Nuclear Physics B227 (1983) 121-133 ;~ North-Holland Publishing Company

GAUGE H I E R A R C H Y IN S U P E R G R A V I T Y GUTS* Pran NATH

Department of Physics, Northeastern University, Boston, MA 02115 R. A R N O W I T T * *

Lyman Laboratory of Pto'sies, Harvard University, Cambridge, MA 02138 A. H. C H A M S E D D I N E

Department of Physics, Northeastern University, Boston, MA 02115 Received 17 December 1982 (Revised 13 June 1983)

The gauge hierarchy at the tree level in supergravity GUTS is examined. General criteria on the superpotential are given which guarantee the hierarchy to all orders in the gravitational coupling constant ~. Thus for theories obeying these criteria the light Higgs VEVs are protected from both the Planck and GUT masses. An effective potential for these light fields is constructed.

1. Introduction The gauge hierarchy problem [1] has been a long standing and difficult one in grand unified theories. Thus all G U T models must possess both a superheavy G U T mass M - 1 0 1 4 - 1 0 a6 GeV and a low-energy electro-weak mass scale me_~ - 300 GeV simultaneously. N o simple explanation for the remarkable disparity of these two mass scales exists in conventional G U T models without prior adjustment of mass parameters of the theory by hand. Supersymmetry has offered new approaches to this problem due to the "no renormalization" theorem [2]. Thus if the gauge hierarchy is achieved at the tree level in global supersymmetry, it will be maintained to all l o o p orders, and there have been a number of suggestions towards resolving the gauge hierarchy problem within the framework of global SUSY GUTS [3]. More recently, an extension of the ideas of supersymmetry G U T to local supersymmetry has been proposed, and a number of models have been considered based * Research is supported in part by the National Science Foundation under Grant No. PHY80-08333 and Grant No. PHY77-22864. ** On Sabbatical leave from Department of Physics, Northeastern University, Boston, MA 02115.

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P. Nath et al. / Gaugehierarchy

122

on the gauge group ( N = 1 supergravity) × G where G is a grand unification group, e.g., G = SU(5) [4-8]. These theories lead to realistic G U T models which incorporate the successes of global SUSY G U T , naturally deduce "soft breaking" interactions with no undesirable light scalar bosons, and include gravity into the unification. Most significant is the fact that models can be constructed in which the spontaneous breaking of supergravity (generated by a super-Higgs mechanism [9]) simultaneously leads to the breaking of SU(2)× U(1), showing the interconnection of the electroweak and gravitational interactions [5]. Any theory which includes gravitational interactions automatically leads to a new gauge hierarchy problem. This is due to the closeness of the G U T mass M and the Planck m a s s K - 1 . Thus defining x 2 = 87rG (where G is the Newtonian constant) one has x = 4.11 × 10 -19 (GeV) -1. For a SUSY G U T model, M - 3 × 1016 GeV, and so one has

(1.1)

~ M - 10 2. Thus gravitational corrections to me_w of size (KM)M,(~:M)2M .... (xM)6M

(1.2)

must all be suppressed if the low-energy scale of the theory is not to be overwhelmed by the high scales. In general, for any acceptable theory, the low-energy electro-weak mass scale must be "protected" from both the Planck mass and the G U T mass scales. In previous work [5,10,11] the SU(5) model based on the superpotential

g( ZA) = gl( Z~,) + g2( Z ) ,

(1.3)

where gl = )~1(-~TrX3 + ½M T r X 2 ) + ~ 2 H ' ( X v + 3M'3,,~) Hy + ~3UHf~ HX u

+e ...... v H M g2 = m2( Z + B )

ow

flM-

xk'

!

+ H£M

):~,

¢

f2Mv. + B1,

(1.4) (1.5)

has been extensively analyzed. Here X~, H x, H i are chiral fields in the 24, 5 and representations and U and Z are gauge singlets. M xy and M~' are the matter 10 and superfields and fl and f2 are matrices in generation space, g 2 ( Z ) is the super-Higgs potential [9] and m - 101° GeV is the intermediate mass scale governing supersymmetry breaking. It was seen that for a range of coupling constants (i.e.,)~2/)~1 > 1.05

P. Nath et al. / Gauge hierarchy

123

[10]) the fields of eqs. (1.3)-(1.5) can be divided as follows: { Z, } = {diagZ,~, },

{Z[}={Z;,x4=y;It~,HX,

x=l,2,3},

( Z,~}= { H,~,HX, x = 4 , 5 ; U }

(1.6) (1.7)

and the super-Higgs field Z. U p o n minimizing the effective potential, one finds that Z grows a VEV of Planck mass size K 1, Zi grows a VEV of G U T mass size M and Z~ grows a VEV of electro-weak (me_w) size: m s - g m 2 - 300 GeV. The VEV's of Z,' are zero (showing that color is preserved to all orders in ~ [10]). In contrast, the masses Z i and Z i" are of size M, while those of Z,~ and Z are of size ms. Thus the gauge hierarchy is achieved to all orders in K at the tree level. (The Higgs doublet is light and the Higgs triplet is heavy.) The fact that a tree level gauge hierarchy can be achieved which protects the low-energy Higgs fields from both the Planck and G U T masses is not a unique feature of the superpotential of eqs. (1.4) and (1.5). We will see in this paper that it holds for a wide class of models which guarantee [10] that to first order in ~ a low mass Higgs doublet arises. In sect. 2 we will examine the equations that minimize the tree effective potential, and show under what circumstances the hierarchy is maintained to higher orders in K so that corrections of size eq. (1.2) do not materialize. We will also calculate the actual size of the corrections to the lowest order result. For models of the type considered in sect. 2, where a low-mass sector is guaranteed, it should be possible to obtain an effective potential governing the low-energy sector alone [12]. In sect. 3, we carry out the elimination of the heavy fields, and so obtain a general expression for this low-energy effective potential, in accord with the results recently obtained by Hall, Lykken, and Weinberg [12]. In sect. 4, some concluding remarks are given.

2. Conditions for tree-level gauge hierarchy In supergravity GUTS, the expression for the tree effective potential for the minimal theory with arbitrary superpotential g ( Z A) is given by [5,13-16]: V = ½ E [ G A G ~ - ~ K 2 1 g l 2] + ~*re t ,,,( Z A+, , / T "7~ ~ , a , 12 ,

(2.1)

where E =- exp(½~2ZA~Z A), T" are the gauge group generators and e~ the associated coupling constants, and

GA = g,A + ~1K2Z+ A g,

g,a ~- cOg/OZa .

(2.2)

We consider here only real (PC conserving) VEVs for which the second, D-term, in

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P. Nath et al. / Gaugehierarehr

eq. (2.1) makes no contribution. (More general situations may also be examined. For example, the D-term makes important contributions in analyses where radiative corrections to the effective potential are needed to break S U ( 2 ) × U(1). However, when radiative corrections are small, the contributions from the D-term to the minimization equations are small and vanish when the radiative corrections are neglected.) The minimization of eq. (2.1) then leads to the conditions TABGe = 0,

(2.3)

TAe = g, AB + -}K2(ZAGe + ZeGA) - ~K4ZaZBg -- K2a,48g •

(2.4)

where

We now assume that the full superpotential g(ZA), with { Z A } = {Z~, Z} has the form of eq. (1.3) where g l ( Z ~ ) is the superpotential governing the G U T group G interactions depending on the G U T mass M, and g 2 ( Z ) is a generalized super-Higgs potential (for notational simplicity we assume here that there is only one gauge singlet super Higgs field Z, though conceivably there may be more than one). We now generalize the specific model of eqs. (1.4) and (1.5) in the following way. We assume that gz is the origin of supersymmetry breaking. Thus with g2 set to zero, eq. (2.3) implies, for g = gl, that G~ = 0 has solutions for the Z~ fields, gl(Z~) depends only on one mass scale M and we postulate that in the global limit ~ ~ 0, these VEVs can be divided into three classes: (Z~ } with both VEV and mass of order M, (Z,'} with zero VEV and m a s s - M, and (Z,~} whose VEV and mass are both zero. [Compare with eqs. (1.6) and (1.7).] There are in general a number of solutions of G~ = 0 corresponding to different broken gauge group vacua. We consider only the physically interesting one (e.g., with S U ( 3 ) x S U ( 2 ) x U(1) symmetry) and assume the convention that gl possesses an additive constant arranged so that at the minimum ga vanishes g , ( Z u ) , ~ . = 0.

(2.5)

We next consider the converse condition with g~ set to zero. We write g2 as g2=m2[Zf2(KZ)+B],

(2.6)

where m has dimensions of mass. We assume that g ~ g2, eq. (2.3) has a solution with supersymmetry broken. On dimensional grounds Z must have a VEV of order ~-1. (The constant B is chosen to set the cosmological constant to zero and hence B - ~-x also.) Thus at the minimum one has (g2)rnin -- m2/K"

{2.7)

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A s a c o n s e q u e n c e of the above, the s u p e r - H i g g s field m u s t have a m a s s of size m~ =- Krn 2 - 300 GeV. W e n o w r e t u r n to the full p r o b l e m where g = g~ + g2 a n d show that one m a y develop an e x p a n s i o n solution for the VEVs in powers of ~: z , = z } °) + z } 1) + z , ~2) + . . . z.

=

Z-

+

:, +

(2.8a)

,

(2.8b)

...,

Zt-X)+ Zt°)+ • - - ,

(2.8c)

where Z~ ") is a term of O ( K ' ) . H e r e the fields Z, are those with n o n - v a n i s h i n g masses a n d possible n o n - v a n i s h i n g VEVs in the limit K---, 0, while the Z,, are a n y fields whose masses and VEVs b o t h vanish in this limit. (The p u r p o s e of our analysis is to d e d u c e the c o n d i t i o n s on the superpotential, eqs. (2.19) a n d (2.20), for the existence of fields of the type Z,,, i.e., theories which possess a tree level gauge hierarchy.) F r o m our previous discussion we see that the masses Mi~°) a n d VEVs Z/°l are O ( M ) . ( F o r convenience, we have c o m b i n e d the Z / a n d Z i fields into a single group, so that some of the Z~°) m a y be zero.) W e now rescale our fields a c c o r d i n g to

Z i = Mz,,

Z,, = m ~ z ~ ,

Z = -1z ,

B = 1 b,

K

(2.9)

K

where m s -= Krn 2. The lower case variables are all dimensionless a n d have e x p a n s i o n s similar to eqs. (2.8) b e g i n n i n g now at zero'th o r d e r in K. F u r t h e r , eqs. (2.5) a n d (2.6) i m p l y that at the m i n i m u m one can write m 2

g = --~-~,,

G z = g , z + ½KZZg = m2(Tz •

(2.10)

W e will also use rn~ to rescale the r e m a i n i n g (7, a c c o r d i n g to Ga

~

m u2 g- - " .

(2.11)

W e now establish the basic result that for a wide class of theories, the d i m e n s i o n less quantities z i, z,~, z, a n d (7, are actually of o r d e r unity, e.g., (7, c o n t a i n s no large terms of size M / m ~ or K - l / m . It is this result that protects the low-mass Higgs sector from the G U T a n d Planck mass. W e c o n s i d e r first eq. (2.3) for the s u p e r h e a v y sector A = Z i. On e x p a n d i n g one gets 7~,Gj + Ti~G~ + T i z G z = 0,

(2.12)

P. Nath et al. / Gaugehierarclo:

126

where G, is GA for A = Z,, G,, is (2.12) becomes

GA for A = Zo. In terms of the rescaled variables, eq.

~ i j ~ . q _ M [ 1~z,C~-2 ¼,zizgGq_ -~g,oao 1 - + ~Ss i 2z , ( G -2 - ½:oG~) ] = 0,

(2.13)

where

~,, = g,,, + u~ [ 82( z,~ + z,< ) - 12~Ssz,z,V] + m,8,, [ ~ z G - V + ~8~o~o1 (2.14) and 8~ and e are two parameters of smallness: e =

KM

- 10 -2,

8s =

Km~

-- 10

16

(2.15)

Now g.,j is proportional to the mass matrix in the Z, sector. By hypothesis, all the Z, fields have G U T mass M and hence T,j - M and possesses an inverse. Hence if (i) G~ and Gz are O(1) and (ii) ga, is no larger than O ( M ) one may solve for ~ by inverting T,j in eq. (2.13), and learn that G, - rn2~ has size at the minimum of the effective potential of

Gi-O(rn 2)

(2.16)

with corrections to the leading term of very small size eSs and 82. Eq. (2.16) is a remarkable result since away from the minimum, on dimensional grounds one would expect G, - M 2 and hence be quite large. It is this smallness of G~ at the minimum that is essential for the protection of the low-energy electroweak mass scale. We will see later that g,~ must actually obey a more stringent constraint than (ii) above. To verify the validity of (i) we examine eq. (2.3) for the A = Zo and A = Z channels. Proceeding as in the Z, sector one finds

[1

1

~ssg,~, + 8-B( ½ z G - g ) + ½e8sSoaz,G+82½(GG + zSo)-82¼zoz, g G +

g,°i~ + ~zaGz - 14z,zGzg, - ~eSsZaZiGig + 70 sl¢,2Za(l i~2= 0, (2.17)

[ ~sg2, ZZ q-(zG Z - g - 1z2g)q- ½(,~sZi< q- (~2za~)lG Z + ~SsZZ,Od+ ~ss , 2(za, -2 + z O g _ ~ z Z o G g ) = o

(2.18)

P. Nath et al. / Gauge hierarchy

127

From the general form of g 2 ( Z ) in eq. (2.6), we see g2, z z ~ O(m~). Hence if g,~, ~ O ( m ~ ) ,

(2.19)

g.~B- O ( m ~ ) ,

(2.20)

we see that eqs. (2.13), (2.17) and (2.18) are equations which will determine G,, G,~ and G z to be of size O(1) with corrections to this leading term of size eS~ and 6~. Eq. (2.17) is the equation that determines the low mass VEVs Z~ =- m~z~. We see that if eqs. (2.19) and (2.20) hold, the effects of the G U T sector Z~, G~ and the Planck super-Higgs sector Z, G z produce contributions which are of the same size as arising from the global supersymmetric low-mass interactions themselves. Thus while the supergravity interactions do produce couplings between the G U T and Planck sectors and the low-mass sector, the effect of the former on the latter maintains the size of the low-mass VEVs. Since the correction terms in eqs. (2.17) are power series in the parameters eS~ and 82, the solutions of eq. (2.17) would be a series of the type Z =m~z

=m~z(°)+A~m~(KM)(~m~)+B~rn~(Km~)2+

...,

(2.21)

where z,~°~,A,~, B. . . . . are numbers of O(1). Thus the size of the low-energy VEVs calculated to first order in ~ are maintained when one extends the calculation to all orders in K, i.e., no terms of eq. (1.2) appear in the expansion. This is the tree level protection theorem. Conditions (2.19) and (2.20) impose real constraints on the form of the superpotential g~(Z~). Eq. (2.19) forbids an interaction of the type X Z i Z j Z ~ for then g.,~i- 2 t Z s - XMz~ and eq. (2.19) would be violated unless the coupling constant X was very small ( X - m , / M ) . Thus in the SU(5) model of eqs. (1.4), (1.5), the interaction X(Tr2;2)U is forbidden. Condition (2.20) is satisfied in the model of (1.4), (1.5) due to the condition M ' = M. Thus from eq. (1.4) one has &.,~l~ ~ ~ + $6l'~ where S is 3 M + (?~3/X2)U. As shown in ref. [10], eq. (2.20) is indeed obeyed provided )t2/X 1 > 1.05.

3. Effective potential for low-mass fields As stressed by Hall, Lykken and Weinberg [12], it is exceedingly useful for model building to obtain a reduced effective potential which governs only the low mass fields of the theory. Such a potential has been obtained in ref. [12]. In this section we shall rederive these results using the protection theorem and the formalism of sect. 2. For theories obeying eqs. (2.19) and (2.20) which have the low-mass VEVs protected from the G U T and Planck masses, it should be possible to do this by explicit elimination of the superheavy and super-Higgs fields Z, and Z in terms of the low-mass fields by solving for them by means of the extremum equations for

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P. Nath et al. / Gauge hierarcltv

v( z,; zo; z): OV

OV

O-Z = O,

OZ~ - O.

(3.1)

These equations were written down in eqs. (2.13) and (2.18) and may be explicitly solved for Z = Z ( Z , ) and Z,=Z(Z~) in a power series of type eq. (2.8) in e6s = (~¢M)(Kms) and ~2 = (tCms)2. Inserting such an expansion into the effective potential leads to a reduced low-energy effective potential U(Z,0:

u(z ) = v [ z i ( z o ) ; zo; z ( z o ) ] .

(3.2)

The full form of U(Z,) is quite complicated and will be discussed elsewhere. In general it begins with a leading term of O(m 4) with small e8s and 62 etc. corrections. The leading term contains the major physical content of U since it is what governs whether SU(2) × U(1) correctly breaks at mass scale m s [5]. A much simpler way of obtaining the O(m 4) parts of U is as follows. Clearly, the conditions that minimize the U of eq. (3.2) must lead to eq. (2.17) for the VEV of Z,~ with the solutions Z~(Z), Z(Z~) of eqs. (2.13) and (2.18) inserted there. Thus after such an elimination, it must be possible to integrate eq. (2.17) (order by order in K) to reconstruct U(Z~). The advantage of proceeding in this fashion is that a large number of cancellations of terms between Zi(Z~) and Z(Z,) due to the extremum conditions have already taken place in eq. (2.17), but if one dealt directly with the effective potential, one would have to discover them explicitly. Thus if we wish to obtain the effective potential to O(m4), one may discard all the e6~ and 82 terms in eq. (2.17) and write:

+ 8o ( Z z-g2)]

+ [(,,s)

+

=0

(3.3) Further, one need only keep the zero'th order parts of z ~ ~Z in the super Higgs factors of eq. (3.3) and the zero'th order part of ~ (i.e., the O(m~) part of G i = m2(7,). In addition, the heavy fields enter into gi(Z~, Z~) of eq. (3.3). We will see that one needs the solutions of eq. (2.13) for Z~ only up to first order in ~ [i.e., O(m~)]. The convenience of these results is that Z (°), ~(0), Z}0) and Z¢ 1) are all "constants" i.e., independent of Z,~ and so the integration of eq. (3.3) to O(m 4) becomes trivial. [In contrast, in doing the reduction directly in the effective potential, one would need to calculate z up to terms of O((Krns) 2) and O(~:2Mms) and Z, up to O(m 2) to obtain U ( Z , ) up to O(m4).] We begin by calculating explicit expressions of Z} 1) and ~(0). As discussed in the analysis following eq. (2.4), the zero'th order quantities Z}°) and z (°) are obtained by

P. N a t h et al. /

129

G a u g e hierarcltv

solving the minimization equations for the two special cases g = g~ (with g2 set to zero) and g = g2 (with g, set to zero). We will assume that these quantities are known and of course will differ from model to model. [Thus in the model of eqs. (1.4) and (1.5) one finds [5] that Z[ °) = diag(2J})= m ( 2 , 2, 2, - 3 , - 3 ) and z (°) = _+(,/2 - ¢6).] Condition (2.19) now implies that eq. (2.13) reduces, in lowest order, to

(3.4)

Mis~m'= 1-7'°)~'°' 4 ~i ~ Z [zm)g~°'- 2~-~o)1 _ where Msj -= ~,l.°(°),j-Here g2~°) = gz(Z (°)) and G-~z °) = Gz(z(°)), i.e.

(3.5)

G~°'=m-2[g2. z + ½K2Zg2]z=z,,,

and are all quantities of O(1). The quantity M,9 is proportional to the mass matrix in the superheavy sector, and by hypothesis has an inverse. Thus <(0)

1 (M--1"~] i j z(O)~(o) [ z (o)g2(o) - - 2 ~(zO) ] t Z

= 4~

"

(3.6)

N o t e that since Z} °) - M and ( M - 1 ) o - M 1,
c~1, = o~1, &l,i + (~ ~2z, g).,

= O.

(3.7)

We expand ga,, in powers of x,

gl,i

=

(gl , i ) (0) ~- tgl,,Sl I k(O) 7 ( 1 ) %

+ "'"

(3.8)

and since the first term of eq. (3.8) vanishes (it is just the condition that determines Z~ °) in the global supersymmetric limit), eq. (3.7) reduces to

MijZ) 1) + ~"'s~il ,,,,, 7(o~a(o),~2-- 0

(3.9)

and hence

z?~ = - !~.2,,,~2,~0,(M 1) j)0~.

(3.10)

Eqs. (3.6) and (3.10) represent explicit solutions for ~(0) and Z, °) for an arbitrary model. Since they depend only on the lowest order quantities, they are "constants," independent of the low-mass fields Z~. Thus it is straightforward now to integrate eq. (3.3) and find the effective tow-mass potential which yields it. Thus one has to lowest order

g,.o~aB = g,.o.( gl.~ + ~msZ~g~ °' ) = ½ [ ( g l . . ~ ) 2 + m s~,-, ~,(o, l, o~ , l . ~=,, - g , ) ] . o ,

(3.11)

P. Nath et al. / Gauge hierarct9'

130

showing that the first term is indeed a total derivative with respect to Z~. In a similar fashion, the other parts of eq. (3.11) can also be shown to be total derivatives, leading to the total low-energy effective potential on the real manifold. These results can be easily generalized to complex fields. In this case eqs. (2.3) and (2.4) b e c o m e

)

(<

OZJ + ~2ZAG~ GB - x2gG~ = 0.

(3.12)

Integrating these equations in the light sector we obtain (including the D-term) U(Za, Z +)

=

½E0[gl,agl,a

_~_

m l 2L~,+~~ L a +

(6) -I-

+)

+mZ(~l,i<'°)+h.c.)] + ½[e,~(Z+,(T'~Z),~)] 2, E o --- exp(4 - 2x/3),

(3.13a) (3.13b)

w = m2g I + m3Z~gt, ~

(3.14a)

gl(Z~, Z~) = gl(Z,, Z , ) - g l ( Z , , 0) - b.

(3.14b)

(The subtraction gl(Z~, 0) in eq. (3.14) is a convenient constant of integration which arranges the cosmological constant for the symmetric vacuum, U(0), to be zero. One m a y then choose b - O ( m ] ) to make the total cosmological constant vanish.) The constants m, are given by m 2 = 21',a42 [/~(0)~(0)* __ ~.(0)~(0)*] "s [vZ ~Z ~52 52 J ,

(3.15)

m 2 = 2 ±m s [ Z ( ° ' ( ~ .,, -- 3g(2°)]

(3.16)

,

a(o) " m 3 = 71,. " ' s8,2

(3.17)

(For zero cosmological constant, m 3 = [m3[ ). In eq. (3.13a), Z, is to be kept only up to first order, i.e. ~1 = g l (Z(O) -[- Z~ 1), Z a )

(3.18)

and one is to expand this quantity only to first order since U(Z,) is accurate only to O ( m 4). Thus one m a y write g l = , g l ( z ( 1 ) , Z a ) + ( gl, j ) Z _ z:o,Z; O, .

(3.19)

]Note (~t,j)z} °) is O ( m 2) as a consequence of eq. (2.19).] Results equivalent to the above have previously been obtained in ref. ]12].

P. Nath et al. / Gauge hierarchy

131

Eq. (3.12) [combined with eqs. (3.6), (3.10) and (3.19)] gives the general low-energy effective potential to leading order for an arbitrary model specified by ga and g2 obeying eqs. (2.6), (2.19) and (2.20). The form of the potential, of course, depends upon the model. Thus for the model of eqs. (1.4) and (1.5) one finds

U=lEo[mZlVlZ+m~(l+9X2)(-"

H,,H "+H£H'-'=)-6Am3(H(, , , + H , H , ~ )

-a~_2 m3z(°)Tt3( UH:H <~+ U*I4,
+ (x_,)21UI'(/7oH o +/-/'/7 '°) + (X,)~(HIES)(/7~,/7")] + U.

+½[e:(Z+,(T"Z)")] 2,

(3.25a)

H:=-H ~+ etc.,

where E 0 = e x p ( 4 - 2 ( 3 ) and UM is the part involving the matter (squark and slepton) fields and is given by

UM= ½Eo[{(X3U- 3Xrn3)(s~
-a(~m3z(°)H~( M~XfaM/¢ ) + (()t3U* - 37tm3)/7 ' ~ - a~_2m3z(°)H~)e ........vM~'~f2M~>'+ h.c.} +flijfli'j"

MC')'imi'~>'"M'JH'>/'+-',y

MtlI/fa.vi~-tBAAit~j]"

.......

l~rv

+ H'H"~M,'iM 'vi'8 i;'+ H2~,'~Mf.~,v i'6ji')

+ f~,,k,'j'( 8,:7,;7'"'M"W'M""'~'~'~L," + 88:,',~,;w'~"'H°,% M""'J~.'.,. +2f2ijilij'(17 ...... .vn°tMZ'wTntc~M ')7 + h.c.)] ,

(3.25b

where t/w*x*V" 6~,,.x). - e~,,w~,~.-~"'""~')",

etc.

(3.25c

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P. Nath et al. / Gauge hierarchy

For the linear Z model of eq. (1.5), the parameters m 3 and Z (°) in eq. (3.25) have the values m 3 = - a 2 1/2m s and Z (°) = - a((6- - v/2) where a = _+1, in accord with the results previously obtained in ref. [10] on the real manifold. An effective potential for a non-GUT model, with linear g z ( Z ) but with the super-Higgs field Z eliminated, has also been discussed by Barbieri et al. [17]. One may obtain higher order corrections to U by solving eqs. (2.13) and (2.18) for Z~ and Z to higher order and inserting into eq. (2.17). By arguments given at the beginning of this section, eq. (2.17) must be integrable order by order. In general, however, these corrections to eq. (3.12) will be quite small.

4. Conclusions

In the preceding discussion we examined the conditions under which a tree level gauge hierarchy will occur for a general supergravity G U T model. The class of G U T models shown to have their low-mass VEV's protected from both the G U T and Planck masses are those whose superpotentials are of the form g ( Z ~ ) = g l ( Z ~ ) + g z ( Z ) , where the super-Higgs potential g2 is of the form of eq. (2.6), and the G U T part gl(Z~) obey eqs. (2.19) and (2.20). Eq. (2.6) implies that g2 -= (~://mZ)g2 is a function of only •Z, i.e., a single mass, the Planck mass ~ 1 scales the VEV of Z. Of course, the nature of physics beyond the Planck mass is completely unknown, and there might be more than one mass scale at such ultra high energies. Our low energy results, of course, are independent of this, and remain valid provided only all such scales are > K-1. Similarly, the assumption that the super Higgs field has no direct coupling to the G U T fields in the superpotential g ( Z A) can be relaxed provided such couplings are sufficiently weak. Thus a gauge hierarchy would still be maintained if g( Z A) contained a term of the form ?~~jZ i Zj Z provided ?~~j - icm s or smaller. (Such couplings might in fact be induced at the loop order.) For theories which possess a low-energy protection phenomena, it is possible to obtain a tree effective potential which governs the low-energy sector alone by eliminating the G U T fields and the super-Higgs field. A general procedure, based on the protection equations (and equivalent to results obtained by Hall, Lykken, and Weinberg [12]) was given in sect. 3, and the leading O(m 4) part calculated in eq. (3.12). This formula can be used to examine the low-energy Bose mass matrix. One may also extend the above ideas of eliminating the heavy and super-Higgs fields to calculate an effective low-energy action including the Fermi interactions. Here the form of the dimension-five operators are of particular importance as they contribute to proton decay, and it is of considerable interest to see if general estimates of their size can be made. Finally, we mention that criteria for protection of the low-energy VEVs from the Planck and G U T masses can also be derived at the one-loop level [7]. Higher loops are more complicated to consider as they involve graviton exchange, and one must define the theory in some appropriate way to actually calculate such diagrams. If

133

P. Nath et al. / Gauge hierarcl~v

o n e c o n s i d e r s j u s t t h e s i m p l e s t c a s e of a l i n e a r s u p e r H i g g s p o t e n t i a l ( w i t h n o G U T fields p r e s e n t ) i.e., g = g2 = rn2 ( Z + B), t h e n u n d e r c e r t a i n a s s u m p t i o n s it is p o s s i b l e to s h o w t h a t the l o o p c o r r e c t i o n s , to all orders, p r o d u c e n e g l i g i b l e m o d i f i c a t i o n s o f the tree results. T h i s is d u e to the fact t h a t the m a s s o f the Z field is m s a n d t h u s the m o m e n t a r u n n i n g a r o u n d Z l o o p s is s c a l e d by this s m a l l m a s s so t h a t the c o r r e c t i o n s are o n l y o f O(Kms).

The inclusion

of the G U T

matter

into

the h i g h e r l o o p

c a l c u l a t i o n s is c u r r e n t l y b e i n g i n v e s t i g a t e d . We

should

like to t h a n k

L. H a l l

and

S. W e i n b e r g

for several s t i m u l a t i n g

c o n v e r s a t i o n s , a n d for e m p h a s i z i n g to us the s i g n i f i c a n c e of the l o w e n e r g y e f f e c t i v e potential.

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