The Backreaction of Anti-M2 Branes on a Warped Stenzel Space
aa r X i v : . [ h e p - t h ] N ov IPhT-T10/169
The Backreaction of Anti–M2 Braneson a Warped Stenzel Space
Iosif Bena † , Gregory Giecold † and Nick Halmagyi † ∗ † Institut de Physique Th´eorique,CEA Saclay, CNRS URA 2306,F–91191 Gif–sur–Yvette, France ∗ Laboratoire de Physique Th´eorique et Hautes Energies,Universit´e Pierre et Marie Curie, CNRS UMR 7589,F–75252 Paris Cedex 05, France [email protected], [email protected]@lpthe.jussieu.fr
Abstract
We find the superpotential governing the supersymmetric warped M–theory solutionwith a transverse Stenzel space found by Cvetiˇc, Gibbons, L¨u and Pope in hep–th/0012011,and use this superpotential to extract and solve the twelve coupled equations underlyingthe first–order backreacted solution of a stack of anti–M2 branes in this space. These anti–M2 branes were analyzed recently in a probe approximation by Klebanov and Pufu, whoconjectured that they should be dual to a metastable vacuum of a supersymmetric 2+1dimensional theory. We find that the would–be supergravity dual to such a metastable vac-uum must have an infrared singularity and discuss whether this singularity is acceptable ornot. Given that a similar singularity appears when placing anti–D3 branes in the Klebanov–Strassler solution, our work strengthens the possibility that anti–branes in warped throatsdo not give rise to metastable vacua. Introduction and discussion
The recent revival of interest in metastable supersymmetry breaking in quantum field theoryis largely due to the work of Intriligator, Seiberg and Shih [1] (ISS). This work presents amechanism to naturally circumvent some of the problems afflicting other models for dynamicsupersymmetry breaking (DSB) [2–5]. A natural question that was posed immediatelyafter [1] is whether metastable vacua also exist in string realizations of supersymmetricfield theories.For type IIA brane–engineering models of supersymmetric field theories, the answer tothis question is negative [6]. Indeed, these models are constructed using D4 branes endingon codimension–two defects inside NS5 branes [6–8], which source NS5 worldvolume fieldsthat grow logarithmically at infinity. In supersymmetric vacua this logarithmic growthencodes the running of the gauge theory coupling constant with the energy [9–12], butthese logarithmic modes are different in the candidate metastable brane configuration andin the supersymmetric one. This implies that the candidate metastable brane configurationand the supersymmetric one differ by an infinite amount, and hence cannot decay into eachother. Hence, the type IIA brane construction does not describe a metastable vacuum of asupersymmetric theory, but instead a nonsupersymmetric vacuum of a nonsupersymmetrictheory.Another arena where one might try to find string theory realizations of metastablevacua are IIB holographic duals of certain supersymmetric gauge theories. The best–knownexample in this class was proposed by Kachru, Pearson and Verlinde [13, 14], who arguedthat a background with anti–D3 branes at the bottom of the Klebanov–Strassler warpeddeformed conifold [15] is dual to a metastable vacuum of the dual supersymmetric gaugetheory. Since the Klebanov–Strassler solution has positive D3 brane charge dissolved influx, the anti–D3 branes can annihilate against this charge (this annihilation happens viathe polarization of the anti–D3 branes into an NS5 brane [16, 17]), and this bulk process isargued to correspond to the decay of the metastable vacuum to the supersymmetric one inthe dual field theory.Another proposal for a metastable vacuum obtained by putting anti–branes at the bot-tom of a smooth warped throat with positive brane charge dissolved in flux has recentlybeen made by Klebanov and Pufu [18], who argued that probe anti–M2 branes at the tipof a supersymmetric warped M–theory background with transverse Stenzel space [19], giverise to a long–lived metastable vacuum. The supersymmetric solution, first found by Cvetiˇc,Gibbons, L¨u and Pope (CGLP) in [20] has M2 charge dissolved in fluxes and a large S in the infrared. The anti–branes can annihilate against the charge dissolved in fluxes bypolarizing into M5 branes [21] wrapping three–spheres inside the S .The probe brane analyses described above, while indicative that a metastable vacuummight exist, are however not enough to establish this. One possible issue which can causethe backreacted solution to differ significantly from the probe analysis is the presence ofnon-normalizable modes. If the anti-branes indeed source such modes then the candidatemetastable configuration is not dual to a non–supersymmetric vacuum of a supersymmetrictheory, but to a non–supersymmetric vacuum of a non–supersymmetric theory, and thesupersymmetry breaking is not dynamical but explicit. The existence of non–normalizablemodes is not visible in the probe approximation (much like the existence of type IIA log–growing modes was not visible in g s = 0 brane constructions [7,8]), but only upon calculatingthe backreaction of the probe branes – a not too easy task. n [22] two of the authors and M. Gra˜na found the possible first–order backreactedsolution sourced by a stack of anti–D3 branes smeared on the large S at the bottom ofthe Klebanov–Strassler (KS) solution, and found two very interesting features: first, of the14 physical modes describing SU (2) × SU (2) × Z –invariant perturbations of the warpeddeformed conifold, only one mode enters in the expression of the force that a probe D3 branefeels in this background. Hence, since anti–D3 branes attract probe branes, if the perturbedsolution is to have any chance to describe backreacted anti–D3 branes, this mode must bepresent . The second feature of this solution is that if the force mode is present, the infrared must contain a certain singularity, which has finite action . Note that having a finite actiondoes not automatically make a singularity acceptable – negative–mass Schwarzschild is anobvious counterexample [27]. As discussed in [22], if this singularity is unphysical, then thesolution sourced by the anti–D3 branes cannot be thought of as a small perturbation ofthe KS solution, and therefore does not describe a metastable vacuum of the dual theory.If this singularity is physical, the first–order solution does describe anti–D3 branes at thebottom of the KS solution, and work is in progress to determine what are the features of thissolution, and whether the perturbative anti–D3 brane solution describes or not metastablevacua of the dual theory.The purpose of this paper is to calculate the first–order backreaction of the other pro-posed metastable configuration with anti–branes in a background with charge dissolved influxes: the anti–M2 branes in the Stenzel–CGLP solution [20]. In order to do this we smearthe anti–M2 branes on the large S at the bottom of the Stenzel–CGLP solution, and solvefor all possible deformations of this background that preserve its SO(5) symmetry. Weconsider an ansatz for these deformations ; the space of deformations is parameterized by6 functions of one variable satisfying second–order differential equations. However, whenperturbing around a supersymmetric solution, Borokhov and Gubser [28] have observedthat these second–order equations factorize into first–order ones, that are much easier tosolve. Nevertheless, in order to apply the Borokhov–Gubser method, one needs to findthe superpotential underlying the supersymmetric solution, which for the warped fluxedStenzel–CGLP solution was not known until now. The first result of this paper, presentedin Section 2, is to find this superpotential , and derive two sets of first–order equationsgoverning the space of deformations.We then show in Section 3 that the force felt by a probe M2 brane in the most generalperturbed background depends on only one of the “conjugate–momentum” functions thatappear when solving the first–order system, and hence on only one of the 10 constantsparameterizing the deformations around the supersymmetric solution. We then solve inSection 4 the two sets of first–order differential equations. Amazingly enough, the solutionsfor the first set of equations (for the conjugate–momentum functions) can be found explicitlyin terms of incomplete elliptic integrals (a huge improvement on the situation in [22]). Wealso find the homogeneous solutions to the other equations and give implicitly the fullsolution to the system in terms of integrals. We also provide the explicit UV and IR The asymptotic behavior of the force matches the one argued for in [23], and the existence of this mode wasfirst intuited in [24] which set out to study the UV asymptotics of the perturbations corresponding to anti–D3branes in the KT background [25]. An IR analysis of some of the non–supersymmetric isometry–preserving perturbations of the Klebanov–Strassler background can also be found in [26]. This was first observed by I. Klebanov. This is the equivalent of the Papadopoulos–Tseytlin superpotential for the KS solution [29–31]. xpansions of the full space of deformations, and find which deformations correspond tonormalizable modes and which deformations correspond to non–normalizable modes.In Section 5 we then use the machinery we developed to recover the perturbative ex-pansion of the known solution sourced by BPS M2 branes smeared on the S at the tip ofthe Stenzel–CGLP solution [20], and analyze the infrared of the possible solution sourcedby anti–M2 branes. After removing some obviously unphysical divergences and demandingthat in the first–order backreacted solution a probe M2 brane feels a nonzero force, we findthat the only backreacted solution that can correspond to anti–M2 branes must have aninfrared singularity, coming from a four–form field strength with two or three legs on thethree–sphere that is shrinking to zero size at the tip of the Stenzel space.Hence, the first–order backreacted solution for the anti–M2 branes has the same two keyfeatures as the anti–D3 branes in KS: the force felt by a probe M2 brane in this backgrounddepends only on one of the 10 physical perturbation modes around this solution, and thesolution where the force–carrying mode is turned on must have an infrared singularitycoming from a divergent energy in the M–theory four–form field strength. Nevertheless,unlike in the “anti–D3 in KS” solution, the action of this infrared singularity also diverges.Again, if this singularity is physical, our first–order backreacted solution describes anti–M2branes in the CGLP background, and, to our knowledge, would be the first backreactedsupergravity solution dual to metastable susy–breaking in 2+1 dimensions since the workof Maldacena and N˘astase [32]. This may be of interest both in the programme of usingthe AdS/CFT correspondence to describe strongly-interacting condensed–matter systems,and also in view of the relevance of three–dimensional QFT’s at strong coupling to a recentholographic model of four–dimensional cosmology [33]. On the other hand, if the singularityis not physical then the backreaction of the anti–M2 branes cannot be taken into accountperturbatively; this indicates that the only solution with proper anti–M2 brane boundaryconditions in the infrared is the solution for anti–M2 branes in a CGLP background withanti–M2 brane charge dissolved in flux, and hence the anti–M2 branes flip the sign of theM2 brane charge dissolved in flux.Given the similarity of the results of the “anti–D3 in KS” and of the “anti–M2 in CGLP”analyses and the drastically–different calculations leading to them, it is rather natural toexpect that the underlying physics of the two setups is the same: either both singularitiesare physical, which indicates that anti–branes in backgrounds with charge dissolved in fluxesgive rise to metastable vacua, or they are both unphysical, which supports the idea thatanti–branes in such backgrounds cannot be treated as a perturbation of the original solution,and may flip the sign of the charge dissolved in flux. Furthermore, our analysis suggeststhat one cannot use the finiteness of the action as a criterion for accepting a singularity.This would allow the anti–D3 singularity and exclude the anti–M2 one, which would berather peculiar, given the striking resemblance of the two systems.There are a few possible explanations for the singularities we encounter in the anti–M2 and anti–D3 solutions. One is that these singularities are accompanied by stronger,physical singularities, coming from the smeared anti–M2 or anti–D3 sources, and one canhope that whatever mechanism renders the stronger singularities physical may cure thesubleading ones as well. Another explanation is that the subleading singularities are aresult of smearing the antibranes. This is a difficult argument to support with calculationalevidence, as the unsmeared solution is a formidable problem even for BPS branes in Stenzelspaces [34, 35]. Furthermore, a naive comparison of the anti–M2 and anti–D3 solutions ndicates that the stronger the physical singularity associated with the brane sources is,the stronger the subleading singularity will be. Hence, it is likely that unsmearing willmake things worse, not better. Note also that one cannot link the divergent four–form fieldstrength with the M5 branes into which the anti–M2 branes at the tip of the Stenzel–CGLPsolution polarize – they have incompatible orientations.It is also interesting to remember that when one attempts to build string realisations offour-dimensional metastable vacua, either via brane constructions [6] or via AdS-CFT [22],the non–normalizable modes one encounters are log–growing modes, which one could inhindsight have expected from the generic running of coupling constants of four–dimensionalgauge theories with the energy.For anti–M2 branes there is no such link. There exist both AdS/CFT duals of metastablevacua of 2+1 dimensional gauge theories [32], as well as brane–engineering constructions ofsuch metastable vacua (using D3 branes ending on codimension–three defects inside NS5branes) [36]. The nonexistence of an anti–M2 metastable vacuum could only be seen insupergravity, and comes from the way the fields of the anti–M2 brane interact with themagnetic fields that give rise to the charge dissolved in fluxes. This may indicate there is aproblem with trying to construct metastable vacua in string theory by putting antibranesin backgrounds with charge dissolved in fluxes. In an upcoming paper [37] we will alsoargue that anti–D2 branes in backgrounds with D2 brane charge dissolved in fluxes [38],that one of us investigated in [39], have similar problems. We are interested in the backreaction of a set of anti–M2 branes spread on a four–sphereat the bottom of the warped Stenzel geometry [19] with nontrivial fluxes. Smearing theanti–M2’s is necessary in order for the perturbed solution to have the same SO(5) globalsymmetry as the supersymmetric solution of Cvetiˇc, Gibbons, L¨u and Pope (CGLP) [20].The perturbed metric and flux coefficients are then functions of only one radial variable,and generically satisfy n second–order differential equations.However, when perturbing around a supersymmetric solution governed by a superpoten-tial, Borokhov and Gubser [28] have observed that these n second–order equations factorizeinto n first–order equations for certain momenta and n first–order equations for the metricand flux coefficients, and that furthermore the n equations for the momenta do not containthe metric and flux coefficients, and hence can be solved independently. This technique hasbeen used in several related works [22, 28, 40] and we consider this to be the technique ofchoice for deformation problems that depend on just one coordinate. While the following summary can be found by now in several sources, we include it herefor completeness. When the equations of motion governing the fields φ a of a certain super-symmetric solution come from the reduction to a one–dimensional Lagrangian L = − G ab dφ a dτ dφ b dτ − V ( φ ) (2.1) hose potential V ( φ ) comes from a superpotential, V ( φ ) = 18 G ab ∂W∂φ a ∂W∂φ b , . (2.2)The Lagrangian is written as L = − G ab (cid:18) dφ a dτ − G ac ∂W∂φ c (cid:19) (cid:18) dφ a dτ − G ac ∂W∂φ c (cid:19) − dWdτ , (2.3)and the supersymmetric solutions satisfy dφ a dτ − G ab ∂W∂φ b = 0 . (2.4)We now want to find a perturbation in the fields φ a around their supersymmetric back-ground value φ a φ a = φ a + φ a ( X ) + O ( X ) , (2.5)where X represents the set of perturbation parameters in which φ a is linear. The deviationfrom the gradient flow equations for the perturbation φ a is measured by the conjugatemomenta ξ a ξ a ≡ G ab ( φ ) (cid:18) dφ b dτ − M bd ( φ ) φ d (cid:19) , (2.6) M bd ≡ ∂∂φ d (cid:18) G bc ∂W∂φ c (cid:19) . (2.7)The ξ a are linear in the expansion parameters X , hence they are of the same order as the φ a . When all the ξ a vanish the deformation is supersymmetric.The main point of this construction is that the second–order equations of motion gov-erning the perturbations reduce to a set of first–order linear equations for ( ξ a , φ a ): dξ a dτ + ξ b M ba ( φ ) = 0 , (2.8) dφ a dτ − M ab ( φ ) φ b = G ab ξ b . (2.9)Note that equation (2.9) is just a rephrasing of the definition of the ξ a in (2.6), while (2.8)implies the equations of motion. Since one considers these perturbations in a metric ansatzin which the reparametrization invariance of the radial variable is fixed, in addition to theseequations one must enforce the zero–energy condition ξ a dφ a dr = 0 . (2.10) Using the analysis of the CGLP solution in [18], one can easily see that the ansatz for theSO(5)–invariant eleven–dimensional supergravity solution we are looking for is ds = e − z ( r ) dx µ dx µ + e z ( r ) h e γ ( r ) dr + e α ( r ) σ i + e β ( r ) e σ i + e γ ( r ) ν i = e − z ( r ) dx µ dx µ + dτ + a ( τ ) σ i + b ( τ ) e σ i + c ( τ ) ν , (2.11) G = dK ( τ ) ∧ dx ∧ dx ∧ dx + m F , (2.12) here F = dA and A = f ( τ ) e σ ∧ e σ ∧ e σ + h ( τ ) ǫ ijk σ i ∧ σ j ∧ e σ k (2.13) ⇒ F = ˙ f dτ ∧ ˜ σ ∧ ˜ σ ∧ ˜ σ + ˙ h ǫ ijk dτ ∧ σ i ∧ σ j ∧ ˜ σ k + 12 (4 h − f ) ǫ ijk ν ∧ σ i ∧ ˜ σ j ∧ ˜ σ k − h ν ∧ σ ∧ σ ∧ σ . (2.14)Our notation for the one–forms on the Stenzel space is by now standard [18], in the sensethat with the definitions σ i = L i , e σ i = L i , ν = L , (2.15)they satisfy dσ i = ν ∧ e σ i + L ij ∧ σ j , (2.16) d ˜ σ i = − ν ∧ σ i + L ij ∧ e σ j , (2.17) dν = − σ i ∧ ˜ σ i , (2.18) dL ij = L ik ∧ L kj − σ i ∧ σ j − e σ i ∧ e σ j . (2.19)Integrating one particular component of the equation of motion for the flux d ∗ G = 12 G ∧ G (2.20)gives K ′ = 6 m h h ( f − h ) − i e − α + β ) − z , (2.21)where we have chosen the integration constant such that the BPS solution [20] is regular,i.e. there are no explicit source M2 branes. We refer to Appendix A for more details.Performing a standard dimensional reduction on this ansatz down to one dimension, weobtain the following Lagrangian L = ( T gr + T mat ) − ( V gr + V mat ) (2.22)with the gravitational and matter sectors given by T gr = 3 e α + β ) (cid:20) α ′ + β ′ − z ′ + 3 α ′ β ′ + α ′ γ ′ + β ′ γ ′ (cid:21) , (2.23) V gr = 34 e α + β h e α + e β + e γ − e α +2 β − e α +2 γ i (2.24)and T mat = − m e α + β − z (cid:16) f ′ e − β + 12 h ′ e − α (cid:17) , (2.25) V mat = 3 m e α +3 β − z (cid:20) h e − α + 14 (4 h − f ) e − β (cid:21) +9 m e − α + β +2 z ) (cid:20) h ( f − h ) − (cid:21) . (2.26) he superpotential is given by W = − e α +2 β ( e α + e β + e γ ) − m e − z (cid:20) h ( f − h ) − (cid:21) . (2.27)It is worth noting that equation (2.2) only defines the superpotential up one independentminus sign which can then be absorbed in (2.8) and (2.9) by changing the sign of the radialvariable and the ξ a . However, with the wisdom of hindsight, we choose a radial variablesuch that fields decay at infinity and not minus infinity, thus simultaneously fixing the signof the superpotential. Here we summarize the expressions that the fields in our ansatz take when specialized tothe zeroth–order CGLP solution [20] around which we endeavor to study supersymmetricand non–supersymmetric perturbations.We should note that the CGLP solution with transverse Stenzel geometry is to thewarped M–theory solution with transverse Stiefel space [41] what the IIB Klebanov–Strasslersolution [15] and the deformed conifold [42] are to the Klebanov–Tseytlin solution [25] andthe singular conifold. The Stenzel space is a higher–dimensional generalization of the de-formed conifold. A useful summary of many details of the supergravity solution can befound in [43] and proposals for the dual field theory can be found in [43, 44]The supersymmetric solution around which we will perturb was found in [20]. It can besummarized in our ansatz by e α = 13 (2 + cosh(2 r )) / cosh( r ) , (2.28) e β = 13 (2 + cosh(2 r )) / sinh( r ) tanh( r ) , (2.29) e γ = (2 + cosh(2 r )) − / cosh ( r ) , (2.30) f = 13 / (cid:0) − ( r ) (cid:1) cosh ( r ) , (2.31) h = − / r ) , (2.32) e z ( y ) = 2 / m Z ∞ y du ( u − / , (2.33)where y ≡ r ) . (2.34)With this change of coordinate we can write e z = √ m y (cid:0) − y (cid:1) ( y − / + 5 √ m F (cid:18) arcsin (cid:18) y (cid:19) | − (cid:19) , (2.35)where the incomplete elliptic integral of the first kind is F ( φ | q ) = Z φ (cid:0) − q sin( θ ) (cid:1) − / dθ (2.36) nd we have fixed the integration constant (denoted c in [20]) by requiring e z → r → ∞ . We now write out explicitly the two sets of equations (2.8) and (2.9). In both cases aparticular field redefinition simplifies things substantially. ξ a equations The ξ a equations (2.8) simplify in the basis˜ ξ a = ( ξ + ξ + ξ , ξ − ξ + 3 ξ , ξ + ξ − ξ , ξ , ξ , ξ ) . (2.37)In the order which we solve them, the equations are˜ ξ ′ = 6 m e − α + β + z ) (cid:18) ( f − h ) h − (cid:19) ˜ ξ , (2.38)˜ ξ ′ = 12 m e − α + β + z ) (cid:18) ( f − h ) h − (cid:19) ˜ ξ , (2.39)˜ ξ ′ = 12 e α − β ˜ ξ − m h e − α + β + z ) ˜ ξ , (2.40)˜ ξ ′ = 6 e − α − β ) ˜ ξ − e α − β ˜ ξ − m e − α + β + z ) ( f − h ) ˜ ξ , (2.41)˜ ξ ′ = 29 e − α + β + z ) h e α + β + γ )+3 z ˜ ξ + m (54 h ( f − h ) −
1) ˜ ξ i (2.42)˜ ξ ′ = 12 e − α − β h e α + β ) ˜ ξ − e α + γ ) ˜ ξ − h e β ˜ ξ + e α (cid:16) − ξ + 2 ˜ ξ + 3 ˜ ξ + 2 ( f − h ) ˜ ξ (cid:17) i , (2.43)where we remind the reader that a prime denotes a derivative with respect to r not y (2.34). φ a equations The φ a equations benefit from a field redefinition as well, φ a = ( α, β, γ, z, f, h ) , (2.44)˜ φ a = ( φ − φ , φ + φ − φ , φ , φ , φ , φ ) (2.45) nd we find˜ φ ′ = 112 e − α + β ) h − ξ + 4 ˜ ξ + 3 (cid:16) ˜ ξ − e α + β ) (cid:16) e α + e β (cid:17) ˜ φ (cid:17)i , (2.46)˜ φ ′ = 112 e − α + β ) h − ξ + 7 ˜ ξ + 12 e α + β ) (cid:16) (cid:16) e β − e α (cid:17) ˜ φ − e γ ˜ φ (cid:17)i , (2.47)˜ φ ′ = 112 e − α + β ) h ˜ ξ − (cid:16) ˜ ξ + 6 e α + β ) (cid:16)(cid:16) e β − e α (cid:17) ˜ φ − e γ ˜ φ (cid:17)(cid:17)i , (2.48)˜ φ ′ = 2 m e − α − β ) h e z ˜ ξ + 3 m (3 h ˜ φ − ˜ φ ) i , (2.49)˜ φ ′ = 16 m e α − β h e z ˜ ξ − m ( f ˜ φ − h ˜ φ + ˜ φ − φ ) i , (2.50)˜ φ ′ = 19 e − α + β + z ) h e z ˜ ξ + m (cid:16) [1 − h ( f − h )] ˜ φ + 18 f ˜ φ + ˜ φ + 2 ˜ φ + 18 h h ˜ φ − φ − f − h ) ( ˜ φ + 2 ˜ φ ) i (cid:17)i . (2.51) Before solving the above equations, we compute the force on a probe M2–brane in theperturbed solution space. As was found in the analogous IIB scenario [22], the force turnsout to benefit from remarkable cancellations and is ultimately quite simple.The membrane action for a probe M2 brane (which by abusing notation we refer to asthe DBI action) is V DBI = √− g g g , = e − z (3.1)and, in the first–order approximation, its derivative with respect to r is F DBI = − dV DBI dr + 3 e − z (cid:16) ˜ φ ′ − z ′ ˜ φ (cid:17) . (3.2)We next consider the derivative of the WZ action with respect to r , which gives the forceexerted on the M2–brane by the G (4) field : F W Z = − dV W Z dr , = G (4)012 r , = − m (cid:20) h ( f − h ) − (cid:21) e − α + β ) − z . (3.3)The zeroth–order and first–order WZ forces thus are F W Z = − m (cid:20) h ( f − h ) − (cid:21) e − α + β ) − z (3.4) nd F W Z = − m h h ( ˜ φ − φ ) + ˜ φ ( f − h ) − φ + 2 ˜ φ + 2 ˜ φ ) (cid:16) h ( f − h ) − (cid:17)i e − α + β ) − z . (3.5)Combining these two contributions to the force we see that the zeroth–order contribu-tions cancel as expected. Then using the explicit φ a equations from section 2.4.2 we findthe beautiful result F = F DBI + F W Z = 23 e − α + β + z )( r ) ˜ ξ ( r ) . At this point it is worthwhile to preemptively trumpet the result (4.3) from Section 4 wherethe exact solution for the mode ˜ ξ is found: F = 23 e − α + β )( r ) Z X = 18 Z X (2 + cosh 2 r ) / sinh r , (3.6)where Z is some numerical factor which we found convenient not to absorb into the X integration constant, Z ≡ e − z (0) . (3.7)So, the UV expansion of the force felt by a probe M2 brane in the first–order perturbedsolution is always F r ∼ X e − r/ + O ( e − r/ ) . (3.8)In terms of ρ , the “standard” radial coordinate , this force comes from a potential propor-tional to ρ − , which agrees with a straightforward extension of the brane–antibrane forceanalysis of [23] to this system. This will be further discussed in a forthcoming publica-tion [45]. In this section we find the generic solution to the system (2.38)–(2.51). This solution spacehas twelve integration constants of which ten are physical. We have managed to solve the˜ ξ a equations exactly whereas for the φ a equations we have resorted to solving them in theIR and UV limits. Related to r via cosh(2 r ) ∼ ρ / . .1 Analytic solutions for the ˜ ξ ’s The first equation (2.38) is solved by˜ ξ = X exp (cid:18) m Z r dr ′ e − α + β + z ) (cid:20) ( f − h ) h − (cid:21)(cid:19) , (4.1)which appears to be a double integral. However, using a standard notation for the warpfactor H = e z , since we have dH dr = − m e γ sinh r tanh r , (4.2)we actually find ˜ ξ = X exp (cid:18)Z r dr ′ H dH dr ′ (cid:19) , = X e z ( r ) − z (0)) . (4.3)It immediately follows that ˜ ξ = X + 2 X e z ( r ) − z (0)) . (4.4)We find convenient not to include e − z (0) into the integration constant X , and will usethe notation Z ≡ e − z (0) . (4.5)We were also able to find exact analytic expressions for ˜ ξ and ˜ ξ , , in term of y ≡ r ) :˜ ξ = y (cid:0) y − (cid:1) X − m Z X √ y (cid:0) y − (cid:1) ( y − / " −
96 + 599 y − y + 119 y − y p y − (cid:0) − y + y (cid:1) (cid:18) F (cid:18) arcsin (cid:18) y (cid:19) | − (cid:19) + 22 (cid:20) Π (cid:18) −√ − arcsin (cid:18) y (cid:19) | − (cid:19) + Π (cid:18) √ − arcsin (cid:18) y (cid:19) | − (cid:19)(cid:21) (cid:19) , (4.6)where F ( φ | q ) is given in (2.36) and Π( n ; φ | m ) is an incomplete elliptic integral of thethird kind Π( n ; φ | m ) = Z φ dθ (cid:16) − n sin ( θ ) (cid:17) q − m sin ( θ ) . (4.7) he expressions for ˜ ξ , are as follows :˜ ξ = 14 √ y − p y − " √ Z X m y (cid:0) − y (cid:1) p y −
1+ 4 h(cid:0) y − (cid:1) X + (cid:0) y − (cid:1) (cid:0) y (cid:1) X i + √ Z m X (cid:20) (cid:0)
19 + 7 y (cid:0) y − (cid:1)(cid:1) F (cid:18) arcsin (cid:18) y (cid:19) | − (cid:19) − (cid:0) y − (cid:1) (cid:0) y (cid:1) (cid:18) Π (cid:18) −√ − arcsin (cid:18) y (cid:19) | − (cid:19) + Π (cid:18) √ − arcsin (cid:18) y (cid:19) | − (cid:19)(cid:19) (cid:21) , (4.8)˜ ξ = √ y −
3) ( y − / " (cid:0) y − (cid:1) (cid:0) y − (cid:1) (cid:20) X + r Z m X (cid:16) y − y ( y − / + 5 F (cid:18) arcsin (cid:18) y (cid:19) | − (cid:19) (cid:17)(cid:21) + 14 (cid:0) y − (cid:1) " − √ Z m X y p y −
1+ 4 (cid:0) y − (cid:1) X − √ Z m X (cid:0) y − (cid:1) F (cid:18) arcsin (cid:18) y (cid:19) | − (cid:19) + 2 (cid:18) Π (cid:18) −√ − arcsin (cid:18) y (cid:19) | − (cid:19) + Π (cid:18) √ − arcsin (cid:18) y (cid:19) | − (cid:19)(cid:19) ! . (4.9)Lastly, ˜ ξ is given by the zero–energy condition (2.10) but its explicit form does not appearto be too enlightening.In Appendix B we provide the IR and UV series expansions of the above solutions for˜ ξ i . φ i equations We now solve the system of equations for φ i (2.46)–(2.50) using the Lagrange method ofvariation of parameters.Equation (2.46) is solved by ˜ φ = ˜ λ ( r )sinh(2 r ) , (4.10)with ˜ λ = 92 Z cosh( r )sinh( r ) (2 + cosh(2 r )) / h − ξ + 4 ˜ ξ + 3 ˜ ξ i + Y IR . (4.11)˜ ξ and ˜ ξ are given in Section 4.1 above and sinh(2 r ) − is the homogeneous solution to the˜ φ equation. he same Lagrange method is used for ˜ φ , which is given by˜ φ = ˜ λ ( r )sinh( r ) (2 + cosh(2 r )) , (4.12)where˜ λ = 94 Z sinh( r ) (2 + cosh(2 r )) / (cid:20) − ξ + 7 ˜ ξ −
43 sinh( r ) cosh( r ) (2 + cosh(2 r )) / ˜ φ (cid:21) + Y IR . (4.13)From this, we obtain an integral expression for ˜ φ :˜ φ = 94 Z h ˜ ξ − ξ +
23 sinh( r ) cosh( r ) (2 + cosh(2 r )) / ˜ φ + 2 sinh( r ) cosh( r ) (2+cosh(2 r )) / ˜ φ i sinh( r ) (2 + cosh(2 r )) / + Y IR . (4.14)The fluxes (cid:16) ˜ φ , ˜ φ (cid:17) = ( f, h ) are given by (cid:18) ˜ φ ˜ φ (cid:19) = (cid:18) cosh( r ) tanh( r ) cosh( r ) (cid:2) − r ) (cid:3) (cid:2) sech( r ) − cosh( r ) (cid:3) cosh( r ) (cid:19) (cid:18) ˜ λ ˜ λ (cid:19) , (4.15)where the derivatives of ˜ λ and ˜ λ are given by (cid:18) ˜ λ ′ ˜ λ ′ (cid:19) = (cid:18) cosh( r ) coth( r ) [cosh( r ) − r ) csch( r )] [3 + cosh(2 r )] sech( r ) sinh( r ) tanh( r ) (cid:19) (cid:18) b b (cid:19) , (4.16)and b , b are the right–hand side of (2.49) and (2.50) respectively. The 2 × Y and Y the constants arising from integrating (4.16), even though the twofunctions ˜ φ and ˜ φ depend on both of them.Finally, relying on the same method, the equation for ˜ φ is solved to˜ φ = e − z ( r ) ˜ λ , ˜ λ = Z e z ( r ) b ( r ) + Y IR , (4.17)where b ( r ) is the right–hand side of (2.51) (setting ˜ φ to zero). We now give the IR expansions of the φ i ’s. We only write the divergent and constantterms since terms which are regular in the IR do not provide any constraint on our solutionspace. Z is defined in (3.7). The X i integration constants are those appearing in the exactsolutions for the ˜ ξ i ’s (4.3)–(4.9) :˜ φ = − r " X + 30 X − √ X / + 12 r Y IR + " X + (cid:0) −
198 3 / Z m (cid:1) X + 80 √ X
12 3 / + O ( r ) , (4.18) φ = Y IR r + 1 r (cid:20)
94 3 / X + 32 3 / X − √ / X − Y IR (cid:21) − r Y IR − (cid:20) / X + 232 3 / X − √ Z m X − / X − Y IR (cid:21) + O ( r ) , (4.19)˜ φ = − Y IR r − r " / X −
12 3 / X − Y IR + " Y IR + 3 / (cid:16) −
18 3 / Z m X + 21 X + 48 X + 4 √ X (cid:17) log( r ) + O ( r ) , (4.20)˜ φ = − r " X − √ X + Z m (cid:0) Y IR − √ Y IR (cid:1) / − (cid:20) Z m √ X − X ! − Z Y IR ! + 148 Z m (cid:16) √ Y IR − Y IR (cid:17) + (cid:20)
32 3 / X − X / + 136 Z m (cid:16) √ X + 78 √ X − X + 11 3 / Y IR −
72 3 / Y IR (cid:17)(cid:21) log( r ) (cid:21) + O ( r ) , (4.21)˜ φ = 2 Y IR + "
98 3 / X + 34 3 / X − / X + 12 Z m X + √ X ! r + O ( r ) , (4.22)˜ φ = 1 r X + √ X Z m + " / X − X + √ X Z m −
772 3 / X −
518 3 / X + 12 Y IR + O ( r ) . (4.23)Note that in the ˜ φ expansion we have also displayed the term of order r – this termwill be relevant for the singularity analysis in Section 6. .2.3 UV behavior We provide the UV asymptotics for all six ˜ φ i ’s, incorporating terms which decay not fasterthan e − r/ . However, as appears in Table 1 below, a few modes have leading behavior inthe UV which is even more convergent than this.˜ φ = 182 / X e − r/ + 2 Y UV e − r − / (cid:20) X − X + 8 √ X + X ) (cid:21) e − r/ − (cid:20) / X − / √ X + X ) (cid:21) e − r/ + 2 Y UV e − r + O ( e − r/ ) , (4.24)˜ φ = 215 2 / X e r/ − / e − r/ X − Y UV e − r + 4 2 / (cid:20) X − X + 64 √ X + X ) (cid:21) e − r/ + 32 Y UV e − r + O ( e − r/ ) , (4.25)˜ φ = − / X e r/ + Y UV + 9693280 2 / X e − r/ + 154 Y UV e − r − / (cid:20) X − X + 256 √ X + X ) (cid:21) e − r/ − Y UV e − r + O ( e − r/ ) , (4.26)˜ φ = 316 2 / Y UV m e r/ + 2726 2 / Y UV m e r/ + 95 2 / X e r/ + 350271183872 2 / Y UV m e r/ − h Y UV + √ (cid:0) Y UV − Y UV (cid:1)i + 216325 2 / X e − r/ + 484605298792 2 / Y UV m e − r/ + 14413 √ Y UV e − r + 398595300314077700 2 / X e − r/ + 797837388321130570240 2 / Y UV m e − r/ + " Y UV + 78912 √ Y UV e − r − / (cid:20) X − X + 4 2563 √ X + X ) (cid:21) e − r/ + 473729599251995778122560 2 / Y UV m e − r/ + O ( e − r ) , (4.27) φ = 18 (cid:0) Y UV − Y UV (cid:1) e r − (cid:0) Y UV − Y UV (cid:1) e r + 18 (cid:0) Y UV + 9 Y UV (cid:1) e − r + 19 4 2 / √ X e − r/ + (cid:20) √ Y UV − (cid:0) Y UV + Y UV (cid:1)(cid:21) e − r − / (cid:20) √ X + 14765 √ X + 2 30839 ( X + X ) (cid:21) e − r/ + 10 (cid:20) − √ Y UV + 3 Y UV (cid:21) e − r + 561105 2 / " √ X − √ X + 187163 ( X + X ) e − r/ + O ( e − r/ ) , (4.28)˜ φ = − (cid:0) Y UV − Y UV (cid:1) e r − (cid:0) Y UV − Y UV (cid:1) e r + 116 (cid:0) Y UV + 3 Y UV (cid:1) e − r + 10 √ / X e − r/ + (cid:20) √ Y UV − (cid:0) Y UV − Y UV (cid:1)(cid:21) e − r − / " √ X + 9 √ X + 116117 ( X + X ) e − r/ − (cid:20) √ Y UV − Y UV (cid:21) e − r + 41105 √ / (cid:20) X − X + 2932 √ X + X ) (cid:21) e − r/ + O ( e − r/ ) . (4.29)To understand the holographic physics of the ˜ φ i modes, we tabulate the leading UVbehavior coming from each mode. To each local operator O i of quantum dimension ∆ inthe field theory, the holographic dictionary associates two modes in the dual AdS space,one normalizable and one non–normalizable [46, 47]. These two supergravity modes aredual respectively to the vacuum expectation value (VEV) h | O i | i and the deformationof the action δS ∼ R d d x O i :normalizable modes ∼ ρ − ∆ AdS ↔ field theory VEV’snon–normalizable modes ∼ ρ ∆ − AdS ↔ field theory deformations of the action . Here we refer to the standard
AdS radial coordinate ρ AdS , to be distinguished from theradial coordinate on the cone, ρ . In the UV, we have ρ ∼ e r/ and ρ AdS ∼ ρ /m / withthe factor of m / taken with respect to the conventions of [18].In Table 1 we have summarized which integration constants correspond to normalizableand non–normalizable modes. As stated in a previous section, the X i are integration con-stants for the ξ i modes and break supersymmetry, while the Y i are integration constants for he modes φ i . It is very interesting to note that in all cases a normalizable/non–normalizablepair consists of one BPS mode and one non–BPS mode.As already mentioned, the mode ˜ ξ , whose integration constant is X and which is theonly mode accountable for the force felt by a probe M2–brane in the first–order perturbationto the CGLP background [20], is the most convergent mode in the UV, though this cannotbe seen from the expansions we have provided but is apparent at higher order in theasymptotics that we have computed. dim ∆ non–norm/norm int. constant6 ρ AdS /ρ − AdS Y UV /X ρ AdS /ρ − AdS Y UV − Y UV /X − X ρ AdS /ρ − AdS X /Y UV ρ AdS /ρ − AdS Y /X ρ − / AdS /ρ − / AdS Y UV + Y UV /X + X ρ − / AdS /ρ − / AdS Y UV /X Table 1: The UV behavior of the twelve SO (5)–invariant modes in thedeformation space of the CGLP solution. As discussed below, only tenof these modes are physical, and the mode of dim. 3 is a gauge artifact. Taking into account a rescaling which culls Y and the zero energy condition whicheliminates X , we are left with a total of ten integration constants or five modes. Theabsence of a physical mode behaving as ρ AdS is related to the quantization of the level ofthe Chern–Simons matter theory. This is unlike in four–dimensional gauge theories, wherewe expect a dimension–four operator corresponding to the dilaton. Note also that we seeexplicitly the dimension ∆ = 7 / X − X or Y + Y . The numerical factors in the combination of those integrationconstants are actually different, but can be rescaled to the shorthand notation we use. Within the space of solutions that we have derived in Section 4 we now proceed to findthe modes which arise from the backreaction of a set of anti–M2 branes smeared on thefinite–sized S at the tip of the Stenzel-CGLP solution ( r = 0). For describing them it isnecessary to carefully impose the correct infrared boundary conditions.The gravity solution for a stack of localized M2–branes in flat space has a warp factor H ( ρ ) = 1 + Q/ρ and as ρ → n –dimensions, the warp factor scales as ρ − n as ρ → d = 8 − n . This isthe IR boundary condition that we will impose on the solution.We must furthermore bring to bear appropriate boundary conditions on the variousfluxes. This is rather simple for M2 branes in flat space, where the energy from G (4) is thesame as that from the curvature. In the presence of other types of flux, the IR boundaryconditions are more intricate. When the background is on–shell, contributions to the stress ensor from all types of flux taken together cancel the energy from the curvature: this is thebasic nature of Einstein’s equation but this is too wobbly a criterion to signal the presenceof M2 branes. Instead, the right set of boundary conditions for M2 branes should enforcethat the dominant contribution to the stress–energy tensor comes from the G (4) flux. The M2 brane charge varies with the radial coordinate r of a section of the Stenzel space [19]: Q M ( r ) = 1(2 π ℓ p ) Z M ⋆G , = − m Vol ( V , )(2 π ℓ p ) (cid:18) h ( r ) ( f ( r ) − h ( r )) − (cid:19) , (5.1)with ℓ p the Planck length in eleven dimensions, M a constant r section of the transverseStenzel space of volume Vol ( V , ) = π [48]. The number of units of G flux through the S is q ( r ) = 1(2 π ℓ p ) Z S G , = − π m (2 π ℓ p ) h ( r ) . (5.2)In the smooth solution their IR values ( r →
0) are Q IRM = 0 , q IR = 1(2 π ℓ p ) π m / , (5.3)reflecting the fact that all M2 charge is dissolved in fluxes. One can obtain a BPS solutionin which smeared M2 branes are added at the tip of the Stenzel space [19] simply by shifting ⋆G in such a way that f − h does not change . Under shifts of f → f +2 N and h → h + N ,the IR M2 brane charge changes to Q M → Q M + ∆ Q M , (5.4)where we define ∆ Q M = − m Vol ( V , )(2 π ℓ p ) (cid:18) N − / N (cid:19) , (5.5)whereas the variation in the units of flux through the S amounts to π m N (2 π ℓ p ) . This intro-duces in the IR a − ∆ Q M /r singularity in the warp factor H ( r ) = 162 m Z r h ( f − h ) − sinh( r ′ ) (2 + cosh(2 r ′ )) / dr ′ . (5.6) This combination multiplies a four-form field strength with one leg along ν , one along σ i and two legs alongtwo of the ˜ σ j directions which shrink in the IR ( e β ∼ r ) his singularity is to be expected as we have smeared BPS M2 branes (whose harmonicfunction diverges as 1 /r near the sources) on the S of the transverse space. It is interestingto see how this BPS solution arises in the first–order expansion around the BPS CGLPbackground [20] in the context of our perturbation apparatus. Given that the ξ i modes areassociated to supersymmetry–breaking, all the X i must be set to zero : X i = 0 . (5.7)Since all the ˜ ξ i are zero, Y IR = Y UV . (5.8)In the IR and the UV, e z +2 α , e z +2 β and e z +2 γ do not blow up but reach constant orvanishing values instead. So we impose Y IR = 0 , Y IR = 0 , Y UV = 0 . (5.9)As a result of (5.9) and (5.8), the mode ˜ φ is identically zero. This yields Y IR = Y UV , Y IR = Y UV .Since BPS M2 branes do not change the geometry of the Stenzel space but only the warpfactor (much like BPS D3 branes also only change the warp factor and not the transversegeometry [49]) we expect the first–order perturbation to e z +2 β to vanish both in the UVand in the IR, and thus2 Y + e − z (0) Y IR + 32 m e − z (0) Y IR = 0 , Y UV = Y UV . (5.10)The constant Y IR is in turn determined by Y UV . Furthermore, the fields ˜ φ , ˜ φ now obeythe corresponding homogeneous equations and the solution is found by replacing ˜ λ , by Y , .The mode ˜ φ corresponds to the first–order perturbation of the warp factor. We allowan 1 /r IR divergence, which means that Y IR doesn’t necessarily need to vanish. We willsee in a moment that this mode is related to the number ∆ Q M of added M2 branes.But first, we note that this does not give rise to a singularity that would be associatedwith ˜ φ − φ , the perturbation to the term in F (2.14) with legs on ν ∧ σ i ∧ e σ j ∧ e σ k .Indeed, the conditions we have imposed render this term harmless and independent of Y IR :˜ φ − φ = 2 Y − Y + O ( r ) = O ( r ).Given that Y IR first shows up in the O ( r ) part of the IR expansion of ˜ φ there is norestriction on it. Moreover, Y does not arise in any of the divergent or constant pieces inthe ˜ φ i IR expansions, but requiring no exponentially divergent terms in the UV imposes Y = Y , in agreement with (5.10).As a result, the perturbation corresponding to adding ∆ Q M M2 branes at the tip isobtained by just setting Y = Y ∼ − ∆ Q M . This perturbation causes the warp factor todiverge in the infrared as − ∆ Q M /r while all the other φ i change by sub–leading termsapart from φ and φ which shift by some N related to ∆ Q M through (5.5).The UV expansion of the new warp factor is H = e z (cid:16) φ (cid:17) , = 163 2 / m e − r/ (1 − Y ) + O ( e − r/ ) , = 163 2 / m e − r/ (cid:18) e − z (0) Y IR + 92 m e − z (0) Y (cid:19) + O ( e − r/ ) , (5.11) here in the last line we used (5.10), and one can see that Y multiplies a 1 /ρ term, asexpected from the exact solution. In order to construct a first–order backreacted solution sourced by anti–M2 branes at thetip of the CGLP solution, the first necessary condition is that the force a probe M2 branefeels be nonzero, which implies: X = 0 . (6.1)Furthermore, since the infrared is that of a smooth solution perturbed with smeared anti–M2 branes, we require that no other field except those sourced by these anti–M2 braneshave a divergent energy density in the infrared.Requiring no r or stronger divergences in ˜ φ , ˜ φ , ˜ φ and ˜ φ immediately implies: X = − √ X ,Y IR = 0 , (6.2) X = − X , Barring any r divergence in ˜ φ , results in Y IR = 0 . (6.3)The divergence in ˜ φ is now˜ φ = 3 / √ Z m Y IR − X r + O ( r ) (6.4)and this is the proper divergence for the warp factor of anti–M2 branes spread on the S in the infrared. The energy density that one can associate with this physical divergence is ρ ( E ) ∼ d ˜ φ dr ∼ r (6.5)Another more subtle divergence in the infrared comes from the M–theory four–formfield strength, which is G = dK ( τ ) ∧ dx ∧ dx ∧ dx + m F , (6.6)where (2.14) F = ˙ f dτ ∧ ˜ σ ∧ ˜ σ ∧ ˜ σ + ˙ h ǫ ijk dτ ∧ σ i ∧ σ j ∧ ˜ σ k + 12 (4 h − f ) ǫ ijk ν ∧ σ i ∧ ˜ σ j ∧ ˜ σ k − h ν ∧ σ ∧ σ ∧ σ . (6.7)The unperturbed metric in the IR is regular and is given by ds = Z / ds + 13 / Z − / (cid:2) dr + ν + σ i + r ˜ σ i (cid:3) , (6.8) ith the constant Z given in (3.7). The vanishing metric components g e σ e σ lead to a diver-gent energy density from the four–form field strength components: F νσ ˜ σ ˜ σ F νσ ˜ σ ˜ σ g νν g σσ g ˜ σ ˜ σ g ˜ σ ˜ σ = 9 √ Z / X r + O ( r − ) (6.9) F r e σ ˜ σ ˜ σ F r e σ ˜ σ ˜ σ g rr g e σ e σ g ˜ σ ˜ σ g ˜ σ ˜ σ = 81 √ Z / X r + O ( r − ) . (6.10)Unlike the analogous computations in IIB [22], when integrating these energy densitiesthe factor of √− G ∼ r − is not strong enough to render the action finite. Hence, thissingularity has both a divergent energy density, and a divergent action.As discussed in the Introduction, if this singularity is physical then the perturbativesolution we find corresponds to the first–order backreaction of a set of anti–M2 branesin the Stenzel-CGLP background. If this singularity is not physical, then our analysisindicates that anti–M2 branes cannot be treated as a perturbation of this background, andhints towards the fact that antibranes in backgrounds with positive brane charge dissolvedin fluxes do not give rise to metastable vacua. Acknowledgments
We would like to thank Mariana Gra˜na and Chris Herzog for interesting discussions. Thiswork is supported in part by a Contrat de Formation par la Recherche of CEA/Saclay,the DSM CEA/Saclay, the grants ANR–07–CEXC–006 and ANR–08–JCJC–0001–0, andby the ERC Starting Independent Researcher Grant 240210 – String–QCD–BH. Subtleties in Section 2.
To justify our choice of integration constant in (2.21), we derive the expression for thenon–dynamical scalar K ′ in two different ways. First of all, we use the expression (2.21)for K ′ that arises from its algebraic equation of motion. Inserting the zeroth–order expres-sions (2.28) of the fields appearing in this expression, we find K ′ = − m sinh( r )cosh( r ) e − z ( r ) (2 + cosh(2 r )) / . (A.11)On the other hand, let us proceed to see if this agrees with the expression obtained fromthe condition that the zeroth–order CGLP solution K ′ has to satisfy K ′ = e − z ( r ) dH dr , (A.12)with H solving ∇ H = − m | F | . (A.13)This reduces to dH dr = 3 2 m e γ sinh(2 r ) (cid:0) ℓ − tanh( r ) (cid:1) (A.14)and one must set ℓ = 0 in order for the solution to be regular. As a result, K ′ = − m sinh( r )cosh( r ) e − z ( r ) (2 + cosh(2 r )) / , (A.15)in agreement with the expression for K ′ found above from the equation of motion for thisnon–dynamical field determined in term of f and h (2.28). B Behavior of ˜ ξ We collect here the infrared and ultraviolet asymptotic expansions of the exact solutionsfor ˜ ξ i which we have derived in Section 4.1. B.1 IR behavior of ˜ ξ The IR behavior of the ˜ ξ a ’s is the following :˜ ξ IR = X + 2 X " − / m e − z (0) r + O ( r ) , ˜ ξ IR = (cid:20) X − √ X + 73 X (cid:21) + h X + 83 √ X + 13 X (cid:16) −
10 3 / e − z (0) m (cid:17) i r + O ( r ) , ξ IR = 3 / e − z (0) m X r + O ( r ) , (B.16)˜ ξ IR = X " − / m e − z (0) r + O ( r ) , ˜ ξ IR = 1 r " X + X √ − / e − z (0) m ! + "
16 (7 X + 12 X ) + X (cid:20) √ − / e − z (0) m − √ e − z (0) m Π (cid:18) −√ − arcsin (cid:18) / (cid:19) | − (cid:19) (cid:21) − / e − z (0) m X log( r ) + " X + 148 X (cid:18) √ / e − z (0) m (cid:19) r + O ( r ) , ˜ ξ IR = − r h X + √ X i + (cid:20) X + X (cid:18) √ / e − z (0) m (cid:19)(cid:21) + (cid:20) X + X (cid:18) √ − / e − z (0) m (cid:19)(cid:21) r + O ( r ) . .2 UV behavior of ˜ ξ The UV behavior of the ˜ ξ a ’s is as follows :˜ ξ UV = X + 323 2 / m X e − z (0) e − r + O ( e − r/ ) , ˜ ξ UV = − X e r + 316 X e r + (cid:20) X + 332 X + 23 √ X + X ) (cid:21) e r + (cid:20) X − X − √ X + X ) (cid:21) + (cid:20) X + 332 X + 23 √ X + X ) (cid:21) e − r + (cid:20) X + 643 √ X (cid:21) e − r + 327 2 / e − z (0) m X e − r/ − (cid:20) X + 2563 √ X (cid:21) e − r + O ( e − r/ ) , ˜ ξ UV = 18 X e r − X e r + 2 X − X e − r + 327 2 / e − z (0) m X e − r/ + 18 X e − r + O ( e − r/ ) , ˜ ξ UV = 163 2 / m X e − z (0) e − r + O ( e − r/ ) , (B.17)˜ ξ UV = 12 ( X + X ) e r + 52 ( X + X ) e − r + 2 (3 X − X ) e − r + 2 (5 X + X ) e − r − / √ e − z (0) m X e − r/ + O ( e − r/ ) , ˜ ξ UV = ( X + X ) e r − X + X ) e − r −
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