A sensible finite SU(5) SUSY gut?

Volume 156B, number 1,2 PHYSICS LETTERS 1 3 June 1985

A SENSIBLE FINITE SU(5) SUSY GUT? "~

J. LEON, J. PEREZ-MERCADER, M. QUIROS

Instituto de Estructura de la Materia, Serrano 119, 28006 Madrid, Spain

and

J. RAMiREZ-MITTELBRUNN

Departamento de Mbtodos Maternltticos, Universidad Cornplutense, 28040 Madrid, Spain

Received 5 March 1985

We construct a one-loop finite grand unified SU(5) model with softly broken SUSY. Doublet- tr iplet splitting requires two fine-tunings and electroweak breaking goes through dimensional transmutation with a 40 GeV top quark. The superparmer spectrum is very tightly constrained and may be parametrized in terms of only one free parameter; it displays the properties that the gravitino mass is unobservable at low energy and the heavier squarks and the gluino are quasi-degenerate. For a slepton of 20 GeV, the photino has a mass of 15 GeV and there is a radiative Higgs with 0(5 GeV) mass. We also find that in this type of .scenario there is an absolute lower bound on the mass of the top quark of 32 GeV.

Softly broken supersymmetric models if compared to non-supersymmetric models, show a substantial improvement in the UV properties of the theory: the parameters in the lagrangian are at most logarithmical- ly divergent. It seems natural in this light to study models without divergencies,i.e, finite models. In fact, the realization several years ago that N = 4 is f'mite [1 ] and more recently that N = 2, if finite to one loop, is finite to all orders of perturbation theory [2] prompt one to construct finite models with softly brokenN= 1 supersymmetry [3,4]. F o r N = 1 supersymmetry, all one knows however is that imposing one-loop finiteness in the SUSY sector is sufficient to ensure it to two loops; no similar statement ex- ists yet for the softly broken sector, where the finiteness conditions are known only at the one- loop level [5].

Now, since finiteness requires that non-trivial relations be satisfied among the parameters in the lagrangian, the predictability of the models improves and the hope dawns that one might be able to answer

Supported in part by "Comisibn Asesora de Investigacibn Cientifica y T6cnica".

66

questions such as the origin of the quark mass matrix, or somehow reduce the number of free parameters that the model requires.

One might ask several questions: where do finite models come from? how does one actually construct a phenomenologically sensible finite grand unified model? which low energy spectra do they predict? In this letter we will address ourselves to the latter two questions, whereas for the first question we will just assume that the models are the low energy limit of some extended supergravity theory or superstring theory that has undergone spontaneous compactifi- cation to 4 dimensions and produced a finite theory [3,61.

The restrictions on the theory for one-loop finiteness of the supersymmetric parameters [3-5 ] are :

(i) that the chiral multiplets be in irreducible repre- sentations, Ro, of the gauge group satisfying

3C2(G ) = ~ T(Ro) , (1) o

(ii) that the cubic couplings fabc that appear in the superpotential

0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 156B, number 1,2 PItYSICS LETTERS 13 June 1985

f= paO a + mab c~adpb + (1/3!)fabccaqbbqb c (2)

be restricted by a

fai/fbij = 4g2C2(Ra)6 b . (3)

Conditions (I) and (3) ensure that the set Oa, mab and fabe remain finite at one-loop, and are sufficient for two-loop finiteness also [7].

Notice that (3) forbids the presence of cubic couplings involving singlets and requires that faiJfbq be "diagonal" in its two free indices.

For the soft-breaking parameter sector one-loop finiteness imposes some extra relations in addition to (1) and (3) * l . For a d = 3 soft-breaking parametrized as

B =gabzaz b + (l[3!)GabcZazbz c (4)

where the z a are the scalar components of the chiral superfields C a, and with a scalar potential

V = ~ f a f a + m2/20~a zaz b + m3/2(B + B*) a

1 ~ D 2 + ~ ~ , ( 5 )

one finds that the two conditions

Gi/k = ~fijk (6)

and

Oab 1 2 a = ~ 6 b (7)

are sufficient for one-loop finiteness. Condition (1) restricts the multiplet contents of

the theory, (3) imposes severe restrictions on the al- lowed couplings and (6) and (7) have very interesting consequences in the low-energy behaviour of a finite model as we will see later on.

A classification of models satisfying (1) has already been given by several authors [8] who l~md that for an SU(5) gauge group, there are three possible models with 5 ,5 , 10, 10 and 24 chiral multiplets with multi- plicities of(2, 3, 3, 2, 1), (3, 5, 3, 1,1) and (4, 7, 3, 0, 1) respectively * 2

~1 Notice that since scale evolution for the gaugino mass m l / 2 = ~ m a / 2 is controlled by ~]~ = &/e~, ro l l 2 will be finite if ( l ) is fulfilled.

,2 Eq. (1) allows for a four-family model, but it turns out that it cannot be made realistic.

We will work with the last model, taking its contents to be one 24 of Higgs for the SU(5) breaking, 4(5 +5 ) of Higgs some of which will be used for electroweak breaking and the remaining 3(5-+ 10) which we will identify as matter. With this contents, the most general superpotential that may be written, consistent with renormalizability, SU(5) invariance and R-parity conservation is of the form

f = q Tr y3 +M Tr ~2 + Xxy-~x Z Hy

+ Q~ xyFIxHy + k gijx( l Oi)aa( lOj)VS(Hx)reaavS r

+~;:xO0i~'~(5/)dHx)~ ~w~o~ - G 6 D , (8)

in self-explanatory notation. The index field is x , y = 1, . . . ,4; i , ] = 1,2, 3 and ct,/3,/a, u= 1 .... ,5.

The f'miteness conditions (3) for this type of models have been written down by Hamidi and Schwartz [4] and by Jones and Raby [3], who explored models with a single light Higgs coupled to ordinary matter and two color triplets at an intermediate scale of 109 GeV which will predict too fast proton decay through d = 6 operators. Furthermore, Raby and Jones only f'md a solution to the finiteness equations with an unrealistically heavy top quark mass of order ra t = 150 GeV. There is a model though that restricts the Yukawas at grand unification scales to be such that each generation couples to a different Higgs and give a realistic low energy phenomenology. The remaining part of this letter is devoted to it.

Consider the following ansatz for the matter couplings in eq. (8):

[g11112-u, Ig22212 - c , 1933312 ---t,

I g - l l l l 2 = d , Ig-22212=s, Ig33312=b, (9)

different from zero and all othergijk,gi/k equal to zero .

For the Higgs 5-plets coupling to the adjoint we take them to be of the form

I~iil2=xi ( i=1 , . . . , 4 ) , ;~0.=0 ( i ~ / ) . (10)

The solution to the finiteness equations (3) corresponding to the ansatz (9), (10) is unique and given by

d = s = b = ~g2, (I1)

X l = X Z = X 3 = O , x4 =g2 , (12a,b)

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Volume 156B, number 1,2 PHYSICS LETTERS 13 June 1985

u = c = t = ~ g 2 , q 2 = S g 2 , (12c,d)

where g is the SU(5) gauge coupling. At face value this solution seems doomed to fail

also, because the Yukawas for same-charge quarks in different generations are the same and the quarks would be mass-degenerate; but there is a way out of this conundrum, given that each generation couples to different Higgses which are in no way constrained to have the same VEVs. One needs however to make sure that achieving this is compatible with the doublet-triplet mechanism and other low energy phenomenology.

To implement the above solution we rotate the Higgs sector via orthogonal matrices R and S, from bases (H~,H~} to {K~, g'c,} defined as

Ka = Rat3nt3, g,~ = HaSa¢. (13)

The idea is that only the Higgses K 4 and K 4 will be light [O(Mw) ] and only they will acquire appro- priate VEVs so as to give the right fermion and gauge boson spectrum. Taking VEV in (13) and inverting we get

(Ha) = (R)aa(K4) , (1~ a) = (K.4XS)c~4 . (14)

This determines the 4th row of R and independent- ly the 4th column of S. We now choose the (orthogonal) matrices R and S as

-~lr2/rl eQr2/r 1 -a3r l / r2 -a4r l / r2

R= 0 0 a4r 1 -a3r l

a2r 2 --alr 2 0 0

L~I ~2 a3 ~4

(15)

S=

~ls2/sl 0 ~2s2 ~1

~2S2/Sl 0 --~lS2 ~2 , !

--[J3Sl/S2 fl4Sl 0 /~31 '

J ---~4Sl/S2 -/~3Sl 0 ~4

(16)

where

i t

With this choice of R and S one now makes sure that there is one light doublet in the theory and that all triplets have mass of order the grand unification mass, M G. This is readily established for the supersymmetric part of the mass matrix; for by looking at the doublet and triplet parts of the relevant terms in the superpotential, after the SU(5) breaking has been done (~ -- CC)),

f (2) = H(2)[_3hxy w + =Tl~ xy ] H(2),

and

f(3) = ~(3) [2Xxy ~o + CYgxy I H(y3) ;

performing rotation (13) and insisting that in the K - K, basis the doublet mass matrix be of the form (one zero eigenvahie)

yields for C~xy

(18)

where we have used eq. (12b). Notice that eq. 0 7 ) carries with it one ffme-tuning.

The triplet mass matrix follows immediately, without any additional fine-tunings, as

One may readily check that det M (3) - - 56o4~4~4 and therefore there are no light triplets if we choose a 4 and ~4 of O(I).

Below M c , our model is then the supersymmetric standard model with one pair of light WS doublets and solft-breaking terms determined by the finiteness conditions (6), (7) and hierarchy stability as we will comment later on.

To determine ~i and/3/is now straightforward. All we need to do is write down the SU(3)c X SU(2)L X U(1)r Yukawas for the quarks and leptons,

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Volume 156B. number 1.2 PIIYSICS LETFERS 13 June 1985

to find (at MG)

a 3 = ht(MG),

%/~a2 = hc(MG) ,

X/ru a 1 = hu(MG) ,

N/rb/33 = hb(MG) ' (20a)

%/s/32 = hs(MG), (20b)

N/d/31 = hd(MG) , (20c)

where t, b, etc. are given in eq. (12c) and (11). To give the numerical values of a i and/3i we need

to RG-run the Yukawa couplings from M G to M w. This is straightforwardly done [9]; using, for example for the top quark, mt(Mw) = ht(Mw) (K) one may write

ct3 = [ E l / 2 g v / ~ (K)] - 1 m t(Mw) (21 )

via (20a). Here E u is the Yukawa RG enhancement factor [10] for the charge 2/3 quark, and mt(Mw) is the effective top quark mass evaluated at M w [ 11 ].

Since we have a light top quark, electroweak breaking proceeds via radiative corrections [9] and (K) ~-- (I(), so that (gO = 128.5 GeV. The values for ai,/3i are easily computed to be

a 1 =6 .57X 10 -6 ,

a 3 = 9 . 8 5 x 10 - 2 ,

/31 = 1.39 X 10 -5 ,

133 = 1.13 X 10 -2 ,

a 2=2.5x 10 -3,

ot 4 = 0.9951 ,

/32 = 2.79 X 10 -4 ,

/34 = 0.9999. (22)

The ansatz we have made for Yukawa couplings, eq. (9), corresponds to diagonal mass matrices without intergenerational mixing. A realistic Kobayashi- Maskawa mixing can be easily introduced by means of the unitary transformation

5j = Uje-5'e, (23)

where U is related to the KM matrix K via

~, il4 U/e = Ki/g/e4 " (24)

The corresponding piece of the superpotential (8) then reads

Kifi/e" 410iY' e H4 " (25)

Let us remark that (23) does not alter the structure of the neutral weak and electromagnetic currents and, in particular, no flavor changing neutral currents are generated.

In the remainder of this letter we will analyze the

radiative SU(2) × U(1) breaking. In ordinary (i.e. non- finite) theories the supersymmetric and soft-breaking parameters at M G are free; this is to be contrasted with the situation in finite models, where the supersymmetric parameters at M G are fixed by the finiteness solution (12) and the soft breakings are determined by the one-loop finiteness conditions (6) and (7) to be

A (MG) = ml/2/M312, °(MG) = ~A 2 (MG),

B(MG) arbitrary, (26)

where m 1 I2 is the gaugino mass at scale M G. However, due to the doublet-triplet fine-tuning

mechanism, there is a contribution to the light doublets mass squared that comes from the soft° breaking terms and is of the form (B - A)m3/2 X MGH2H 2 ; the gauge hierarchy will therefore be spoiled unless we also impose as an initial condition

B(MG) = A(MG). (27)

In this case the gauge hierarchy which already ex- ists at the tree level will be preserved under radiative corrections, since the only communication between the light (doublets) sector and the heavy (triplets) sector is via the heavy SU(3) X SU(2) X U(1) singlet in the adjoint E.

It is worth noticing that, since in (26) and (27) A(MG) °: ml /2 and B(MG) ¢ A(MG) -- l , finite models require non-minimal supergravity and a spe- cific super-Higgs effect [12].

Since we will take a top quark with mass in the expected experimental range [13] of 30-50 GeV, electroweak breaking will proceed through the dimensional transmutation scenario and the results of ref. [9] will apply here. Because of boundary conditions (26) and (27), both the electroweak breaking and the superpartner spectrum are very tightly constrained and in particular one can easily check that the only free parameters left are m 4 (the supersymmetric mass for higgsinos) and m 1/2- Another amusing feature of the spectrum is that the gravitino mass is unobservable at low energies, a property shared with another class of interesting non-minimal supergravity models: no-scale models [ 14].

The condition for radiative breaking provides a relation between (m4/ml/2) and m t which may be translated into a relation between (m4/m" ~ R) and mt

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Volume 15613, number 1,2 PIIYSICS LETTERS 13 June 1985

I I I I

7o

6o

4O

I L ~ L - 0.5 0 0.5 1 1.5

(m4/m'~R ) Fig. l . Relation between m4/m~R and m t provided by radiative SU(2) × U(I) breaking, which may take place at any point on the curve. The apex corresponds to m t = 32 GeV.

displayed in fig. 1 ; here ~R is the supersymmetric partner of the fight lepton and the lightest slepton, so that once it is detected experimentally the model fixes m 4 uniquely from fig. 1, and with it the superpartner spectrum. An interesting feature of the model, also displayed in fig. 1, is that the condition for SU(2) X U(1) breaking predicts a lower bound for the top quark mass given by

4(1/3 + CL) -- (1 + CB) 2 2 m 2 > 3.64

1 + CQ + CU+ C L + (1 + CA)(CA-CB) Mw

(32 GeV) 2

where the C i are the renormalization coefficients of ref. [9].

We have also analyzed the conditions necessary to avoid the appearance of a global minimum breaking color and/or electric charge and found that this minimum never develops.

A typical spectrum for a physical top quark mass of 40 GeV which is consistent with some of the cur- rent interpretations [15] of the UA1 monojet events and the PETRA [16] and MAC [17] bounds is displayed in table 1 ,a

It also is worth mentioning that because of the

finiteness constraints that are operative at Mx, the ratios of masses of negative R-parity particles are parameter-free; typically, one has relations such as for example

2 msq = [(1 +3CD)a2UM/3Ot2(Mw)]m ~ .

This situation, genuine to finite models is to be contrasted with the one in non-finite models where, for example the ratio msq/m ~ is not fixed.

In summary, we have introduced without offering any explanation as to the origin of the finiteness conditions at the pre-SU(5) level, a finite SU(5) model with three generations of matter, which at scales below M x reproduces the minimal supersymmetric standard model with a fight [0(40 GeV)] top quark and radiative electroweak breaking. The triplet- doublet splitting mechanism requires two consecutive fine-tunings. However, the "classical" successes of the minimal model such as mb/m r are kept, but with the bonus that the predictability of the model is con- siderably enhanced; in particular the superpartner spectrum may be parametrized in terms of only one free parameter which can be conveniently c osen to be the gaugino mass m 1/2. Any allusion to the gravitino mass disappears from the spectrum and the lowest minimum is the SU(2) X U(1) breaking minimum. Furthermore, for a light top quark and within the dimensional transmutation scenario, the model sets a lower bound on its mass of 32 GeV.

Note added. After completing this work, we ba- came aware of a new preprint by Bj6rkman, Jones and Raby [18] where they introduce an ansatz equiv- alent to ours and obtain a model with a Ught top quark. We thank L. Ibgt~ez for bringing this paper to our attention.

, 3 This spec t rum shows a gluino that is nearly degenerate with the heavier squarks; the tightest squark mass m~" R is of 0 ( 5 0 GeV) for rn~R ~ 20 GeV, and m~" ~ 0 ( 7 0 GeV). The phot ino has a mass o f O(15 GeV) a n d t h e r e is a slight- ly less massive radiative Higgs which for rn~" R ~ 20 GeV is predicted to have a mass o f around 5 GeV.

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Volume 156B, number 1,2 PIIYSICS LETTERS 13 June 1985

Table 1 The low energy superpartner and Higgs spectrum predicted by the model discussed in the text, for a top quark of 36 GeV at M w. We have used the notation rn~R = m i , rh~ ~ m~/m 2, A = mt/m, e = m/M w and 6 = e - 1/4. The two columns correspond to the

two possible values of rn24 obtained from fig. 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.908 m 2 0.484 m 2

all families th{~ L 1.719 1.719

- z 1 . 0 0 0 1 . 0 0 0 rn~R

rhg first and second families 11.669 11.669

rh ~L 10.844 10.844 ,~R ~1, R 10.739 10.739

third family thL 11.162 11.162

rhbR 10.739 10.739

rh ~'~L 10.496 + A 2 + 4.493A 10.496

tR 10.496 + A 2 - 4.493A 10.496

charged Higgses ^ 2 _ ~ 2 -2 rnH+ - m t l - 5.737 + e 2.890

neutral Higgses rh 2 5.737 2.890 t t l

th~t 5.737 + 1.309 e -2 2.890 2 . 2

tara d 0.058 0.066

gluino th ~- 1 3 . 3 3 6 1 3 . 3 3 6 R

photino rn~/M~ 0.035 + 0.2736 + 0.78962 0.035

ZH-ino rnzH/M~v 1.501 + 1.1786 + 1.67262 1.254 2 2 HZ-ino rnllZ/M~v 1.313 + 0.4086 + 1.53562 1.471

WH-ino they H 3.349 + 0.181e -1 + e -2 0.963

ttW-ino t h a w 3.349 - 0 . 1 8 1 e -I + e -2 0.963

axino th42 1.908 0.484

+ / 2 + 3.596A

+ A 2 - 3.596A

+ e - 2

+ 1.309 e -2

+ 0.0716 + 0.56362

+ 0.0206 - 0.73062

+ 0.8936 + 1.04062

+ 0.504~ -1 + E -2

-- 0.504e -1 + e -2

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[21

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Volume 156B, number 1,2 PHYSICS LETTERS 13 June 1985

[3] D.R.T. Jones and S. Raby, Phys. Lett. 143B (1984) 137; Los Alamos preprint LA-UR-84-2692.

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A sensible finite SU(5) SUSY gut?