Localization of the Graviphoton and the Graviscalar on the Brane

Journal of the Korean Physical Society, Vol. 43, No. 5, November 2003, pp. 663 669 ∼

Y. S. Myung∗ Relativity Research Center and School of Computer Aided Science, Inje University, Gimhae 621-749

(Received 29 July 2003)

The question of whether the Kaluza-Klein (KK) graviphoton h5µ and graviscalar h55 are localized or not on the brane is an important issue. In this letter, we address this problem in five dimensions. Here, we consider the massless (zero-mode) propagations without requiring the Z2-symmetry on h5µ. We obtain the graviton hµν , the graviphoton, and the graviscalar exchange amplitudes on the shell. We find that the graviscalar has a tachyonic mass. It turns out that h5µ admits localized zero-modes on the brane while h55 does not have a localized zero-mode. This is contrasted to the fact that the bulk spin-0 field has a localized zero-mode on the brane, but the bulk spin-1 field does not have a localized solution in five dimensions.

PACS numbers: 04 Keywords: Brane world scenario

In the conventional Kaluza-Klein (KK) approach to for zero-modes can be obtained after integrating the 5D five-dimensional (5D) gravity, the spacetime manifold is pure gravity action over x5 = z S1. However, in the 1 ∈ factorized as M4 S . Here M4 is the Minkowski space- non-factorizable compactification, additional terms that time, and S1 is⊗ the circle. The spectrum of 5D pure are function of x5 = z R exist in the linearized equa- gravity is split into four-dimensional (4D) massless fields, tion. In general, we cannot∈ obtain a consistent zero- such as graviton, graviphoton, graviscalar, and an infi- mode solution only by requiring ∂5f = 0 at the level of nite tower of massive spin-two fields [1,2]. In particular, the equation of motion. The integration of the 5D action the U(1) gauge symmetry of the graviphoton in the 4D over z is a good starting point to obtain the zero-mode effective action originates from the translational isome- solution for the non-factorizable compactification [8–11]. tries in the extra dimension. Also, the study of this issue is very important for phe- On the other hand, there has been much interest in the nomenological purposes because its zero modes (massless phenomenon of localization of gravity proposed by Ran- modes) correspond to the standard model particles local- dall and Sundrum (RS) [3,4] and a large number of brane ized on the brane. If the brane world scenario is correct, world models have been developed [5–7]. RS assumed a the various fields we observe are the zero modes of KK single positive tension 3-brane and a negative bulk cos- [12] and bulks fields [13] which are trapped on the brane mological constant in the 5D spacetime [4]. They ob- by the gravitational interaction.1 Here, we consider the tained a 4D localized gravity by fine-tuning the tension KK fields only. of the brane to the cosmological constant. The intro- A simple choice for the KK fields is a part (h5µ = 2 duction of branes usually gives rise to a warping of the h55 = 0) of the RS gauge to obtain the localized 4D extra dimensions, resulting in a non-factorizable space- gravity on the brane [4]. Th brane-bending effect ap- time. Apparently, the presence of the brane breaks the pears under the RS gauge with a localized matter source translational isometries in the extra dimensions. Hence, [16]. Here, the bending of the wall ξˆ5 acts like a new we worry about obtaining the graviphoton in the RS ap- scalar under the RS gauge.3 Ivanov and Volovich [18] and proach. Myung and Kang [19] have discussed the propagation of The zero-mode propagations are not easy to derive in h5µ with h55 = 0. The propagation of all metric com- the non-factorizable compactification. In the conven- ponents, including h5µ = 0, h55 = 0, was investigated in 6 6 tional KK approach, one can find the zero mode f(x) Ref. 20. It turned out that the massive modes of h5µ and only by requiring ∂5f = 0 in the equation of motion. h55 with uniform external sources could not propagate This is possible because it is in the factorizable compactification. In this case, the 5D Laplacian is split 1 2 The field theoretic mechanism for gauge field localization on a into the 4D Laplacian and ∂5 , the latter producing brane was first suggested in Ref. 14. Also, the localization of 2 a (mass) -term [2]. Equivalently, the 4D effective action quantum fields on the brane was recently discussed in Ref. 15. 2 This is composed of the Gaussian-Normal gauge (h5µ = h55 = 0) µν µ and the 4D transverse traceless gauge (∂µh = 0, hµ(h) = 0). ∗E-mail: [email protected] 3 For the other approach, see Ref. 17. -663- -664- Journal of the Korean Physical Society, Vol. 43, No. 5, November 2003

2 2 on the branes. Recently, the case of h55 = 0, h5µ = 0 Here, H = k z + 1, Φ = g55, and κΦ Aµ = g5µ. The with a localized source was discussed [21].6 However, it above corresponds| | to the standard KK− decomposition as was pointed out that at long distance where we can ob- 2 2 2 tain 4D gravity, the propagation of h is not allowed. γµν + κ Φ AµAν κΦ Aµ 55 gMN = 2 − 2 (3) κΦ Aν Φ The next question is whether or not the massless modes − of the graviphoton h and the graviscalar h propagate 5µ 55 with Aµ = γµν A and A A = A Aµ. Here, κ is intro- on the brane. ν µ duced for the small gauge· coupling constant. It is known for h ϕ on the RS brane that con- 55 Under a specific class of coordinate transformations, sistency with hρ = h ∼= 0 requires h = 0 without ρ 5µ 55 such as an external source [11]. Concerning the massless prop- µ µ µ agation of h5µ aµ, we expect that the breaking of x x˜ =x ˜ (x), z z˜ = z + ξ(x), (4) isometries in the∼ extra dimension by the brane would → → not allow the 4D effective action to be invariant mani- ∂xP ∂xQ fromg ˜MN = ∂x˜M ∂xÑ gPQ we obtain festly under U(1) gauge transformations [3]. Explicitly, that fact comes from the Z -symmetry argument based ∂xα ∂xβ 2 γ˜ = γ , on the fact that the RS ground state solution is sym- µν ∂x˜µ ∂x˜ν αβ α metric under z z. If one requires that this sym- ∂x 1 ∂ξ → − A˜µ = Aα + κ− , metry be preserved up to the linearized level, hµν (x, z) ∂x˜µ ∂x˜µ and h55(x, z) will be even with respect to z, but h5µ Φ(˜˜ x, z˜) = Φ(x, z). (5) will be odd: h5µ(x, z) = h5µ(x, z). This implies that − − h5µ(x, 0) = 0 on the brane. Thus, we do not expect to We observe that γµν transforms like a 4D metric tensor have zero modes of the gravivector. Here, we do not re- and Φ a scalar field under diffeomorphisms in Eq. (4). quire such a Z2-symmetry in the linearized calculation. Also, we point out that the zero modes have 4D diffeo- 4 Then, the analysis of the linearized equation around the morphisms plus U(1) gauge transformations for Aµ. RS background reveals that the graviphoton possesses In this work, we are mainly interested in the zero-mode the U(1) gauge symmetry [12]. effective action. It is a non-trivial problem to determine In this paper, we clarify whether or not the gravipho- what the zero mode is if the full spacetime is not fac- ton h5µ and the graviscalar h55 are localized on the brane torizable. In order to obtain the zero modes, we assume with the matter sources. This is one of the important is- that γµν , Aµ, and Φ are functions of x-coordinates only. sues about the brane-world scenario [8, 9, 22–24]. The Plugging Eqs. (2) and (3) with γµν (x), Aµ(x), and Φ(x) naive condition that the zero mode be localized on the into Eq.(1) and integrating over z leads to [12] brane is equivalent to the normalizability of the ground- state wave function on the brane [9]. This requires that 2 the 5D action after its integration of over z be finite 1 4 κ 3 2 Izero m = Z d x√ γ ΦR(γ) Φ F [9, 13]. However, this is valid for the bulk fields. Ac- − 16πG4 − h − 4 tually, further work is necessary for a complete study 2 1 µ 2 2 µ 2 + 6k Φ− + Φ 2 δν + κ Φ A Aν + κ ΦA A . of massless propagations, including the graviphoton and − q| | · i the gravisacalar, on the brane. As a definite criterion, (6) we introduce the local sources to calculate the graviton, graviphoton, and graviscalar exchange amplitudes on the We observe that the zero-mode gravitational degrees shell. of freedom in the 5D spacetime are split into the 4D We start from the second RS model with a single brane graviton γµν , a graviphoton Aµ, and a graviscalar Φ, as at z = 0 [4,12] usual. However, the properties of the vector field and the scalar field are very different from those in the con- 4 ∞ √ gˆ ˆ 4 ventional KK reduction. The first two terms in Eq. (6) I = Z d x Z dz − (R 2Λ) Z d x gˆBσ.(1) 16πG5 − − p− are the same form as in the conventional KK compacti- −∞ fication; thus, they have the U(1) gauge symmetry. The Here, G5 is the 5D Newtonian constant, Λ is the bulk difference from the conventional KK approach is the last 2 cosmological constant,g ˆB is the determinant of the in- term which is proportional to k . If we start from the KK duced metric for the 3-brane, and σ is the tension of metric decomposition with Aµ = 0 and Φ = 1 in Eq. (3) the brane. We assume that the value of σ is fine-tuned as in the RS approach, this “potential” term disappears, 2 such that Λ = 6k (< 0) with k = 4πG5σ/3. Let us and one obtains the pure 4D gravity without the cos- introduce the domain-wall− metric mological constant on the brane. The zero cosmological constant arises because of the fine-tuning between the 2 M N 2 M N ds =g ˆMN dx dx = H− (z)gMN dx dx

2 µ ν 2 µ 2 4 = H− (z)hγµν dx dx + Φ (dz κAµdx ) i. This is because in 5D general relativity, one can always choose − the case of Aµ = 0 and Φ = 1 by using the general covariance (2) [25]. Localization of the Graviphoton and the Graviscalar on the Brane – Y. S. Myung -665- brane tension σ and the 5D bulk cosmological constant ysis around the RS solution. As we will see later, the Λ. nonlinear term and A A cannot generate any mass-like µ 2 2 µ · Apparently the non-linear term ( δν + κ Φ A Aν ), term. as well as the squared term A A, implyp| not only that the| In order to see explicitly how the dynamical aspect of 4D effective action no longer has· the U(1) gauge symme- Φ arises, let us conformally transform the metric as try but also that the graviphoton does not exist. This phenomenon arises mainly from the presence of the brane γµν γ¯µν = Φγµν . (7) in the 5D Anti-de Sitter spacetime. However, this obser- → vation is not a complete one. Actually, the propagation The zero-mode effective action in Eq. (6) is then written of fields should be determined using a perturbation anal- as

2 E 1 4 κ 3 2 3 2 ¯ µ ¯ Izero m = Z d x√ γ¯ R(¯γ) Φ F Φ− Φ µΦ − 16πG4 − h − 4 − 2 ∇ ∇

2 2 1 µ 2 3 µ 2 2 +6k Φ− Φ− + Φ 2 δν + κ Φ A Aν + κ Φ A A (8) − q| | · i

in the Einstein frame. Here, all contractions are done Even if we use this instead of Eq. (10), we never lose the µν 2 µν αβ using the metricγ ¯ as F =γ ¯ γ¯ FµαFνβ and A generality for analyzing the RS background. To check it, µν µν · A =γ ¯ AµAν . Hence,γ ¯ corresponds to the canonical we have the relations metric. Now, we wish to derive the equations from the eﬀective action in Eq. (8). First of all, we have to change µ µ δ δν + Nν 1 1 µ µ 2 3 2 3 µ 2 3 µ p| | = κ Φ AµAν 1 κ Φ A A + , the nonlinear term of δν + Nν with Nν = κ Φ A Aν δγ¯µν 2 − 2 · ··· into a manageable form.p| Considering| the small gauge µ µ (12) coupling constant (κ < 1), we assume that δν > Nν . Using the formula as 1 1 1 det[1 + x] = 1 + tr(x) tr(x2) + (tr(x))2 + , µ µ δ δν + Nν 3 1 p 2 − 4 8 ··· p| | = κ2Φ2A A 1 κ2Φ3A A + , (9) δΦ 2 · − 2 · ··· (13) one ﬁnds

µ µ 1 2 3 1 2 3 2 q δν + Nν = 1 + κ Φ A A (κ Φ A A) + . | | 2 · − 8 · ··· µ µ δ δν + Nν 2 3 1 2 3 (10) p| | = κ Φ Aµ 1 κ Φ A A + . δAµ − 2 · ··· For simplicity, we use here the relation of (14) Making use of Eq. (11), we obtain the truncated equa- µ µ 1 2 3 δν + Nν 1 + κ Φ A A. (11) tions of motion q| | ' 2 ·

2 κ 3 α 1 2 2 2 3 2 2 1 Rµν = Φ FµαFν γ¯µν F 6k κ (1 Φ)AµAν + Φ− ¯ µΦ ¯ ν Φ 2k γ¯µν (Φ− + Φ 2) , (15) 2 − 4 − − 2 h∇ ∇ − − i µ 2 2 3 1 µ ¯ Fµν + 12k κ Φ− (1 Φ)Aν = 3Φ− ¯ ΦFµν , (16) ∇ − − ∇ 2 µ 1 µ 2 1 1 2 2 κ 2 ¯ ¯ µΦ Φ− ¯ Φ ¯ µΦ + 2k Φ− (4 Φ 3Φ− ) κ Φ A A = F . (17) ∇ ∇ − ∇ ∇ h − − − · i 4

It is easily checked that Aµ = 0 and Φ = 1 satisfies Eq. (10) for the nonlinear term, counting Eqs. (13) and Eqs. (16) and (17). Also, if we use the full expression in (14) with Aµ = 0 leads to the same situation. In this -666- Journal of the Korean Physical Society, Vol. 43, No. 5, November 2003 case, Eq. (15) leads to Rµν = 0. Considering Eqs. (12) ηµν ,Aµ = 0, Φ = 1). Let us introduce small fluctuations and (10), we find Rµν = 0. Thus, any 4D Ricci-flat met- around the RS solution: ricγ ¯µν is a solution to the 4D effective action in Eq. (8). In particular,γ ¯µν = ηµν =diag( + ++) corresponds to γµν = ηµν + κhµν ,Aµ = 0 + aµ, Φ = 1 + κϕ.(18) − the RS solution with Z2-orbifold symmetry. It is noted that only for the case of Aµ = 0, we can find a consis- Considering gMN = ηMN + κhMN , we have the relations tent solution because the case of Aµ = 0 results in an h5µ = aµ and h55 = 2ϕ. We note that the non-zero aµ 6 − unwanted situation. That is, it is not easy to find a so- breaks the Z2-symmetry. From Eq. (7), we find lution to the equations which include many terms like 2 ¯ ¯ A A,(A A) , . This is why in the RS approach, they γ¯µν = ηµν + κhµν , hµν = hµν + ϕηµν . (19) · · ··· set Aµ = 0 at the beginning [4]. Now we are in a position to consider the perturba- Then, the bilinear action of Eq. (8), which governs the tion analysis around the RS ground-state solution (¯γµν = perturbative dynamics, is given by [2,26]

2 bil κ 4 1 µ¯αβ ¯ µ¯ ¯ µ¯ ν ¯ µ¯ ν ¯α Izero m = Z d x ∂ h ∂µhαβ ∂ h∂µh + 2∂ hµν ∂ h 2∂ hµα∂ hν − 16πG4 n − 4h − − i 1 µ ν µ ν 3 µ 2 2 1 µν µ (∂µaν ∂ a ∂ν aµ∂ a ) ∂µϕ∂ ϕ + 6k ϕ + hµν T + aµJ + ϕJϕ , (20) −2 − − 2 2 o

¯ µν ¯ where h = η hµν = h + 4ϕ. Here, we introduce in Eq. (20), we obtain the equations of motion: the 4D external sources of (T µν (x),J µ(x),J (x)) to ϕ ¯ ¯ α¯ α¯ hµν + ∂µ∂ν h ∂µ∂ hαν + ∂ν ∂ hαµ obtain the correct physical propagations. Originally, − these all belong to localized sources on the brane as ¯ α β ¯ µν µ ηµν h ∂ ∂ hαβ = Tµν , (21) (T (x),J (x),Jϕ(x))δ(z) [16, 27]. However, after the − − − ν integration over z, they lead to the last line of Eq. (20). aµ ∂µ(∂ν a ) = Jµ, (22) − − Surprisingly, it turns out that although the Z2-symmetry 1 along the z-axis is broken at the linearized level, the bi- ϕ + 4k2ϕ = J . (23) 3 ϕ linear effective action is invariant under the U(1) gauge − transformation. Our previous observation about the Here, we find the 4D diffeomorphisms plus the U(1) U(1) symmetry breaking caused by the nonlinear term gauge symmetry. Hence, these can be taken into account and A A is not correct, at least for the RS ground-state by the source conservation laws · µ µ solution. Here, a nice combination of the nonlinear term ∂ Tµν = 0, ∂ Jµ = 0. (24) and A A in Eq. (8) does not generate any mass term This means that Eqs. (21) and (22) are compatible with like a ·a. This appears as a term of higher order than the source conservation laws. By taking the trace of the squared· order: 6k2κ2(1 φ)A A 6k2κ3ϕa a. Eq. (21), we have We expect that this may contribute− · to the→ − quantum cor-· rection. However, if ϕ = 0, this term does not appear. ¯ α β ¯ 1 h ∂ ∂ hαβ = T (25) The known method to obtain the localization of zero- − 2 ρ modes is find the bilinear action without the external with T = Tρ . Hence, Eq. (21) becomes sources [9,13,24]. The action in Eq. (20), obtained af- ¯ ¯ α¯ ter the integration over z and the perturbation around hµν + ∂µ∂ν h ∂µ∂ hαν the RS background, is finite. Also, it seems to have a − α¯ 1 canonical form for all field fluctuations. Hence, following + ∂ν ∂ hαµ = (Tµν ηµν T ). (26) the conventional criterion, the zeromodes of the gravi- − − 2 ¯ ton, graviscalar, and graviphoton all are localized on the So far we have not chosen any gauge for hµν and aµ. brane. However, this may be wrong because it misses Now, let us choose the harmonic gauge and the Lorentz the roles of the potential terms and the external source. gauge, respectively: The actual propagation of the physical zero-modes on the 1 ∂µh¯ = ∂ h,¯ ∂ aµ = 0. (27) brane can be determined by calculating their exchange µν 2 ν µ amplitudes for the sources [2,26,28–30]. If these gauge conditions are used, Eq. (26) and Eq. (22) In order to understand what physical states there are, reduce to let us derive the linearized equations. From the action ¯ 1 hµν = (Tµν ηµν T ), aµ = Jµ. (28) − − 2 − Localization of the Graviphoton and the Graviscalar on the Brane – Y. S. Myung -667-

The first equation is exactly the same equation as was To obtain the on-shell exchange amplitude induced by derived for the graviton zero-mode by using the brane- the sources, let us plug Eqs. (21), (22), and (23) into bending effect [16]. The brane-bending calculation re- Eq. (20) [2]. Then, we find quired h = 0. Furthermore from Eqs. (25) and (28), we get the trace equation ¯ h = T. (29)

2 ampx κ 4 1 ¯ µν µ Izero m = Z d x hµν (x)T (x) + aµ(x)J (x) + ϕ(x)Jϕ(x) . (30) − 32πG4 n2 o We wish to take the Fouriertransform which makes the calculation easy [28]:

2 ampp κ 4 1 ¯ µν µ Izero m = Z d p hµν (p)T (p) + aµ(p)J (p) + ϕ(p)Jϕ(p) . (31) − 32πG4 n2 o

From Eqs. (28) and (23), the Fourier-transformed ﬂuc- Jϕ(p) ϕ(p) = 2 2 (34) tuations are given by 3(p + mϕ) ¯ 1 1 hµν (p) = Tµν (p) ηµν T (p) , (32) p2 − 2 2 2 J (p) with mϕ = 4k . Substituting these into Eq. (31) leads a (p) = µ , (33) to − µ p2

κ2 1 1 J µ(p)J (p) J 2(p) Iampp = d4p T µν (p)T (p) T 2(p) + µ + ϕ . (35) zero m Z n 2 µν 2 2 2 o − 32πG4 2p − 2 p 3(p + mϕ)

If the massless states are to be studied, it is best to the source conservation law pµJ µ = 0 leads to choose the light-cone frame of pµ = (p1, 0, 0, p4) [2]. µ Then, the source conservation law p Tµν = 0 in Eq. (24) J1 = J4. (38) gives the relations T = T = T ,T = T ,T = T . (36) Making use of this, one has the spin-1 exchange ampli- 11 41 44 12 42 13 43 tude Using this, we ﬁnd the spin-2 exchange amplitude µ 2 2 2J Jµ = J+1 + J 1 (39) µν 1 2 2 2 | | | − | T (p)Tµν (p) T (p) = T+2 + T 2 , (37) − 2 | | | − | with J 1 = J2 iJ3. Finally, plugging this information 1 ± ± where T 2 = (T22 T33) iT23. On the other hand, into Eq. (35) leads to ± 2 − ±

κ2 1 2 J 2 Iampp = d4p T 2 + T 2 + J 2 + J 2 + ϕ , (40) zero m Z n 2 +2 2 +1 1 2| | 2 o − 62πG4 p | | | − | | | | − | 3(p + mϕ)

indicating a total of four massless states: the gravitons pears to be massive. Unfortunately, it has the tachyonic 2 2 with helicities λ = 2 and the graviphotons with helici- mass mϕ = 4k , which may indicate an unstable, mass- ± ¯ − ties λ = 1 [2]. Therefore, hµν and aµ, indeed, represent less mode on the brane. This means that the graviscalar massless± spin-2 propagations and massless spin-1 prop- cannot be localized on the brane. This can be easily read agations on the brane, respectively. Of course, these all off from the 4D effective action in Eq. (6). Essentially, are localized on the brane . we wish to treat the graviscalar as a massless freedom. On the other hand, the graviscalar ϕ in Eq. (40) ap- However, the brane with tension σ k generates the un- ∼ -668- Journal of the Korean Physical Society, Vol. 43, No. 5, November 2003 wanted tachyonic terms for the graviscalar Φ, and those ACKNOWLEDGMENTS terms are proportional to k2.5 In the absence of the sources, the consistency between the linearized equation The author thanks Gungwon Kang and Hyungwon with h = 0 leads to ϕ = 0 [11]. At this stage, we com- Lee for helpful discussions. This work was supported in ment on the role of the trace of hµν . We assume that part by the Korea Science and Engineering Foundation, in the presence of the sources, this is the pure gauge de- Project No. R02-2002-000-00028-0. gree of freedom, as it was in the brane-bending approach [16]. Therefore, we can choose h = 0, which implies that Eq. (29) leads to ϕ = T/4. This leads to a contradic- tion to the graviscalar equation, Eq. (23). We have two REFERENCES equations, one massless and the other massive, for the same field ϕ. 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