field both in CFT(3) and in HS(4). In the latter case its two-point function is given relying on AdS/CFT correspondence. We present in Section 4 some remarks on representation theory and close with some conclusions.
At present the status of AdS/CFT correspondence is that all field correspondences predicted have been established, but an access to the anomalous dimensions (equivalent to the masses) of the quasiprimary fields within HS(4) by a perturbative algorithm is still lacking.
2. THE O(N) INVARIANT CRITICAL SIGMA MODEL
We start summarizing the stable conformal sigma model. Its basic field is a scalar O(N)-vector field
v(x) and an O(N)-scalar field a(x) with the La-grangian
d3xdßp(x)dß p(x) +
{a(x! )a(x2 ))cft = (xj2)-s(a),
(6)
xi(x2))cFT = ôij (x212)-^), (7)
where the field dimensions 5 are
5(a) = 2 + n(a),
and the anomalous parts n are expandable as
<x £
n
r=1
Nr '
(8) (9)
(10)
interest in the present context. Moreover, the critical coupling constant can be expanded in the same way:
+ zc/2 J d3xa(x)p(x)p(x).
The auxiliary field a(x) serves as a Lagrangian multiplier for the constraint
p(x)p(x) = const. (2)
All conformal fields in a CFT can be obtained by an operator-product expansion of appropriate fields, and any field is either conformal (quasiprimary) or a derivative of a conformal (secondary) field. The latter fields can be ordered in classes according to the number n of derivations. An element of the conformal group maps a field of CFT into another one by a coordinate change
x —> Ax (3)
and a multiplier depending on x in general. Therefore, a derivative field transforms such that derivatives of the multiplier appear besides the multiplier itself. On the other hand, a local field on AdS(4) is covariant with respect to the isometry group of AdS space which is the same conformal group as in CFT but the multipliers are independent of the coordinate w in the conventional representation. The coordinate transforms as
w —► Aw. (4)
However, the difference between quasiprimary and secondary fields on AdS is visible in their two-point functions.
We are interested in the O(N) invariant sub-CFT generated by a(x) and the bilocal field [4]
6(xi ,x2)= N-1/2 ). (5)
We normalize fields by (x12 = x1 — x2)
£
r=1
Nr
with
zi
n4
(11)
(12)
The even-rank currents that tend to conserved currents in the free-field limit can be derived by operator-product expansion (which for four-point functions is almost the same as a conformal partial-wave expansion) of the four-point function
(b(xi,x3)b(x2 ,X4 ))cft (13)
in the limit
xi3 ^ 0,x24 ^ 0.
(14)
The 1/N expansion of (13) can be obtained from a graphical expansion so that to each order 1/Nr there contributes a finite number of graphs. This has been done in the literature a long time ago [5]. The order r = 0 can be evaluated exactly and is most conveniently expressed in terms of variables
xx
u
13^24 2 ™2 '
xx
xx
14^23 2 ™2
12 34
xx
12 34
as
(x22 x34)-0(")
+
(15)
(16)
+ C1Fa(u,v) + C2Fv2 (u,v)}
The first two terms arise from the disconnected graphs and the last two terms arise from an exchange graph of the a field in the channel (1,3) ^ (2,4). Both functions F are series in u and 1 — v. Whereas Fa represents a single conformal partial wave for the representation of the field a, the remaining three terms give by harmonic analysis conformal partial waves for the currents jl (x) for all l e 2N and no other fields [4]. That l = 0 is excluded follows from the fact that
1+v-S(v) + C2FV2 =0 for u = 1 — v = 0. (17)
The currents are traceless symmetric tensors of even rank (spin) l and at this order (i.e., r = 0) of 1/N they are conserved with dimension
5(jl) = d + l — 2 (= l + 1 at d = 3). (18)
At the next order 1/N the four-point function (13) involves box graphs which can still be analyzed. One group of terms (logu terms) correct the dimensions of the fields found at leading order [6, 4]
Only for a few leading powers the coefficients nr are known. We shall give them only for the fields of
5(a) =2 + n(a), m(a) =
16
3^2'
(19)
z
c
1
ö(jl ) = l + i + n(jl ),
(20)
ni (jl ) =
16 (l - 2) 3vr2(21 - 1)'
The currents jl are no longer conserved except for l = 2.
At the same order another group of terms can be analyzed in terms of conformal partial waves for the representations of the "twist currents" jl,t(x), l e e 2N, t e N with dimensions at leading order
5(jl't) = d + l — 2 + 2t (= l + 1 + 2t at d = 3),
(21)
which are also traceless symmetric tensors of even rank (spin) l. They are never conserved.
In order to determine their anomalous dimensions n(jl,t) one has to go to the order 1/N2 and extract the logu terms. This is rather involved and has not been done yet. It may be simpler to start from a (2t + 4)-point function, since these operators can also be obtained as normal products of
jl(xi)a(x2)a(x3) • • • a(xt+i).
(22)
For the later applications we need some technical tools. Contracting the tensor jl(x) with l d-vectors a (d = 3 of course), the result is denoted jl(x; a). From representation theory we know that its two-point function is
(jl(xi;a)jl(x2;b))cFT = N(x\2)-S(j> x (axi2 )(bxi2 )'
(23)
(ab) - 2-
i2
— trace terms
U u o (bxi2) 0=0 — 2-s-Xi2,
2
x
i2
(b')2 = b2,
j (xi; a)jl (x2 ; b))cFT = N (x^ )-S(j > x (25)
(24)
x(a2b2)1/2-
l! r<d/2-i / ab' \ 2l(d/2 — l)i 1 \(a2b2)1/2 )
(b(xi,x3)b(x2,x4))cFT = Nl(x2i2) s[jl\ab')1 x
(26)
2 2
2' (ab')2
(see [7]).
In the O(N) invariant sub-CFT of the conformal sigma model, which we can call the "minimal" sigma model, and after a normalization of all basic fields, N is a positive free parameter that we can assume to vary over the reals. Because we handle this model with a 1/N perturbative expansion, maybe only large N are relevant. By AdS/CFT this model is mapped on a local AdS field theory with coupling constant zc(N). Relevant are only small coupling constants, and these appear in an infinite number of coupling terms. Since the sigma model is renormal-izable, it seems that this field theory H S(4) is also well defined. However, the correspondence may be only classical and not apply directly to n-point functions including non-tree graphs.
3. THE GOLDSTONE FIELD IN HS(4)
We define the higher spin field hl(w) on AdS(4) as a symmetric double-traceless tensor field with respect to the tangential planes of AdS(4) and contract it therefore with l (d + 1)-vectors a resulting in
hl (w; a) : hl (w; a) = 0.
(27)
We decompose hl into two traceless fields [8]
a2
hl(w-a) =ipl(w;a) + — dl~2(w,a), (28)
2ao
where
ao =
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