INTRODUCTION motivation 5d supergravity models An important question in supersymmetric theories is how supersymmetry is broken in the low energy world

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1 Peggy Kouroumalou Univ. of Athens, Greece The soft scalar sector of supergravity compactified on S 1 /Z 2 orbifolds G. Diamandis, V. Georgalas, P. Kouroumalou, A.B. Lahanas [hep-th: ] [hep-th: ] [hep-th ] [hep-th 09/////] (to appear) () December 15, / 28

2 INTRODUCTION motivation 5d supergravity models An important question in supersymmetric theories is how supersymmetry is broken in the low energy world. New possibilities arise in the context of theories with extra dimensions. The majority of extra dimensional models include two boundaries (membranes) embedded in a higher dimensional bulk. On the one boundary lives the visible sector of the 4 dimensional theory, while the hidden sector lives on the other. Depending on how supersymmetry breaking is transmitted to the visible brane: gauge mediated models. Not (large) FCNC. Inelegant complication to µ problem. gravity mediated models Guidice- Masiero mechanism resolves µ problem. In trouble with flavor at the UV. () December 15, / 28

3 INTRODUCTION motivation 5d supergravity models An extra compact dimension can change the situation by introducing an extra scale R (compactification radius). In that case, visible and hidden sectors are seperated by distance R. Potential dangerous FC tree level couplings between visible and hidden brane operators are not present 1 (locality) for R >>. Quantum corrections M 5Planck turn out to be 1 and they are governed by the πr good flavour behaviour of gravity in the IR. [Chacko et al. (1999), Randall-Sundrum (1998)] () December 15, / 28

4 INTRODUCTION motivation 5d supergravity models The Horava-Witten model provides the prototype model in which to study the transmission of supersymmetry breaking via an extra dimension. In this model, an E 8 set of gauge fields was assumed on each of two 10 dimensional planes fixed by the Z 2 orbifold and seperated by an extra S 1 dimension. Relation between the orbifold and compactification radius suggests that universe appears five dimensional in a certain regime. The effective 5- dimensional theory of the model is a gauged N = 1 supergravity with 4 dimensional boundaries [Lucas et. al. (1998)] () December 15, / 28

5 N =2, D =5 Sugra THE MODEL In our work we study the transmission of the supersymmetry breaking in the context of N = 2, D = 5 Supergravity compactified on S 1 /Z 2 orbifold. Visible and hidden branes are at, the fixed by Z 2 points, x 5 = 0 and x 5 = πr respectively. () December 15, / 28

6 THE MODEL extra dimension S 1 :[ πr,πr] N =2, D =5 Sugra Supergravity multiplet {e m µ, Ψĩ µ, A µ} µ =(µ, 5) curved m =(m, 5) flat fermions are symplectic Majorana, obeying: Ψ i = ɛ ij (Ψ j ) T C i =1, 2 is the symplectic SU(2) R index. The N = 2, D = 5 bulk Lagrangian is [Gunaydin Zagermann (2000)]: e ( 1) L = 1 2 R(ω) 1 2 Ψ ĩ µγ µ ν ρ ν Ψ ρi 1 4åĨ J F Ĩ µ νf J µ ν g xỹ ( µ φ x )( µ φỹ )+(fermion terms) + (Chern Simons terms) () December 15, / 28

7 THE MODEL fields parity assignment Z 2 even : eµ m, e 5 5, Ψ 1 µl, Ψ2 5L, A0 5, Aa µ, λ 1a L Z 2 odd e 5 µ, e5 m, Ψ 2 µl, Ψ1 5L, A0 µ, λ 2a L, φ x N =2 Z 2 N =1supersymmetry on the branes. Next, we consider a chiral multiplet (ϕ, χ) living on the visible brane at x 5 = 0. Working in the on-shell scheme and using Nöther procedure we derived the coupling of the chiral multiplet to the bulk fields, ignoring at first the radion multiplet. [ Diamandis, Georgalas, Kouroumalou, Lahanas (2004) ]. Our result agrees with other approaches using off-shell formulation of supergravity [Rattazzi et al. (2003)]. () December 15, / 28

8 the radion multiplet Kähler function COUPLING TO THE RADION MULTIPLET The radion multiplet propagates in the bulk but since it consists of even fields, couples also to the brane fields. Following Nöther procedure to find the coupling of the brane fields to the radion multiplet, turns out to be cumbersome. Alternatively, we can use the standard knowledge of N = 1, D = 4 supergravity. The main observation, coming out from the supersymmetric transformation of the fields,is that the restriction of the radion on the brane forms a chiral multiplet: T ( 1 2 [e 5 5 i 2 3 A 0 5],χ (T ) ψ 2 5) () December 15, / 28

9 COUPLING TO THE RADION MULTIPLET the radion multiplet Kähler function We can seek for a Kähler function F to describe the coupling in the usual manner in N =1D =4 supergravity [Zumino (1979), Bagger Witten (1982)] We can split F as: F = N (T, T )+K(T, ϕ, T,ϕ ) N (T, T ) is the restriction of 5D supergravity (survives in the absence of brane multiplets) K(T, ϕ, T,ϕ ) (5) K(ϕ, ϕ ), (5) e 5 5 δ(x 5) F turns out to be: F = 3 ln T + T 2 + δ(x 5 ) 2 T + T K(ϕ, ϕ ) () December 15, / 28

10 INTERACTION RADION- BRANE FIELDS Lagrangian up to 4 fermion terms i, j runs both for chiral AND radion fields L (0) = e (4) [ F ij µ ϕ i µ ϕ j + i ( 2 χ i σ µ D µ χ j + χ j σ µ D µ χ i)] i 4 e(4) [(F ij F m µ ϕ m F m µ ϕ m ))] [(2(F mij µ ϕ m F ij m µϕ m )]χ i σ µ χ j 1 2 e (4) ( F ij ν ϕ j χ i σ µ σ ν ψ µ + h.c. ) + e(4) 4 ɛκλµν (F m κ φ m F m κ φ m )ψ λ σ µ ψ ν + e(4) 4 F ij (iɛκλµν ψ κ σ λ ψ µ + ψ µ σ ν ψ µ )χ i σ ν χ j (F ij F kl 2R ij kl ) χi σ µ χ j χ k σ µ χ l () December 15, / 28

11 SUPERPOTENTIAL W (ϕ) AT x 5 = πr Including a superpotential W (ϕ), gives rise to Yukawa and potential terms: L w e (4) = (x 5)exp( F 2 )[W ψ µ σ µν ψ ν + i (D i W )χ i σ µ ψ µ ] 2 + ( (x 5 )) 2 exp(f)[f ij D i W D j W 3 W 2 ] L p with: D i W W i + F i W W i W ϕ i. The extra power of (x 5 ) multiplying the potential term is cancelled in the first term but NOT in the term W 2. Singularities of this kind, arise from the propagation of the bulk fields in order to maintain supersymmetry and are also present in the flat case. [ Mirabelli Peskin (1998)] () December 15, / 28

12 TRANSMISSION OF SUPERSYMMETRY BREAKING W = c at x 5 = πr mass terms for ψ µ, ψ 2 5 We consider a constant superpotential c on the hidden brane. The resulting Lagrangian is of the form: L w (h) e (4) = e N 2 (x5 πr)[c ψ m σ mn ψ n + 3i 2 cψ2 5 σm ψ m + 3 ] 2 cψ2 5 ψ2 5 +(h.c) Mass terms for the gravitino ψ µ and the spinor field of the radion multiplet ψ appear only on the hidden 2 5 brane. In the absence of brane chiral multiplets the corresponding K ähler function is that of a no-scale model. () December 15, / 28

13 gauge fixing term TRANSMISSION OF SUPERSYMMETRY BREAKING Our gauge choice will be the one in which the bulk kinetic terms of ψµ 1 ψ µ, ψµ 2 and ψ 1,2 become 5 diagonal [Diamandis, Georgalas, Kouroumalou, Lahanas (2007)]. We add to the bulk the gauge fixing term: i ξ 2 ˆΨ i mˆγ mˆγ r ˆγñ r ˆΨ ĩ n ˆΨ i m, i =1, 2 denote the gravitinos of the original N = 2 Lagrangian: ( ) ( ) ˆψ ˆΨ 1 m 1 m ˆψ = ˆψ 1 m, ˆΨ 2 m 2 m = ˆψ 1 m () December 15, / 28

14 gauge choice Dirac spinors TRANSMISSION OF SUPERSYMMETRY BREAKING We distinguish between the hatted fields and the unhatted, since in order to have the canonical form of Einstein-Hilbert action on the brane, we had to Weyl rescale and shift the f ünbein and the gravitinos. The choice ξ = 3 and an additional shift to ψ1,2 4 5 eliminates the m, 5 mixings. From further on, the following Dirac spinors are used: Ψ= ( ψ 2 5 ψ 1 5 ), Ψ m = ( ψ 1 m ψ 2 m The orbifolded propagators in the mixedmomentum representation [Puchwein Kunszt 2004] for Ψ, Ψ m with the specific gauge choice are found to be [Diamandis, Georgalas, Kouroumalou, Lahanas (2007)]: () December 15, / 28 ).

15 TRANSMISSION OF SUPERSYMMETRY BREAKING orbifolded propagators G mn (orb.) (p, y, y ) = ( 1 2 γ n pγ m + in mn γ 5 y )F (p, y, y ) Ψ m G (orb.) (p, y, y ) = 2i 9 ( p + iγ5 y )F ( p, y, y ) Ψ. 1 F 2qsin(qπR) {cos[q(πr y y )] iγ5 cos[q(πr y y )]} y, y are along the fifth dimension and q = p 2 + iɛ. The above are the massless propagators for the gravitino and the spinor field of the radion multiplet. Mass terms on the hidden brane are treated as interactions. () December 15, / 28

16 THE SOFT SUPERSYMMETRY TERMS relevant terms for scalar masses m 2 ϕ gravitino mass scale Next we will calculate the soft scalar masses, trilinear couplings and gaugino masses induced on the visible brane due to the transmission of supersymmetry breaking at the hidden brane. Scalar masses m 2 ϕ For simplicity we assume one chiral multiplet on the visible brane. K(ϕ, ϕ )=ϕϕ The relevant Lagrangian terms for the calculation of scalar masses on the visible and hidden branes, (in terms of the Dirac spinors) are respectively: L visible = ik 2 5 ϕ 2 δ(x 5 )[ ΨP R γ n n P L Ψ+ 1 9 Ψ m P R γ m γ s s γ r P L Ψ r + i 3 (Ψ mp R γ m γ n n C Ψ T +Ψ T mcp L γ m γ n n Ψ) ] () December 15, / 28

17 THE SOFT SUPERSYMMETRY TERMS relevant terms for scalar masses m 2 ϕ gravitino mass scale L hidden ˆB mn {}}{ = e (4) δ(x 5 πr)k5 2 c [ΨT mcp L (γ mn 1 3 γm γ n )Ψ n (ΨT P L CΨ) + i 2 ( Ψ m P R γ m P L Ψ) + (h.c)] P L,R = 1 2 (1 ± iγ 5 ) are the chiral projection operators, C the charge conjugation matrix. gravitino mass scale signaling supersymmetry breaking is m 3/2 = k5 2 c πr () December 15, / 28

18 THE SOFT SUPERSYMMETRY TERMS Feynman graphs for m 2 ϕ (a) (b) (c) (d) (e) (f) Figure: Diagrams relevant for the calculation of the induced scalar masses. Curly lines denote the gravitinos, solid lines denote the spinor field of the radion multiplet and dashed lines stand for the scalar fields lying on the visible brane () December 15, / 28

19 soft scalar mass m 2 ϕ non tachyonic THE SOFT SUPERSYMMETRY TERMS The contribution of each graph is of the form d 4 p (2π) 4 Tr [V G(p, 0,πR)V 1G(p,πR,πR)V 2 G(p,πR, 0)] Summing all the contributions of the Feynman graphs, yields the following finite mass correction m 2 ϕ = k2 5 ζ(3) 16 π 5 R 3 m2 3/2 = ζ(3) m 2 3/2 π 3 R 2 M Planck 1 GN = GeV k M 3 5Planck and M 2 Planck = 16πRM2 5 M 2 Planck Scalar mass turns out to be non-tachyonic as a result of the right treatment of ψ which cannot be 2 5 removed globally from the Lagrangian.[Bagger Belyaev (2006), Benakli Moura] (2008)] () December 15, / 28

20 THE SOFT SUPERSYMMETRY TERMS trilinear couplings W (Φ) : λ 6 Φ3 gravitino contribution [diagram (f)] is negative, also, in our approach. Trilinear couplings A 0 To calculate the trilinear soft scalar couplings induced on the visible brane due to the supersymmetry breaking on the hidden brane, we add on the visible brane a cubic superpotential: W (Φ) = λ 6 Φ3 relevant Yukawa-type terms for the computation are: L (vis.) w = k5 2e(4) δ(x 5 )W (ϕ){ Ψ m P R (γ mn 1 3 γm γ n )P R C Ψ T n ΨT CP L Ψ i 2 Ψ m γ m P L Ψ+( h.c)} () December 15, / 28

21 THE SOFT SUPERSYMMETRY TERMS correction to the cubic potential Feynman graphs (a) (b) (c) Figure: Diagrams relevant for the calculation of the trilinear soft scalar masses the correction to the cubic potential is calculated to be 3 16 m 3/2k5 2 ζ(3) π 5 R 3 W (ϕ)+(h.c). () December 15, / 28

22 THE SOFT SUPERSYMMETRY TERMS so the trilinear soft scalar coupling is: trilinear coupling U(1) gauge multiplet at x 5 =0 A 0 = 3 16 m 3/2k 2 ζ(3) 5 π 5 R 3 = 3 ζ(3) m 3/2 π 3 R 2 Gaugino masses M 2 Planck In order to study the induced gaugino masses, we assume a U(1) gauge multiplet on the visible brane. We start from the globally supersymmetric Lagrangian L 0 = e (4) δ(x 5 )[ 1 4 F µνf µν + i λ σ µ D µ λ D2 ] Following Nöther procedure, we find the relevant terms for the computation of the gaugino masses. () December 15, / 28

23 GAUGINO MASSES lagrangian for gaugino masses δm (a) δm (b) elimination of the auxiliary fields D, Fierz rearrangement and use of the Dirac spinors yields : k2 5 4 e(4) δ(x 5 )( Ψ m P R CP R Ψ m Λ T CP L Λ+ h.c) k 5e (4) δ(x 5 )(i Ψ m P R γ ab γ m P L Λ+ h.c)f ab Λ denotes the Majorana spinor for the gaugino. the contributions of the two diagrams (a) and (b) to the gaugino mass are: δm (a) = 2 ζ(3) π 3 R 2 m 3/2 M 2 Planck δm (b) = 3 ζ(3) m 3/2 2 π 3 R 2 M 2 Planck () December 15, / 28

24 GAUGINO MASSES Feynman graphs for gaugino mass gaugino mass m 1/2!!!!!!!!!!!!!!!!!"#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!$#! Figure: Diagrams relevant for the calculation of the gaugino mass. Solid lines denote the gaugino and springlike lines the photon. Blobs stand for mass insertions on the hidden brane. Summing the contributions of the two diagrams : m 1/2 = 7 ζ(3) m 3/2 2 π 3 R 2 M 2 Planck () December 15, / 28

25 PHENOMENOLOGY OF THE SOFT SUSY BREAKING SECTOR First results -work in progress [ Diamandis, Georgalas, Kouroumalou, Lahanas] to appear. as free parameters of the model can be considered to be the gravitino mass m 3/2 and the volume of the extra dimension R or alternatively, gravitino mass m 3/2 and gaugino mass m 1/2. the value of gaugino mass turns out to be negative with magnitude given by m 3/2 m 1/ R 2 MPlanck 2 if the gravitino mass is in the TeV range so is m 1/2 provided R is close to the Planck scale : R 1 3M Planck () December 15, / 28

26 PHENOMENOLOGY OF THE SOFT SUSY BREAKING SECTOR values of gravitino larger than about 10 TeV as demanded by nucleosynthesis, result to R M Planck with the gaugino mass in the TeV range so that gravitino crisis may be evaded. () December 15, / 28

27 COMMENTS - CONCLUSIONS In our toy model we have studied the transmission of supersymmetry breaking from the hidden to the visible sector, in the context of N = 2, D =5 supergravity compactified on S 1 /Z 2 orbifolds. a constant superpotential on the hidden brane was assumed, that was enough to trigger supersymmetry breaking. we calculated the soft scalar mass, the trilinear scalar coupling and the gaugino mass up to order O(k 2 5 ). soft scalar mass turns out to be non-tachyonic as a result of the inclusion of the spinor field of the radion multiplet and also that results to non- zero gaugino mass at 1 loop. () December 15, / 28

28 COMMENTS - CONCLUSIONS other approaches [Rattazzi et al. (2003), Gherghetta Riotto (2002)] work in the unitary gauge where ψ5 2 = 0 and that results to tachyonic scalar masses and vanishing gaugino masses at one loop. The use of this gauge can be implemented only conditionally [ Bagger et al. (2001), (2002) ] and taking ψ5 2 = 0 all over 5-dimensional space-time is rather questionable. Future Work more realistic superpotential on the hidden brane additional supersymmetry breaking mechanism Fayet- Iliopoulos term hybrid model Thank you! () December 15, / 28