UPR/920-T January 2001 hep-th/0101057 Localized Massive Gravity from B fields J. F. V´azquez-Poritz 1 Dept. of Phys. and Astro., University of Pennsylvania, Philadelphia, PA 19104 0 0 2 n a ABSTRACT J 4 2 We consider five-dimensional domain-wall solutions which arise from a sphere reduc- tion in M-theory or string theory and have the higher-dimensional interpretation of the 2 v near-horizon region of various p-branes in constant, background B fields. We analyze the 7 5 fluctuation spectrum of linearized gravity and find that there is a massive state which is 0 1 localized and plays the role of the four-dimensional graviton. 0 1 0 / h t - p e h : v i X r a Research supported in part byDOEgrant DOE-FG02-95ER40893 1 Introduction Randall and Sundrum [1, 2] have shown that, with fine tuned brane tension, a flat 3-brane embedded in AdS can have a single, massless bound state. Four-dimensional gravity is 5 recovered at low-energy scales. It has also been proposed that part or all of gravitational interactions are the result of massive gravitons. For example, in one model, gravitational interactions are due to the net effect of the massless graviton and ultra-light Kaluza-Klein state(s) [3, 4, 5, 6]. In another proposal, there is no normalizable massless graviton and four-dimensionalgravity isreproducedatintermediatescales fromaresonance-like behavior of the Kaluza-Klein wave functions [5, 6, 7, 8, 9]. Recently, it has been shown that an AdS brane in AdS does not have a normalizable 4 5 massless graviton. Instead, there is a very light, but massive bound state graviton mode, which reproduces four-dimensional gravity [10, 11, 12, 13]. The bound state mass as a function of brane tension, as well as the modified law of gravity, were explored in [14, 15]. Abraneworldinwhichtwoextrainfinitespatialdimensionsdonotcommutehasrecently beenusedinordertolocalizescalarfieldsaswellasfermionicandgaugefields[19]. Also,the cosmological evolution of the four-dimensional universe on the probe D3-brane in geodesic motion in the curved background of the source Dp-brane with nonzero NS B field has been explored [20]. In this paper, we consider brane world models which arise from a sphere reduction in M-theory or string theory, as the near-horizon of p-branes [16] with a constant, background B-field on the world-volume. The dual field theory is non-commutative Yang-Mills [17, 18]. We consider string theoretic p-brane solutions for p = 3,4,5 with 0,1,2 world-volume dimensions wrapped around a compact manifold, respectively. For p > 5, the space of the extra large dimension has finite-volume, with a naked singularity at the end, for which case the localization of gravity is trivial. For allofthecases westudy,wefindthatthereisnonormalizablemasslessgraviton, but thereisamassiveboundstategravitonwhichplaystheroleofthefour-dimensionalgraviton. This yields another class of examples for which gravitational interactions are entirely the result of massive gravitons. As we will see, the bound mass increases continuously from zero as a constant, background B or E-field is turned on, except for the case p = 5, for which there is a mass gap. This paper is organized as follows. In section 2, we find the graviton wave equation in the background of five-dimensional domain-walls that originate in string theory as p-branes in constant background B and E-fields. In section 3, we show that the massless graviton 1 wave function is not normalizable for nonzero background B or E fields, indicating that the massless graviton is not localized on the brane. In section 4, we show that there is a massive graviton localized on the brane for nonzero B and E-fields. Lastly, an appendix shows how the graviton wave equation can be solved exactly for p = 5. 2 The graviton wave equation 2.1 With background B field Consider the metric of a p-brane (expressed in the string frame) in a constant NS B field in the 2,3 directions [18, 21]: ds2 = H−1/2( dt2+dx2+h(dx2 +dx2)+dy2)+l4H1/2(du2+u2dΩ2), (2.1) 10 − 1 2 3 i s 5 R7−p H = 1+ , h−1 =H−1sin2θ+cos2θ, l4u7−p s where u = r/l2 is a dimensionless radial parameter and i = 1,..,p 3. The B field and s − dilaton are given by B = tanθH−1h, e2ϕ = g2H(3−p)/2h, (2.2) 23 7 p cosθR7−p = (4π)(5−p)/2Γ − g lp−3N, (2.3) (cid:16) 2 (cid:17) s s where N is the number of p-branes and g is the asymptotic value of the coupling constant. s θ is known as the non-commutativity parameter, and is related to the asymptotic value of the B-field: B∞ =tanθ. 23 Consider the following decoupling limit, in which we take the B field to infinity [18]: l 0, l2tanθ = b, (2.4) s → s b x˜ = x , 2,3 l2 2,3 s where b,u,x˜ and g lp−5 stay fixed. Thus, µ s s u R l−2ds2 = ( )(7−p)/2( dt2+dx2+h˜(dx˜2+dx˜2)+dy2)+( )(7−p)/2(du2+u2dΩ2 ), (2.5) s R − 1 2 3 i u 8−p where h˜−1 = 1+b2(u/R)7−p. The dual field theory is Yang-Mills with noncommuting 2,3 coordinates [18]. For small u, the above solution reduces to AdS S5, corresponding to 5 × ordinary Yang-Mills living in the IR region of the dual field theory, which is also the case if we had taken the decoupling limit with finite B-field. From this point on, we drop the˜on the coordinates. For p = 5, we change coordinates to u/R (1+k z )2/(p−5) [16]: 6 ≡ | | l−2ds2 = (1+k z )7p−−p5( dt2+dx2+h˜(dx2+dx2)+dy2+dz2)+(1+k z )pp−−35dΩ2 , (2.6) s | | − 1 2 3 i | | 8−p 2 For p = 5, we change coordinates to u/R e−k|z|/2 [16]: ≡ l−2ds2 = e−k|z|/2( dt2+dx2+h˜(dx2+dx2)+dy2+dz2+dΩ2). (2.7) s − 1 2 3 i 3 After dimensional-reduction over S8−p, theabove metric correspondsto a p+2-dimensional domain wall at z = 0. We will consider the case where y are wrapped around a compact i manifold, so that there is a 1+ 3-dimensional dual field theory on the remaining world- volume coordinates. The equation of motion for the graviton fluctuation Φ = g00h is 01 ∂ √ ge−2ϕgMN∂ Φ= 0, (2.8) M N − whereh is associated with the energy-momentum tensor component T of the Yang-Mills 01 01 theory. For p = 5, consider the ansatz 6 9−p Φ = φ(z)eip·x = (1+k z )2(5−p)ψ(z)eip·x. (2.9) | | Thus, ψ′′ +Uψ = m2ψ, (2.10) − (p 9)(3p 19)k2 p 9 α2 U = − − + − kδ(z)+ , (2.11) 4(5 p)2(1+k z )2 5 p (1+k z )2(7−p)5 p − | | − | | − where α b (p2+p2), (2.12) ≡ q 2 3 and m2 = p2. − For p = 5, consider the ansatz Φ = φ(z)eip·x = ek|z|/2ψ(z)eip·x. (2.13) Thus, ψ satisfies (2.10) with 1 U = k2 kδ(z)+α2e−k|z|. (2.14) 4 − For more general directions of the world-volume B field, α b p2, where p are the ≡ q i i momenta along the large, non-commuting directions in the world-volume. Thus, if only the wrapped y coordinates are non-commuting, then α = 0. i 2.2 With background B and E fields Consider the metric of a p-brane in a constant two-component NS B field background [18, 21]: ds2 = H−1/2(h ( dt2+dx2)+h (dx2 +dx2)+dy2)+l4H1/2(du2 +u2dΩ2 ), (2.15) e − 1 m 2 3 i s 8−p R7−p H = 1+ , h−1 = H−1sin2θ +cos2θ , h−1 = H−1sinh2θ +cosh2θ . l4u7−p m m m e − e e s 3 The B and E field components and dilaton are given by B23 = tanθmH−1hm, E01 = tanhθeH−1he, e2ϕ =g2H3−2phehm. (2.16) − Consider the following decoupling limit [18]: ′ b l 0, coshθ = , cosθ = fixed, (2.17) s e m → l s ′ b x = x˜ , x = l cosθ x˜ , 0,1 l2 0,1 2,3 s m 2,3 s where b′,u,x˜ and g lp−7 remain fixed. Thus, µ s s u u l−2ds2 = H1/2 ( )7−p( dt˜2+dx˜2)+( )7−ph˜ (dx˜2+dx˜2)+H−1dy2+du2+u2dΩ2 , s (cid:16) R − 1 R m 2 3 i 8−p(cid:17) (2.18) where we define u ˜h−1 =1+( )7−pcos−2θ . (2.19) m R m The dual field theory is Yang-Mills with noncommuting 0,1 and 2,3 coordinates [18]. Once again, we drop the˜on the coordinates. For p = 5, we change coordinates to u/R 6 ≡ (1+k z )2/(p−5): | | ls−2ds2 = H1/2(cid:16)(1+k|z|)2(5p−−p7)(−dt2+dx21+h˜m(dx22+dx23)+dz2)+H−1dyi2+(1+k|z|)p−45dΩ28−p(cid:17). (2.20) For p = 5, we change coordinates to u/R e−k|z|/2: ≡ l−2ds2 = H1/2(e−k|z|( dt2+dx2+h˜ (dx2 +dx2)+dz2+dΩ2)+H−1dy2). (2.21) s − 1 m 2 3 3 i As in the case of a one-component B-field, we consider a dimensional-reduction over S8−p and y wrapped around a compact manifold. One effect of the E-field, as opposed to the i B-field, is to add a breathing-mode to S8−p. For p = 5, we insert this metric into the 6 graviton equation of motion (2.8) with the ansatz (2.9) and obtain (2.10) with U given by (2.11) and with p2+p2 q 2 3 α . (2.22) ≡ cosθ m For p = 5, we use the wave function ansatz (2.13) and obtain (2.10) with U given by (2.14). Again, in general α p2/cosθ , where p are the momenta along the large, ≡ q i m i non-commuting directions. In this paper, we will focus on the p = 3 case, for which the volcano potential U is plotted in Figure 1, for α = 0 (solid line) and α = 2 (dotted line). As can beseen, the effect of the B and E fields is the raise the peak of the potential. This is the case in general for p 5. ≤ 4 8 6 4 2 -4 -3 -2 -1 0 1 2 3 4 Figure 1: U(z) for α= 0 (solid line) and α = 2 (dotted line) for p = 3. 3 The massless graviton Consider the coordinate transformation u/R (1+k w )−1 (3.1) ≡ | | and the wave function transformation p−3 Θ(w) = (1+k w ) 4 ψ(z). (3.2) | | applied to the graviton wave equation (2.10). Thus, ′′ m2 Θ +VΘ = Θ, (3.3) − (1+k w )p−3 | | where (6 p)(8 p)k2 α2 V = − − (6 p)kδ(w)+ . (3.4) 4(1+k w )2 − − (1+k w )4 | | | | For the massless case, this is a Schro¨dinger-type equation. The wave function solution can be written in terms of Bessel functions as iα iα Θ(w) = N (1+k w )1/2[J +A(α)Y ], (3.5) 0 | | ν(cid:16)1+k w (cid:17) ν(cid:16)1+k w (cid:17) | | | | where ν = (7 p)/2 and N is the normalization constant, in the event that the above 0 − solution is normalizable. A(α) is solved by considering the δ(w) boundary condition: 2νJ (iα)+iα[J (iα) J (iα) ν ν+1 ν−1 A(α) = − , (3.6) 2νY (iα)+iα[Y (iα) Y (iα)] ν ν−1 ν+1 − − 5 1 0.8 0.6 0.4 0.2 1 2 3 4 5 6 7 Figure 2: A2 versus α for p = 3. | | whereweused2J′(x) = J (x) J (x),andlikewise forY′(x). Inorderfor themassless n n−1 − n+1 n gravitontobelocalizedonthebrane,thecorrespondingwavefunctionmustbenormalizable, i.e., ∞ ψ(w,m = 0)2dw < . (3.5) can be written in the form R0 | | ∞ Θ(w) = (normalizable part)+A(non renormalizable part), (3.7) − as can be seen from the asymptotic forms of the Bessel functions. Thus, the massless graviton is only localized on the brane if A = 0. In this limit, there are no background B or E fields and we recover the Randall-Sundrum model. As shown in Figure 2, for the case of the D3-brane, as α increases, the modulus of A approaches unity. A has the same characteristics for p 5 in general. The issue arises as ≤ to whether there is a massive graviton state bound to the brane for nonzero A. 4 Localization of the massive graviton We will first concentrate on the case of the brane world as the dimensionally-reduced near- horizon of the D3-brane. For this case, the massive wave equation (2.10) with U given by (2.11) is amodifiedMathieu’s equation [22], supplementedby theδ(w) boundarycondition. The exact solution of the modified Mathieu’s equation is known, and has been applied to absorption by D3-branes and six-dimensional dyonic strings [23, 24, 25, 26, 27]. However, it is rather laborious to use the exact solution in this context and so we content ourselves with plotting numerical solutions– except for the case p = 5, for which the exact solution is a sum of Bessel functions (see appendix A). 6 1 0.8 0.6 0.4 0.2 0 0.05 0.1 0.15 0.2 Figure 3: ψ(z = 0)2 versus m for α = 0 (dotted line) and α = .02 (solid line) for p = 3. | | 1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 Figure 4: ψ(z)2 for m/k = 0 for p = 3. | | In order to solve for the unnormalized wave function, we input the δ(z) boundary con- dition and solve the Schro¨dinger-type equation outwards. We then numerically integrate in order to find the correct normalization factor. As can be seen from the dotted line in Figure 3, for the case of the D3-brane with no background B and E fields, there is a resonance in the modulus of the wave function on the brane for m = 0, which implies that the massless graviton is localized on the brane. This behavior can also be seen from Figure 2, by the fact that A = 0 and the massless solution is normalizable. We may arrive at the same conclusion via Figures 4 and 5, where the massless wave- function has a peak on the brane at z = 0 whereas the wavefunction of mass m = 1.5k oscillates without feeling the presence of the brane [10]. 7 1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 Figure 5: ψ(z)2 for m/k = 1.5 for p = 3. | | In the case of nonzero B or E fields, the resonance in the modulus of the wave function on the brane is at a nonzero mass. Thus, for a nonzero, constant background B or E field, a massive graviton is localized on the brane. This behavior is shown by the solid line in Figure 3, where the ψ(z = 0)2 is plotted for α = .02, and there is a large resonance | | in the wave function on the brane for m/k = .0822. Again, this behavior may also be ascertained from plots of ψ(z)2 for various values of m/k. For a mass significantly less | | than the resonance mass, the wave function has a peak at z = 0 that is not relatively high, while a wave function with a mass significantly greater than the resonance mass oscillates without feeling the brane. It is straight-forward to repeat the above analysis for p = 4,5. Note that a D4-brane reduced on S1 S4 yields the same graviton wave equation as a M5-brane reduced on × T2 S4. Likewise, the D5-brane reduced on T4 S3 and a M5-brane intersection reduced × × on T4 S2 or K3 S2 yield the samegraviton wave equation. This was initially observed in × × [16] in the case of zero B-field. Figure 6 shows the resonance mass versus α for p = 3,4,5. We classify them by the corresponding near-horizon brane configurations in ten or eleven dimensions. Thesolid line correspondsto the D3-brane. Thedotted line corresponds to the D4 orM5-brane. TheboldlinecorrespondstotheD5-braneor anM5-brane intersection. It is rathercuriousthatthethreelines appearto intersectat thesamepoint, atapproximately α = 2.688 and m/k = .976. 8 1.2 1 0.8 0.6 0.4 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 6: Resonance mass versus α for p = 3 (solid line), 4 (dotted line) and 5 (bold line). A p = 5 as an exactly solvable model Consider the equations (3.3) and (3.4). Although this is not in Schro¨dinger form, for p = 5 this equation is easily solvable for the massive case. The solution for Θ is iα iα Θ = N(1+k w )1/2[J +A(α)Y ], (A.1) | | −iγ(cid:16)1+k w (cid:17) −iγ(cid:16)1+k w (cid:17) | | | | where 2J (iα)+iα[J (iα) J (iα) −iγ −iγ+1 −iγ−1 A(α) = − , (A.2) 2Y (iα)+iα[Y (iα) Y (iα)] −iγ −iγ−1 −iγ+1 − − and γ 4(m/k)2 1. Transforming back to the z coordinate and ψ wave function using ≡ p − (3.1) and (3.2) we obtain the wave function solution ψ(z) = N [J (iα e−k|z|/2)+A(α)Y (iα e−k|z|/2)]. (A.3) 0 −iγ −iγ For the massless graviton ψ(z) e−k|z|/2+A(α)ek|z|/2, (A.4) ∼ and so the massless graviton wave function is not normalizable for nonzero B or E fields. As in the zero B-field case, there is a mass gap and so we must have m2 1k2 for the ≥ 4 massive wave functions. Expanding the Bessel functions for large k z yields | | eπ/2 ψ(z) = N [1+iA(α)csch(πγ)cosh(πγ)]eiq|z|−iγln(α/2) (A.5) (cid:16) Γ(1 iγ) − − Γ(1 iγ) iA(α)csch(πγ)e−π − e−iq|z|+iγln(α/2) , (cid:17) Γ(1+iγ) whereq m2 k2/4. Wenumerically normalizethewave function,andfindtheresonant ≡ p − mass by solving for the maximum of ψ(z = 0)2 for a given α. The resonant mass versus | | 9