On the stabilization of modulus in Randall Sundrum model by R 2 interaction

Transcription

1 PRAMANA c Indian Academy of Sciences Vol. 86, No. 3 journal of March 206 physics pp On the stabilization of modulus in Randall Sundrum model by R 2 interaction A TOFIGHI Department of Nuclear Physics, Faculty of Basic Science, University of Mazandaran, P.O. Box , Babolsar, Iran A.Tofighi@umz.ac.ir MS received 6 January 204; revised 20 September 204; accepted 5 December 204 DOI: 0.007/s ; epublication: 5 August 205 Abstract. A solution to the problem of modulus stabilization is to couple a massless bul scalar field non-minimally to five-dimensional curvature. We present an exact treatment of the stabilization condition. Our results show that the square of effective mass of this scalar field is necessarily negative. We also find the existence of a closely spaced maximum near the minimum of the effective potential. Keywords. Field theories in higher dimensions; Randall Sundrum model; modulus stabilization. PACS Nos h;.0.K. Introduction To explain the large hierarchy between the wea scale and the Planc scale, many theories such as supersymmetry and higher-dimensional theories have been proposed. One of these attempts, Randall Sundrum I [, explains this hierarchy in terms of a small extra dimension. This proposal involves a Planc brane and a TeV brane and the space between the branes is a slice of anti-de Sitter space. By solving the five-dimensional Einstein equations, one obtains the metric for this space as ds 2 = e 2σ η μν dx μ dx ν r 2 dϕ 2, ) σ = r ϕ and η μν = diag[,,,, 2) π < ϕ < πis the extra-dimensional coordinate, r is the compactification radius and is a parameter which is assumed to be of order 5d Planc scale, M. The problem of stability of this extra dimension was addressed by Goldberger and Wise GW) in ref. [2. Their solution involved a massive bul scalar field with the usual inetic term in the bul and quartic interactions localized on the two branes. Since then, many Pramana J. Phys., Vol. 86, No. 3, March

2 A Tofighi studies have appeared on this subject [3 2. The studies of refs [3,4 consider models for the stabilization of the modulus containing a bul scalar field interacting with the space-time curvature R. Grzadowsi and Gunion [3 considered a class of generalizations of the Randall Sundrum model containing a bul scalar field, interacting with the curvature R through the general coupling Rf ). They showed that by choosing a non-trivial bacground for the bul scalar field it is possible to neglect the effect of the metric bac-reaction, and they obtained the general form of the scalar potential V ). Granda and Oliveros [4 considered the case of a massless scalar field but with nonminimal interaction with the curvature R. In this wor, by a suitable choice of the parameter, one can neglect the effect of bac-reaction of the scalar field on bacground geometry. Their wor essentially corresponds to the wor of ref. [3, but with V ) = 0. However, the discussion in refs [4,5 are related to infinitely large quartic coupling. In this wor, we present an exact treatment of the Granda Oliveros model. An exact analysis of the GW mechanism is discussed in ref. [6. The plan of this paper as follows: In 2, we describe the model, obtain the effective potential, express the extremization condition for this effective potential and obtain the value of the stabilized modulus. We also show that the square of the effective mass of the bul scalar field is negative. In 3, we study the stability of the modulus r. In the limit of infinite quartic coupling our results are in agreement with previous results [5. We also investigate the case the quartic coupling is finite but very large, and finally in 4, we present our conclusions. 2. Effective potential The action of the model is of the form: S = S gravity + S vis + S hid + S, 3) S gravity = S vis = π d 4 x G[2M 3 R, 4) π π d 4 x gs [L s V s, S hid = π π d 4 x gp [L p V p, 5) π π S = dx 4 dφ GG MN M N ξr 2 ) π dx 4 g s λ s 2 vs 2 )2 dx 4 g p λ p 2 vp 2 )2, 6) is the five-dimensional cosmological constant, V s, V p are the visible and hidden brane tensions, G = det[g MN, R is the bul curvature for the metric ) and is given by R = 20σ 8σ, 7) r 2 σ = φ σ, σ = 2r[δφ) δφ π). 538 Pramana J. Phys., Vol. 86, No. 3, March 206

3 Stabilization of modulus in Randall Sundrum model The φ-dependent vacuum expectation value φ) is obtained from the equation of motion φ e 4σ φ ) = ξrr 2 e 4σ + 4e 4σ λ s r 2 vs 2 )δφ π) + 4e 4σ λ p r 2 vp 2 )δφ). 8) Away from the boundaries φ = 0,π) the solution is φ) = Ae ν+2)σ + Be ν+2)σ, 9) ν = ξ. If we insert this solution in eq. 6) and integrate over φ, we obtain the effective four-dimensional potential, V r), for the modulus r, which is given by V r) = ν + a)a 2 e rπ ) + ν a)b 2 e rπ ) +λ s e 4rπ 2 π) v 2 s )2 + λ p 2 0) v 2 p )2. 0) Here a = 2 + 8ξ. The coefficients A and B are determined by imposing appropriate boundary conditions on the 3-branes. We obtain these boundary conditions by inserting eq. 9) into the equations of motion and matching the delta functions. The results are and [a + ν)a + a ν)b 2λ p 0)[ 2 0) vp 2 =0 ) e 2rπ [a + ν)e vrπ A + a ν)e vrπ B+2λ s π)[ 2 π) v 2 s =0. 2) In a previous wor [5, we considered the limit of λ p, λ s. In this limit 0) = v p and π) = v s. In order to investigate the case of finite quartic coupling, we must calculate the first and second derivatives of the effective potential. By using eqs ), 2) and after a lengthy calculation we get dv r) = 4 2 π[a + ν)e rπ A 2 + a ν)e rπ B 2 + 2a ν 2 )AB dr 4πe 4rπ 2 π) vs 2 )2. 3) From eq. 3) we obtain a simple form for the second derivative of the potential which is given by dv 2r) [ = 4 2 πν 2 + ν)a db da + ν 2)B. 4) dr 2 dr dr In obtaining the above result we used the extremization condition dv r)/dr) = 0. If we denote φ = 0) = Q p r) and φ = π) = Q s r), then from eq. 9) we can express the coefficients A and B as A = Q sr)e 2σ Q p r)e νσ 2sinhνσ ) B = Q pr)e νσ Q s r)e 2σ 2sinhνσ ), 5). 6) Pramana J. Phys., Vol. 86, No. 3, March

4 A Tofighi By substituting these expressions in eqs ), 2), we get ν e 2σ 2sinhνσ ) ν Qp 2sinhνσ ) Q s a + ν e νσ + ν a a + ν = 2λ p eνσ )) Q p Q 2 p Q v2 p ), 7) s eν 2)σ + ν a )) 2v e ν+2)σ = 2λ s Q2 s v2 s )e 2σ. 8) By inserting eqs 5), 6), 8) into eq. 3) under extremization condition we get for λ s = 0) [ x 4ξ 2 λ s Q 2 s ν eν 2)σ e ν+2)σ ) + x 2 = C 2, 9) x = Q p 2 + ν Q s eν 2)σ ν 2 e ν+2)σ, [ 2 + ν C = eν 2)σ + ν 2 e ν+2)σ C 20) and C = 4[a + ν)e2ν 2)σ e 4σ 2a ν 2 ) + a ν)e 2ν+2)σ. 2) [2 + ν)e ν 2)σ + ν 2)e ν+2)σ 2 It is easy to obtain the variable x from the quadratic eq. 9) and by some manipulation we obtain r = 2 + ν π2 ν) ln + ν 2 ) e σ + b ± λ s Q 2 s C2 b 2 ) + λ 2 s Q4 s C2 + λ s Q 2 s ) ) ) Qs r), 22) Q p r) 8ξe ν 2)σ e ν+2)σ ) b =. 23) 2 + ν)e ν 2)σ + ν 2)e ν+2)σ Expression for r in eq. 22) is valid for any value of the quartic coupling constant. We note that in the large r limit C 2ξ/ν + 2)). Therefore, in order to have a meaningful result, the coupling constant ξ must be negative. This in turn implies that the square of the effective mass of the bul scalar field is negative. To compare this result with the corresponding result for GW mechanism, we note that from ref. [6 ν GW 2 C GW ν GW + 2, ν GW = 4 + m2. 24) Pramana J. Phys., Vol. 86, No. 3, March 206

5 Stabilization of modulus in Randall Sundrum model Hence in the GW mechanism, the effective mass squared of the bul scalar field is strictly positive. Utilizing the above results, we calculate the second derivative of the effective potential which is given by dv 2 r) dr 2 [ = 4πνe 2σ λ p Q 2 p sinhνσ ) v2 p ) 4ξ)Q pq s + λ sq 2 s v2 s ) + 4ξ)Q s Q p + 2πλ pλ s Q 2 s v2 s )Q2 p v2 p )Q pq s + 42 νπξ sinhνσ ) e 2σ Q 2 s e2σ Q 2 p ) + 62 πξ + 2ξ)Q p Q s. 25) Here prime denotes derivative with respect to r. In order to have a negligible bacreaction of the scalar field on the bacground geometry, we require v s,v p M 3/2 and ξ. Hence we can neglect the stress tensor for the scalar field in comparison to the stress tensor induced by the bul cosmological constant [4. 3. Stability of the modulus To investigate the stability of the modulus r we consider two different cases. Case I. λ p,λ s. In the discussion of this case for the GW mechanism of ref. [6, the value of second derivative of the effective potential for Q s = v s, Q p = v p, Q s = 0 and Q p = 0, was identically zero, hence they had to resort to an asymptotic analysis. But for our case eq. 25) has a more complex structure than its GW counterpart. So a direct analysis is possible. In this limit from eq. 22), we obtain in the large r limit) [ ) r = πν 2) ln vp v s ν + 2 ±. 26) 2ξ Moreover, by using eq. 26), the second derivative becomes dv 2r) = 63 π 2 νξv p v s 3ξe 2σ [ 3ξ ± 2. 27) dr 2 sinhνσ ) In order to have meaningful results for the modulus, v p and v s must have similar signs. Hence, for [ ) r = πν 2) ln vp ν ) 2ξ v s dv 2 r)/dr2 )>0. That means r corresponds to the value of the stable modulus. This result agrees with our first-order calculations reported in ref. [5. For the configuration v p = 0.35, v s =, = 4 and ξ = 0.0, the value of r = 2.2. Pramana J. Phys., Vol. 86, No. 3, March

6 A Tofighi Case II. λ p and λ s are finite but very large. In this case, from eq. 7) we find that the value of Q p is lower than v p and in the limit of λ p approaches v p. Similarly, if v p /v s > then from eq. 8) we find that the value of Q s is higher than v s and in the limit of λ s approaches v s. Hence it is appropriate to consider a /λ expansion of boundary scalar field. From eqs 7), 8) we get νe 2σ Q p r) = v p + λ p v p 4sinhνσ ) vs 2 + ν v p e2 ν)σ + ν 2 eν+2)σ νe 2σ )), 29) Q s r) = v s + λ s v s 4sinhνσ ) vp 2 + ν v s eν 2)σ + ν 2 )) e ν+2)σ. 30) Now by using eqs 29), 30) we obtain a modified expression for the modulus in the large r limit) r = πν 2) ln n 2 + ν ± 2ξ ν+2 q 2 ) tν 2) qν + 2) + qνn e2 ν)πr ) ), 3) n = v p, t =, q =. 32) v s λ p vp 2 λ s vs 2 4. Conclusions We have utilized a massless bul scalar field with non-minimal coupling to fivedimensional Ricci scalar to stabilize the size of extra dimension in the Randall Sundrum model. We have assumed the value of the coupling ξ. Hence there is no need to consider the bac-reaction of the scalar field on the bacground geometry. So, we have presented an alternative formulation for stabilizing the modulus. In this framewor the large value of r is due to the small value of the coupling constant ξ while in the Goldberger Wise [2 mechanism the large value of r is due to a small bul scalar mass. For finite quartic couplings we have obtained analytical expression for the size of the stabilized modulus. Our analysis shows that the value of coupling ξ must be negative. We have made a /λ expansion in the large r limit and obtained an analytical expression for this case. The parameters in this case are ξ, n, q and t. Similar to GW case [6, our study also reveals the existence of a very closely spaced maximum along with the minimum. It remains a problem to investigate the physical consequences of this result. 542 Pramana J. Phys., Vol. 86, No. 3, March 206

7 Stabilization of modulus in Randall Sundrum model The issue of the stability of Randall Sundrum brane-world with a tachyonic scalar has been dealt with in refs [9,0. In our model, the presence of an effective negative mass term in the five-dimensional Lagrangian is due to the negative value of the parameter ξ. However, this will not induce an instability provided the tachyonic modes do not appear in the four-dimensional effective theory [0. It will be interesting to study the parameter space of the model. Instead of brane potential with quartic coupling, it is also possible to consider brane potential of the quadratic form. We plan to report on these issues in future. References [ L Randall and R Sundrum, Phys. Rev. Lett. 83, ) [2 W D Goldberger and M B Wise, Phys. Rev. Lett. 83, ) [3 B Grzadowsi and J F Gunion, Phys. Rev. D68, ) [4 L N Granda and A Oliveros, Europhys. Lett. 74, ) [5 A Tofighi and M Moazzen, Mod. Phys. Lett. A 28, ) [6 A Dey, D Maity and S SenGupta, Phys. Rev. D75, ) [7 T Tanaa and X Montes, Nucl. Phys. B 582, ) [8 C Csai, M Graesser, L Randall and J Terning, Phys. Rev. D62, ) [9 K Ghorou and A Naamura, Phys. Rev.D64, ) [0 J L Lesgougues and L Sorbo, Phys. Rev.D69, ) [ J M Cline and H Firouzjahi, Phys. Rev.D64, ) [2 P Kanti, K A Olive and M Pospelov, Phys. Lett. B 538, ) Pramana J. Phys., Vol. 86, No. 3, March