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Gravity induced symmetry breaking and ground state of local supersymmetric GUTs
Volume 121B, number 1
PHYSICS LETTERS
20 January 1983
GRAVITY INDUCED SYMMETRY BREAKING AND GROUND STATE OF LOCAL SUPERSYMMETRIC GUTS Pran NATH, R. ARNOWITT and A.H. CHAMSEDDINE Department of Physics, Northeastern University, Boston, MA 02113, USA
Received 20 August 1982
It is shown that local supersymmetry does not necessarily imply existence of anti-de Sitter vacua in grand unified theories. A locally supersymmetric SU(5) grand unified example is presented in which the supergravity interactions make the SU(3) × SU(2) X U(1) vacuum be the absolute minimum of the effective potential for a class of solutions whose Higgs and matter multiplets have no vacuum expectation values in the global limit (~ ~ 0). Supersymmetry breaking occurs at a scale m s ~ 1 TeV through super Higgs effect and degeneracy is broken by O(ms4) rather than O(K2M6).
Supersymmetric grand unified theories represent an interesting and important alternative to standard grand unified models. Theories based on global supersymmetry [ 1] shed light on the mass hierarchy problems, correctly account for sin20 w and the mass ratio m b / m ~ as well as yielding a proton decay rate rp ~ 1031 yr. Further, if supersymmetry breaks at a mass scale m s ~ 1 TeV, a whole new array of physics is predicted, some of which should become accessible in future experiments. However, breaking global supersymmetry is not easy for theories of this type, for if supersymmetry does not break at the tree level, it will not break perturbatively at the loop level [2]. Thus while internal symmetry [e.g., SU(5)] may spontaneously break, the different possible vacua [e.g., SU(5), SU(4) × U(1), and SU(3) × SU(2) × U(1)] will remain degenerate. In recent letters by the authors [3] and by Weinberg [4], an alternate point of view was taken in which global supersymmetry was promoted to the local supersymmetry o f N = 1 supergravity [5] and coupled to left-handed chiral multiplets [6] and a gauge vector multiplet [7]. It was shown there [3,4] that models could be constructed in which the gravitational interactions of supergravity can remove the degeneracy of the different vacua. Further, it was shown in ref. [3] that by including a super-Higgs term [8,6] in the superpotential, one could induce a spontaneous 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
breaking of supersymmetry which simultaneously breaks the electro-weak SU(2) × U(1) symmetry down to SU(3) c × Uy(1) at the same mass scale ms, allowing the evaluation m s ~ 300 GeV. One feature of these models is that if the cosmological constant of any of the vacua is adjusted to be zero, the other vacua correspond to anti-de Sitter spaces at lower energies. While the decay into these lower vacua by finite size bubble formation is most likely forbidden [4,9], we view this as an "unattractive" feature of the model. In this paper we will show that the previous result that the Minkowski vacuum (i.e., vacuum with zero cosmological constant) lies higher than the other vacua is not a general consequence of locally supersymmetric GUT models. We will then illustrate this general result with a special model (arising from a generalization of a global superpotential introduced by Witten [1 ]) where SU(4) × U(1) lies above the SU(3) × SU(2) × U(1) vacuum. [The SU(5) vacuum is forbidden for this model after spontaneous breaking.] The higher vacuum would thus correspond to a de Sitter space and naturally decay by bubble formation to the Minkowski vacuum. Models of this type are of particular interest not only because of their inherent stability, but because they will produce a rather strikingly different cosmological picture of the early Universe. We start by writing the scalar potential corre33
Volume 121B, number 1
PHYSICS LETTERS
sponding to the coupling of chiral scalar multiplets and a gauge vector multiplet to N = 1 supergravity in the special case of the normalized scalar field kinetic energy [3,4]: 1 2 ZaZa)(GaG + + A - ~K2[g12) V = ~1 e exp(~t<
1 + + ~ [ea(zA ' (TaZ)A)] 2,
(1)
where z A =-A A + iB A are all the scalar fields present in the theory, K2 - 87rG, T ~ are the gauge group generators and ea their associated charges, g(zA) is the superpotential and
G A = ag/az a + lt~ 2z~g(z).
(2)
The second term in eq. (1), arising from the elimination of the D auxiliary field of the vector multiplet is positive definite and vanishes at the minimum of V for all examples considered here. We will thus ignore this term in the future. The condition that extremizes eq. (1) on the real (PC conserving) manifold reads TABG B = 0,
(3)
where TAB = 02g/ag AaZB + ½t~Z(zAag/azB + zBag/az A) 1 4 + -~t~ ZAZBg -- t~26ABg.
(4)
It is clear that if det TAB 4: 0, eq. (3) implies G A = 0 which leads to a set of solutions z(~) for the minimum. Then at the minimum eq. (1) implies Vmin(Z(i) ) = -~K2e exp(½ t~2z(~) 2)ig(z(i))t2 <~O. (5) Thus, if by adding a constant to g(z) one adjusts g(z (I)) to vanish, then all the other minima for i 1> 2 will lie lower, characteristically at a distance ~ r 2M6 lower, where M is the GUT mass. We now show under what circumstances it is possible to avoid this result. The condition that all the G A vanish implies that supersymmetry has not been broken. (This may be seen from the fact that the supersymmetry transformation for the left spinor components ×L reads [6] - ~ exp(~K ZBeB)GAe L + and so vanishes at the minimum.) It is therefore possible to evade the above theorem on Vmi n if two conditions are met. Writing g(z) = gl(z) + g2(z) we require (1) gl(z (i)) vanish for Z (i) at least in the global limit K ~ 0. [This eliminates the ~2M6 splitting and re-establishes the degeneracy of the vacua to O(K 2).] (2) 34
20 January 1983
We add to gl the super-Higgs potential [8,6] g2(Z) = m2(Z + B),
(6)
where m and B are constants and Z a scalar field component o f an additional (real) scalar multiplet, g2(Z) breaks supersymmetry so that TAB now possesses one zero eigenvalue. (Indeed, the fact that det TAB vanishes is in general the signal that supersymmetry has broken.) Thus, if one were to diagona!ize the matrix TAB , there would be one GA, call it GO, which will not vanish, while the remaining G A still vanish. At the minimum eq. (1) now reads in the diagonalized representation Vmi n =~elexP(½t~2z(iA) 2)[G2(z(i) ) --~t< 3 2 ~(z(i))12]. (7) It is clear that the right-hand side of eq. (7) is no longer negative definite. On account of condition (1)above the splitting must be O(• 4) and vanish as m ~ 0. One finds in fact that the different vacua are separated by ~K4m 8 -- m 4, where m s ~ 1 TeV is the scale of supersymmetry breaking. Clearly which minima lies lowest now depends upon the detailed nature o f the model considered. As an example, we consider a generalization of a superpotential considered by Witten [1 ] where gl(ZA) = ~0X(M 2 - tr ~2) + ~ltr ~22A + X4Yt r ~A
+ x2n'x(Y.Xy + 3oL~XyY)ny + X3UH'xI-1x + fiyM~xM] XyH; + gi]Mi xyMj zuItVexyzuv,
(8)
where we chose ~ = X4/X 1 to guarantee that the Higgs doublet remains light + 1. Here ,~Xy and AXyare two 24 representations of SU(5),H x and/_/x are 5 and 5 Higgs fields, X, Y and U are singlet fields and_Mx and Mxy are the scalar components of the quark 5 and 10 multiplets (i and j are generation indices). In the limit with g2 set to zero, v has the minima with H x = H x = X = U=M'x =MXy = AXy = 0 and (i)
~oXy = ( M / N / ~ ) ( 6 X y
-- 56x5~5y),
(9)
,1 As recently pointed out by Bucella et al. [10], there are additional solutions for the global theory with non-zero Higgs and matter vacuum expectation values (some which even break color). A full treatment of the theory of eq. (8) would require an examination of these possibilities as well.
Volume 121B, number 1
PHYSICS LETTERS and the el, 2 are determined as follows
YO = 3M/x/20. (ii)
~-,oXy = (M/x/~) [2¢5Xy -- 5(~x464y + 8Sx~Y5)],
The two solutions correspond to a residual symmetry of SU(4) X U(1), and SU(3) X SU(2) X U(1) respectively. Both have vanishing gl(z(~ )) and hence obey condition (1) above. [This arises due to the linear dependence on A and X in eq. (8).] The super-Higgs potential eq. (6) considered by itself with gl set to zero leads to the following minima [3].
KB(o ) = ---a(2x/2 - X/~),
a=+l,
(10)
B 0 has been chosen, for convenience, so that Vmi n = 0. Note Z(0 ) and B(0 ) are scaled by the Planck mass. We now consider the full theory where g = gl + g2Clearly it should be possible to solve the minimization conditions eq. (3) in a perturbative power series around the zeroth order solutions eqs. (9) and (10). For this purpose we expand B and all the vacuum expectation values z A in a perturbation series in ~. This expansion takes the form ZA = z(A0) + :~)) where for z h 4: Z, z~,0) is zeroth order in K, and begins at first order. For Z and B expansions eq. (10) shows that expansions begin with K-1 so that z(1 ) and B(1 ) = 13(1)/K begin at O(1). We find then for the solution of eq. (3) generated by case (i) of eq. (9) the following result to lowest order:
Case {i): Hx :Hx:U=O
e 1 = (e2/8~)(1 -X/3)F1,
e 2 = (3a/2X/1-O)(X1/X2)e.
(9 con'd)
YO = (M/N/~O)~'I/X4"
KZ(o ) = a(x/~ - x/6),
20 January 1983
,
X = -lx/2(ams/XO)Fl,
(11)
(13)
Z(1 ) = -~Vt2(1 --N/r3)aK3m 4 X [F2(8~2) -1 + 9~,2Gk4)-4], F 1 -= 1 + 9X2(X4) -2,
These solutions break SU(5) to SU(4) X U(1) and after the breaking Vmi n becomes to leading order Vmin = 1E0 ((N/~ - i )3[ F 1 82( ~ - 0) 2 - 1 +T6XI(~, 9 2 4) -4 ]m 4s + x/~am4/3(l) },
(20)
where E 0 - exp(4 - 2x/~). Case (ii): For case (ii) o f e q . (9) we find the following solution in the presence of the super-Higgs effect: Hx
=H x :U=0,
(21)
X = -(1/2x/2)(ams/Xo)F2,
(22)
y = (M/%/~)(~.11~.4) X {1 + (1 -v~)e2[(8X02)-iF2 + (2X2) -1] },
(14) (15)
where e = m2/M = ms/M. The solutions for EXy and AXy are of the form
ZXy = (M/x/~)(1 + el)(6Xy -- 56x585y),
(16)
AXy = ( M / ~ O )e2(~Xy - 5¢5x565y),
(17)
(23)
Z(1) = - [( 1 - X/3)/Zv r~] a K3m4 X [(8X2)-lF2 + X2(60X4)-l], F 2 -= 1 + X2/30X 2.
(24) (25)
ZXy and AXy are given by
ZXy : (M/v~-0)(1 + el)[28Xy -- 5(~x464y + ~x5~5y)] , (26)
AXy = (Ml~O)e212~Xy - 5(~x4~4y + ~x5~5y)] ,(27) e I : (e218~k2)(a -X/3)F2;
e2 =
(28)
(12)
y = (3M/v~O)(;kI/X4) X(1 + (1 - Vt3)e2[Fl(8~,2) -1 + ½(~4) -2] },
(18, 19)
In this case SU(5) is broken to SU(3) X SU(2) X U(1) and Vmi n becomes to lowest order 2 4 - 1 ]ms4 Vmi n = ½E0{(~/r3-3)[(8X0)2 - 1 F 22 + ~.1(60~) + x/~am413(1) }.
(29)
If we now adjust the constant/3(1 ) so that Vmi n for the SU(3) X SU(2) X U(1) solution is zero (i.e., no cosmological constant), one obtains
v~am4~(1) = - ( x / ~ - -~) 2 1F 22 + X2(60X44)-l]m 4. X [(8XO)-
(30) 35
Volume 121B, number 1
PHYSICS LETTERS
Eqs. (20), (29) and (30) then imply the following result for the two minima of the effective potential _1 4 s 3 V[SU(4) X U(1)] - i m s E o ~ (%/r~ _ ~) X [X0-2X~-2(1 + ~X4/XI ) 62 2 +2X2(X4)-4] , V[SU(3) X SU(2) × U(1)] = 0.
(31) (32)
From eqs. (31) and (32)we see that while the SU(3) × SU(2) X U(1) symmetric vacuum is minkowskian, the other SU(4) X 15(1) symmetric vacuum is de Sitter and lies higher. This result is in sharp contrast to the analysis of eq. (5) [3,4] where the vacua other than the minkowskian vacuum were necessarily anti-de Sitter. Thus unlike the case o f eq. (5) where one needed the C o l e m a n - D e L u c c i a - W e i n b e r g arguments for stability of the minkowskian vacuum against bubble formation, the minkowskian vacuum of eq. (32) is automatically stable. This example explicitly illustrates that the problem of eq. (5) is not a general consequence o f local supersymmetric grand unification models. We conclude with a few additional comments concerning this model. There are no light scalar fields in the model as they all acquire a mass m s through the 1 2+ term ~msZ~hZh in the potential. An interesting and important feature o f the model is the fact that having started with the global solutions eq. (9) that preserve SU(3) c, the color gauge invariance is not broken by the supergravity interactions to O(~), i.e., (Hx) = (Hx) = 0 for x = l , 2, 3. (It is possible that color is broken at O(g 2). Then the color gluons would gain a mass of size eg2m4/M ~ 10 - 2 eV.) Finally we mention a unique feature of this model that the different vacua of eq. (26) are separated by O(m 4) where m s ~ 1 TeV. Theories of this type may thus have interesting cosmological properties as the universe will not choose the physical vacuum until relatively late, i.e., until the temperature o f the early universe has dropped to k T <~m s. This is strikingly different from the models of refs. [3,4] where the energy gaps of the different vacua are scaled by (k2M6) 1/4 ~ 1014 GeV. Our formulation of locally supersymmetric grand unification contains a new non-gravitational mass
36
20 January 1983
scale m. The theory as currently formulated relates m ~ 1011 GeV to the SU(2) X U(1) breaking mass ,2 m s ~ 103 GeV by m s = r m 2. One possibility is that this super-Higgs mass scale itself arises by dynamical breaking at the loop level, which would then make m ~ r2M3[ln(M2/la2)] 1/2,.. 1011 GeV [11]. Research for this paper was supported in part by the National Science Foundation under Grant No. PHYS0-08333. ¢2 One may break SU(2) × U(1) gauge invariance to Uy by adding a term XsH'xAXyHy to the superpotential. The mechanism for breaking SU(2) × U(1) is the same as discussed in ref. [3]. However, for this model the SU(3) c × Uy(1) vacuum does not lie below the SU(3) × SU(2) × U(1) but can still lie below SU(4) × U(1). Models where SU(3) c × Uy(1) vacuum lies lowest can most likely be constructed.
References [1] E. Witten, Nucl. Physics B177 (1981) 477;B185 (1981) 513; M. Dine, W. Fischler and M. Srednicki, Phys. Lett. 104B (1981) 199; S. Dimopoulous and H. Georgi, Nucl. Physics B193 (1981) 150; N. Sakai, Z. Phys. C l l (1981) 153; D.V. Nanopoulos and K. Tamvakis, CERN preprint TH3227; Phys. Lett. l13B (1982) 151. [2] S. Weinberg, Phys. Lett. 62B (1976) 111. [3] A.H. Chamseddine, R. Arnowitt and P. Nath, Northeastern University preprint NUB #2559 (1982), submitted to Phys. Rev. Lett. [4] S. Weinberg, Phys. Rev. Lett. 48 (1982) 1176. [5] D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13 (1976) 3214; S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335. [6] E. Cremmer et al., Nucl. Phys. B147 (1979) 105. [7] K.S. Stelle and P.C. West, Nucl. Phys. B145 (1978) 175. [8] J. Polony, Budapest Preprint KFKI-1977-93 (1977), unpublished. [9] S. Coleman and F. DeLuccia, Phys. Rev. D21 (1980) 3305. [10] F. Bucella, J.P. Derendinger, S. Ferrara and C.A. Savoy, Phys. Lett. l15B (1982) 375. [ 11 ] R. Arnowitt, A.H. Chamseddine and Pran Nath, Electroweak interaction mass scale from loop corrections in locally supersymmetric GUTS, in preparation.
PHYSICS LETTERS
20 January 1983
GRAVITY INDUCED SYMMETRY BREAKING AND GROUND STATE OF LOCAL SUPERSYMMETRIC GUTS Pran NATH, R. ARNOWITT and A.H. CHAMSEDDINE Department of Physics, Northeastern University, Boston, MA 02113, USA
Received 20 August 1982
It is shown that local supersymmetry does not necessarily imply existence of anti-de Sitter vacua in grand unified theories. A locally supersymmetric SU(5) grand unified example is presented in which the supergravity interactions make the SU(3) × SU(2) X U(1) vacuum be the absolute minimum of the effective potential for a class of solutions whose Higgs and matter multiplets have no vacuum expectation values in the global limit (~ ~ 0). Supersymmetry breaking occurs at a scale m s ~ 1 TeV through super Higgs effect and degeneracy is broken by O(ms4) rather than O(K2M6).
Supersymmetric grand unified theories represent an interesting and important alternative to standard grand unified models. Theories based on global supersymmetry [ 1] shed light on the mass hierarchy problems, correctly account for sin20 w and the mass ratio m b / m ~ as well as yielding a proton decay rate rp ~ 1031 yr. Further, if supersymmetry breaks at a mass scale m s ~ 1 TeV, a whole new array of physics is predicted, some of which should become accessible in future experiments. However, breaking global supersymmetry is not easy for theories of this type, for if supersymmetry does not break at the tree level, it will not break perturbatively at the loop level [2]. Thus while internal symmetry [e.g., SU(5)] may spontaneously break, the different possible vacua [e.g., SU(5), SU(4) × U(1), and SU(3) × SU(2) × U(1)] will remain degenerate. In recent letters by the authors [3] and by Weinberg [4], an alternate point of view was taken in which global supersymmetry was promoted to the local supersymmetry o f N = 1 supergravity [5] and coupled to left-handed chiral multiplets [6] and a gauge vector multiplet [7]. It was shown there [3,4] that models could be constructed in which the gravitational interactions of supergravity can remove the degeneracy of the different vacua. Further, it was shown in ref. [3] that by including a super-Higgs term [8,6] in the superpotential, one could induce a spontaneous 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
breaking of supersymmetry which simultaneously breaks the electro-weak SU(2) × U(1) symmetry down to SU(3) c × Uy(1) at the same mass scale ms, allowing the evaluation m s ~ 300 GeV. One feature of these models is that if the cosmological constant of any of the vacua is adjusted to be zero, the other vacua correspond to anti-de Sitter spaces at lower energies. While the decay into these lower vacua by finite size bubble formation is most likely forbidden [4,9], we view this as an "unattractive" feature of the model. In this paper we will show that the previous result that the Minkowski vacuum (i.e., vacuum with zero cosmological constant) lies higher than the other vacua is not a general consequence of locally supersymmetric GUT models. We will then illustrate this general result with a special model (arising from a generalization of a global superpotential introduced by Witten [1 ]) where SU(4) × U(1) lies above the SU(3) × SU(2) × U(1) vacuum. [The SU(5) vacuum is forbidden for this model after spontaneous breaking.] The higher vacuum would thus correspond to a de Sitter space and naturally decay by bubble formation to the Minkowski vacuum. Models of this type are of particular interest not only because of their inherent stability, but because they will produce a rather strikingly different cosmological picture of the early Universe. We start by writing the scalar potential corre33
Volume 121B, number 1
PHYSICS LETTERS
sponding to the coupling of chiral scalar multiplets and a gauge vector multiplet to N = 1 supergravity in the special case of the normalized scalar field kinetic energy [3,4]: 1 2 ZaZa)(GaG + + A - ~K2[g12) V = ~1 e exp(~t<
1 + + ~ [ea(zA ' (TaZ)A)] 2,
(1)
where z A =-A A + iB A are all the scalar fields present in the theory, K2 - 87rG, T ~ are the gauge group generators and ea their associated charges, g(zA) is the superpotential and
G A = ag/az a + lt~ 2z~g(z).
(2)
The second term in eq. (1), arising from the elimination of the D auxiliary field of the vector multiplet is positive definite and vanishes at the minimum of V for all examples considered here. We will thus ignore this term in the future. The condition that extremizes eq. (1) on the real (PC conserving) manifold reads TABG B = 0,
(3)
where TAB = 02g/ag AaZB + ½t~Z(zAag/azB + zBag/az A) 1 4 + -~t~ ZAZBg -- t~26ABg.
(4)
It is clear that if det TAB 4: 0, eq. (3) implies G A = 0 which leads to a set of solutions z(~) for the minimum. Then at the minimum eq. (1) implies Vmin(Z(i) ) = -~K2e exp(½ t~2z(~) 2)ig(z(i))t2 <~O. (5) Thus, if by adding a constant to g(z) one adjusts g(z (I)) to vanish, then all the other minima for i 1> 2 will lie lower, characteristically at a distance ~ r 2M6 lower, where M is the GUT mass. We now show under what circumstances it is possible to avoid this result. The condition that all the G A vanish implies that supersymmetry has not been broken. (This may be seen from the fact that the supersymmetry transformation for the left spinor components ×L reads [6] - ~ exp(~K ZBeB)GAe L + and so vanishes at the minimum.) It is therefore possible to evade the above theorem on Vmi n if two conditions are met. Writing g(z) = gl(z) + g2(z) we require (1) gl(z (i)) vanish for Z (i) at least in the global limit K ~ 0. [This eliminates the ~2M6 splitting and re-establishes the degeneracy of the vacua to O(K 2).] (2) 34
20 January 1983
We add to gl the super-Higgs potential [8,6] g2(Z) = m2(Z + B),
(6)
where m and B are constants and Z a scalar field component o f an additional (real) scalar multiplet, g2(Z) breaks supersymmetry so that TAB now possesses one zero eigenvalue. (Indeed, the fact that det TAB vanishes is in general the signal that supersymmetry has broken.) Thus, if one were to diagona!ize the matrix TAB , there would be one GA, call it GO, which will not vanish, while the remaining G A still vanish. At the minimum eq. (1) now reads in the diagonalized representation Vmi n =~elexP(½t~2z(iA) 2)[G2(z(i) ) --~t< 3 2 ~(z(i))12]. (7) It is clear that the right-hand side of eq. (7) is no longer negative definite. On account of condition (1)above the splitting must be O(• 4) and vanish as m ~ 0. One finds in fact that the different vacua are separated by ~K4m 8 -- m 4, where m s ~ 1 TeV is the scale of supersymmetry breaking. Clearly which minima lies lowest now depends upon the detailed nature o f the model considered. As an example, we consider a generalization of a superpotential considered by Witten [1 ] where gl(ZA) = ~0X(M 2 - tr ~2) + ~ltr ~22A + X4Yt r ~A
+ x2n'x(Y.Xy + 3oL~XyY)ny + X3UH'xI-1x + fiyM~xM] XyH; + gi]Mi xyMj zuItVexyzuv,
(8)
where we chose ~ = X4/X 1 to guarantee that the Higgs doublet remains light + 1. Here ,~Xy and AXyare two 24 representations of SU(5),H x and/_/x are 5 and 5 Higgs fields, X, Y and U are singlet fields and_Mx and Mxy are the scalar components of the quark 5 and 10 multiplets (i and j are generation indices). In the limit with g2 set to zero, v has the minima with H x = H x = X = U=M'x =MXy = AXy = 0 and (i)
~oXy = ( M / N / ~ ) ( 6 X y
-- 56x5~5y),
(9)
,1 As recently pointed out by Bucella et al. [10], there are additional solutions for the global theory with non-zero Higgs and matter vacuum expectation values (some which even break color). A full treatment of the theory of eq. (8) would require an examination of these possibilities as well.
Volume 121B, number 1
PHYSICS LETTERS and the el, 2 are determined as follows
YO = 3M/x/20. (ii)
~-,oXy = (M/x/~) [2¢5Xy -- 5(~x464y + 8Sx~Y5)],
The two solutions correspond to a residual symmetry of SU(4) X U(1), and SU(3) X SU(2) X U(1) respectively. Both have vanishing gl(z(~ )) and hence obey condition (1) above. [This arises due to the linear dependence on A and X in eq. (8).] The super-Higgs potential eq. (6) considered by itself with gl set to zero leads to the following minima [3].
KB(o ) = ---a(2x/2 - X/~),
a=+l,
(10)
B 0 has been chosen, for convenience, so that Vmi n = 0. Note Z(0 ) and B(0 ) are scaled by the Planck mass. We now consider the full theory where g = gl + g2Clearly it should be possible to solve the minimization conditions eq. (3) in a perturbative power series around the zeroth order solutions eqs. (9) and (10). For this purpose we expand B and all the vacuum expectation values z A in a perturbation series in ~. This expansion takes the form ZA = z(A0) + :~)) where for z h 4: Z, z~,0) is zeroth order in K, and begins at first order. For Z and B expansions eq. (10) shows that expansions begin with K-1 so that z(1 ) and B(1 ) = 13(1)/K begin at O(1). We find then for the solution of eq. (3) generated by case (i) of eq. (9) the following result to lowest order:
Case {i): Hx :Hx:U=O
e 1 = (e2/8~)(1 -X/3)F1,
e 2 = (3a/2X/1-O)(X1/X2)e.
(9 con'd)
YO = (M/N/~O)~'I/X4"
KZ(o ) = a(x/~ - x/6),
20 January 1983
,
X = -lx/2(ams/XO)Fl,
(11)
(13)
Z(1 ) = -~Vt2(1 --N/r3)aK3m 4 X [F2(8~2) -1 + 9~,2Gk4)-4], F 1 -= 1 + 9X2(X4) -2,
These solutions break SU(5) to SU(4) X U(1) and after the breaking Vmi n becomes to leading order Vmin = 1E0 ((N/~ - i )3[ F 1 82( ~ - 0) 2 - 1 +T6XI(~, 9 2 4) -4 ]m 4s + x/~am4/3(l) },
(20)
where E 0 - exp(4 - 2x/~). Case (ii): For case (ii) o f e q . (9) we find the following solution in the presence of the super-Higgs effect: Hx
=H x :U=0,
(21)
X = -(1/2x/2)(ams/Xo)F2,
(22)
y = (M/%/~)(~.11~.4) X {1 + (1 -v~)e2[(8X02)-iF2 + (2X2) -1] },
(14) (15)
where e = m2/M = ms/M. The solutions for EXy and AXy are of the form
ZXy = (M/x/~)(1 + el)(6Xy -- 56x585y),
(16)
AXy = ( M / ~ O )e2(~Xy - 5¢5x565y),
(17)
(23)
Z(1) = - [( 1 - X/3)/Zv r~] a K3m4 X [(8X2)-lF2 + X2(60X4)-l], F 2 -= 1 + X2/30X 2.
(24) (25)
ZXy and AXy are given by
ZXy : (M/v~-0)(1 + el)[28Xy -- 5(~x464y + ~x5~5y)] , (26)
AXy = (Ml~O)e212~Xy - 5(~x4~4y + ~x5~5y)] ,(27) e I : (e218~k2)(a -X/3)F2;
e2 =
(28)
(12)
y = (3M/v~O)(;kI/X4) X(1 + (1 - Vt3)e2[Fl(8~,2) -1 + ½(~4) -2] },
(18, 19)
In this case SU(5) is broken to SU(3) X SU(2) X U(1) and Vmi n becomes to lowest order 2 4 - 1 ]ms4 Vmi n = ½E0{(~/r3-3)[(8X0)2 - 1 F 22 + ~.1(60~) + x/~am413(1) }.
(29)
If we now adjust the constant/3(1 ) so that Vmi n for the SU(3) X SU(2) X U(1) solution is zero (i.e., no cosmological constant), one obtains
v~am4~(1) = - ( x / ~ - -~) 2 1F 22 + X2(60X44)-l]m 4. X [(8XO)-
(30) 35
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Eqs. (20), (29) and (30) then imply the following result for the two minima of the effective potential _1 4 s 3 V[SU(4) X U(1)] - i m s E o ~ (%/r~ _ ~) X [X0-2X~-2(1 + ~X4/XI ) 62 2 +2X2(X4)-4] , V[SU(3) X SU(2) × U(1)] = 0.
(31) (32)
From eqs. (31) and (32)we see that while the SU(3) × SU(2) X U(1) symmetric vacuum is minkowskian, the other SU(4) X 15(1) symmetric vacuum is de Sitter and lies higher. This result is in sharp contrast to the analysis of eq. (5) [3,4] where the vacua other than the minkowskian vacuum were necessarily anti-de Sitter. Thus unlike the case o f eq. (5) where one needed the C o l e m a n - D e L u c c i a - W e i n b e r g arguments for stability of the minkowskian vacuum against bubble formation, the minkowskian vacuum of eq. (32) is automatically stable. This example explicitly illustrates that the problem of eq. (5) is not a general consequence o f local supersymmetric grand unification models. We conclude with a few additional comments concerning this model. There are no light scalar fields in the model as they all acquire a mass m s through the 1 2+ term ~msZ~hZh in the potential. An interesting and important feature o f the model is the fact that having started with the global solutions eq. (9) that preserve SU(3) c, the color gauge invariance is not broken by the supergravity interactions to O(~), i.e., (Hx) = (Hx) = 0 for x = l , 2, 3. (It is possible that color is broken at O(g 2). Then the color gluons would gain a mass of size eg2m4/M ~ 10 - 2 eV.) Finally we mention a unique feature of this model that the different vacua of eq. (26) are separated by O(m 4) where m s ~ 1 TeV. Theories of this type may thus have interesting cosmological properties as the universe will not choose the physical vacuum until relatively late, i.e., until the temperature o f the early universe has dropped to k T <~m s. This is strikingly different from the models of refs. [3,4] where the energy gaps of the different vacua are scaled by (k2M6) 1/4 ~ 1014 GeV. Our formulation of locally supersymmetric grand unification contains a new non-gravitational mass
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scale m. The theory as currently formulated relates m ~ 1011 GeV to the SU(2) X U(1) breaking mass ,2 m s ~ 103 GeV by m s = r m 2. One possibility is that this super-Higgs mass scale itself arises by dynamical breaking at the loop level, which would then make m ~ r2M3[ln(M2/la2)] 1/2,.. 1011 GeV [11]. Research for this paper was supported in part by the National Science Foundation under Grant No. PHYS0-08333. ¢2 One may break SU(2) × U(1) gauge invariance to Uy by adding a term XsH'xAXyHy to the superpotential. The mechanism for breaking SU(2) × U(1) is the same as discussed in ref. [3]. However, for this model the SU(3) c × Uy(1) vacuum does not lie below the SU(3) × SU(2) × U(1) but can still lie below SU(4) × U(1). Models where SU(3) c × Uy(1) vacuum lies lowest can most likely be constructed.
References [1] E. Witten, Nucl. Physics B177 (1981) 477;B185 (1981) 513; M. Dine, W. Fischler and M. Srednicki, Phys. Lett. 104B (1981) 199; S. Dimopoulous and H. Georgi, Nucl. Physics B193 (1981) 150; N. Sakai, Z. Phys. C l l (1981) 153; D.V. Nanopoulos and K. Tamvakis, CERN preprint TH3227; Phys. Lett. l13B (1982) 151. [2] S. Weinberg, Phys. Lett. 62B (1976) 111. [3] A.H. Chamseddine, R. Arnowitt and P. Nath, Northeastern University preprint NUB #2559 (1982), submitted to Phys. Rev. Lett. [4] S. Weinberg, Phys. Rev. Lett. 48 (1982) 1176. [5] D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13 (1976) 3214; S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335. [6] E. Cremmer et al., Nucl. Phys. B147 (1979) 105. [7] K.S. Stelle and P.C. West, Nucl. Phys. B145 (1978) 175. [8] J. Polony, Budapest Preprint KFKI-1977-93 (1977), unpublished. [9] S. Coleman and F. DeLuccia, Phys. Rev. D21 (1980) 3305. [10] F. Bucella, J.P. Derendinger, S. Ferrara and C.A. Savoy, Phys. Lett. l15B (1982) 375. [ 11 ] R. Arnowitt, A.H. Chamseddine and Pran Nath, Electroweak interaction mass scale from loop corrections in locally supersymmetric GUTS, in preparation.