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Low energy effective potential for N = 1 supergravity with non-minimal scalar couplings
Volume 136B, number 3
PHYSICS LETTERS
1 March 1984
LOW ENERGY EFFECTIVE POTENTIAL FOR N = 1 SUPERGRAVITY WITH NON-MINIMAL SCALAR COUPLINGS ¢r
Stephane OUVRY 1
Department of Physics, City College of the City University of New York, New York, NY 10031, USA Received 31 October 1983 Revised manuscript received 1 December 1983 We derive a low energy effective theory for a new class of models in N = 1 supergravity with non-minimal scalar couplings.
Recently a great deal o f interest has been devoted to N = 1 supergravity. The extensive work of Cremmer et al. [1] on the one hand, of Bagger and Witten [2] on the other hand has provided the most general formulation o f the N = 1 supergravity lagrangian in the component formalism. Several authors [3] have then attempted to build realistic models using the so-called minimal couplings of matter to supergravity. The minimal couplings have the virtue o f giving directly canonical kinetic terms in the lagrangian. However, apart from this rather technical aspect, there is no obvious reason why one should use these couplings rather than others. Furthermore it has been [4] shown that problems related to the SU(2) × U(1) breaking arise in this class of models. This is partly due to the fact that positive scalar mass terms of order m2/2~ appear in the effective theory, whether GUT fields are present or not, thus making the breaking rather cumbersome. These mass terms are certainly appealing because they automatically give Weinberg-Salam mass to the scalar quarks and leptons, but as it has been shown, both in global [5] and local SUSY GUTs [6], one can also generate radiatively these masses. In order to make the breaking possible, one has to introduce light sin-
Supported by the National Science Foundation, Grant # NSF-PHY-82-15364. I On leave from Institut de Physique Nucl6aire, Orsay, France, and LPTPE, Universit6 Pierre et Marie Curie, Paris, France. Address after February 1, 1984: IPN, Divison de Physique Th6orique, 91406 Orsay Cedex, France. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
glet fields; this results [7] in jeopardizing the hierarchy at the perturbative level. Another possibility is to break SU(2) × U(1) radiatively [8], but a very heavy top quark (m t ~> 60 GeV) may be needed. In a recent letter [9] we derived a new class of models in N = 1 supergravity corresponding to a particular choice of non-minimal scalar couplings. This choice is intended [6] to reproduce the Wess-Zumino N = 1 supergravity [10] as originally formulated in the superspace. Thus, even though these couplings are called non-minimal they lead to the simplest and most natural formulation o f N = 1 supergravity in the superspace. In the language of K/ihler manifolds [2] for non-linear (r-models, they correspond to an underlying spherical geometry. It has been shown [9] that in this new formulation o f N = 1 supergravity, the Polony term, or equally the O'Raifeartaigh mechanism, are unable to break supersymmetry. A new hidden superpotential g2(z) has been introduced: it breaks the supersymmetry everywhere inside the circle o f radius x/"6/K in the complex (z) plane, where z represents the hidden complex scalar singlet field. An effective theory has been also derived by brutal force setting K = 0. However it is obvious that in the presence of GUT fields a better approximation is needed. In this letter we derive a low energy effective theory for this new class o f N = 1 supergravity models. As it will appear, the derivation is simpler and more transparent than in the minimal case [11 ]. Furthermore this new effective theory presents interesting features: the above mentioned mass terms disappear, 165
Volume 136B, number 3
PHYSICS LETTERS
the cosmological constant is exactly zero, the VEV of the hidden field (and thus its mass) has to be determined depending on the particular model cbnsidered. Our notations are the same as in ref. [9] and our assumptions concerning the super GUT potential gl (v), where y represents the matter fields, are the same as in ref. [11]. As far as the gauge sector is concemed (D term), the analysis is the same as in ref. [11 ] and will not b'e reproduced here. We assume the existence of a set of scalar field VEVs YO for which, in the absence of the hidden scalar sector, supersymmetry is unbroken and spacetime is flat:
~gl(y)/ayaly=y ° = 0,
gl(YO ) = 0,
y g ' = 0.
gab = a2gl /~ya ~yb [Y=Yo~ M G U T '
(4)
for a = A and b ---B and zero otherwise. The matrix [g] = (gAB) is assumed to be non-singular. Finally
gabc = a3gl ]aya ayb~yCly=y o ~ 1.
(5)
In the presence of supergravity the GUT fields are 2 m 33/2 expected to get corrections in order m 3/2' m3/2' and so on. Thus Y =Y0 + ¢,
(6)
where
(Yi)~mt3/2 •
(7)
(8) where G1 = g l - ~ Y
agl/aY ,
g2 = ao(1 + gz/x/6)3, 166
= ([1 - -~K2(.Vay*a + zz*)] -1 ] K 2 y , . B ( 2 e - I V) + [1 -- IK2(yay*a +g2*)] -2
x [(agl/oya)*aZy~/ay%y8 - ~KZcfaci/ay B - ~K2(aGi/ayB)G~]).
(11)
We are going to expand eq. (11) in a power series with m 3/2 for expansion parameter. To do this we need to use the following identities: 1 _
.t b . t ¢
~gl/OY a =gab (~b + "~8abcqJ q) ,
(12)
a2gl / Oy aoyB = gaB + gaBc gpc,
(13)
1
a
-b
G 1 = c - ~yogab qJ + ~(gabgpaqjb -y~gabc(~b(oc), (14) aG1/ayB = -sYogaB 1 a 1 a -YogaBc~ a c ), + ~(gaB~
G2 = g2 - ]z ~g2/az, (9)
(15)
where gl is assumed to be at most cubic and c is a constant that we are free to add to the complete superpotential g = gl + g2. c can also be expanded:
c = c 1 + c 2 + ....
(16)
where c i ~ m~/2. G 2 is assumed to be of order m3/2, as it will be verified later. Let us first consider the first term of eq. (11), which is in order m 3/2 :
3
X [lagl/aya[ 2 - ~ K 2 I G l l 2 _ ~K a 2 (G1G 2• +h.c.)],
a
(aV/ayB)=o
1 [g]Yl) * ( - g1[ g ] Y 0 ) - gY0
- i K ( G 2 ) ( - $ [g]YO)"
2e -1 V = [1 - ~K2(yay*a + zz*)] - 2
a
(10)
is stationary in the heavy scalars YA in order to integrate them out:
0 = [g]([g]yl)*_~K2(Cl
We have now to express the fact that the potential V given by [9] :
1
= - ( 3 / K 2 ) I n [1 - ~K2(yay .,7 + zz*)]
(3)
Furthermore
q~=Yl +Y2 + ....
and corresponding to a K~hler potential of the form
(1,2)
the scalar index a runs over values A labelling complex superheavy chiral scalar and a' labelling complex light scalars (a' = a) and real scalars degenerate with superheavy gauge bosons in the absence of the hidden sector (a' = K). We have y0A ~ MGUT ,
1 March 1984
2
*
1
(17)
Using the fact that [g] is non-singular, eq. (17) has for solution [g]Yl = - { K 2 [(Cl + (G2))/(1 - bZ)]Y~ ,
(18)
where b 2 = ~ K 2 y o y ~. The factor 1/(1 - b2), if to be expanded, leads to a power series expansion with parameter e = (MGuT/MpI) 2 . c 1 can now be adjusted in order to cancel the first term in the power expan-
Volume 136B, number 3
PHYSICS LETTERS
1 March 1984
sion of V, which is in order m~/2 :
This condition leads to
0 = I[g]y 112 - {K2[Cl - ~Y0 [g]Y112
c2 = --[Yl [glYl'
-- {K2 {(G~')(el - ~Y0 ' [g]Yl) + h.c.).
(19)
and the solution o f e q . (28) is finally: u* = }K 2 [((G 2) + Cl)/(1 - b2)] *
This equation has as solution: c 1 =P(G2),
p = - 1 -+ (1 - b2)1/2. •
(20,21)
We can now check that G 2 is indeed in order m3/2. The mass of the gravitino is given by [1 ] : m ~/2 = (K -2 exp ( - q)),
X D, 1 + ~K2yoO, ly~)/(1 - b2)].
(31)
Let us now compute Veff in order m4/2: We have 2e -1 Veff = (1 - b 2 - la12) -2
(22)
X [uu*-~K2(YoU)(YoU)*+ ~[ga'bcYl 1 b Ylc ]2
with = 3 In (-½K2~b) - In (¼K61g['2).
(23)
At the lowest order
m2/2 = (-~K41Cl +g212)(1 - b21a12) -3
(24)
+ {([glYl) A [([glY3)A +gAbcyby~] * _ {K2(Cl - S Yl 0 [g]Yl + (G2)) l
= ¼K4(G2G~)(p + 1 + a)2(1 - b 2 - [a12) -3, (25) where a = Kzo/x/6. This equation tells us that indeed G 2 is in order m 3/2. We have now to use the stationarity condition at the order m ~/2 :
0 = ([g]yl)*4gABeY ~ + [glu* _ {K2(Cl _ {_K2(c2
--
(30)
aK2
1
*1
A
b c
+h.c.)]
•
(32)
= (1 - b 2 - la12) -2 X ,L[ u u * - log 2 ( vv0 u ) (,,~ v 0 u ' ) * + l ga,bcY lb Y lc2 -- ] K 2 ( [((G 2) + Cl)1(1 - b2)]
a
¢
-- 3Y0 [g]Yl ) a([g]Yl - Y o gABcYl ) __1aYo
y
X [c 3 -$(Y0 [g] 3 - Y l [g]Y2 +Y0 gAbcYlY2 )]
u + ~Yl [g]Yl) * (-ff1 [g]Y0 )
* I A c (G2)ff([g]y 1 - Y O gABcY l )'
(26)
where u = [g]Y2 + 2gAbcYl i b Yl" c
(27)
The terms of the formgABcY ~ disappear because of eq. (17), and we finally get:
X (c 3 + ~Yl
[glY2)*+ h.c.)].
(33)
In the last eq. (33) the Y3 and gAbcybly~ contributions have automatically disappeared. Using eqs. (20), (21) and (31), and adjusting c 3 to cancel Veff at the minimum, we get: 2e -1veff = ( 1 - b 2 - lal 2) -2 {~lga'beYlYll 1 b c2 + ~K 2 [(G~)(1 - b 2)-l/2gAbcY: y by~ + h.c.] ) + o 0,
Let us adjust c 2 so as to cancel the term in order m3/2 in the power expansion of V:
(34) where o 0 is a constant in order m4/2 and z 0 has been determined by minimizing V with respect to z. Using the results of ref. [11 ] coming from the stationarity of V with respect to the y K,s (i.e. y f = gKafl = gKaA y A = 0), eq. (34) becomes:
O = u * [ g ] y l _ ~ K 2 ( c 2 - ~YoUl + [ y l [ g ] y l ) .
1 a b2I 2e -1 ~ f f = (1 - b 2 - lal2)-2(r~lgc~abYlYl
0 =u*-- ½K 2 [((G2)+ Cl)/(1 -- b 2 ) ] * y l + ½K2(c 2 _ ~YoUl + -~Yl [g]Yl)*Y0 •
X (c 1 - ~Y0 [g]Yl + (G2))"
(28)
(29)
-+I K 2 [(G;)(1 -~h2a-1/2~,,6Aab.,"" A 1 " al.r " b1
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PHYSICS LETTERS
where the indices a and b run now on A or a. As it should be, only the light fieldsy~ appear explicitly: eq. (35) is the low energy effective potential for nonminimal N = 1 supergravity models. As stated above, the derivation is transparent, the cosmological constant is exactly zero (at least in order m4/2), z 0 has to be determined depending on g l ,1, there is no automatic mass term of the form m~12y~y~*. The SU(2) X U(I) breaking is thus likely t'o be more natural in this class of models. In this respect some analysis has already been made [6] in the limit K = 0. Another paper is in preparation using eq. (35) as effective theory. Two remarks have finally to be made: (1) the kinetic terms for the scalars are not canonical in this class of model. In order to put them in a canonical form one has to perform the following rescaling: yg ~ y ~ ( 1 - b 2 - lal2) 1/2, (G2) -+ (G2)(1 - b 2 - [a12) 1/2.
(36)
(2) the only scale parameter which enters eq. (35) is the gravitino mass related to G 2 through eq. (25). I would like to thank the High Energy Group at the City College for their warm hospitality and Dr. J. Lykken and Dr. X. Wu for discussions. :~1 The minimization of V with respect to z is non-trivial in this class of non-minimal models. One possibility is to invoke quantum loop corrections (see for example AlvarezGaume et al. [8] ). Another possibility consists in coupling directly z to the GUT fields at the tree level.
168
1 March 1984
References [1] E. Cremmer, B. Julia, J. Scherk, S. Ferrara, L. Girardello and P. Van Nieuwenhuizen, Nucl. Phys. B147 (1979) 105; E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Phys. Lett. 116B (1982) 231 ;Nucl. Phys. B212 (1983) 413. [2] E. Witten and J. Bagger, Phys. Lett. l15B (1982) 202; J. Bagger, Nucl. Phys. B211 (1983) 302. [3] A.H. Chamseddine, R. Arnowitt and Pran Nath, Phys. Rev. Lett. 49 (1982) 970; H.P. Nilles, M. Srednicki and D. Wyler, Phys. Lett. 120B (1983) 346; R. Barbieri, S. Ferrara and C.A. Savoy, Phys. Lett. l19B (1982) 343; S. Weinberg, Phys. Rev. Lett. 50 (1983) 387. [4] M. Dine and W. Fischler, UPR report No. 01215T (1983). [5] G: Farrar, Rutgers University preprint (1982); J. Polchinski and L. Susskind, Phys. Rev. D26 (1982) 3661. [6] X. Wu, Thesis (1983) unpublished. [7] H.P. Nilles, M. Srednicki and D. Wyler, Phys. Lett. 124B (1983) 337; J.M. Fr~re, D.R.T. Jones and S. Raby, UMR report No. UMHE-82-58,to be published. [8] L.E. lb~'~ez,Phys. Lett. l18B (1982) 73; L. Alvarez-Gaume,J. Polchinski and M.B. Wise, Nucl. Phys. B221 (1983)495; C. Kounnas, A.B. Lahanas, D.V. Nanopoulos and M. Quiros, Phys. Lett. 132B (1983) 95;CERN report 3657 (1983). [9] N.P. Chang, S. Ouvry and X. Wu, Phys. Rev. Lett. 51 (1983) 327. [ 10] J. Wess and Zumino, Phys. Lett. 74B (1978) 51. [11] L. Hall, J. Lykken and S. Weinberg, Phys. Rev. D27 (1983) 2359.
PHYSICS LETTERS
1 March 1984
LOW ENERGY EFFECTIVE POTENTIAL FOR N = 1 SUPERGRAVITY WITH NON-MINIMAL SCALAR COUPLINGS ¢r
Stephane OUVRY 1
Department of Physics, City College of the City University of New York, New York, NY 10031, USA Received 31 October 1983 Revised manuscript received 1 December 1983 We derive a low energy effective theory for a new class of models in N = 1 supergravity with non-minimal scalar couplings.
Recently a great deal o f interest has been devoted to N = 1 supergravity. The extensive work of Cremmer et al. [1] on the one hand, of Bagger and Witten [2] on the other hand has provided the most general formulation o f the N = 1 supergravity lagrangian in the component formalism. Several authors [3] have then attempted to build realistic models using the so-called minimal couplings of matter to supergravity. The minimal couplings have the virtue o f giving directly canonical kinetic terms in the lagrangian. However, apart from this rather technical aspect, there is no obvious reason why one should use these couplings rather than others. Furthermore it has been [4] shown that problems related to the SU(2) × U(1) breaking arise in this class of models. This is partly due to the fact that positive scalar mass terms of order m2/2~ appear in the effective theory, whether GUT fields are present or not, thus making the breaking rather cumbersome. These mass terms are certainly appealing because they automatically give Weinberg-Salam mass to the scalar quarks and leptons, but as it has been shown, both in global [5] and local SUSY GUTs [6], one can also generate radiatively these masses. In order to make the breaking possible, one has to introduce light sin-
Supported by the National Science Foundation, Grant # NSF-PHY-82-15364. I On leave from Institut de Physique Nucl6aire, Orsay, France, and LPTPE, Universit6 Pierre et Marie Curie, Paris, France. Address after February 1, 1984: IPN, Divison de Physique Th6orique, 91406 Orsay Cedex, France. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
glet fields; this results [7] in jeopardizing the hierarchy at the perturbative level. Another possibility is to break SU(2) × U(1) radiatively [8], but a very heavy top quark (m t ~> 60 GeV) may be needed. In a recent letter [9] we derived a new class of models in N = 1 supergravity corresponding to a particular choice of non-minimal scalar couplings. This choice is intended [6] to reproduce the Wess-Zumino N = 1 supergravity [10] as originally formulated in the superspace. Thus, even though these couplings are called non-minimal they lead to the simplest and most natural formulation o f N = 1 supergravity in the superspace. In the language of K/ihler manifolds [2] for non-linear (r-models, they correspond to an underlying spherical geometry. It has been shown [9] that in this new formulation o f N = 1 supergravity, the Polony term, or equally the O'Raifeartaigh mechanism, are unable to break supersymmetry. A new hidden superpotential g2(z) has been introduced: it breaks the supersymmetry everywhere inside the circle o f radius x/"6/K in the complex (z) plane, where z represents the hidden complex scalar singlet field. An effective theory has been also derived by brutal force setting K = 0. However it is obvious that in the presence of GUT fields a better approximation is needed. In this letter we derive a low energy effective theory for this new class o f N = 1 supergravity models. As it will appear, the derivation is simpler and more transparent than in the minimal case [11 ]. Furthermore this new effective theory presents interesting features: the above mentioned mass terms disappear, 165
Volume 136B, number 3
PHYSICS LETTERS
the cosmological constant is exactly zero, the VEV of the hidden field (and thus its mass) has to be determined depending on the particular model cbnsidered. Our notations are the same as in ref. [9] and our assumptions concerning the super GUT potential gl (v), where y represents the matter fields, are the same as in ref. [11]. As far as the gauge sector is concemed (D term), the analysis is the same as in ref. [11 ] and will not b'e reproduced here. We assume the existence of a set of scalar field VEVs YO for which, in the absence of the hidden scalar sector, supersymmetry is unbroken and spacetime is flat:
~gl(y)/ayaly=y ° = 0,
gl(YO ) = 0,
y g ' = 0.
gab = a2gl /~ya ~yb [Y=Yo~ M G U T '
(4)
for a = A and b ---B and zero otherwise. The matrix [g] = (gAB) is assumed to be non-singular. Finally
gabc = a3gl ]aya ayb~yCly=y o ~ 1.
(5)
In the presence of supergravity the GUT fields are 2 m 33/2 expected to get corrections in order m 3/2' m3/2' and so on. Thus Y =Y0 + ¢,
(6)
where
(Yi)~mt3/2 •
(7)
(8) where G1 = g l - ~ Y
agl/aY ,
g2 = ao(1 + gz/x/6)3, 166
= ([1 - -~K2(.Vay*a + zz*)] -1 ] K 2 y , . B ( 2 e - I V) + [1 -- IK2(yay*a +g2*)] -2
x [(agl/oya)*aZy~/ay%y8 - ~KZcfaci/ay B - ~K2(aGi/ayB)G~]).
(11)
We are going to expand eq. (11) in a power series with m 3/2 for expansion parameter. To do this we need to use the following identities: 1 _
.t b . t ¢
~gl/OY a =gab (~b + "~8abcqJ q) ,
(12)
a2gl / Oy aoyB = gaB + gaBc gpc,
(13)
1
a
-b
G 1 = c - ~yogab qJ + ~(gabgpaqjb -y~gabc(~b(oc), (14) aG1/ayB = -sYogaB 1 a 1 a -YogaBc~ a c ), + ~(gaB~
G2 = g2 - ]z ~g2/az, (9)
(15)
where gl is assumed to be at most cubic and c is a constant that we are free to add to the complete superpotential g = gl + g2. c can also be expanded:
c = c 1 + c 2 + ....
(16)
where c i ~ m~/2. G 2 is assumed to be of order m3/2, as it will be verified later. Let us first consider the first term of eq. (11), which is in order m 3/2 :
3
X [lagl/aya[ 2 - ~ K 2 I G l l 2 _ ~K a 2 (G1G 2• +h.c.)],
a
(aV/ayB)=o
1 [g]Yl) * ( - g1[ g ] Y 0 ) - gY0
- i K ( G 2 ) ( - $ [g]YO)"
2e -1 V = [1 - ~K2(yay*a + zz*)] - 2
a
(10)
is stationary in the heavy scalars YA in order to integrate them out:
0 = [g]([g]yl)*_~K2(Cl
We have now to express the fact that the potential V given by [9] :
1
= - ( 3 / K 2 ) I n [1 - ~K2(yay .,7 + zz*)]
(3)
Furthermore
q~=Yl +Y2 + ....
and corresponding to a K~hler potential of the form
(1,2)
the scalar index a runs over values A labelling complex superheavy chiral scalar and a' labelling complex light scalars (a' = a) and real scalars degenerate with superheavy gauge bosons in the absence of the hidden sector (a' = K). We have y0A ~ MGUT ,
1 March 1984
2
*
1
(17)
Using the fact that [g] is non-singular, eq. (17) has for solution [g]Yl = - { K 2 [(Cl + (G2))/(1 - bZ)]Y~ ,
(18)
where b 2 = ~ K 2 y o y ~. The factor 1/(1 - b2), if to be expanded, leads to a power series expansion with parameter e = (MGuT/MpI) 2 . c 1 can now be adjusted in order to cancel the first term in the power expan-
Volume 136B, number 3
PHYSICS LETTERS
1 March 1984
sion of V, which is in order m~/2 :
This condition leads to
0 = I[g]y 112 - {K2[Cl - ~Y0 [g]Y112
c2 = --[Yl [glYl'
-- {K2 {(G~')(el - ~Y0 ' [g]Yl) + h.c.).
(19)
and the solution o f e q . (28) is finally: u* = }K 2 [((G 2) + Cl)/(1 - b2)] *
This equation has as solution: c 1 =P(G2),
p = - 1 -+ (1 - b2)1/2. •
(20,21)
We can now check that G 2 is indeed in order m3/2. The mass of the gravitino is given by [1 ] : m ~/2 = (K -2 exp ( - q)),
X D, 1 + ~K2yoO, ly~)/(1 - b2)].
(31)
Let us now compute Veff in order m4/2: We have 2e -1 Veff = (1 - b 2 - la12) -2
(22)
X [uu*-~K2(YoU)(YoU)*+ ~[ga'bcYl 1 b Ylc ]2
with = 3 In (-½K2~b) - In (¼K61g['2).
(23)
At the lowest order
m2/2 = (-~K41Cl +g212)(1 - b21a12) -3
(24)
+ {([glYl) A [([glY3)A +gAbcyby~] * _ {K2(Cl - S Yl 0 [g]Yl + (G2)) l
= ¼K4(G2G~)(p + 1 + a)2(1 - b 2 - [a12) -3, (25) where a = Kzo/x/6. This equation tells us that indeed G 2 is in order m 3/2. We have now to use the stationarity condition at the order m ~/2 :
0 = ([g]yl)*4gABeY ~ + [glu* _ {K2(Cl _ {_K2(c2
--
(30)
aK2
1
*1
A
b c
+h.c.)]
•
(32)
= (1 - b 2 - la12) -2 X ,L[ u u * - log 2 ( vv0 u ) (,,~ v 0 u ' ) * + l ga,bcY lb Y lc2 -- ] K 2 ( [((G 2) + Cl)1(1 - b2)]
a
¢
-- 3Y0 [g]Yl ) a([g]Yl - Y o gABcYl ) __1aYo
y
X [c 3 -$(Y0 [g] 3 - Y l [g]Y2 +Y0 gAbcYlY2 )]
u + ~Yl [g]Yl) * (-ff1 [g]Y0 )
* I A c (G2)ff([g]y 1 - Y O gABcY l )'
(26)
where u = [g]Y2 + 2gAbcYl i b Yl" c
(27)
The terms of the formgABcY ~ disappear because of eq. (17), and we finally get:
X (c 3 + ~Yl
[glY2)*+ h.c.)].
(33)
In the last eq. (33) the Y3 and gAbcybly~ contributions have automatically disappeared. Using eqs. (20), (21) and (31), and adjusting c 3 to cancel Veff at the minimum, we get: 2e -1veff = ( 1 - b 2 - lal 2) -2 {~lga'beYlYll 1 b c2 + ~K 2 [(G~)(1 - b 2)-l/2gAbcY: y by~ + h.c.] ) + o 0,
Let us adjust c 2 so as to cancel the term in order m3/2 in the power expansion of V:
(34) where o 0 is a constant in order m4/2 and z 0 has been determined by minimizing V with respect to z. Using the results of ref. [11 ] coming from the stationarity of V with respect to the y K,s (i.e. y f = gKafl = gKaA y A = 0), eq. (34) becomes:
O = u * [ g ] y l _ ~ K 2 ( c 2 - ~YoUl + [ y l [ g ] y l ) .
1 a b2I 2e -1 ~ f f = (1 - b 2 - lal2)-2(r~lgc~abYlYl
0 =u*-- ½K 2 [((G2)+ Cl)/(1 -- b 2 ) ] * y l + ½K2(c 2 _ ~YoUl + -~Yl [g]Yl)*Y0 •
X (c 1 - ~Y0 [g]Yl + (G2))"
(28)
(29)
-+I K 2 [(G;)(1 -~h2a-1/2~,,6Aab.,"" A 1 " al.r " b1
(35) 167
Volume 136B, number 3
PHYSICS LETTERS
where the indices a and b run now on A or a. As it should be, only the light fieldsy~ appear explicitly: eq. (35) is the low energy effective potential for nonminimal N = 1 supergravity models. As stated above, the derivation is transparent, the cosmological constant is exactly zero (at least in order m4/2), z 0 has to be determined depending on g l ,1, there is no automatic mass term of the form m~12y~y~*. The SU(2) X U(I) breaking is thus likely t'o be more natural in this class of models. In this respect some analysis has already been made [6] in the limit K = 0. Another paper is in preparation using eq. (35) as effective theory. Two remarks have finally to be made: (1) the kinetic terms for the scalars are not canonical in this class of model. In order to put them in a canonical form one has to perform the following rescaling: yg ~ y ~ ( 1 - b 2 - lal2) 1/2, (G2) -+ (G2)(1 - b 2 - [a12) 1/2.
(36)
(2) the only scale parameter which enters eq. (35) is the gravitino mass related to G 2 through eq. (25). I would like to thank the High Energy Group at the City College for their warm hospitality and Dr. J. Lykken and Dr. X. Wu for discussions. :~1 The minimization of V with respect to z is non-trivial in this class of non-minimal models. One possibility is to invoke quantum loop corrections (see for example AlvarezGaume et al. [8] ). Another possibility consists in coupling directly z to the GUT fields at the tree level.
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