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Nuclear Physics B 804 (2008) 70-89

www. elsevier. com/locate/nuclphysb

Small extra dimensions from the interplay of gauge and supersymmetry breaking

Wilfried Buchmüllera, Riccardo Catena b'*, Kai Schmidt-Hobergc

a Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany b ISAS-SISSA, 34013 Trieste, Italy c Physik Department T30, Technische Universität München, 85748 Garching, Germany

Received 4 April 2008; accepted 9 June 2008

Available online 21 June 2008

Abstract

Higher-dimensional theories provide a promising framework for unified extensions of the supersym-metric Standard Model. Compactifications to four dimensions often lead to u(1) symmetries beyond the Standard Model gauge group, whose breaking scale is classically undetermined. Without supersymmetry breaking, this is also the case for the size of the compact dimensions. Fayet-Iliopoulos terms generically fix the scale m of gauge symmetry breaking. The interplay with supersymmetry breaking can then stabilize the compact dimensions at a size 1 /m, much smaller than the inverse supersymmetry breaking scale 1 /u.. We illustrate this mechanism with an 50(10) model in six dimensions, compactified on an orbifold. © 2008 Elsevier B.V. All rights reserved.

1. Introduction

Higher-dimensional theories provide a promising framework for unified extensions of the supersymmetric Standard Model [1]. Interesting examples have been constructed in five and six dimensions compactified on orbifolds [2-7], which have many phenomenologically attractive features. During the past years it has become clear how to embed these orbifold GUTs into the heterotic string [8-10], separating the GUT scale from the string scale on anisotropic

* Corresponding author.

E-mail addresses: wilfried.buchmueller@desy.de (W. Buchmüller), catena@sissa.it (R. Catena), kschmidt@ph.tum.de (K. Schmidt-Hoberg).

0550-3213/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2008.06.012

orbifolds [11]. A class of compactifications yielding supersymmetric Standard Models in four dimensions (4D) have been successfully constructed [12-14].

For a given orbifold compactification of the heterotic string, one can consider different orb-ifold GUT limits where one or two of the compact dimensions are larger than the other five or four, respectively [10]. One then obtains an effective five-dimensional (5D) or six-dimensional (6D) GUT field theory as intermediate step between the full string theory and the supersymmetric Standard Model. We shall focus on 6D field theories compactified on T2/Z2 with two Wilson lines. These models have four fixed points where quantum corrections generically induce Fayet-Iliopoulos terms [15,16]. In the case of the heterotic string the magnitude of these local terms is O(MGUT), which suggests that they may lead to a stabilization of the compact dimensions at R — 1/Mgut [16].

Quantum corrections to the vacuum energy density, the Casimir energy, play a crucial role in the stabilization of compact dimensions [17]. Various aspects of the Casimir energy for 6D orbifolds have already been studied in [18-20]. Stabilization of the volume can be achieved by means of massive bulk fields, brane localized kinetic terms or bulk and brane cosmological terms [18]. Alternatively, the interplay of one- and two-loop contributions to the Casimir energy can lead to a stabilization at the length scale of higher-dimensional couplings [21]. In addition, fluxes and gaugino condensates play an important role [22,23].

In this paper we consider orbifold GUTs, which generically have two mass scales: M — MGUT, the expectation value of bulk fields induced by local Fayet-Iliopoulos terms, and x ^ MGUT, the scale of soft supersymmetry breaking mass terms. As we shall see, the interplay of 'classical' and one-loop contributions to the vacuum energy density can stabilize the extra dimensions at small radii, R — 1/MGUT ^ 1/x with bulk energy density O(p2M^UT). We shall illustrate this mechanism with an 50(10) model in six dimensions [24] which together with gaugino mediation [25,26] is known to lead to a successful phenomenology [27,28].

The paper is organized as follows. In Section 2 we briefly describe the relevant features of the 6D orbifold GUT model. The Casimir energies of scalar fields with different boundary conditions are discussed in Section 3. These results are used in Section 4 to evaluate the Casimir energy of the considered model. In Section 5 the stabilization mechanism is described. Appendices A and B deal with the mode expansion on T2/z3 and the evaluation of Casimir sums, respectively.

2. The model

As an example, we consider a 6D N = 1 50(10) gauge theory compactified on an orbifold T2/Z2, corresponding to T2/Z2 with two Wilson lines [24]. The model has four inequivalent fixed points ('branes') with the unbroken gauge groups 50(10), the Pati-Salam group GPS = SU(4) x 5U(2) x 5U(2), the extended Georgi-Glashow group Ggg = 5U(5) x U(1)X and flipped 5U(5), Gfl = SU(5)' x U(1)', respectively. The intersection of these GUT groups yields the Standard Model group with an additional U(1) factor, GSM = SU(3)c x SU(2)l x U(1)Y x U(1)X, as unbroken gauge symmetry below the compactification scale. At the fixed points only 4D N = 1 supersymmetry remains unbroken. Gauge and supersymmetry breaking are realized by assigning different parities to the different components of the 45-plet of S0(10), which is a 6D N = 1 vector multiplet containing 4DN = 1 vector (V) and chiral (S) multiplets (cf. Table 1).

The model has three 16-plets of matter fields, localized at the Pati-Salam, the Georgi-Glashow, and the flipped SU(5) branes. Further, there are two 16-plets, $ and and two 10-plets, H5 and H6 of bulk matter fields. Their mixing with the brane fields yields the characteristic flavor structure of the model [24,28].

Table 1

Decomposition of the 45-plet of 50(10) into multiplets of the extended Standard Model gauge group G'SM = SU(3)c x SU(2)l x U(1)y x U(1)x and corresponding parity assignments. For later convenience we also give the Kaluza-Klein masses Mi n

^SM V £

Z2 ZGG ZPS M2 Z2 ZGG ZPS

(8,1)0,0 + + + 2 2 4(Ly + nT ) - - -

(3 2)-5,0 + + - 4(m2 + (n+12/2)2) (r2 r2 - - +

(3,2) 5,0 + + - 4(m2 + (n+1/2)2) (r2 r2 - - +

(1,3)0,0 + + + 22 ) - - -

(1,1)0,0 + + + 22 ) - - -

(3, 2) 1,4 + - - 4 ( (m+1/2)2 4( R2 , (n+1/2)2 ) + R22 ) - + +

(3; 1)_2 4 3 ,4 + - + 4( (m+1/2)2 4( R2 + n2) + R2) - + -

(1,1)1,4 + - + ,( (m+1/2)2 4( R2 + nL) + R2) - + -

(3,2)-1,-4 + - - ,( (m+1/2)2 4( R2 , (n+1/2)2 ) + R2 ) - + +

(3,1) | ,-4 + - + 4( (m+1/2)2 4( R2 2 _i_ nL) + R2) - + -

(1,1)-1,-4 + - + 4( (m+1/2)2 4( r2 2 _i_ nL) + R2) - + -

(1,1)0,0 + + + 22 4(Lt + nT ) - - -

The Higgs sector consists of two 16-plets, $ and $c, and four 10-plets, H1,...,H4, of bulk hypermultiplets. Each hypermultiplet contains two 4D N = 1 chiral multiplets, the first of which we denote by the same symbol as the hypermultiplet. The Higgs multiplets have even R-charge and the matter fields have odd R-charge.

The hypermultiplets H1 and H2 contain the two Higgs doublets of the supersymmetric Standard Model as zero modes, whereas the zero modes of H3 and H4 are color triplets (cf. Table 2). The zero modes of the 16-plets are singlets and color triplets,

$: Nc, Dc; $c: N,D. (1)

The color triplets Dc and D, together with the zero modes of H3 and H4, acquire masses through brane couplings.

Equal vacuum expectation values of $ and $c form a flat direction of the classical potential,

<$} = (Nc) =<N ) = ($c). (2)

Non-zero expectation values can be enforced by a brane superpotential term or by a Fayet-Iliopoulos term localized at the GG-brane where the U(1) factor commutes with the Standard Model gauge group.

Table 2 _

Decomposition and parity assignments for the bulk 16- and 10-plets of S0(10). The 16-plets 0c,0c have the same parities as 0 and 0 and conjugate quantum numbers with respect to the extended Standard Model gauge group. Only fields with all parities positive remain in the low energy theory

S0(10)

SM' (1,2) 1 2 2 > 2 (1,2) 1 2 2,2 (3,1) 1 2 3 > 2 (3,1) 1 2 3,2

Hc H Gc G

ZPS zps z22 ZPS Z22 ZPS Z22

H1 + + + - - + - -

H2 + - + + - - - +

H3 - + - - + + + -

H4 - - - + + - + +

H5 - + - - + + + -

H6 - - - + + - + +

S0(10) 16

SM' (3,2) 1 . 6 ,-1 (1,2) 1 3 (3,1) 2 , (1,1)1,-1 (3,1) 1 3 (1,1)0,-5

Q L Uc,Ec Dc,Nc

ZPS Z2GG zps Z22 ZPS Z22 ZPS z22

0 - - - + + - + +

0 + - + + - - - +

The expectation values (2) break S0(10) ^ SU(5), and therefore also the additional U(1)X symmetry, as is clear from the decomposition

16 ^ 10i © 5-3 © 15, (3)

16 ^ 10-1 © 53 © 1-5, (4)

where 15 and 1-5 correspond to Nc and N, respectively. The decomposition of the 45 vector multiplet reads

45 ^ 24o © 10-4 © 104 © 1o- (5)

The expectation values (2) generate for the 10- and 10-plets and the singlet in Eqs. (3)-(5) the bulk mass

M2 ~ g2^)2, (6)

where g6 is the 6D gauge coupling. Hence, the fields (3,2) 1 4, (3, 1)_2 4, (1,1)1,4, (1,1)0,0 and

6 , 3,

their complex conjugates contained in the vector multiplet as well as the corresponding fields in 0 and 0c obtain bulk masses from the Higgs mechanism in addition to their Kaluza-Klein masses. Since the spontaneous breaking of S0(10) preserves 6D N = 1 supersymmetry, one obtains an entire massive hypermultiplet for each set of quantum numbers.

Supersymmetry breaking is naturally incorporated via gaugino mediation [27]. The non-vanishing F-term of a brane field S generates mass terms for vector- and hypermultiplets. In the considered model, S is localized at the S0(10) preserving brane, which yields the same mass for all members of an S0(10) multiplet. For the 45 vector multiplet and the 10 and 16 hypermul-

tiplets of the Higgs sector one has

AS =j d4x d2yS 2(y^j d20 2^3 S Tr[WaWa ] + h.c.

+ / SfS(HjH1 + H2H + AStS(H3H + HJH4)

+ A SfS(0f0 + 0ct 0c J. (7)

Here Wa(V), H1,...,H4 and 0,0c are the 4D N = 1 multiplets contained in the 6D N = 1 multiplets, which have positive parity at y = 0; A is the UV cutoff of the model, which is much larger than the inverse size of the compact dimensions, A ^ 1 V. For the zero modes, the corresponding gaugino and scalar masses are given by

M 2 2 2 XV2 2 X"^2

mg =—o—, =--o—, =--o—, m0 =--^—, (8)

g A2V H1,2 A2V H3,4 A2V 0 A 2 V

where V = (2n)2R1R2 is the volume of the compact dimensions, and m = FS/A. Note that the gaugino mass is stronger volume suppressed than the scalar masses.

3. The Casimir energy

The zero-point energies of bulk fields depend on size and shape of the compact dimensions. Their sum, the Casimir energy, is a quantum contribution to the total energy density whose minimum determines the size of the compact dimensions in the lowest energy state, the vacuum. As long as supersymmetry is unbroken, the Casimir energy vanishes since bosonic and fermionic contributions compensate each other. In the following we shall evaluate the Casimir energy for the different boundary conditions which occur in T2/Z^ orbifold compactifications.

3.1. Bulk, brane and Kaluza-Klein masses

Consider a real scalar field in 6D with bulk mass M and brane mass m. As discussed in the previous section, in gaugino mediation m is due to supersymmetry breaking on a brane whereas M is generated by the Higgs mechanism in 6D. From the action

S = 2f d4xd2y0(x,y)(^-92 - dy2 + M2 + AS2(y^0(x,y) (9)

and the mode decomposition

0(x, y) = J24i(x)b(y), f d2yHi(y)Hj(y) = Sij,

one obtains

S = 2 f d4x Z 0i (x) (-dx2 + M2 + M2)0i (x) + A E 0i (x)Cij0j (x)

where Mi are the Kaluza-Klein masses and

Cij = 0)j(0). (12)

On the orbifold T2/Z|, one has for all modes (cf. Appendix A),

6(0) = J2 = , 1 , (13)

V V ^¡2n2R1R2

except for the zero mode, where 60(y) = 1/W.

The one-loop contribution to the vacuum energy density depends on the Kaluza-Klein mass matrix MKK, the universal mass M and the brane mass matrix C,

V(1) = 2lndet(-dX + mKK + M2 + ^ Cj. (14)

For small supersymmetry breaking, p2 ^ M2 + M2, the effective potential can be expanded in powers of the small off-diagonal terms of the mass matrix,

V(1) = 2 E ln(-9x + Mf + M2 + £Ci

1 (1 1 ni 6\

2A KL r a2 , ^2 , Cih a2 , u2 , 1 + ° (15)

+ T ^ ) , ... C

2\A2) ^.(-d2 + M2 + M2) 1 (-d2 + M22 + M2)

In the following we shall only keep the diagonal terms of C, which contribute to V(1) at leading order in p2.

The Casimir energy of gauge fields and gauginos can be directly obtained from the Casimir energy of a real scalar field. After appropriate gauge fixing this essentially amounts to counting the physical degrees of freedom (cf. [18]). Thus, it is enough to perform the vacuum energy calculation for a real scalar field.

3.2. Casimir energy of a scalar field

The geometry of the orbifold T2/Z2 contains as free parameters the radii R1 and R2 of the torus. The Casimir energy of a scalar field on the orbifold is then given by the quantum corrections to the corresponding effective potential. At one-loop order, this is obtained by summing over the continuous and discrete spectrum corresponding to the four flat and two compact dimensions,

vm=IE U^«++M 2

with Elm,« shorthand for the double sum and ML« denoting the Ka1uza-K1ein masses; the mass M now stands for bulk and brane mass terms.

The Kaluza-Klein masses ML«« depend on the possible boundary conditions on T2/Z2 and can be read off from the mode expansion listed in Table 1. Generically they can be written as

(m + a) (n + P)

= — [e2(m + a)2 + (n + P)2], R22

where (a, P) = (0, 0), (0,1/2), (1/2,0), (1/2, 1/2) and e2 = R^R For simplicity, we restrict our discussion to 'rectangular tori'. The general case will be discussed elsewhere [29]. Clearly, the contributions for the different boundary conditions satisfy the relations,

VM0(R1,R2) = VM0(R2,R1),

v!/2,1/2(r1,r2) = vM/2,1/2 (R2,Rx),

v°;1/2(R1 ,R2) = V1/2,° (R2,R1).

1 / 2, 0

The expression (16) for the Casimir energy is divergent. Following [18,30], we extract a finite piece using zeta function regularization,

z(s)=Me is2, f +R2 \-e2(m+a)2+(n+p)2]+m 2)

Note that, as in dimensional regularization, a mass scale Mr is introduced. The momentum integration can now be carried out and one obtains

Z(s) = T

2 (2n)4

^2 r(s - 2)| ' r(s)

\J2] 4 [e2(m + a)2 + (n + P)2] + M:

32n2R\-2s (s - 2)(s - 1)

R2 \ 2-s

[e2(m + a)2 + (n + P)2} + -2 M2 m,n\ 4

The boundary conditions of fields on the orbifold T2/Z^ are characterized by three parities. For positive (negative) parity the field is nonzero (zero) at the corresponding fixed point. For the Casimir energy only those chiral and vector multiplets are relevant which are nonzero at the fixed point where supersymmetry is broken. Hence one parity, chosen to be the first one, has to be positive. Inspection of the mode expansion in Appendix A shows that for the fields 0+-±, corresponding to (a, j) = (1/2,0), (1/2, 1/2), one has to perform the sum

EL = £ <22)

m=0 n=-œ

whereas the boundary conditions 0++±, with (a, P) = (0, 0), (0,1 /2), requires the sum

— œ œ œ ~

&0,mj2 +j2 j2

n=0 m = 1 n=-œ,

The two summations (22) and (23) are carried out in Appendix B. The result can be expressed in the following form, which is suitable for numerical analysis,

M6RlR2(11 ^ (M\\ . . M4 (3 (M\\ VM (R1,R2) = M7681TVÎ2 - lo4M ) ) - Sa°se°M-2{-4 -log(M ) )

1 M3R2 ^ cos(2npa)

8n4 R2 ^ p3 1 p=1

K3(npMR1)

- ~ cos(2npP) ~ ÎR2 m + a)2 + M!Ri

n4 R4^ p5/2 ^2Sa0SmA R^ (m + a) + 4 j

2 p=1 m=0

îr2 -, m2R2\

x K5/^2npR^J (m + a)2 + j. (24)

We have checked numerically that this expression satisfies the symmetry relations (18). As a good approximation, where the symmetries are manifest, one can derive [29]

-6RR2 ( 11 . ( M\\ . - M4 (3 ( M \

V- = —76R8jr \Î2 -log(-)) -^p^ U -log(-j

1 -3R2 ^ cos(2npa)K3(npMR1)

8n4 R2 ^ p3 1 p=1

1 M3R1 ^ cos(2npP)

8n4 R2 ^ p3

-K3(npMR2). (25)

The term a Sa0Sp0, which is independent of R1 and R2, is precisely the contribution of the 'zero' mode in (21), with a = p = m = n = 0.

The dependence of the first two terms in (24) on the regularization scale pr is a remnant of the subtraction of divergent bulk and brane cosmological terms, as in dimensional regularization [20]. The corresponding contributions to the anomalous dimensions of the 6D and 4D cosmolog-ical terms read

d M 6R1R2 d M4

Y6 = pr-^6 =--, Y4 = pr-^4 =-. (26)

dpr 768n dpr 64n

The presence of these terms demonstrates that the renormalization of the divergent energy density (24) requires counter terms for the bulk and brane cosmological terms.

In general, the Casimir energy is a sum of the four possible terms,

Vm = AVM° + BV°M1/2 + CVM20 + DVM/2A/2, (27)

where the coefficients A,...,D depend on the field content of the model and we have assumed equal masses for simplicity. The four functions ,...,V^2,l/2 are shown in Fig. 1. For small R1,2, VM° is attractive and vM/2,1/2 is repulsive, whereas the other two have mixed behavior.

In supersymmetric theories there is a cancellation between bosonic and fermionic contributions, and the expression (27) for the Casimir energy is replaced by

,0,0 T/0,0\ , u(\r0,1/2 T/0>1/2\

V = a(V-0 - V-0) + b(V-1'2 - V-

M2«0 - V-/2«0) + d{V™ -,-

+ C(VMA° - VD + d(VH2«1/2 - V-/2«1/2), (28)

where M' = VM2 + m2, with supersymmetric mass M and supersymmetry breaking mass m; the coefficients a,...,d again depend on the field content of the model. Compared to the non-supersymmetric case (27), the behavior at small R1,2 is inverted. For bulk vector- and hy-permultiplets only the 4D N = 1 vector and chiral multiplets are relevant, which couple to the brane where supersymmetry is broken.

The qualitative behavior of Fig. 1 is easily understood by evaluating explicitly the Casimir energy (25) at small radii Ri ,R2 ^ 1 /M. Expanding the Bessel function K3 for small arguments

Fig. 1. The four different contributions to the Casimir energy in units of the supersymmetric mass M. From top left to bottom right we have vM°, v}^'2'<° , VÎÎ,1/2 and vM/2'1/2 as defined in the text.

and performing the summations over p, one obtains

vM°(Ri,R2) =--— 1 - — m2r2 + •

M ( 2) 945n rA 16 1 +

+ R1 ^ R2,

vM/2,0(r1 ,R2) =

(R1 ,R2) = -

C1 1 - — M 2r2 + • 945n R5 V 16 1

31 R1 147 2 2

--U 1--M2-2 + •

30240n r5\ 124 2

C1 1 - — M 2-2 + • 30240n R5 V 124 1

1 R1 ( 21 2 2 --1 1--m2r2 + •

945n R5A 16 2

1/2,1/2

(-1,-2) =

c. 1 - — m2-2 + •

30240n -H 124 1

+ -1 ^ -2.

From these equations one immediately reads off the behavior of VM at small radii. For Ri,2 ^ 0, with Ri/R2 fixed, one obtains the behavior of the Casimir energy for 5D orbifolds.

For supersymmetric models, the mass independent terms cancel, and with M'2 - M2 = m2 the second terms in the expansion yield the inverted behavior at small R1,2.

4. Casimir energy of the orbifold model

Given the results of the previous section we can now easily evaluate the Casimir energy of the orbifold GUT model described in Section 2. At the branes, only 4D N = 1 supersymmetry is preserved. A multiplet contributes to the Casimir energy if its bosonic and fermionic degrees of freedom have different masses. This only happens if its first Z2 parity is positive so that it can couple to the singlet S at the 50(10) brane, whose non-vanishing F-term breaks 4D N = 1 supersymmetry spontaneously. Hence, from the 6D N = 1 vector multiplet only V contributes (cf. Table 1). Also for the hypermultiplets only one 4D N = 1 chiral multiplet is relevant. The corresponding chiral multiplets with positive Z2 parity are listed in Table 2.

4.1. Contribution from the vector multiplet

The expectation values (2) break 50(1°) spontaneously to SU(5). This generates the mass M for the 21 vector multiplets of the coset S0(1°)/SU(5).1 Since the Higgs mechanism preserves 6D N = 2 supersymmetry, also 21 hypermultiplets become massive. In addition all gauginos acquire a supersymmetry breaking mass mg.

From Tables 1 and 2 and from the mode decomposition we can now read off the total Casimir energy of the massive vector multiplet on T2/l?2,

Vg = 24( V00 - V«m0) + 24( V °,1/2 - Vmf2) + 2(V0M° - O

+ 16(</2,° - V^0) + 24(</2/ - V™2), (33)

where M' = ^M2 + mg. Using the expansion (29) and mg = \/(À2V) one finds at small radii, Vg = 71 R + ■■), (34)

48n A4 V2 V R3

where the dots denote terms of relative order O(MiR1,2), with Mi = mg,M,M', which have been neglected.

4.2. Contributions from hypermultiplets

The contribution of hypermultiplets to the Casimir energy again depends on the symmetry breaking, i.e., the choice of parities. Consider the 10-plets H1,2 which contain the Higgs doublets as zero mode. From Table 2 one reads off,

Vh = 8(vmh - v°,°) + 8(Vm0,H1/2 - V°,1/2)

+ 12(VoH2,0 - v1/20) + 12(v0H2,1/2 - V 1/2,1/2), (35)

1 We shall ignore the 0(1) factors for the masses of different SU(5) representations as they will not be important in the following discussion.

which, together with (29) and m2° = -Xp /(A V), yields 1 Xp2 ( R2 5 R1 \

V° = -72S?Aipv(-5R3 + 5 R3 + "} (36)

For the 10-plets H3,4 the choice of parities is different, leading to color triplets as zero modes. The corresponding Casimir energy is given by

vh=12«° - v 0,0)+12(vm,1/2 - V 0,1/2)

+ 8(V-m1H2,0 - V 1/2,0) + 8(Vm/H2,1/2 - V 1/2,1/2). (37)

Here we have neglected the supersymmetric brane masses (cf. [24]) which cancel in the behavior at small R1,2,

1 X'p2 ( R2 5 R1 \

V° =---^ 10 r2 +--1 + ••• . (38)

720nA2VV R3 2 R2 /

In the same way one obtains for the 16-plets,

T/ lA/0,0 T/0,0^ i isA/0,1/2 T/0,1/2^ i 0W1/1/2,1/2 1/1/2,1/2) V* = 2(VM - VM ) + 16)V - VM ) + 2nVM' - VM )

+ 8(V„1*2,0 - V1/2,0) + 14(Vm0*0 - V0,0), (39)

with M' = ^M2 + ~m2*, which yields for small radii

1 X"p2{ R2 R1 \

V* = -72SiApv(4Ri-11R3 + • •} (40)

The four contributions to the Casimir energy, Vg, VH, V° and V* are displayed in Fig. 2. Note that features at larger radii, like the profile in the R2-direction for V*, can be lost in the simplified expression where we keep only the leading term in p2. The behavior at small radii however is unchanged and obvious from the analytic expressions given above. Note that only V° is repulsive in all directions at small radii.

To leading order in 1 /A, the Casimir energy is determined by the contribution from hypermul-tiplets since the gaugino mass is stronger volume suppressed than the scalar masses. Depending on signs and magnitude of X, X' and X", the resulting behavior at small radii can be attractive or repulsive. As an example, we shall assume in the following X' < 0, |X'| > |X|, |X"| which yields a repulsive behavior at small radii.

5. Stabilization of the compact dimensions

In the previous section we have calculated quantum corrections to the effective potential at small radii and we have seen that, depending on the supersymmetry breaking parameters, the behavior can be attractive or repulsive. In the latter case a bulk cosmological term can lead to stabilization of the compact dimensions [18]. As we shall show in this section, stabilization can also follow from the interplay of the Higgs mechanism in 6D, which generates bulk mass terms, and supersymmetry breaking on the brane.

Consider the mass M generated by spontaneous symmetry breaking as discussed in Section 2 (cf. (6)),

M2 ~ g2(*f = g2V)2, (41)

Fig. 2. The different contributions to the Casimir energy from the bulk vector multiplet and the hypermultiplets of the Higgs sector (see text). From top left to bottom right we have the contributions from the vector multiplet, the 10-plets H12, the 10-plets H3,4, and the 16-plets Ф, Фс.

where g6 has dimension length and g4 = g6/\fV is dimensionless. For simplicity, we shall assume that M is small compared to the Kaluza-Klein masses and approximately constant.

In orbifold compactiflcations of the heterotic string expectation values ф) can be induced by localized Fayet-Iliopoulos terms. Vanishing of the D-terms then implies

V [ф° )2 = СЛ2, (42)

where С ^ 1 is a loop factor and Л is the string scale or, more generally, the UV cutoff of the model. For instance, in the 6D model of [16] one finds for the localized anomalous ^(1)'s, СЛ2 - gMp/(384n2).

Supersymmetry breaking by a brane field S, with / = Fs/Л, leads to a 'classical' vacuum energy density,

V(0) = —"J d2y j d4eA82(ур^(ф^ + Ф^ФС)) --X"Фс f

= -к" ^-C, (43)

with V = (2n)2RiR2. For к" > 0, V(0) is attractive at large radii. Note that this supersymmetry breaking mass term does not lead to a negative mass squared for Ф and Фс since these fields are

Fig. 3. Casimir energy of the 10-plets H3 and H4 together with the classical energy density from the supersymmetry breaking brane.

assumed to be stabilized by much larger supersymmetry preserving masses at the minimum. We assume that no tachyonic mass terms are generated for fields whose expectation values are not fixed by the D-term potential.

The classical energy density V(0) together with the Casimir energy V(1) = V'H yields the total energy density,

VotRlR = V(0)(Ri,R2) + V(1)(Ri,R2)

1 p2X' 288n3 ~Ar

1 1 R4 + 4R4

X" p2C An2 R1R2 '

The effective potential is attractive at large radii and, for X' < 0, i.e. m2H}4> 0, repulsive at small radii. One easily verifies that the effective potential Vtot has a stable minimum at

Rfn = V2R:

21/4 /-V 1_

!2vnV—M'

Here M is the mass given by Eq. (41) at the minimum, and we have assumed ^4(Vmin) ~ 1 /V2, as it is the case for Standard Model gauge interactions. As Fig. 3 illustrates, the total energy density Vtot is very flat for large radii.

In orbifold compactifications of the heterotic string one typically has M ~ MGUT. It is very remarkable that the interplay of gauge and supersymmetry breaking has lead to a stabilization at Rmin ~ 1 /Mgut, independent of the scale p of supersymmetry breaking. The reason is that both, the classical vacuum energy density as well as the one-loop Casimir energy are proportional to p2 which therefore does not affect the position of the minimum. Another interesting implication of the potential is that for p ^ MGUT,

AVtot(Rmin) = Vtot(œ) - Vto^R

' V2MGUT

Note that the energy density Vtot is negative at the minimum. It has to be tuned to zero by means of a brane cosmological constant. In a full supergravity treatment of stabilization also the interactions of the supersymmetry breaking brane field with the radion fields have to be taken into account.

The fact that the energy density difference Vtot(^) - Vtot(Rmin) is much smaller than MGUT has important cosmological consequences. In the thermal phase of the early universe, the volume

of the compact dimensions and, correspondingly, the value of 4D coupling constants begins to change already at temperatures T ~ V\MGUT ^ MGUT (cf. [31]).

6. Conclusions

We have calculated the one-loop Casimir energy for bulk fields on the orbifold T2/I?2. As expected, depending on the boundary conditions, the behavior at small radii can be attractive or repulsive. For the considered supersymmetric model, the Casimir energy is proportional to the scale of supersymmetry breaking. The relative strength of the couplings of the different bulk fields to the supersymmetry breaking brane field then determines whether the behavior of the total energy density is repulsive or attractive at small radii.

Quantum corrections also modify the behavior at large radii. In orbifold compactifications with U(1) gauge factors, generically Fayet-Iliopoulos terms are generated locally at the orbifold fixed points. This leads to a breaking of these U(1) gauge symmetries by the Higgs mechanism. Since the symmetry breaking is induced by local terms, the generated masses scale like M ~ 1/W with the volume of the compact dimensions.

The coupling of the bulk Higgs field to the supersymmetry breaking brane field gives rise to a classical contribution to the total energy density which scales like 1/V with the volume. Depending on the sign of the coupling, the behavior of the energy density at large radii can be attractive or repulsive. An attractive behavior at large radii, together with a repulsive behavior due to the Casimir energy at small radii, can stabilize the compact dimensions. Since the supersymmetry breaking scale factorizes, the vacuum size of the compact dimensions is determined by the remaining mass scale, the mass M generated by the Higgs mechanism, Rmin ~ 1/M ~ 1/MGUT. At the minimum the energy density Vtot is negative and has to be tuned to zero by adding a brane cosmological term.

The characteristic feature of the described stabilization mechanism is a potential well much smaller than the GUT scale, AVtot(Rmin) ~ /2MGUT < MGUT. Clearly, this has important cos-mological consequences, both for the thermal phase of the early universe as well as a possible earlier inflationary phase.

Acknowledgements

We would like to thank L. Covi, K. Fredenhagen, G. von Gersdorff, A. Hebecker, J. Moller, S. Parameswaran, M. Peloso, E. Poppitz, M. Ratz and J. Schmidt for helpful discussions. This work has been supported by the SFB-Transregio 27 "Neutrinos and Beyond" and by the DFG cluster of excellence "Origin and Structure of the Universe".

Appendix A. Mode expansion on T2/Z|

The orbifold T2/Z2 has four fixed points which we denote by yO = (0,0), yPS = (nR1/2,0), yGG = (0,nR2/2) and yfl = (nR1/2,nR2/2) (cf. [32]). The possible boundary conditions of functions on this orbifold are characterized by three parities (a,b = +, -),

0±ab(yO - y) = ±0±ab(yO + y),

0a±b(yPS - y) = ±<Pa±b(yPS + y),

0ab±(yGG - y) = ±<Pab±(yGG + y). (A.1)

It is straightforward to define an orthonormal basis on the torus. The mode expansion of functions with the boundary conditions (A.1) then reads explicitly,

0+++(x, y) =

V2n 2R1R22sn,osm

œ œ œ

So,mJ2 + E E

n=0 m=l n=-œ.

'2my i 2ny 2\

x cos--1--,

Ri R2 )

0++-(x, y) =

V2n2RlR2

œ œ œ

&0,mJ2 + E E

n=0 m = l n=-œ.

0+2+m-2n+l)(x)

'2my i (2n + l)y2 \ x cos|--1--,

Ri R2 )

0+-+(x, y) =

V2n2RlR2

. m=0n=-œ.

0+2m^l,2n)(x)

/(2m + l)yi i (2n)y2\

0+—(x, y ) =

V2n2RlR2

. m=0n=-œ.

'(2m + l)yi (2n + l)y2 x cos| ----+

0-++(x, y) =

. m=0n=-œ.

0-2++l,2n+l)(x)

V2n2RlR2

. /(2m + l)yi (2n + l)y2 \ x sin--1--,

V Ri + R2 )

0-+-(x, y) =

V2n2RlR2

. m=0 n=-œ.

0-2++l,2n)(x)

'(2m + l )y i 2ny 2 x sin -::--+

0—+(x, y) =

V2n2RlR2

— œ œ œ

5o,^E +E E

n=0 m = l n=-œ.

0(2m,2n+l)

'2my i (2n + l)y2 x sin| —--+

0---(X, y ) =

œ œ œ

5o,^E +E E

n=0 m = l n=-œ.

V2n2RlR2

.^(2my i (2n)y2\

~rT + R2 / '

0{mn)(x)

(A.2a)

(A.2b)

(A.2c)

(A.2d)

(A.2e)

(A.2f)

(A.2g)

(A.2h)

Appendix B. Evaluation of Casimir sums

Our evaluation of the Casimir double sums requires two single sums which we shall now consider. The first sum reads

F(s; a, c) -^-t— . (B.l)

( + a)2 + c2]s ' ;

This is a series of the generalized Epstein-Hurwitz zeta type. The result can be found in [30] and is given by

F(s; a,c) = C—t {-irrm + S) c~2mzH(-2m, a) + ^ ~ 2 ) -

r(j) ^ m! 2V(s)

2ns TO

+ — cl/2-sY ps-l/2 cos(2npa)Ks-i/2(2npc), (B.2)

r(s) ^-f p=i

where ZH(s,a) is the Hurwitz zeta-function. Note that this is not a convergent series but an asymptotic one. In the following it will be important that ZH(—2m, 0) = ZH(—2n, 1/2) = 0 for m e N and n e N0. In our case, the first sum in F(s; a, c) thus reduces to a single term. For a = 1/2 the sum vanishes, and for a = 0 only the first term contributes; with ZH (0, 0) = 1/2 one obtains c-2s/2.

The second, related sum is given by

F(s; a, c) = V -^-t— . (B.3)

( ; , ) ^ [(m + a)2 + c2]s J

l=—to

Using the two identities (m e N)

ZH(-2m,a) = -ZH(-2m, 1 - a), (B.4)

F(s; a, c) = F(s; a, c) + F(s; 1 — a, c), (B.5) one easily obtains, in agreement with [18],

' ' ' " "2

F(s' a, c) = |c|1-2s

/ 1\ ^ -1 ^ - 2j + 4 E cos(2npa)(nplcQs 2 K- i(2np|c|)

These two sums provide the basis for our evaluation of the Casimir sums. B.1. Casimir sum (I) on T2/Z2 We first consider the summation

EL, £. (B,7)

m=0n= to

In this case the Casimir energy (cf. (21)) is obtained from

J2 J2 [e2(m + a)2 + (n + j3)2 + k2](B.8)

m=0n= to

where we have shifted s ^ s + 2 and defined k2 = R2M2. Using the expression for F(s; a, c), we can perform the sum over n,

J2 E [e2(m + «)2 + (n + P)2 + k2] '

m=0 n= to

— ( _ 1) to

= ^^Trr E {e2(m+«)2+k 2)1/2-s

+ Vcos(2npP)Y(np)s- 1Ue2(m + a)2 + k2)2

p=1 m=0

x 1 (2np^fe2(m + a)2 + k2) = f1(s) + f2(s). (B.9)

Let us consider fi(s) first. The sum over m can be performed with the help of F(s; a, c),

F( — 1) to

f1(s) = —^ J2(e2(m + a)2 + k2)1/2-s

_F(s - 1/2) , n k2-2s

= ,/n—--—~k zh(0,u) +--

V r(s) ( , 2(s - 1) e

+ —s)e-SK1-S Eps-1cos(2npa)Ks-^2n^Kjj- (B.10)

Recalling the shift in s, we can now write Z(s) (21) as

1 /4\-s p2s+4 i ^ — (s - 1/2) , n k

Z(s) = ^ -to -T^a^n - ) k Zh(0,o.) +

32n2\R2J s(s + 1) v r(s) S 2(s - 1) e

2ns s x s to

+--e-sK1-s V ps-1 cos(2npa)Ks-i \ 2np,

r(s) \ \e

4 ,--+TO TO _ 1

+ ^T E cos(2npP) V (np)s-1Ue2(m + a)2 + k2)1

— (s) in p = 1 m = 0

x Ks-1 (2npVe2(m + a)2 + k2)|. (B.11)

Now we have to differentiate with respect to s and set s = -2. Since r(-2) = to, the derivative has only to act on r(s) if the corresponding term is inversely proportional to T(s). Performing the differentiation, using

ds r(s)

_ —'(s) s=-2 —(s)2

= 2, (B.12)

and Ka(z) = K—a(z), and substituting again e = R2/R1 and k2 = R2M/2, we finally obtain for the Casimir energy,

^(I) _ M5r2

M6RiR2/11

G2-log(

Zn(0,a) + ,

120n 768n V12

1 M3R2 ^ cos(2npa)

8n4 ^Y.-1^ K3(nPMR1) i p=1

^ t ^ t( R2 (m + a)2 + M2Ri)

--2 , -, m2R2\

x K5/2[2np—J (m + a)2 + 1

(B.13)

The second term corresponds to a finite part of the 6D cosmological constant. The dependence on the regularization scale \xr shows that an infinite contribution has been subtracted.

B.2. Casimir sum (II) on T2/Z2

The second relevant summation is

œ œ œ

&m,oj2 +j2 j2

. (B.14)

n=0 m = 1 n=-œ _

For the corresponding boundary conditions one has a = 0. The Casimir sum can then be written

œ œ œ

[e2m2 + (n + P)2 + k2] S

n=0 m=1 n=-œ

œ œ œ

n=0 m=0 n=-œ n=-œ.

[e2m2 + (n + P)2 + k2] s

(B.15)

R2 2 2

where we again shifted s ^ s + 2 and set -R2 M = k . The double sum is the sum (I) which we have already evaluated. Using

— 1 to

£ [(n + P)2 + K2]-s = £[(n + 1 - P)2 + k2]-s,

(B.16)

one easily finds for the remaining piece2

f3(s) = -J2[(n + 1 - P)2 + k2] s

= -k-2szn(0,1 - P) -Vn^S—^ k 1-2s

2 Note that ZH(0, 1) =

-1/2, and ZH(-2m, 1) = 0for m e N.

- —* 1/2-SI>i-1/2cos(2np(1 - ß)Ks-1/2(2npK). (B.17)

Differentiating the corresponding contribution to Z(s), setting s = -2, and substituting k = R2M/2 yields the Casimir energy,

<ß(,I) = VT» + £(2 - 2log(£)){,(«W - ß) - 2^MR

- ¡14R4 £ (MR fK^pMR) (B.18)

R2 p=1 P V 7

The first of the additional terms does not depend on the radii. It represents a finite contribution to the brane cosmological term. The dependence on the regularization scale again shows that a divergent contribution has been subtracted.

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