Scaling dimensions of monopole operators in the CP N b 1 theory in dimensions

Transcription

1 PUPT-479 arxiv: v [hep-th] 9 Feb 06 Scaling dimensions of monopole operators in the CP N b theory in + dimensions Ethan Dyer, Márk Mezei, Silviu S. Pufu, 3 and Subir Sachdev 4,5 Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA 3 Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA 4 Department of Physics, Harvard University, Cambridge, MA 038, USA 5 Perimeter Institute for Theoretical Physics, Waterloo, Ontario NL Y5, Canada Abstract We study monopole operators at the conformal critical point of the CP N b theory in + spacetime dimensions. Using the state-operator correspondence and a saddle point approximation, we compute the scaling dimensions of these operators to next-to-leading order in /N b. We find remarkable agreement between our results and numerical studies of quantum antiferromagnets on two-dimensional lattices with SU(N b )globalsymmetry,using the mapping of the monopole operators to valence bond solid order parameters of the lattice antiferromagnet.

2 Contents Introduction Setup 6 3 N b = theory 8 4 /N b corrections 0 4. E ective action at quadratic level Eigenvalues of the integration kernels Kernels at q = The D kernel at q = The K kernels at q = Kernels for general q The D kernel The K and H kernels Numerical Results The large q limit 5 6 Conclusions 9 A Derivation of integration kernels 30 A. Evaluation of Fourier transforms A. Check of gauge invariance B Formulas for I B and I H 35 B. Results for q =0, / B. Results for arbitrary q C Asymptotic expansions 4 C. Small distance expansion of the Green s function C. UV asymptotic of the scalar kernel C.3 The mixed kernel C.4 Gauge field kernel C.5 Ultraviolet expansion in the CP N model C.6 Fourier transforms on S R

3 Introduction In + dimensions, pure U() gauge theory confines []. One can prevent confinement by introducing a su ciently large number N of massless matter fields, in which case the infrared dynamics is believed to be governed by a non-trivial interacting conformal field theory (CFT). Such CFTs arise quite frequently in the description of quantum critical points of condensed matter systems in two spatial dimensions [ 5]. They also serve as useful toy-models for more intricate four-dimensional dynamics, as they can be studied perturbatively in the /N expansion, where the gauge interactions are suppressed [6 9]. Our goal in this paper is to study monopole operators in one such CFT, namely the CP N b theory tuned to criticality. This theory is a nonlinear sigma-model with CP N b target space, and can be equivalently described as a U() gauge theory coupled to N b complex scalars of unit charge that satisfy a length constraint; see [0] for a textbook treatment. The action is where =...N b, S = N b g Z h i d 3 x (@ µ ia µ ) + i ( ), (.) is a Lagrange multiplier imposing the length constraint, and g is a coupling constant. This theory becomes critical provided that one tunes the coupling to g = g c for some g c. The interest in monopole operators in this theory is motivated by their interpretation as order parameters for the valence bond solid (VBS) order of quantum antiferromagnets [ 3]. The quantum antiferromagnets are defined on bipartite lattices in two spatial dimensions, and have a global SU(N b ) symmetry. Each site of the first (second) sublattice has states transforming under the fundamental (anti-fundamental) of SU(N b ). The sites interact via short-range exchange interactions with SU(N b )symmetry.thereisnoexplicitreference to a gauge field in the lattice Hamiltonian. Nevertheless, when the spin states on each site are represented in terms of parton degrees of freedom, a U() gauge field, A µ,emergesin the path integral formulation in the /N b expansion. The partons become the matter fields in this gauge theory. As their exchange constants are varied, such antiferromagnets can exhibit ground states with two distinct broken symmetries. First, there is the state with antiferromagnetic order, in which the SU(N b )symmetryisbrokenbythecondensationof : this is the Higgs phase of the U() gauge theory. Second, we have the state with VBS order in which SU(N b )symmetryispreservedbutalatticerotationsymmetryisbroken.in the U() gauge theory, this state appears initially as a Coulomb phase; however, the con-

4 finement of the U() gauge theory by the proliferation of monopoles leads to the appearance of VBS order in the lattice antiferromagnet. This is a consequence of subtle Berry phases associated with the monopole tunneling events in the lattice antiferromagnet, which endow the monopole operators with non-trivial transformations under lattice symmetry operations, identical to those of the VBS order. The quantum phase transition between these two states of the lattice antiferromagnet has been argued to be continuous [8, 9], and described by the CP N b theory in (.), with monopoles suppressed at the quantum critical point; the critical point is therefore deconfined. From the perspective of the CP N b theory, this connection to the VBS order of the antiferromagnet is powerful because it allows the monopole operators to be expressed as simple, local, gauge-invariant operators of the lattice model. Moreover, the couplings of the lattice model can be chosen to avoid the sign problem of quantum Monte Carlo, and this allows e cient studies on large lattices of the CP N b CFT at the deconfined critical point between the Higgs and VBS phases [4 8]. Block et al. [7, 8] have obtained the scaling dimensions of VBS operators on a number of lattice antiferromagnets, and here we will compare their results with the /N b expansion for the scaling dimensions of the monopole operators in the CP N b CFT. Unlike the lattice antiferromagnet, the monopole operators of the CP N b field theory are not defined simply as products of fields that appear in the Lagrangian (.). Instead, the monopole operators appear as singular boundary conditions that these fields must obey at the point where the monopole operator is inserted [3,9,30]. One way of defining monopole operators is the following. The CP N b the action of SU(N b ), the charged scalar fields model has SU(N b ) U() top global symmetry. Under transform in the fundamental representation. The U() top factor in the global symmetry group is a topological symmetry whose conserved current is j µ = 8 µ F. (.) The Dirac quantization condition implies that the conserved charge q = R d xj 0 satisfies q Z/. It should be noted that our definition of q here di ers by a factor of from earlier work [3, 7, 3]. One can define the monopole operators as operators that have non-vanishing U() top charge q. For each q, wewillfocusonthemonopoleoperatorm q with the lowest scaling dimension. The other operators in the same topological charge sector can be thought intuitively as products between the monopole operator with lowest scaling dimension and more 3

5 conventional operators from the q =0sector. Throughoutthispaper,weassumewithout loss of generality that q 0. The physical quantities that we compute depend only on q. We aim to determine the scaling dimension of the monopole operator M q with U() top charge q to next-to-leading order in /N b.themostconvenientwayofperformingthiscomputation is to use the state-operator correspondence, under which a local operator inserted at the origin of R 3 is mapped to a state of the CFT on the conformally-flat background S R. The scaling dimension of the monopole operator M q is therefore mapped to the ground state energy on S in the sector with magnetic flux R F =4 q through the S [9 3]. (See also [3] for an earlier approach to computing scaling dimensions of monopole operators.) As is standard in thermodynamics, this ground state energy q can be related to the partition function on S R via q = lim log Zq ( ) log Z S R q F q, (.3) where in the middle equality we regularized the S R partition function by compactifying the R direction into a large circle of circumference. In the case at hand, the S ground state energy in the presence of 4 q magnetic flux is easily computed at leading order in N b,wherethelagrangemultiplierfield and the gauge field don t fluctuate and assume a saddle point configuration that minimizes this energy. It is reasonable to assume that the large N b saddle point (for both the magnetic flux through S and the value assumed by the Lagrange multiplier field) is rotationally-symmetric. At leading order in N b,thegroundstateenergyons comes from performing the Gaussian integral over the matter fields [3]. The /N b correction to this result takes into account the Gaussian fluctuations of the gauge field as well as of those of. We perform an analysis of these fluctuations around the rotationally-invariant saddle point; by computing their determinant, we extract the /N b correction to the scaling dimension of M q. Our results are listed in Table below. In Figure we compare our result for F / to numerical studies of the lattice antiferromagnet and find a remarkable agreement. This agreement suggests that the next correction to F / in /N b is probably quite small. From comparing the scaling dimensions collected in Table to 3, we can also estimate the upper bound on N b below which the monopole operators are expected to be relevant; these bounds are also presented in Table. There is inherently some uncertainty in these estimates, as they come from extrapolating the large N b expansion to small values of N b. Nevertheless, our relevance bounds come close to what Ref. [7] found from numerics, as can be seen from Table I in [7]. 4

6 q q N b for which q < < / N b O(N b ) apple N b O(N b ) apple 6 3/ N b O(N b ) apple N b O(N b ) none 5/.467 N b O(N b ) none Table : Results of the large N b expansion of the monopole operator dimensions q obtained through calculating the ground state energy in the presence of q units of magnetic flux through S.Inthelastcolumnofthetablewelistedourestimatesforwhenthemonopole operators are relevant. From Table, it can be noticed that we give an argument for this behavior, and find that q grows with q approximately as q 3/.InSection5 q = q 3/ N b + c + O(N b ) + O(q / ) (.4) with a constant c 0.3 that can be deduced from the numerical values presented in Table. In Section 5, we take the first steps towards deriving the value of c analytically. It would be very interesting to develop a more complete understanding of the large q behavior. Similar calculations were performed in theories with fermionic matter in [3, 33]. also [35] where only the fluctuations of the Lagrange multiplier field were calculated with the purpose of studying monopole insertions in theories with global U() symmetry.) In [36, 37], monopole operators were studied holographically. supersymmetric theories, see, for instance, [38 43]. (See For studies of monopole operators in The rest of this paper is organized as follows. In Section, we set up our computation. In Section 3, we review the leading order analysis at large N b.insection4,weexaminethe /N b corrections around the spherically-symmetric saddle points of the e ective action for the gauge field and Lagrange multiplier. In Section 5 we analyze the behavior at large q. We end with concluding remarks in Section 6. Several technical details of our computation are included in the Appendices. The calculation in the case with scalar matter performed in this paper is technically more challenging than that for fermionic matter due to additional UV divergences. One can renormalize these divergences using zeta-function techniques, as we do here, but the same result can be derived in other renormalization schemes [34]. 5

7 F / /N b /N square honeycomb rectangular /N b Figure : The scaling dimension of the q =/ monopoleoperator,f /. The solid line is the N b = result (Ref. [3]), and the dashed line is the leading /N b correction computed in the present paper (see Table ). The quantum Monte Carlo results are for lattice antiferromagnets with global SU(N b ) symmetry on the square (Refs. [4, 5]), honeycomb (Ref. [7]), and rectangular (Ref. [7]) lattices. Setup In order to study the large N b limit of the CP N b theory, it is convenient to rescale the fields such that the action (.), appropriately generalized to that on an arbitrary conformally-flat space with metric tensor g µ,takestheform S = g Z d 3 x p h g(x) g µ [(r µ + ia µ ) ][(r ia ) ]+ R i 8 + i ( N b ), (.) where R is the Ricci scalar. In this paper we will work on S R, whichweparameterize by coordinates x (,, ). The monopole scaling dimension is equal to the ground state energy F q on S R in the presence of a magnetic flux R F =4 q through the S. Our main task is to determine the We could have absorbed the conformal mass term R 8 by shifting i, but we chose not to. 6

8 /N b expansion of this ground state energy, which we write as F q = N b F q + F q + O(/Nb ). (.) N b When q = 0, the corresponding ground state energy F 0 is nothing but the scaling dimension of the unit operator. We therefore must have F 0 =0. It is not hard to see that at large N b, the fluctuations of point configuration are suppressed. Indeed, upon integrating out the scalars and of A µ around any saddle in the action (.), one obtains an e ective action for the gauge field and Lagrange multiplier given by S e [A µ, i ]=N b appletr log (r µ ia µ ) i i g Z d 3 x p g. (.3) Let s expand A µ and around a saddle point by writing 3 A µ = A q µ + a µ, i = µ q + i, (.4) where a µ and are fluctuations around the saddle point configuration A µ = A µ and i = µ q. As can be easily seen from (.3), the e ective action for these fluctuations is proportional to N b,sotheirtypicalsizeisoforder/ p N b and are therefore suppressed at large N b. To leading order in /N b,itisthereforecorrecttoseta µ = =0,providedthatthebackground values A q µ and µ q are such that the saddle point conditions S e [A µ, i ] = S e [A µ, i ] A µ =a µ=0 =a µ=0 =0 (.5) are obeyed. One can then develop the /N b expansion to higher orders by integrating over the fluctuations a µ and using the e ective action (.3). In this paper we will focus only on saddles that are rotationally-invariant on S and translationally-invariant along R. Theseconditionsimplythatµ q is a constant and that, in the sector of monopole flux R F =4 q, thebackgroundmagneticfieldf q = da q is uniformly distributed over S : 3 In terms of the quantity a q introduced in [3], we have a q = µ q q. F q = q sin d ^ d. (.6) 7

9 One can choose a gauge where the background gauge potential A q can be written as A q = q( cos )d. (.7) (This expression is well-defined everywhere away from the South pole at =.) The saddle point condition (.5) is satisfied provided that the constant µ q is chosen such that it minimizes the value of the e ective action evaluated when a µ = =0. Inotherwords,the equation that determines µ q e [A q µ,µ q =0. (.8) This equation depends non-trivially on q, andhencesodoesµ q. In the next section, we calculate the coe cient Fq by simply evaluating S e at the saddle point, while in Section 4, we compute the correction F q from the functional determinant of the fluctuations around this saddle point. 3 N b = theory At leading order in N b,onecanidentify N b F q = S e [A q µ,µ q], (3.) evaluated for the value of µ q that solves (.8), or equivalently at large N q =0, (3.) and with the coupling g tuned to the critical value g = g c.inotherwords, F q =trlog (r µ ia q µ) + µ q g c µ q. (3.3) Using the fact that the eigenvalues of the gauge-covariant Laplacian on S in the presence of magnetic flux 4 q are j(j +) F q = Z d X j=q q [44, 45], we obtain (j +)log +(j +/) + µ q q 4 g c µ q. (3.4) 8

10 The first term in this expression is divergent and requires regularization. The second term is also divergent because, as we explain shortly, the inverse critical coupling /g c diverges linearly, and so the second term in (3.4) cancels part of the divergence in the first term. To be explicit, let us deduce an expression for /g c.thesaddlepointcondition(3.)atq =0 can be written as Z 4 d g = X (j +). +(j +/) + µ (3.5) 0 j=0 The theory is critical when the correlators on S R are those obtained by conformally mapping the power-law correlators on R 3. For a scalar field,thisprescriptionyieldsa conformally coupled scalar on S R, forwhichµ 0 = 0. Hence criticality is achieved when, to leading order in N b, we have g = g c and µ 0 =0,with Z 4 d = g c X (j +) +(j +/). (3.6) j=0 After substituting (3.6) into (3.4), the resulting expression is still divergent, but can be rendered finite using, for instance, zeta-function regularization as in [35], or Pauli-Villars regularization as in [3,3]. We will not repeat that calculation here. The regularized ground state energy coe cient F q is F q = X j=q apple (j +/) (j +/) + µ q q / qµ q + q( + q ) 6. (j +/) (µ q q ) (3.7) This expression can easily be evaluated numerically for any µ q. Note that the same procedure gives F0 =0,asrequiredbyconformalsymmetry. As mentioned above, the value of µ q can be obtained from the saddle-point equation (.8), which yields 0 j +/ A q =0. (3.8) q(j +/) + µ q q j=q Note that one can obtain this equation directly by di erentiating (3.4) with respect to µ q without the need of zeta-function regularization. Upon substituting the solution of (3.8) 9

11 into (3.7), one obtains the values of F q given in Table. These values agree precisely with q µ q Fq / / / Table : A few values for the parameters µ q and F q. those obtained in [3] by a very di erent method. 4 4 /N b corrections In this section we compute the next to leading oder correction to the dimensions of monopole operators. The systematics of the calculation are presented in the first four subsections, with the numerical results presented in Section E ective action at quadratic level To obtain the leading /N b correction F q to the result of the previous section, one should consider the quadratic fluctuations of the gauge field and of the Lagrange multiplier around the saddle (.4). Expanding (.3) at small a µ and,onecanwritethequadratictermin the e ective action as S () e = S() + S aa () + S a (), S () = N Z b d 3 xd 3 x 0p g(x) p g(x 0 ) (x)d q (x, x 0 ) (x 0 ), S aa () = N Z b d 3 xd 3 x 0p g(x) p g(x 0 )a µ (x)k q,µµ0 (x, x 0 )a µ 0(x 0 ), Z = N b d 3 xd 3 x 0p g(x) p g(x 0 ) (x)h q,µ0 (x, x 0 )a µ 0(x 0 ), S () a (4.) where D q, K q,andh q are integration kernels whose expressions will be given shortly. This e ective action is non-local because it was obtained after integrating out the fields, which 4 The quantity Fq should be identified with q in [3], while our µ q should be identified with C q in [3]. 0

12 are massless. The kernels appearing in (4.) can be written in terms of correlators of and of the current J µ = i (r µ ia q µ) (r µ + ia q µ), (4.) as N b D q (x, x 0 )= g h (x) (x 0 ) i q, N b K q,µµ0 (x, x 0 )= g hj µ (x)j µ0 (x 0 )i q + g gµµ0 (x x 0 )h (x) i q, N b H q,µ0 (x, x 0 )= i g h (x) J µ0 (x 0 )i q, (4.3) where the delta-function contains a factor of / p g(x) initsdefinition. Thecorrelatorsin (4.3) are evaluated under the assumption that the gauge field and Lagrange multiplier are non-dynamical and fixed at their background values A µ = A q µ and i = µ q.thesubscriptq on the angle brackets in the expressions above serves as a reminder of these assumptions. Performing the Gaussian integral over a µ and,wecanwritethecoe cient F q in (.) as F q = log det 0 M q, (4.4) where we defined the matrix of kernels 0 D q (x, x 0 ) H q (x, x 0 ) H q 0 i (x, x 0 ) 0 M q (x, x 0 ) H q (x, x 0 ) K q 0(x, x0 ) K q i 0(x, x0 ) C A, (4.5) H qi (x, x 0 ) K q i 0(x, x0 ) K qi i 0(x, x0 ) with i =, and the primed indices contracting with the index of the field at x 0. The prime on the determinant in (4.4) means that when computing the functional determinant we should ignore the zero eigenvalues that are required to be present due to gauge invariance. 5 Our goal in the rest of this section is to calculate a regularized version of this determinant, thus obtaining F q. 5 For a more detailed treatment of gauge fixing, see [33].

13 4. Eigenvalues of the integration kernels One can start evaluating the expressions in (4.3) in terms of the N b = limit of the Green s function G q (x, x 0 )forthecomplexscalars,whichisdefinedby h (x) (x 0 )i = g G q (x, x 0 ). (4.6) Performing the required Wick contractions in (4.3), we obtain D q (x, x 0 )=G q (x, x 0 )G q (x, x 0 ), K q µµ 0 (x, x 0 )=D µ G q (x, x 0 )D µ 0G q (x, x 0 ) G q (x, x 0 )D µ D µ 0G q (x, x 0 ) + D µ G q (x, x 0 )D µ 0G q (x, x 0 ) G q (x, x 0 )D µ D µ 0G q (x, x 0 ) (4.7) +g µ (x x 0 )G q (x, x), H q µ (x, x 0 )=G q (x, x 0 )D 0 µ 0G q (x, x 0 ) G q (x, x 0 )D µ 0G q (x, x 0 ), where D µ µ ia q µ(x) andd µ 0 µ 0 + ia q µ(x 0 ) denote the gauge-covariant derivatives in the presence of the background gauge field. In order to calculate the eigenvalues of the matrix of kernels (4.5) required for (4.4), we make use of the S rotational symmetry and the translational symmetry along R. These symmetries imply that the eigenvectors of this matrix are of the form e i times an appropriate (scalar or vector) spherical harmonic on S. We will need the usual spherical harmonics Y jm (, ), as well as the vector harmonics X i,jm = Y i jm = p j(j i Y jm, p j(j +) ik p k Y jm, (4.8) where i, k =,,and = =. We can decompose the Lagrange multiplier fluctuation and the gauge field fluctuation a µ in terms of these modes. Because, is a scalar field, we only need the usual spherical harmonics for its mode expansion. a is also decomposed in terms of Y jm,whilea i is decomposed using the vector harmonics X i,jm and Yjm. i Wewill refer to the former vector harmonics modes as E modes and to the latter as B modes, as they are the S analogs of E and B modes familiar from other contexts: E and B modes have vanishing curl and divergence, respectively, and they transform like the E and B field under parity.

14 We can expand each of the kernels in Fourier modes as D q (x, x 0 )= Z d X jm D q j ()Y jm(, )Y jm( 0, 0 )e i( 0 ) (4.9) Z d X K (x, q x 0 )= jm Z d Xh K q ii (x, x 0 )= 0 jm K q, j ()Y jm (, )Yjm( 0, K q,ee j ()X i,jm (, )X i 0,jm( 0, + K q,eb j ()X i,jm (, )Yi,jm( 0, Z 0 d X h K q i (x, x 0 )= K q, E 0 j jm Z d X H q (x, x 0 )= 0 jm Z d X h H q i (x, x 0 )= 0 jm ()Y jm (, )X i 0,jm( 0, H q, j ()Y jm (, )Yjm( 0, H q,e j and form the matrix of coe cients 6 0 M q j () = ()Y jm (, )X i 0,jm( 0, 0 )e i( 0 ) 0 )+K q,bb j ()Y i,jm (, )Y i 0,jm( 0, 0 )+K q,eb j ()Y i,jm (, )Xi,jm( 0, 0 i 0 ) 0 )+K q, B j ()Y jm (, )Y i 0,jm( 0, 0 )e i( 0 ) 0 )+H q,b j ()Y jm (, )Y i 0,jm( 0, 0 ) e i( 0 ) i 0 ) e i( 0 ) (4.0) i 0 ) e i( 0 ) (4.) D q j () Hq,B j () H q, j () H q,e j () H q,b j () K q,bb j () K q, B j () K q,eb j () H q, j () K q, B j () K q, j () K q, E C j () A. (4.) H q,e j () K q,eb j () K q, E j () K q,ee j () Note that S rotational symmetry implies that these coe cients do not depend on the quantum number m. Note that the matrix M q j () is not Hermitian, which can be traced to the fact that in the action (.) the Lagrange multiplier field appears multiplied by a factor of i. The entries of this matrix are related by gauge invariance and CP symmetry. Gauge 6 We are indebted to Nathan Agmon whose work revealed that the original version of this equation contained a minus sign error [46]. 3

15 invariance of the kernels in position space imply that at separated points 7 r µ K q µµ 0 (x, x 0 )=0, r µ0 K q µµ 0 (x, x 0 )=0, r µ0 H q µ 0 (x, x 0 )=0. (4.3) Plugging in the decompositions (4.0) and (4.) in these conservation equations, we obtain that the Fourier space kernel M q j () shouldhavethefollowingeigenvectorswithzero eigenvalue: 0 0 i p j(j +) M q j () =0, Mq j () 0 0 i p j(j +) T =0, (4.4) where the pure gauge eigenvector is written in ( B,, E) components,justlikem q j () in (4.). From (4.4) we can express K q, E j () andk q,ee j () intermsofk q, j (). The second restriction on the entries of M q j () comesfromthecp invariance of the theory and the monopole background around which we are working. Under CP the modes of and the B modes of a µ transform in the same way, while the and E modes acquire arelativeminussign.becausethee ectiveaction(4.)isinvariantundercp,weconclude that there is no mixing between, B and, E modes. These constraints imply that M q j () takesablockdiagonalform.forj>0, 0 M q j () = D q j () Hq,B j () 0 0 H q,b j () K q,bb j () K q, i j () p K q, j(j+) j (). (4.5) C A i 0 0 p K q, j(j+) j () j(j+) Kq, j () This matrix has eigenvalues: q q ± = (Dq j ()+Kq,BB j ()) ± (D q j () Kq,BB j ()) 4 H q,b j (), q E j(j +)+ = K q, j, j(j +) (4.6) 7 There are multiple equivalent ways to see that these equations are true. Using the definitions of the kernels in terms of correlators (4.3) they are the consequences of the Ward identity r µ J µ (x) = 0. Alternatively, (4.) should be zero for a pure gauge configuration a µ = r µ (x), for arbitrary (x). Partial integration readily gives (4.3). 4

16 as well as a zero eigenvalue corresponding to a pure gauge mode. When j =0,theharmonicsX jm and Y jm are not defined, so the matrix M q j () reduces to the matrix M q 0() = Dq 0() 0. (4.7) 0 K q, 0 () In addition, the only remaining vector harmonic Y 00 is a constant on S,andcanbegauged away. Thus gauge invariance imposes K q, 0 () =0,andtheonlynon-vanishingeigenvalue is D q 0(). We will derive expressions for the entries of the matrices (4.5) (4.7) shortly. After doing so, we can calculate F q from (4.4). It is convenient to subtract F 0 =0from F q. 8 The expression we would like to calculate becomes: F q = Z d X j=0 (j +)log det 0 M q det 0 M 0. (4.8) Using the expression for the eigenvalues from (4.6) and that H 0,B j () =0byparitysymmetry, then (4.8) becomes: F q = Z d 6 4 log Dq 0() D0() + X (j +)log 0 j= apple K q, j () D q j ()Kq,BB j ()+ H q,b j () Dj 0()K0, j ()K 0,BB j () (4.9) Explicit expressions for the coe cients in (4.5) can be obtained by inverting (4.9) (4.). Let us explain how to do so for D q j () first,andleavethedetailsofhowtoperformanalogous computations for the K and H kernels to Appendix A. For D q j () weobtain: Z D q j () ( 0 )= d 3 xd 3 x 0 p g(x) p g(x 0 )Y jm(, )D q (x, x 0 )Y jm ( 0, 0 )e i( 0 0). (4.0) Since the LHS is independent of m, we can average the RHS over all possible values of m. After performing the average, the RHS becomes invariant under performing a combined 8 Note that as the gauge fixing condition is independent of the monopole background, any possible contribution from the Faddeev-Popov ghosts cancels after subtracting the vacuum contribution. 5

17 rotation in (, )and( 0, 0 ), so we can take the limit 0 0. We can also use that D q (x, x 0 ) depends only on 0 to set 0 = and remove the 0 integral. The simplified expression is D q j () = 4 Z j + d 3 x p g(x) lim jx m= j Y jm(, )D q (x, x 0 )Y jm ( 0, 0 )e i. (4.) It is only the m = 0 term that contributes to the sum. Analogous formulas for the K and H kernels are given in (A.3). Using explicit formulas for the spherical harmonics, (4.) can be simplified further to Z D q j () = d 3 x p g(x)p j (cos )D q (x, 0)e i, (4.) where by x 0 =0wemeanthelimit 0, 0 0. Similar expressions (albeit more complicated) can be obtained for the other coe cients appearing in (4.5) (4.7). 4.3 Kernels at q =0 When q =0,onecanobtainclosedformformulasfortheentriesofthematrix(4.5) (4.7). In this case, the Green s function G 0 (x, x 0 )canbeobtainedbyconformallymappingther 3 one, namely /(4 x x 0 ), and from G 0 one can construct position-space expressions for all the kernels in (4.7). The conformal mapping from flat space gives G 0 (x, x 0 )= where is the angle between the points on S : 4 p (cosh( 0 ) cos ), (4.3) cos =cos cos 0 +sin sin 0 cos( 0 ). (4.4) Since G 0 (x, x 0 )isreal,eq.(4.7)impliesthath 0 (x, x 0 )=0,andconsequentlyH 0,B j () = The D kernel at q =0 Plugging (4.7) and (4.3) into (4.), we obtain Dj 0 () = Z Z d 8 0 e i P j (cos ) sin d (cosh cos ). (4.5) 6

18 This expression can be evaluated by first performing the integral, which yields Dj 0 () = Z d e i Q j (cosh ), (4.6) 8 where Q j (x) isthelegendrefunctionofthesecondkind. Wecanthenexpandtheremaining integrand at large, Dj 0 () = Z d 8 X n=0 (n + j) (n +/) n (n + j +3/) e (n+j+) e i, (4.7) and perform the integral term by term. The result can be written as [35] D 0 j () = ((j ++i)/) 4 ((j ++i)/). (4.8) Note that in deriving (4.8) we encountered no divergences in the sums and integrals we performed The K kernels at q =0 Next, we aim to find an expression for K 0, j (). While the expression for K 0, j () that follows from (4.0) is UV divergent (as would be its flat space analog), the following di erence is finite: K 0, j () K 0, 0 (0) = Z Z d 8 0 It can be checked that sin d cos cosh e i P (cosh cos ) 3 j (cos ). (4.9) cos cosh (cosh cos ) = 3 r S 4(cosh cos ). (4.30) Substituting (4.30) into (4.9), integrating by parts twice in the sphere directions, and using r S P j (cos ) = j(j +)P j (cos ), one can easily show that K 0, j () K 0, 0 (0) = j(j +) Dj 0 (). (4.3) 7

19 Similarly, it can be shown that K 0,BB j () K 0,BB (0) = ( + j ) Dj 0 () K 0,EE j () K 0,EE (0) = D0 j (), K 0, E j () = i p j(j +) Dj 0 (). D0 0(0), (4.3) These expressions are consistent with the requirements of gauge invariance in (4.5). They also agree with the flat-space limit expected at large and j, whichwasobtainedin[4]. Also, it follows from (4.7) that at j = 0 gauge invariance requires K 0, 0 (0) = 0. From (4.5) we also know that K 0,EE () = K0, (). Taking the 0limitimpliesK 0,EE (0) = 0, as K 0, (0) is finite. Assuming these relations, we have K 0, j () = K 0,BB j(j +) Dj 0 (), j () = ( + j ) Dj 0 ()+C 0, K 0,EE j () = D0 j (), K 0, E j () = i p j(j +) Dj 0 (), (4.33) where the constant C 0 remains to be determined; we will see later around (4.57) that C 0 = Kernels for general q For general q, thereisnosimpleclosedformexpressionforthe Green s function. One can determine an integral expression for it by first expanding the fields in Fourier modes: = Z d X jx j=q m= j,jm()y q,jm (, )e i, (4.34) where Y q,jm are the monopole spherical harmonics introduced in [44,45]. At leading order in N b,theactionforthefields becomes S = g X jx j=q m= j Z d +(j +/) + µ q q,jm(), (4.35) 8

20 from which we can read o h,jm (),j 0 m 0(0 ) i = g ( 0 ) jj 0 mm 0G j(), (4.36) with G j () +(j +/) + µ q q. (4.37) From (4.6), (4.36), (4.34), and (4.37) we can write G q (x, x 0 )as G q (x, x 0 )= = X j=q X j=q Z " jx d e i( 0 ) m= j e iq F q,j ( ) e E qj 0 E qj, where in the second line we defined the polynomial in cos Y q,jm (, )Y q,jm( 0, 0 ) # G j () (4.38) F q,j ( ) r j + 4 Y q,j( q)(,0), (4.39) and is a phase factor discussed in [45] that can be defined through e i cos( /) = cos( /) cos( 0 /) + e i( 0) sin( /) sin( 0 /). (4.40) The angle was defined in (4.4), and the energy E qj is E qj q (j +/) + µ q q. (4.4) 4.4. The D kernel Let us first determine D q j (). Using (4.7), (4.), and (4.38), we have D q j () = 4 j + X j 0,j 00 =q Z d 3 x p g(x)f 0,j ( )F q,j 0( )F q,j 00( ) e (E qj0 +E qj00) +i 4E qj E qj 0. (4.4) 9

21 Performing the integral, we can simplify this expression to D q j () = 8 j + X j 0,j 00 =q apple E qj 0 + E qj 00 E qj 0E qj 00( +(E qj 0 + E qj 00) ) I D (j, j 0,j 00 ), (4.43) where I D (j, j 0,j 00 )= Z 0 The integral can be performed analytically, and we have sin d F 0,j ( )F q,j 0( )F q,j 00( ). (4.44) I D (j, j 0,j 00 )= apple (j +)(j 0 +)(j 00 +) j j 0 j 00. (4.45) q q We can check that this result equals (4.8) for q =0and,forinstance,forj =0 D 0 0() = X j 0 =0 Note that the summation in (4.4) is absolutely convergent. +(j 0 +) = tanh( /). (4.46) The K and H kernels Similarly, for the other kernels we can use (4.7), (4.0), (4.), and (4.38) to obtain H q, j () =H q,e j H q,b j () = K q, j () = 8 j + K q, E j () = K q,ee j () = K q,bb j () = () =K q, B j () =K q,eb j () =0, 6q i X apple (j +) p E qj 0 + E qj 00 I j(j +) E j 0,j 00 qj 0E qj 00( +(E qj 0 + E qj 00) H (j, j 0,j 00 ), ) =q X apple (Eqj 0 + E qj 00)( +4E qj 0E qj 00) X I E j 0,j 00 qj 0E qj 00( +(E qj 0 + E qj 00) D (j, j 0,j 00 (j 0 +) )+, ) 4 E =q j 0 qj 0 =q 8 X apple (j +) p i(e qj 0 E qj 00) I (j, j 0,j 00 ), j(j +) E j 0,j 00 qj 0E qj 00( +(E qj 0 + E qj 00) ) =q 8 X apple (E qj 0 + E qj 00) X I (j +)j(j +) E j 0,j 00 qj 0E qj 00( +(E qj 0 + E qj 00) E (j, j 0,j 00 )+ ) =q j 0 =q 8 X apple (E qj 0 + E qj 00) X I (j +)j(j +) E qj 0E qj 00( +(E qj 0 + E qj 00) B (j, j 0,j 00 )+ ) j 0,j 00 =q j 0 =q (j 0 +) 4 E qj 0 (j 0 +) 4 E qj 0 (4.47),, 0

22 where Z I H (j, j 0,j 00 )= d sin tan 0 F 0,j( )F 0 q,j 0( )F q,j 00( ), h i I (j, j 0,j 00 )= j 00 (j 00 +) j 0 (j 0 +) I D (j, j 0,j 00 ), I E (j, j 0,j 00 )= h j 0 (j 0 +) ID j 00 (j +)i 00 (j, j 0,j 00 ), Z " # I B (j, j 0,j 00 4 )= d sin sin F 0,j( )F 0 q,j 0 0( )F q,j 0 00( ) 4q tan ( /)F0,j( )F 00 q,j 0( )F q,j 00( ). 0 (4.48) A detailed derivation of these formulas is contained in Appendix A. The quantity I D (j, j 0,j 00 ) appearing in (4.48) was given explicitly in (4.44). Similar explicit expressions for I H (j, j 0,j 00 ) and I B (j, j 0,j 00 ) are given in Appendix B. Note that while the expressions for H q, j (), H q,b j (), and K q, E j () aboveareabsolutely () arenotandrequireregularization. convergent, those for K q, j (), K q,ee j (), and K q,bb j In order to regularize the latter, it is convenient to first compute the quantities K q, j () K 0, 0 (0), K q,ee j () K 0,EE (0), K q,bb j () K 0,BB (0), (4.49) which are free of divergences, and then add back the appropriately regularized values for K 0, 0 (0), K 0,EE (0), and K 0,BB (0). As argued in the previous subsection, gauge invariance implies K 0, 0 () =K 0,EE (0) = 0, but does not immediately determine K 0,BB (0) denoted by C 0 in (4.33). We can return now to that issue. Let us first examine K 0, 0 () =0. From (4.47), we have K 0, 0 (0) = X apple (j 0 +) j 0 =0 4 E 0j 0 + X j 0 =0 (j 0 +) 4 E 0j 0 =0. (4.50) (Each sum is divergent individually, but the combined summation is convergent.) Next, we can examine K 0,EE (0). Using (4.47) and doing a bit of algebra, we have K 0,EE (0) = 3 X j 0 j 00 = apple j 00 +/ j 0 +/ j 0 +/ j 00 +/ + X j 0 =0 (j 0 +) 4 (j 0 +/). (4.5)

23 By using the symmetry between the summation in j 0 and j 00,thisexpressioncanbewritten further as K 0,EE (0) = 6 X j 0 j 00 = Summing over j 00 we can write this expression as K 0,EE (0) = 4 apple X j 0 =0 j 00 +/ j 0 +/ + X j 0 =0 (j 0 +) 4 (j 0 +/). (4.5) (j 0 +/) j 0 +/ + X (j 0 +) 4 (j 0 +/). (4.53) Both of the individual sums in this expression as well as the combined summation are divergent, but gauge invariance dictates that K 0,EE (0) = 0. This result should be thought of as a prescription. It can also be justified in zeta-function regularization, in which (4.53) gives K 0,EE (0) = (0, /)/(4 ) =0. Next, from (4.47), we can also write an expression for K 0,BB (0): j 0 =0 K 0,BB (0) = 8 X j 0 =0 apple j 0 (j 0 +)(j 0 +) (j 0 +/) 3 + X j 0 =0 (j 0 +) 4 (j 0 +/). (4.54) Both terms are again divergent, but, using (4.53), we can calculate K 0,BB (0) K 0,EE (0) = 4 X j 0 =0 Since K 0,EE (0) = 0, this equation proves that C 0 =0in(4.33). (j 0 +) = 3 = D0 0(0). (4.55) We can now provide alternate, appropriately regularized formulas for K q, j (), K q,ee j (),

24 and K q,bb j () thataremanifestlyconvergent.theyare: K q, j () = K q,ee j () = K q,bb j () = " X 8 j + j 0 =q X j 00 =q " X 8 (j +)j(j +) j 0 =q " X 8 (j +)j(j +) j 0 =q (E qj 0 + E qj 00)( +4E qj 0E qj 00) E qj 0E qj 00( +(E qj 0 + E qj 00) ) I D(j, j 0,j 00 ) X j 00 =q X j 00 =q # + (j0 +), 4 E qj 0 (E qj 0 + E qj 00) E qj 0E qj 00( +(E qj 0 + E qj 00) ) I E(j, j 0,j 00 ) + (j0 +) 8 E qj 0 # + C q, (E qj 0 + E qj 00) E qj 0E qj 00( +(E qj 0 + E qj 00) ) I B(j, j 0,j 00 ) + (j0 +) 8 E qj 0 # + C q, (4.56) where C q X j 0 =q j E qj 0 X j 0 =0 j E 0j 0 = j 0 +/ A q(j 0 +/) + µ q q j 0 =q 3 q5. (4.57) The expression for K q, j () was obtained by simply combining the two summations in the expression in (4.47). The expressions for K q,ee j () andk q,bb j () wereobtainedbysubtracting K q,ee (0) = 0 from the expressions in (4.47). In (4.57) we discover the saddle point equation for µ q (3.8) (obtained after tuning the coupling g to the critical value g c ), thus C q =0. One can check from (4.47) and (4.48) that the gauge-invariance relations (4.5) are obeyed. Such a check is most simply performed by matching the residues of the functions of at their poles at = ±i(e qj 0 + E qj 00) foreachj 0 and j 00,aftersymmetrizationbetweenj 0 and j 00 see Appendix A.. Now that we have expressions (4.56) we can begin evaluating the order /N b corrections to the free energy, which is the subject of the next subsection. 3

25 4.5 Numerical Results With the regularized formulas for the kernels in hand, we are almost ready to calculate the subleading correction F q to the free energy using (4.9). Let us write this expression as F q = Z d X (j +)L q j (), (4.58) j=0 where L q j () can be read o from (4.9). (See also (C.47) (C.48).) As shown in Appendix C, at large and j the integrand in this expression behaves as L q j () = 8µ q +..., (4.59) +(j +/) thus rendering the integral (4.58) linearly divergent. There are several ways of understanding how to regularize this divergence. One way is to use zeta-function regularization to write Z d X 8µ q (j +) +(j +/) = j=0 X j=0 (j +) 8µ q j + =4µ q (0, /) = 0. (4.60) Then one can subtract (4.60) from (4.58), and evaluate F q = Z d X (j +) j=0 apple L q j () 8µ q +(j +/) (4.6) instead of (4.58). This expression is no longer linearly divergent. Another way of understanding the subtraction in (4.6) is that the critical coupling g c, which was obtained in (3.6) at leading order in N b,receives/n b corrections. A similar phenomenon was encountered in [4] when computing the thermal free energy at subleading order in /N b. Just as in [4], it can be argued that Z 4 d = g c X (j +) +(j +/) j=0 apple + 4 N b + O(/N b ). (4.6) The /N b term in this expression contributes to F q through the last term in (.3) precisely as the subtraction implemented in (4.6). This expression can be derived rigorously by performing a careful renormalization analysis of theory on S R [34]. Even after the linear divergence in F q has been taken care of, this quantity is still potentially logarithmically divergent. This logarithmic divergence cancels when using a 4

26 regularization prescription consistent with conformal symmetry. In practice, we evaluate the integral in (4.6) with a symmetric cuto : (j +/) + <. (4.63) This can be thought of as preserving rotational invariance on R 3,asthehighenergymodesare insensitive to the curvature of the sphere. Given the kernels, (4.56), and the regularization described above, we are able to evaluate F q numerically. To obtain good precision, we first evaluate (4.6) numerically in a region (j +/) + < ( 0 ) ; then in the region ( 0 ) < (j +/) + < we replace L q j () in(4.6)withtheasymptoticexpansion derived in Appendix C (accurate up to terms of order O(/ (j + ) + 7/ )) and evaluate the integral analytically as. We notice that the result converges very rapidly as we increase 0.(Inpractice, 0 =0isalreadysu cientlylarge.) Ourresultsfor F q are given in Table 3. See also Table where we collected the results from Tables and 3 together. q F q 0 0 / / /.8879 Table 3: The coe cient F q in the large N b expansion (.) of the ground state energy in the presence of q units of magnetic flux through S. 5 The large q limit Whereas the analytic computation of M q j () seemstobeahopelessendeavorforfiniteq, we found that the q limit is tractable. The reason for the simplification is that this is essentially a flat space limit: reintroducing the radius R of S we have a strong magnetic field B = q/r on the sphere at large q. quanta move on Landau levels, which are localized on / p B = R/ p q distances, hence they don t feel the e ect of the curvature of the sphere. To leading order in /q, theproblembecomestheanalysisofthecp N b field in flat space. First, we want to calculate F q in a constant magnetic.wewillbemorecavalieraboutdivergencesthaninthe 5

27 rest of the paper, and write, following (3.7), F q = = X (j +) (j +/) + µ q q / j=q X (q +n +) (5.) q(n +/) + µ q +(n +/) /, n=0 where we introduced n j resulting saddle point equation for µ q is (3.8): X n=0 q, andassumedzeta-functionregularizationasimplicit. The q +n + qq(n +/) + µ q +(n +/) =0. (5.) This equation has a solution only provided that we scale µ q correctly with q, namely µ q =q O. (5.3) q Plugging this Ansatz into (5.) and only keeping the leading terms, we obtain 0= X n=0 p (n +/) + 0 =, + 0. (5.4) Note that in obtaining this equation we assumed that n q even though we are summing over all positive n. Thisassumptionisjustifiedbecausethecontributionofn & q is higher order in /q. The constant 0 can therefore be obtained as the root of the transcendental equation (5.4). Going to one higher order we can determine : = 0 + 3, + 0 3, + 0, (5.5) which gives the large q expansion for µ q: µ q q O. (5.6) q As seen from Figure, (5.6) is in excellent agreement with the values of µ q obtained from solving (3.8) at large q. 9 9 In [3] it was noticed that µ q q/5 at large q. 6

28 m q q Figure : A comparison between the values of µ q found by solving (3.8) (blue points) and the large-q analytical approximation (5.6) (solid black line). Plugging back into (5.), we have that at leading order in q F q =q X n=0 = q 3/. p q(n +/) + q 0 =(q) 3/, + 0 (5.7) Quite nicely, this equation can be understood in flat space terms: the Landau levels of a massive scalar field are given by E n = p B(n +/) + m = R p q(n +/) + q 0, (5.8) and have degeneracy N = B Vol(S ) =q, givingexactly(5.)forthefreeenergy(ifweset R = ). Note that the q 3/ scaling of (5.7) follows from flat space dimensional analysis: the free energy density is an intensive quantity of mass dimension 3, hence it has to be independent of R, andwegetf B 3/. While we leave a full evaluation of M q j () atlargeq to future work, as a first step we now derive large q behavior of M q j () when =0andj q. This corresponds in the flat space limit to taking the momentum p BR. At =0,weonlyhavetodetermine D q j,hq,b j,k q,bb j,andk q, j,astherestofthematrixelementsvanish see(4.5). Wecan obtain a closed-form formula for these kernels by taking the explicit expressions for them, and expanding for large q. Thisisquiteatedioustask,especiallyforK q,bb j,wherewehave to expand (B.8) for fixed j, j 0,j 00 and large q. The resulting expressions can be summed over j 0,j 00 analytically using zeta-function regularization. The results are given by the simple 7

29 expression 0 M q j (0) = 3, p q 0 i p j(j +) i p j(j +) 0 j(j +) j(j +)( C 0 + ) 0A. (5.9) One curious subtlety is that while the individual terms in K q,bb j,k q, j are O( p q), this O( p q) contribution vanishes upon summation over j 0,j 00. The subleading terms give the result in (5.9). This implies that we have to know the saddle point value of µ q to first subleading order (5.3) and makes appearance in the final result. We obtain the flat space kernel by the replacement j(j +) p in (5.9). Our expression is valid for p BR. Itwouldbeaninterestingexercisetoobtainthefullquadratice ective action of the CP N b model in flat space in a constant magnetic field. The matrix in (5.9) has one zero eigenvalue corresponding to the pure gauge mode. The nonzero eigenvalues for j =are(4.6): q ± ± i p q, q E p. (5.0) q We can check that (5.0) agrees with the numerical results see Figure λ E Re(λ ± ) Im(λ ± ) q Figure 3: The numerical results for the three eigenvalues, the analytic large q value in black. q E, q +, and q are plotted against 0 For j = 0 the matrix is, and the only nonzero element is D q 0 (0), as in (4.7). 8

30 6 Conclusions We have presented here the leading correction to the large N b result for the scaling dimension of the monopole operator in the CP N b CFT. This correction was obtained by computing the Gaussian fluctuation determinant of the U() gauge field A µ, and the Lagrange multiplier,ons R. Computationofhigherordertermsinthe/N b expansion appears to be possible by the present methods, but will involve considerable e ort. Our computation now opens the possibility of quantitatively testing the most subtle and novel aspects of the theory of deconfined criticality [8, 9] in two-dimensional lattice antiferromagnets. An important feature of this theory is the connection between the monopole operator and the VBS operator of the antiferromagnet [ 3]. This connection allows a Monte Carlo computation of the monopole scaling dimension by measuring correlators of the VBS order in lattice models. We compared our present result with the Monte Carlo studies and found very good agreement both in the dimension of the lowest monopole operator and in the number of scalar fields below which monopole operators become relevant; see Fig. and Table in the Introduction. We conclude by noting that our present study is among the most complex theoretical computations of critical exponents which have been compared to Monte Carlo simulations. Our calculation assumed conformal invariance to realize a framework in which the exponents could be determined, and it did not reduce to identifying poles in a Feynman graph expansion [47]. It would be of great interest to apply the recent progress in bootstrap methods [48,49] to also determine such exponents. Acknowledgments We are very grateful to Nathan Agmon, whose joint work with SSP [46] revealed a crucial typo in the first version of this paper. We thank Ribhu Kaul for valuable discussions and for providing the numerical results in Fig.. We also thank Max Metlitski for useful discussions. The work of ED was supported by the NSF under grant PHY The work of MM was supported by the Princeton Center for Theoretical Science. SSP was supported in part by the US NSF under grant No. PHY The work of SS was supported by the NSF under grant DMR and MURI grant W9NF from ARO. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. 9

31 A Derivation of integration kernels A. Evaluation of Fourier transforms In this Appendix we explain how to obtain the explicit formulas appearing in (4.47). When dealing with vectors on S it is convenient to use the frame e i a,withi =, and a =,, defined by e i =(, 0), e i = 0,. (A.) sin One can then convert the coordinate index of a quantity v i to a frame index by writing v a = e i av i. (A.) The frame index can then be raised and lowered with ab and ab,respectively;inother words, it makes no di erence whether it is upper or lower. Our starting point are equations (4.9) (4.). In the main text, we explained how (4.9) can be inverted to obtain (4.0). The analogous formulas for the other kernels are K q, j () = 4 Z j + K q, E j () = 4 Z j + K q,ee j () = 4 Z j + K q,bb j () = 4 Z j + H q,b j () = 4 Z j + D q j () = 4 Z j + d 3 x lim jx x 0 0 m= j d 3 x lim jx x 0 0 m= j d 3 x lim jx x 0 0 m= j d 3 x lim jx x 0 0 m= j d 3 x lim x 0 0 m= j d 3 x lim jx jx x 0 0 m= j Y jm(, )K q 0 (x, x 0 )Y jm ( 0, Y jm(, )K q a 0 (x, x 0 )X a0 jm( 0, X a jm(, )K q aa 0 (x, x 0 )X a0 jm( 0, Y a jm(, )K q aa 0 (x, x 0 )Y a0 jm( 0, Y jm(, )H q a 0 (x, x 0 )Y a0 jm( 0, Y jm(, )D q (x, x 0 )Y jm ( 0, 0 )e i, 0 )e i, 0 )e i, 0 )e i, 0 )e i, 0 )e i. (A.3) Using the spectral decomposition (4.38), the position-space kernels appearing in this expres- 30

32 sion can be written as D q (x, x 0 )= X F q,j 0( )F q,j 00( ) e (Eqj0 +E ) 0 qj00, 4E j 0,j 00 qj 0E qj 00 K q (x, x 0 )= X F 0 q,j 0( )F q,j 00( ) e (Eqj0 +E qj00 ) 0 (Eqj 0 E qj 00) (E qj 0 + E qj 00) ( ) 4E j 0,j 00 qj 0E qj 00 + X j 0 j E qj 0 (x x 0 ), K q a (x, x 0 )= X e (Eqj0 +E ) 0 qj00 (E 0 qj 0 E qj 00)sgn( 0 ) 4E j 0,j 00 qj 0E qj 00 D a 0e iq F q,j 0( ) e iq F q,j 00( ) e iq F q,j 0( ) D a 0e iq F q,j 00( ) K q aa (x, x 0 )= X j E j 0 qj 0 e (E qj 0 +E qj 00 ) 0 aa 0 (x x 0 )+ X 4E qj 0E qj 00 j0,j00, Da e iq F q,j0( ) D a0e iq F q,j00( ) + Da0e iq F q,j0( ) D a e iq F q,j00( ) e iq F q,j 0( ) D a D a 0e iq F q,j 00( ) Da D a 0e iq F q,j 0( ) e iq F q,j 00( ), H q a 0 (x, x 0 )= X j 0,j 00 e (E qj0 +E qj00) 0 4E qj 0E qj 00 e iq F q,j 0( ) D a 0e iq F q,j 00( ) Da 0e iq F q,j 0( ) e iq F q,j 00( ), (A.4) where D a ia q a(x), D a a 0 + ia q a 0 (x 0 ). (A.5) In taking the x 0 0 limit in (A.3), it is convenient to use the addition formula for the spherical harmonics jx m= j Y jm(, )Y jm ( 0, 0 )=F 0j ( ), F 0j ( ) j + 4 P j(cos ), (A.6) where is the relative angle between the points (, )and( 0, 0 )ons. Taking derivatives 3

33 of this formula and using the definition of the vector harmonics in (4.8), we have lim jx x 0 0 m= j lim jx x 0 0 m= j lim jx x 0 0 m= j lim jx x 0 0 m= j lim jx x 0 0 m= j lim x 0 0 m= j lim jx jx x 0 0 m= j Y jm(, )Y jm ( 0, Y jm(, )X a0 jm( 0, Y jm(, )Y a0 jm( 0, X a jm(, )Y jm ( 0, Y a jm(, )Y jm ( 0, X a jm(, )X a0 jm( 0, Y a jm(, )Y a0 jm( 0, 0 )=F 0j ( ), 0 )= 0 )= 0 )= 0 )= 0 )= 0 )= p j(j +) cos sin F0j( ) 0 0 sin cos cos sin p 0 F0j( ) 0 j(j +) sin cos F p 0j( ) 0, j(j +) 0 0 p, j(j +) F0j( ) 0 j(j +) j(j +) F 00 0j( ) 0 0 F 0 0j ( ) sin F 0 0j ( ) sin 0 0 F 00 0j( ) cos sin cos sin, sin cos sin cos,,. (A.7) We also have h i lim x 0 0 e iq(x,x0) F qj ( ) = F qj ( ), lim x 0 0 D a h i e iq(x,x0) F qj ( ) = h i lim D x 0 a 0 e iq(x,x0) F qj ( ) = 0 h i lim D ad a x 0 0 e iq(x,x0) F qj ( ) = 0 F 0 qj( ) iq tan F qj( ) F 0 qj( ) Fqj( ) 00 iq(f qj ( ) sin Fqj 0 ( )) +cos, iq tan F cos sin qj( ) sin cos iq(f qj ( )+sin Fqj 0 ( )) +cos Fqj 0 ( ) sin + q tan F qj( ) cos sin. sin cos Plugging in (A.7) and (A.8) into (A.3) and performing the integrals over and, (A.8),we 3