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PreprinttypesetinJHEPstyle-HYPERVERSION Gravity and Electroweak Symmetry Breaking in a 9 RSI/RSII Hybrid Model 0 0 2 n a J 6 Brian Glovera and Jong Anly Tana 2 a Particle Theory Group, Department of Physics, College of William and Mary, ] h Williamsburg, VA 23187-8795 p - p E-mail: [email protected],[email protected] e h [ Abstract: We present a hybrid RSI/RSII model in which we both solve the hierarchy 1 problem and produce a continuum of KK graviton modes. In this model, four dimensional v 4 gravity can bereproducedandthe radionmodecan bestabilized. We thenmodify thehybrid 9 gravity model to include SU(2) xSU(2) xU(1) bulk gauge fields. Electroweak symmetry 7 L R B−L 3 isbrokenbythechoiceofappropriateboundaryconditions. Byadjustingthesizeofoneregion . 1 of the extra dimension, we show that the S parameter can be decreased while protecting the 0 9 ρ parameter from corrections. We find that as the S parameter is decreased by 60%, MZ′ ∼ 0 and MW′ stay below 1800 GeV, protecting unitarity. : v i X Keywords: Extra Dimensions, Randall-Sundrum, Radion, Electroweak Symmetry r Breaking, Higgsless, Oblique Corrections, S Parameter. a Contents 1. Introduction 1 2. Gravity in the Hybrid Model 3 2.1 Kaluza-Klein Modes 5 2.2 Radion Stabilization 7 3. Higgsless Symmetry Breaking in the Hybrid Model 7 3.1 Oblique Corrections 8 4. Conclusions 10 1. Introduction Theideaofwarpedextradimensionswasfirstintroducedin1983whenRubakovandShaposh- nikov suggested that a vanishing 4D cosmological constant would result if a 5D bulk vacuum energy was tuned to cancel the large 4D vacuum energy of the Standard Model (SM) fields [1]. This work was popularized in 1999 when Randall and Sundrum introduced two famous examples of warped extra dimensions which led to interesting and distinct phenomenology (hereafter called RSI [2] and RSII [3]). In the first model (RSI), a finite warped extra dimension living between a positive and a negative tension brane was used to solve the hierarchy problem. This model predicts Kaluza-Klein (KK) graviton excitations to have masses on the order of a few TeV which could possibly be detected at the Large Hadron Collider (LHC) in the near future. In the RSII model, Randall and Sundrum considered a warped infinite extra dimension. Although they no longer solved the hierarchy problem, they foundthat four dimensionalgravity canstillbereproducedinaninfiniteextradimensionsincethecorrections to Newton’s Law at large distances are suppressed on the positive tension brane. Since these models were first introduced, many extensions of their work have been proposed. Some of these extensions include adding extra branes to the bulk of RSII [4, 5, 6], localizing gravity on thick branes [7], adding SM fields to the bulk of RSI [8], Higgless models in an RSI background [9], etc. In one of these models [5], an extra negative tension brane was includedinthebulkoftheinfiniteextradimensionofRSII.Thismodel,ifstable,wasdesigned tosolvethehierarchyproblemasinRSIbutwithaninfiniteextradimension. However, itwas found that when the scalar gravity mode (radion) of the five dimensional graviton is carefully considered, the theory becomes unstable [10]. This instability arose since the kinetic term of the radion in these theories was found to be negative [11]. The bulk stress tensor violates – 1 – the positivity of energy condition and the brane is unstable to crumpling. More recently, Agashe et al. [12] pointed out that if one could stabilize a IR-UV-IR model with Z parity 2 about the UV brane, one could address the hierarchy problem naturally. They argue that in an alternate UV-IR-UV model, one would have to add large brane kinetic terms in order to solve the hierarchy problem. In Section 2 we propose a model in which the negative tension brane is placed at an orbifold fixed point with positive tension branes living in the bulk of an infinite, warped extra dimension (see Fig. 1). The metric is given by ds2 = e−A(y)dx2 +dy2 where the warp factor is 2k y if 0 y r 1 A(y) = − | | ≤ | |≤ (1.1) 2k y 2(k +k )r if y > r. ( 2 1 2 | |− | | AsinLykkenandRandall[4],thistheoryhasacontinuousKKspectrumwhilealsosolvingthe hierarchy problem. However, the phenomenology of our model is more of a hybrid between RSI and RSII in which the KK gravitons of RSI become resonances. Placing a negative tension brane at an orbifold fixed point projects out the negative energy mode of the radion and therefore allows the theory to be stabilized. We calculate the gravitational spectrum and show how this theory can be stabilized. HybridGravityModel 80 PositiveTensionBranes 60 NegativeTensionBranes 40 20 Ly HA 0 -20 -40 -60 -r 0 r y Figure 1: The Hybrid RSI/RSII gravity model. The space is orbifolded around y = 0 and extends to infinity. Warped extra dimensions have also proven to be interesting for models of Higgsless Elec- troweakSymmetryBreaking. InCacciapagliaetal. [9],SU(2) xSU(2) xU(1) gaugefields L R B−L were included in the bulk of AdS space. They showed that breaking SU(2) x U(1) down R B−L to U(1) on the Planck brane protects the ρ parameter from corrections since the broken Y SU(2) gauge group shows up as a custodial symmetry in the holographic interpretation [13]. – 2 – It was found, as in technicolor theories, that an order one S parameter is produced in con- flict with experiments. In order to address this problem, a Planck brane kinetic term was added which was found to decrease the S parameter but at the price of destroying unitarity. They also added a U(1) brane kinetic term to the TeV brane which also lowered the S B−L parameter but at the price of making T nonzero. More recently Carone et al. [14] showed that a holographic UV-IR-UV model can be constructed, with SU(2) xU(1) gauge fields L B−L in the bulk, in which a custodial symmetry is generated without introducing a SU(2) gauge R group. They found that like the standard higgsless model, the S parameter is too large. In Section 3 we modify our hybrid model to include gauge fields in the warped extra dimension. Following Csa`ki et al. [9], we include SU(2) x SU(2) x U(1) gauge fields in the bulk and L R B−L use boundary conditions to break the symmetry in order to reproduce the SM on one of our branes. In order to have a normalizable photon, we have brought in another negative tension brane from infinity to cut off the space at an orbifold fixed point (see Fig. 2). We find corrections to the ρ parameter to besuppressed, signaling that an approximate custodial symmetry is preserved. We calculate oblique corrections in this model and find that as the added slice of the extra dimension increases, the S parameter decreases. We stress that this method of reducing the S parameter appears to keep corrections to the ρ parameter suppressed while preserving unitarity for a decrease in S up to 60%. HybridHiggslessModel 80 PositiveTensionBranes 60 NegativeTensionBranes 40 20 Ly HA 0 -20 -40 -60 -Hr +r L -r 0 r Hr +r L 1 2 1 1 1 2 y Figure 2: The Hybrid RSI/RSII higgsless model. The space is orbifolded around y =0 and ends at the location of the outside negative tension branes (y = (r1+r2)). ± 2. Gravity in the Hybrid Model Our theory is defined by placing a negative tension brane at an orbifold fixed point (y = 0) – 3 – in an infinite fifth dimension (the TeV brane). Two additional positive tension branes are added at the points y = r (the Planck branes). It is important to point out that unlike the ± theories proposed in [4] and [5], we place the TeV brane at the orbifold fixed point which (as we will discuss later) stabilizes the Radion mode (see [11]). The Z symmetry demands that 2 the tensions of the two additional Planck branes be equal. The action takes the form. S = d5x g(5) 2M(5)3R Λ g(4)V δ(y) g(4)V δ(y+r)+δ(y r) . − pl − b− − 1 − − 2{ − } Z q (cid:20) q q (cid:21) (2.1) If we assume four-dimensional Poincare invariance, the metric is given by ds2 = g dxMdxN (2.2) MN with e−A(y) 0 0 0 0 − 0 e−A(y) 0 0 0   g (xµ,y) = 0 0 e−A(y) 0 (2.3) MN  0 0 0 e−A(y) 0    0 0 0 0 1     and 2k y if 0 y r 1 A(y) = − | | ≤ | |≤ (2.4) 2k y 2(k +k )r if y > r. ( 2 1 2 | |− | | As in [2], the assumption of four-dimensional Poincare invariance leads one to derive the (5)3 tension of the TeV brane located at y = 0 to be V = 24M k and the cosmological 1 − pl 1 constant between the Planck and TeV branes is Λ = 24M(5)k2. Likewise, the tension on 1 − pl 1 (5)3 the Planck brane located at y = r is found to be V = 24M (k +k ) and the cosmological 2 pl 1 2 constant outside the Planck branes is Λ = 24M(5)k2. It is useful to transform the metric 2 − pl 2 to manifestly conformally flat coordinates, where Einstein’s equations take a simpler form. In these coordinates, the metric takes the form g (xµ,z) = e−A(z)diag( 1,1,1,1,1) (2.5) MN − where 1 if z z e−A(z) = (−k1|z|+1)2 ≤ b (2.6) 1 if z > z . ( (k2|z|+C)2 b Now the Planck branes are located at z = (1 e−k1/r)/k and the constant C = k /k + b 1 2 1 ± − − exp[ k r](1+k /k ) is chosen such that z is the same for the two slices of AdS space. 1 2 1 b − – 4 – 2.1 Kaluza-Klein Modes For now we will just consider the spin-2 fluctuation of the metric. The scalar mode (radion) will be discussed in the following section. Consider a pertubation of the form ds2 = e−A(z) dxµdxν(η +h (x,z))+dz2 . (2.7) µν µν (cid:0) (cid:1) The transverse traceless solution can be written as h (x,z) = e3A(z)/4h˜ (x)ψ(z) where µν µν (cid:3) h˜ (x) = m2h˜ (x) and 4 µν µν ∂2+V(z) ψ(z) = m2ψ(z). (2.8) − z The potential V(z) is found to (cid:2)be [3] (cid:3) 9 3 V(z) = (∂ A(z))2 ∂2A(z) 16 z − 4 z 15k2 1 if z zb 3k = 4(−k1|z|+1)2 | | ≤ + 1 δ(z)  15k22 if z > zb ( k1+1)  4(k2|z|+C)2 | |  − 3 k k 1 2  +  (δ(z z )+δ(z+z )). b b −2 ( k z +1) (k z +C) − (cid:18) − 1| | 2| | (cid:19) As usual, since the equation of motion for the Kaluza-Klein modes can bewritten in the form Qˆ†Qˆψ(z) = m2ψ(z) with Qˆ = ∂ +(3/4)A′(z), there is a zero mode solution that satisfies z Qˆψ (z) = 0: 0 3 ψ (z) = N exp[ A(z)]. (2.9) 0 −4 −1/2 N is found by normalization: N = exp[ 3/2A(z)]dz . − The higher KK modes are found by solving equation [2.8] subject to the following boundary (cid:2)R (cid:3) conditions and normalization: 1) ψ (z) is continuous at the Planck branes (z = z ). m b ± 2) ψ′ (z) is discontinuous at: m a) the TeV brane: ∆(ψ′ (z)) = 3k ψ (0). m |z=0 1 m b)thePlanckbranes: ∆(ψ′ (z)) = 3 k1 + k2 ψ ( z ). m |z=±zb −2 −k1|zb|+1 k2|zb|+C m ± b (cid:16) (cid:17) 3) ψ (z) approaches a normalized plane wave solution for very large z. m The solution is ( z +1/k )1/2[a Y (m( z +1/k ))+b J (m( z +1/k ))] if z zb 1 m 2 1 m 2 1 ψ (z) = −| | −| | −| | | | ≤ m (z +C/k )1/2[a′ Y (m(z +C/k ))+b′ J (m(z +C/k ))] if z > zb. ( | | 2 m 2 | | 2 m 2 | | 2 | | (2.10) – 5 – The boundary conditions and normalization give the following relationships among the coef- ficients: 1/2 k a Y (me−k1r/k )+b J (me−k1r/k )= 1 a′ Y (me−k1r/k )+b′ J (me−k1r/k ) m 2 1 m 2 1 k m 2 2 m 2 2 (cid:18) 2(cid:19) h (2.1i1) a Y (m/k )+b J (m/k )= 0 (2.12) m 1 1 m 1 1 1/2 k a Y (me−k1r/k )+b J (me−k1r/k )= 1 a′ Y (me−k1r/k )+b′ J (me−k1r/k ) m 1 1 m 1 1 k m 1 2 m 1 2 (cid:18) 2(cid:19) h (2.1i3) a′2 +b′2 = m. (2.14) m m KKModeSpectrumatz=0 1.4´10-10 k1=6,k2=7,r=5 1.2´10-10 kRS1=6,rRS1=5 1´10-10 LÈ0 H0 8´10-11 Ψ (cid:144)È HLÈ0 6´10-11 m Ψ È 4´10-11 2´10-11 0. 5. 10. 15. 20. 25. m(cid:144)k 1 Figure 3: Mass Spectrum for both the Hybrid RS (solid) and RS1 (dashed) models. The Hybrid RS model’s spectrum was normalized by the zero mode’s value at z=0. Unlike the RS1 model, there is a continuous spectrum of graviton modes (all m > 0 are allowed). The RS1 spectrum is discrete and given by m = k x , where x denotes the zeros n 1 n n of J (x) [15]1 . In Fig. 3 we compare the Hybrid RS KK spectrum to that of RS1. We have 1 chosen order one parameters such that k r = 30. The resonances in the spectrum correspond 1 nicely to the discrete spectrum found in RS1. Since the modes are suppressed compared to 1Sincewehavenormalized themetrictobe1attheTeVbraneinsteadofthePlanckbraneasdoneinRS1 [2], our spectrum is multiplied by exp[k1r] as compared to thesolution found in [15] – 6 – the zero mode, the corrections to Newton’s Law are small: 1 ψ (0)2 ∞ 1 ψ (0)2e−m|x¯−x¯′| V(x¯,z = 0,x¯′,z′ = 0) = | 0 | + | m | dm (2.15) 2M3 x¯ x¯′ 2M3 x¯ x¯′ pl | − | Z0 pl | − | 1 |ψ0(0)|2 1+ ∞e−m|x¯−x¯′||ψm(0)|2dm . (2.16) ∼ 2M3 x¯ x¯′ ψ (0)2 pl | − | (cid:18) Z0 | 0 | (cid:19) 2.2 Radion Stabilization As mentioned above, placing the TeV brane at the orbifold fixed point will allow the radion mode to be stabilized. To see this we need to include the spin-0 fluctuation of the 5 dimensional graviton. The proper way to include this mode was discussed in [11] and [16]. It was found that the metric can be written in such a way that the spin-2 calculation goes through as done above and is decoupled from the spin-0 radion mode (f(x)). For the metric given in Equation 2.4, Pilo et al. [11] found that the effective four dimensional lagrangian contains the term (5)3 24M k pl 1 e−2k1r 2 √ gd4xf(cid:3)f. (2.17) L ⊃ 2k − k +k − 1 (cid:18) 2 1(cid:19)Z Since the kinetic term is always positive in our model, positivity of energy is not violated. The radion can be stabilized using a mechanism like the one introduced by Goldberger and Wise [17]. 3. Higgsless Symmetry Breaking in the Hybrid Model In this section we will put SU(2) SU(2) U(1) gauge fields in the bulk. The metric L× R× B−L is given by (2.5) (see Fig. 2). However, unlike before, in this section we cut off the infinite extra dimension in order to make the massless mode normalizable 2. This is accomplished by adding a negative tension brane at an orbifold fixed point: y = (r +r ) (or z = z = 1 2 b2 1/k (e(k2r2−k1r1) k /k (e−k1r1 1+k /k e−k1r1)) in z-coordinates). The 5D action for this 2 2 1 1 2 − − model is: 1 1 1 S = d4x dz g(5) Ra RaMN La LaMN B BMN (3.1) − −4 MN − 4 MN − 4 MN Z Z q (cid:20) (cid:21) where Ra , La , and B are the SU(2) , SU(2) , and U(1) field strengths. MN MN MN L R B−L Usingthesameprocedureas[9],wechosetoworkinunitarygaugewhereallKKmodesof thefieldsLa,Ra,B areunphysical. BoundaryconditionswereimposedtobreaktheSU(2) 5 5 5 L× SU(2) U(1) symmetry to the Standard Model at z = z and to SU(2) U(1) R × B−L b2 D × B−L 2We will now use r1 instead of r to denote the distance of the first brane to the origin. Also we will only consider half of the space for most of the discussion since the other half is obtained by orbifolding about the origin – 7 – at z = 0. The boundary conditions are: ∂ (La +Ra) = 0, La Ra = 0, ∂ B = 0, z = 0 : z µ µ µ− µ z µ (3.2) La+Ra = 0, ∂ (La Ra) = 0, B = 0 ( 5 5 z 5 − 5 5 ∂ La = 0, R1,2 = 0 z µ µ z = z : ∂ (g B +g˜R3) = 0, g˜B g R3 = 0, (3.3) b2  z 5 µ 5 µ 5 µ− 5 µ La5 = 0, R5a = 0, B5 = 0  whereg andg˜ arethe5DgaugecouplingforSU(2) andU(1) respectively. Inaddition 5 5 L,R B−L to the boundary conditions we imposed continuity for the wave function at z = z . The bulk b equation of motion for the gauge fields is 1 q2 ∂z2′ − z′∂z′ + k2 ψ(z′) = 0 (3.4) " 1,2# where z′ = k z+1 or k z+C for 0 z z and z z z respectively. The solution to 1 2 b b b2 − ≤ ≤ ≤ ≤ this equation is given by ( k z+1) adJ (q ( z+1/k ))+bdY (q ( z+1/k )) , 0 z z ψd = − 1 i 1 i − 1 i 1 i − 1 ≤ ≤ b (3.5) i (k z+C) a′dJ (q (z+C/k ))+b′dY (q (z+C/k )) , z z z ( 2 (cid:0)i 1 i 2 i 1 i 2 (cid:1) b ≤ ≤ b2 (cid:16) (cid:17) where d labels the corresponding gauge bosons (W , L3, B, R3). Following [9], we expand ± the fields in their Kaluza-Klein modes as follows: ∞ 1 B (x,z) = a γ(x)+ ψB(z)Zj(x) (3.6) µ g˜ 0 j µ 5 j=1 X ∞ 1 L3(x,z) = a γ(x)+ ψL3(z)Zj(x) (3.7) µ g 0 j µ 5 j=1 X ∞ 1 R3(x,z) = a γ(x)+ ψR3(z)Zj(x) (3.8) µ g 0 j µ 5 j=1 X ∞ L±(x,z) = ψL±(z)Wj±(x) (3.9) µ j µ j=1 X ∞ R±(x,z) = ψR±(z)Wj±(x) (3.10) µ j µ j=1 X 3.1 Oblique Corrections In order to calculate the electroweak corrections in our model we ensure that all corrections are oblique. This is done by adjusting the coupling of the fermions localized at z = z so b2 that the zero mode couplings are equal to the SM couplings at tree level. For our model the – 8 – relations are g˜ψ(B)(z ) g′2 5 1 b2 = (3.11) − g ψ(L3)(z ) g2 5 1 b2 (L±) g ψ (z ) = g (3.12) 5 1 b2 (L3) g ψ (z ) = gcosθ (3.13) 5 1 b2 W For the photon kinetic term, we canonically normalize it as follows: Z = (a /g˜)2+(a /g )2 I = 1 (3.14) γ 0 5 0 5 (cid:16) zb2 (cid:17) I = e−A(z)/2dz. (3.15) Z−zb2 Equations (3.12) and (3.13) are used to determine the correct normalization for the W and Z wavefunctions. Given the gauge field’s wavefunctions, we calculated the oblique corrections using the relations between the vacuum polarization and the wavefunction renormalization: Z = 1 γ − Π′ , Z = 1 g2Π′ , and Z = 1 (g2 +g′2)Π′ [18]. The wavefunction renormalizations QQ W − 11 Z − 33 are give by Z = zb2 ψW 2e−A(z)/2dz = zb2 ψL+ 2+ ψR+ 2 e−A(z)/2dz (3.16) W Z−zb2 Z−zb2(cid:16) (cid:17) Z = zb2 (cid:2)ψZ (cid:3)2e−A(z)/2dz = zb2 (cid:2)ψL3 2(cid:3)+ ψ(cid:2)R3 2(cid:3)+ ψB 2 e−A(z)/2dz, (3.17) Z Z−zb2 Z−zb2(cid:16) (cid:17) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) and the zero momentum vacuum polarizations are Π (0) = 1 zb2 ∂ ψL+ 2+ ∂ ψR+ 2 e−A(z)/2dz (3.18) 11 g2 z z Z−zb2(cid:16) (cid:17) Π (0) = 1 (cid:2)zb2 ∂(cid:3)ψL3(cid:2)2+ ∂(cid:3)ψR3 2+ ∂ ψB 2 e−A(z)/2dz. (3.19) 33 g2 +g′2 z z z Z−zb2(cid:16) (cid:17) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (3.20) The Peskin-Takeuchi oblique corrections as a function of vacuum polarization are defined as [18] : S = 16π(Π′ Π′ ) (3.21) 33 − 3Q 4π T = (Π (0) Π (0)) (3.22) sin2θ cos2θ M2 11 − 33 W W Z U = 16π(Π′ Π′ ) (3.23) 11 − 33 Since we are only considering the tree level corrections, Π′ = 0. As an input to our model, 3Q we use the values of the SM electroweak parameters at the Z-pole: M = 80.045 GeV, W sin2θ = 0.231, and α = 127.9. We also assume k r = 30. In the limit r 0, M sets W 1 1 2 W → – 9 –