DCPT-12/47 NSF-KITP-12-234 Boundary conditions for scalars in Lifshitz Tom´as Andradea,b∗ and Simon F. Rossb† a Department of Physics, UCSB 3 Santa Barbara, CA, 93106, USA 1 0 2 b Centre for Particle Theory, Department of Mathematical Sciences n Durham University, South Road, Durham DH1 3LE, U.K. a J Abstract 7 We consider the quantisation of scalar fields on a Lifshitz background, ex- ] h ploring the possibility of alternative boundary conditions, allowing the slow t - falloff mode to fluctuate. We show that the scalar field with alternative bound- p e aryconditionsisnormalizableforalargerrangeofmassesthanintheAdScase. h However, we then find a new instability for alternative boundary conditions, [ implyingthattherangeofmasseswherealternativeboundaryconditionsdefine 2 a well-behaved dual theory is m2 < m2 < m2 +1, analogously to the AdS v BF BF case. The instability is of a novel type, with modes of arbitrarily large momen- 2 7 tum which grow exponentially in time; it is therefore essentially a UV effect, 5 and implies that the dual field theory is simply not defined where it appears. 2 . Wediscusstheinterpretationinthedualfieldtheory, andgiveaproposedlower 2 bound on the dimension of scalar operators. 1 2 1 : 1 Introduction v i X Theholographicdescriptionoffieldtheorieswithanisotropicscalingsymmetrypresents r a an interesting extension of AdS/CFT, which may have valuable applications in con- densed matter theory. The simplest example of such a dual description is the Lifshitz metric originally constructed in [1]. The geometry is dr2 ds2 = −r2zdt2 +r2d(cid:126)x2 +L2 , (1) r2 where L2 represents the overall curvature scale, and the spacetime has d + 1 di- mensions, so there are d = d − 1 spatial dimensions (cid:126)x. The asymptotically Lif- s shitz solutions of a bulk gravity theory are conjectured to provide a dual holographic ∗[email protected] †[email protected] 1 description of a non-relativistic field theory, with the anisotropic scaling symmetry t → λzt, xi → λxi. This duality is interesting both for its potential application in condensed matter, and as an extension of our understanding of holography and the relation between field theory and spacetime descriptions. The holographic dic- tionary, relating bulk spacetime quantities to field theory observables, is now fairly well-developed [1, 2, 3]. As in AdS/CFT, this identifies the leading asymptotic falloff of bulk fields with sources in the dual field theory. An interesting early observation in AdS holography [4] was that there can be more than one conformal field theory associated to a given bulk theory, depending on the choice of boundary conditions. As first noted in [5] for scalar fields, for some parameter values it is possible to introduce an alternative quantisation in the bulk spacetime, where a mode which is subleading in the asymptotic expansion of the field is fixed and the leading piece is allowed to vary. This alternative quantisation leads to a second conformal field theory dual to the same spacetime, with different operator dimensions for the operators dual to the bulk fields whose boundary conditions have been changed. One can also consider mixed boundary conditions, which are dual to renormalisation group flows interpolating between the two conformal field theories, generated by a double-trace deformation of the field theory. For a scalar field, the alternative quantisation is possible when the mass of the field is in the range m2 < BF m2 < m2 + 1, where m2 = −d2 is the Breitenlohner-Freedman bound [5] for d BF BF 4 boundary dimensions. The importance of this possibility for holography was realised in part because the dimension of the operator dual to the scalar with standard boundary conditions, (cid:113) ∆ = d + d2 +m2, was always strictly greater than the unitarity bound, ∆ > + 2 4 d − 1. With the alternative boundary condition, the scalar is dual to an operator 2 (cid:113) with dimension ∆ = d − d2 +m2, which precisely saturates the unitarity bound − 2 4 when m2 = m2 +1. BF ThisanalysisofalternativeboundaryconditionsinAdSdeepenedourunderstand- ing of the correspondence, through an improved bulk understanding of unitarity and an understanding of the relation of double-trace and more general deformations of the field theory and boundary conditions in the bulk [6, 7]. The early work on scalar fields was subsequently extended to consider vector fields in [8, 9], and for metric perturbations in [10]. In the bulk spacetime, the restriction to the range m2 < m2 + 1 comes from BF the fact that the slow fall-off mode is only normalizable with respect to the usual Klein-Gordon norm for masses in this range. Similar restrictions arise for a vector field [11, 9]. A deeper understanding of this restriction was obtained in [10], where it was observed that one can define an alternative norm by adding boundary terms to the Klein-Gordon current, such that the solution with Neumann boundary conditions is always normalizable. However, this norm is not positive for m2 > m2 + 1 for BF generic boundary metrics [10], and for the flat boundary metric, where the norm is positive, an IR divergence appears enforcing the unitarity bound [12]. Inthispaper,weaimtoaddressthesameissuesinLifshitz. Wewillfocusprimarily 2 on investigating when alternative boundary conditions are possible from the bulk spacetime perspective. In the present paper we do this calculation for a scalar field in a fixed Lifshitz background; a companion paper to follow will consider linearised fluctuations in a theory including dynamical gravity. Our intention is to use this study to shed further light on the duality. In particular, the calculation with a scalar field will lead to a new prediction for a bound on operator dimensions in Lifshitz field theories (or at least those with gravitational duals). After some preliminary discussion of the equation of motion in section 2, we first investigate the normalizability for a scalar field on a Lifshitz background in section 3. We find that the slow fall-off mode is normalizable, so that alternative boundary conditions are possible, for a larger range of masses than in the AdS case: m2 < m2 < m2 +z2, (2) BF BF where m2 = −1(z + d )2 and d is the number of spatial dimensions. The lower BF 4 s s bound m2 is the analogue of the Breitenlohner-Freedman bound in this case. The BF immediate source of this wider range is that the measure in the Klein-Gordon inner product includes a factor of r−z, so slower falloffs become normalizable as z increases. More deeply, this can be related to the perspective of [10] by observing that as z increases the dimension of boundary counterterms involving time derivatives of the boundarydataincrease, andtheinnerproductneedstobemodifiedbyaddingbound- ary terms only when the operator dimension is small enough for these terms to be relevant. In the field theory, the wider mass range (2) corresponds to a weaker restric- tion on the possible dimensions of scalar operators. Thus, the spacetime calculation predicts a reduced lower bound for the dimension of scalar operators in the field theory. However, in section 4, we show that the scalar field on a Lifshitz background with alternative boundary conditions can be unstable even when m2 > m2 . In the AdS BF case, an instability for conformally-invariant boundary conditions is kinematically forbidden: conformal symmetry implies that the dispersion relation is ω = ±k. How- ever for Lifshitz boundary conditions, the scaling symmetry only fixes the dispersion relation to be ω = αkz for some dimensionless parameter α. If there is a family of modes where α has a positive imaginary part, they represent an instability. We will first consider the question analytically for z = 2, where the solution of the scalar equation can be written in terms of confluent hypergeometric functions. We show that the theory with Dirichlet boundary conditions is stable, but that the theory with Neumann boundary conditions is unstable if m2 > m2 +1. (3) BF We note that the instability is a UV effect, as if there is any instability then there will be exponentially growing modes for all values of k, and the dominant instability is associated with arbitrarily large values of k. Thus, we would expect this instability to appear for any asymptotically Lifshitz spacetime when we take Neumann boundary conditions for the scalar. We confirm this expectation analytically by considering the scalar field on the z = 2 Lifshitz black hole solution introduced in [13], where the 3 wave equation can be solved in terms of hypergeometric functions. This makes it clear that the instability we are seeing here is not associated with the singularity in the interior of the spacetime for the pure Lifshitz solution [1, 14, 15]; this is a new UV pathology in these solutions. We complete our analytic discussion by giving a general argument for stability of black hole solutions with Dirichlet boundary conditions, following arguments previously given in [16, 13]. Wethenextendourdiscussiontoarbitraryvaluesofz bysolvingthewaveequation numerically. We do numerical analysis both for the pure Lifshitz spacetime and for a spacetime with an IR cutoff, which is easier to treat numerically, using both a spectral method and shooting. We verify that the IR cutoff does not alter the instability for high momentum. We find that the theory with Neumann boundary conditions is unstable if m2 > m2 +1 for all z. BF In the field theory dual to the alternative quantisation, the dimension is given by (cid:112) ∆ = 1(d +z)− 1 (d +z)2 +4m2, so m2 < m2 +1 corresponds to − 2 s 2 s BF 1 ∆ ≥ (d +z)−1. (4) s 2 This generalises the result ∆ ≥ d −1 in the relativistic case (where d = d +1 is the 2 s number of spacetime dimensions). In the Lifshitz field theory, there is no independent derivation of such a bound; in the absence of the usual state-operator map for such non-relativistic theories, there is no direct analogue of the usual argument for the unitarity bound. However, it is certainly surprising that the bound is so high; if we consider a free scalar field theory with kinetic term (∂ φ)2, Lifshitz scaling would t require the scalar to have dimension ∆ = 1(d +z)−z. This matches the bound that 2 s would be obtained from normalizability considerations alone, but the instability we find suggests that for interacting Lifshitz field theories the bound on the dimension is higher. We will discuss the field theory interpretation a little more in the conclusions in section 5. Note added: While this paper was in preparation [17] appeared, which has con- siderable overlap with our analysis of normalizability in section 3. 2 Scalar wave equation We consider a scalar field satisfying the Klein-Gordon equation (cid:3)φ−m2φ = 0 in the Lifshitz spacetime (1) with m2 < 0, and we neglect back-reaction on the spacetime metric1. The wave equation in the Lifshitz geometry (1) is r1−z−ds∂ (rz+ds+1∂ φ)−(r−2z∂2 −r−2∂2 +m2)φ = 0. (5) r r t i As in AdS, the derivatives along the boundary direction have a subleading effect at large r, and the asymptotic behaviour of the solutions is φ ∼ φ r−∆+ +φ r−∆−, (6) + − 1Extending the analysis to include back-reaction might be an interesting project for the future. 4 where (cid:114) 1 1 ∆ = (z +d )± m2 + (z +d )2. (7) ± s s 2 4 The analogue of the Breitenlohner-Freedman bound [5] for Lifshitz spacetimes is thus m2 = −1(z+d )2. We will also write m2 = m2 +ν2, so ∆ = 1(z+d )±ν, with BF 4 s BF ± 2 s ν > 0 by convention. The usual boundary condition is to fix the slow fall-off mode φ − and let φ fluctuate. We will generally refer to this as a Dirichlet boundary condition, + and to the converse condition of fixing φ and letting φ fluctuate as a Neumann + − boundary condition, although this terminology is really only valid for m = 0 where ∆ = 0. − To solve the equation (5) explicitly, we use a boundary plane wave basis, writing φ = e−iωt+i(cid:126)k·(cid:126)xψ(r). (8) The wave equation then becomes r1−z−ds∂ (rz+ds+1∂ ψ)−(−r−2zω2 +r−2k2 +m2)ψ = 0. (9) r r The d dependence here can be simplified by writing ψ(r) = r−ds+zχ(r); then s 2 r∂ (r∂ χ)−(−r−2zω2 +r−2k2 +ν2)χ = 0. (10) r r The asymptotic behaviour of χ is then easily seen to be χ ∼ r±ν as r → ∞, with the plus (minus) sign corresponding to Neumann (Dirichlet) boundary conditions, and χ ∼ e±izωrz as r → 0. For real ω both behaviours at r → 0 are regular. For complex ω one grows exponentially and the other decays; we select the exponentially damped mode. We can also note that either ω or k can be absorbed by a redefinition of r, so the equation can be rewritten as (cid:18) ω2 (cid:19) r∂ (r∂ χ)− r−2 +ν2 −r−2z χ = 0. (11) r r k2z This makes manifest the fact that the physics on the pure Lifshitz background can depend only on the dimensionless combination ω/kz. For generic ω and z, (10) has no solution in terms of known special functions. For ω = 0, we can solve it in terms of Bessel functions, but for both the normalizability and instability discussions our interest is in solutions with non-zero ω. More helpfully, for z = 2, the equation reduces to a confluent hypergeometric equation; this was previously analysed in [1], and we will use this solution in studying the instability in section 4. 3 Normalizability for scalars Inthissection, weconsiderthenormalizabilityoftheprobescalarfield. Wefirstgivea simpleconsiderationofnormalizabilitywithrespecttothenormalKlein-Gordoninner 5 product, showing that the range of masses for which alternative boundary conditions are possible is enlarged as we increase z. We then do a more detailed analysis of the inner product and counterterms, following [10, 12] closely. This allows us to understand the result better from the spacetime point of view, seeing the relation to kinetic counterterms in the action. We verify that inside our mass range, the standard Klein-Gordon inner product without any explicit boundary contributions is an appropriate inner product; in particular it is finite and positive definite for real ω. We assume we use the standard Klein-Gordon inner product, i (cid:90) √ (φ ,φ ) = ddsxdr hnµ(φ∗∂ φ −φ ∂ φ ), (12) 1 2 2 1 µ 2 2 µ 1 Σ where Σ is a spacelike surface, which we will take to be a surface of constant t. The wave equation (9) can be written as a Sturm-Liouville (SL) problem with eigenvalue λ = ω2 for the operator (cid:20) (cid:18) (cid:19) (cid:21) d d L = w(r)−1 − p(r) +q(r) , (13) dr dr with p = r(ds+z+1), w = rds−z−1 and q = r(ds+z−1)(m2 + r−2(cid:126)k2). The inner product (12) then becomes (ω +ω ) (φ ,φ ) = (2π)dsδ(ds)((cid:126)k −(cid:126)k )ei(ω1−ω2)t 1 2 (cid:104)ψ ,ψ (cid:105) , (14) 1 2 1 2 1 2 SL 2 where (cid:104)·,·(cid:105) is the corresponding SL inner product SL (cid:90) ∞ (cid:104)ψ ,ψ (cid:105) = rds−z−1ψ∗ψ dr. (15) 1 2 SL 1 2 0 With the standard Dirichlet boundary conditions φ = 0, the fields fall off as − φ ∼ r−∆+, and the large r behaviour of the integral is (cid:82)∞r−2ν−2z−1dr, so the fast fall off modes are normalizable for any ν. If we consider instead the Neumann bound- ary condition φ = 0, then φ ∼ r−∆−, and the large r behaviour of the integral + is (cid:82)∞r2ν−2z−1dr, so the slow fall off modes are normalizable with respect to this standard inner product if ν < z, that is if m2 < m2 < m2 +z2. (16) BF BF Increasing z thus increases the mass range for which the Neumann boundary condi- tions are allowed. In this range, we could also consider mixed boundary conditions; the flux through infinity vanishes, so that the inner product is conserved, for any linear boundary condition φ = fφ for real f. + − The theory with Neumann boundary conditions is dual to a field theory with Lifshitz scaling with an operator of dimension 1 ∆ = (z +d )−ν. (17) − s 2 6 The more general mixed boundary conditions are dual to non-scale invariant theories, which should interpolate between the Neumann boundary conditions in the UV and Dirichlet boundary conditions in the IR. We note that for z > d , ∆ could formally s − take negative values. We will find an instability in the next section before we reach such values, however, so we will confine ourselves to considering ν < 1(z+d ), where 2 s a linearised analysis of the asymptotics is possible. 3.1 Inner product and counterterms Togainabetterunderstandingoftheoriginsoftheextendedregionofnormalizability, weshouldconsideramorecarefulanalysisoftheinnerproduct, following[10,12]. The argumentaboveassumedthatwecouldusethestandardKlein-Gordoninnerproduct. However, [10] made it clear that the presence of kinetic boundary counterterms in the action for fields in AdS implies that we generally need to add corresponding boundary terms to the inner product to ensure that it is conserved. The correct inner product is always finite, as the boundary terms cancel any divergence from the bulk, but these terms may spoil positivity. The correct question to ask is then when we need to add boundary contributions to the Klein-Gordon inner product, and to check that the inner product is positive and conserved. We will see below that m2 < m2 + z2 BF is precisely where no explicit boundary contribution is required to have a conserved inner product. The terms φ in (6) can be expanded in a double power series in r−2, r−2z, ± φ = φ(0)+r−2φ(1)+r−4φ(2)+r−2zφ(z)+r−2−2zφ(z+1)+..., φ = φ(ν)+..., (18) − + where the terms which involve powers of r−2 are local functions of φ(0) and its spatial derivatives, while the terms which involve powers of r−2z are functions of temporal derivatives of φ(0).2 For example, 1 φ(1) = ∂2φ(0), (19) 4(ν −1) i while 1 φ(z) = ∂2φ(0). (20) 4z(z −ν) t If ν < z, the terms involving temporal derivatives of φ(0) are subleading compared to φ(ν) in the asymptotic expansion of φ: φ ∼ r−21(z+ds)+ν(φ(0) +r−2φ(1) +r−4φ(2) +...+r−2νφ(ν) +r−2zφ(z) +...). (21) Toobtainafiniteon-shellaction, oneneedstoaddcountertermstothebareaction to cancel divergences associated to the terms φ(0) and φ(i) for i < ν. For example, 2We don’t write the subleading terms involving derivatives of φ(ν) explicitly because they don’t enter the calculation. 7 if 1 < ν < 2, there will be divergences in the on-shell action involving both φ(0) and φ(1), 1 (cid:90) √ I = − ddsxdt −γnµφ∂ φ (22) bare µ 2 r=r0 (cid:90) 1 = ddsxdtr2ν[∆ φ(0)2 +(∆ +2)φ(0)φ(1)r−2 +∆ φ(0)φ(ν)r−2ν +...], 2 0 − − 0 + 0 r=r0 The divergences are cancelled by adding appropriate counterterms. The leading di- vergenceiscancelledbyaφ2 counterterm, andtheφ(0)φ(1) divergencecanbecancelled by a (∂ φ)2 counterterm. i In [10], it was shown that derivative counterterms in the action naturally lead to boundary contributions to the symplectic structure. At the level of the inner product, we can understand these terms as being required to ensure conservation of the inner product: we add boundary terms to the Klein-Gordon current so that the flux through the boundary at large r vanishes. The key difference in our case is that the counterterm in the action only involves spatial derivatives. As a result, the usual Klein-Gordon current will be conserved despite the appearance of a φ(1) term in the flux through the boundary. The flux of the Klein-Gordon current through the boundary is i (cid:90) √ F = ddsxdt −γnµ(φ∗∂ φ −φ ∂ φ∗) (23) 2 1 µ 2 2 µ 1 r=r0 (cid:90) i = ddsxdtr2ν[2r−2(φ(1)∗φ(0) −φ(0)∗φ(1))+(∆ −∆ )r−2ν(φ(ν)∗φ(0) −φ(0)∗φ(ν))]. 2 0 0 1 2 1 2 + − 0 1 2 1 2 r=r0 If we work in the plane-wave basis, the divergent term is (cid:90) k2 −k2 i ddsxdtr2ν−2 1 2 ei(ω1−ω2)t−i((cid:126)k1−(cid:126)k2)·(cid:126)xψ(0)∗(r)ψ(0)(r), (24) 0 4(1−ν) 1 2 r=r0 and if we integrate over the region between two surfaces t = constant, the integral over the spatial directions will introduce an overall momentum delta-function, so that this divergent term vanishes.3 As claimed earlier, the finite piece will vanish for any mixed boundary condition φ = fφ with a real coefficient f. Thus, for ν < 2, the + − Klein-Gordon inner product is conserved for any such mixed boundary conditions; in particular it is conserved for both Dirichlet and Neumann boundary conditions, without adding an explicit boundary term to the inner product. This argument can easily be extended to general ν < z. The key point is that the subleading terms appearing in the asymptotic expansion for φ will all involve only spatial derivatives of φ(0), so they make vanishing contributions to the total flux through the boundary at infinity between two surfaces t = constant. Once we know that the inner product is conserved, it is easy to see that it is orthogonal in the plane wave basis φ = e−iωt+i(cid:126)k·(cid:126)xψ(r). We saw already in (14) that 3One can add a local counterterm to cancel this divergence; this counterterm will not affect the value of the inner product. 8 (cid:126) (cid:126) the spatial integral makes the inner product vanish if k (cid:54)= k . Now the fact that it 1 2 is independent of the spatial slice t = constant we choose to evaluate it on implies it must vanish if ω (cid:54)= ω for real ω, as otherwise it would be time dependent. We are 1 2 therefore left with the inner product of the plane wave modes with themselves, (cid:90) ∞ (φ ,φ ) = (2π)dsVol(x)ω drrds−z−1|ψ |2, (25) 1 1 1 1 0 where Vol(x) is the spatial volume. This is manifestly positive if ω > 0. 1 We can therefore understand the condition ν < z as arising from requiring that the free data φ(ν) appears in the asymptotic expansion before the first term which involves time derivatives of φ(0). That is, it is precisely because we do not require counterterms in the action involving time derivatives that the standard Klein-Gordon norm remains appropriate for more general boundary conditions. 4 Instability for Neumann boundary conditions We would now like to consider the spectrum for the different possible boundary con- ditions, to check if there are any instabilities, looking for regular solutions which grow exponentially in time. From the point of view of the dual field theory, instabilities appear as poles in the two-point function in the upper half frequency plane.4 For the AdS case, the symmetries imply that the two-point function is a function of the Lorentz invariant ω2 − k2. Instabilities can then occur only in the case of mixed boundary conditions, where they correspond to tachyonic poles in the two- point function with ω2 −k2 = −m2 .5 For the conformally invariant pure Dirichlet bdy or Neumann boundary conditions, by contrast, no such instability is possible, as there is no scale to provide a value for m2 . bdy In the Lifshitz case however, the symmetry is less restrictive, and the two-point function for general boundary conditions can depend separately on ω and k. For con- formally invariant boundary conditions, the two-point function (up to overall scaling) must be a function of the invariant ω/kz, but this still admits the possibility of in- stability, if the two-point function has a pole at ω = α, Im α > 0. (26) kz WewillseebelowthatsuchunstablemodesappearforNeumannboundaryconditions when ν > 1. The existence of such instabilities in the conformally invariant case is perhaps surprising. Moreover, as a result of the scale invariance, the instability has no associated timescale. Unlike the relativistic case, when an instability occurs, there are exponentially growing modes for all momenta, and the modes of high momenta have 4Since our situation is time-translation invariant, poles at complex ω will appear in complex conjugate pairs. 5Themixedboundaryconditionscorrespondtothefieldtheorydeformedbyamulti-traceopera- tor,andabulkinstabilitycanbeinterpretedasthisdeformationmakingthefieldtheoryHamiltonian unbounded from below [18, 19]. 9 arbitrarily high growth rates. Thus, this instability indicates the complete breakdown of the expansion around the putative background, which is not valid even in an open neighbourhood in time. We therefore interpret this instability as saying that no dual field theory exists for the boundary conditions where the instability is present. Becausethereareunstablemodesofarbitrarilyhighmomentum, thisisessentially a UV effect in the field theory. That is, this instability will affect not just the pure Lifshitz spacetime, but any solution which asymptotically approaches this solution with the Neumann boundary conditions. In particular, this implies that the theories with mixed boundary conditions will alsobeunstable, astheseapproachtheNeumannboundaryconditionintheUV.That is, the field theories dual to the mixed boundary conditions are relevant deformations of the scale-invariant theory dual to the Neumann boundary conditions, so the non- existence of this UV theory implies that the theories dual to the mixed boundary conditions will also not be well-defined. 4.1 Analytic calculation for z = 2 We first consider the case z = 2, where it is possible to write the solution explicitly in terms of confluent hypergeometric functions, as discussed in [1]. For z = 2, the generic solution of (9) is ψ(u) = eiu2ω/2[φ u(ds+2)+ν F (a,b,−iu2ω)+φ u(ds+2)−ν F (a−b+1,2−b,−iu2ω)], + 2 1 1 − 2 1 1 (27) where u = 1/r, 1 k2 a = (1+ν)+i , b = 1+ν, (28) 2 4ω and F (α,γ,z) is the confluent hypergeometric function, whose series expansion is 1 1 F (α,γ,z) = 1+ αz+.... The first term in (27) corresponds to the fast falloff mode 1 1 γ at infinity, and the second term to the slow falloff. For complex ω, the solution which is regular at r = 0 is the one which is ex- ponentially damped. The solution which is regular for Im ω > 0 can be written as ψ = eiu2ω/2u(ds+2)+νU (cid:0)a,b,−iωu2(cid:1) (29) 2 whereu = 1/r,theconstantsa,baregivenin(28),andU(a,b,z)isTricomi’sconfluent hypergeometric function, which is given in terms of the confluent hypergeometric function by Γ(1−b) Γ(b−1) U(a,b,z) = F (a,b,z)+ z1−b F (a−b+1,2−b,z). (30) 1 1 1 1 Γ(a−b+1) Γ(a) Wethenwanttoaskforvaluesofω suchthatthisalsosatisfiestheboundarycondition at u = 0. Using (30), the solution near u = 0 is (−iω)−νΓ(ν) Γ(−ν) ψ = uds+2−ν (1+...)+uds+2+ν (1+...), (31) 2 Γ(cid:0)1+ν +ik2(cid:1) 2 Γ(cid:0)1−ν +ik2(cid:1) 2 4ω 2 4ω 10