A realistic grand unified theory without fine tuning

Volume 122B, number 1

PHYSICS LETTERS

24 February 1983

A REALISTIC GRAND UNIFIED THEORY WITHOUT FINE TUNING ~ B. SATHIAPALAN

California Institute of Technology, Pasadena, CA 91125, USA Received 23 September 1982 Revised manuscript received 4 November 1982

We present a realistic grand unified model whose observable properties are stable against small changes of its bare parameters. It is an O (10) supersymmetric theory with supersymmetry softly broken explicitly through terms of dimensionality <2. It is consistent with all the low-energy phenomenology.

One of the long-standing problems in particle physics is to understand and explain in fundamental rather than existential, terms, the small r a t i o Mw/M P ~ 10-17 of the weak interaction scale to the Planck scale. However, even if, in order to avoid many other well-known problems related to gravity, we do not worry about the Planck scale for the time being and think of it as a natural cut-off, we still have to understand in the context of grand unified theories the small ratio Mw/ M X ~ 10 - 1 4 of the weak interaction to the grand unification scale. This is the well-known hierarchy problem [1]. However, it seems natural to expect that a theory which claims a deeper reasoning about the smallness of this ratio, should first solve the "fine tuning" problem [2]. It should not be dependent upon careful adjustments of parameters ("fine tuning") in order to get the right low-energy phenomenology, i.e. its physical predictions should be stable against small changes of the bare parameters in the lagrangian. It is believed [3] that this is a fundamental property a satisfactory physical theory must have. It is certainly true that in the context of renormalizable field theories, physical masses are usually inputs of the theory and can be kept at their physically observed values by adding appropriate counterterms at each order in perturbation theory. There is no inconsistency in this procedure. But, one should require a phys¢' Work supported in part by the US Department of Energy under Contract No. DE-AC-03-81-ER40050.

ical theory not only to be consistent but also to be natural, i.e. a change 6~/~ ,~ 1 in one or more of its bare parameters to lead to 6PIP ,~ 1 in the predicted value P of any physical quantity, calculated in the presence of a real ultraviolet cutoff. There have been various suggestions in the literature about the dynamical mechanism which generates various mass scales in a theory in which the lagrangian has either one or no mass scale [ 3 - 6 ] . But due to a number of phenomenological requirements a model must meet, it is not yet clear whether any of these can be called fully realistic. Our program here is less ambitious. Leaving aside the question of the exact mechanism which provides us with the two very different mass scales, our starting point will be a lagrangian with large and small mass parameters. The main new feature of our realistic grand unified model is that no fine tuning of its parameters is required for obtaining phenomenologically acceptable predictions. The model is completely stable against any change 6~/~ ~ 1 of its bare parameters. There have been recently some models constructed based on the SU(5) group which address this issue [6,7]. The model we discuss below is a globally supersymmetric [8] grand unified theory based on the gauge group O(10). Its superfield content is: (1) the gauge vector multiplet V a 45 of O(10), (ii) chiral superfieldsM a a 16 of O(10) one for each matter family, "a" being a family index, (iii) one 54 (~0), two

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45's (1~ and El), one 16 (Xl) and one 16 (X2) all chiral, and (iv) three chiral superfields H, H ' and M all 10's. Thus it contains a 45, and a 10 more than the minimal requirements for the correct symmetry breaking O(10) -+ SU(3) X SU(2) X U(1) -+ SU(3) X U(1) [9]. This is the price paid for a solution to the fine tuning problem. We start with the superpotential: t 1 tr so2 +gX t 1 tr tp3 +~M t 2 t r ~2 I~'=[~M + X2 tr Z~o2 + mXlX2 +gXl~X2] + ( ½ M 3 tr E2 + X3 tr ElsoY. l + f H E 1 H ' )

+ #H '2 + ll'H 2 +fabMaPMbH + j ' M 2 . Here M I ,M 2 , M 3 and m ~ 1016 GeV and/a,/a' ~ 102 G e V , / / ' ~ 106 GeV. 1-' stands for the Clebsch-Gordan coefficients for the 10 contained in 16 X 16 of O(10). It should be noted at this point that this superpotential is not the most general that can be written down consistent with supersymmetry and O(10) invariance. For e.g. the term X1E1X2 has been left out. There is no manifest global symmetry that justifies this. However the no-renormalization theorems of global supersymmetry [ 10] ensure that such a term is not generated in any order of perturbation theory. Therefore the requirement of renormalizability does not force such a term upon us, and we do not consider it unnatural to leave out this term. Further the discrete symmetries {21 -+ - E 1 ,H' ~ - H ' } a n d M o - M forbid all SO(10) invariant, off-diagonal wave function renormalizations (other than harmless generation mixing). It should also be pointed out that a variation however small, from zero to something non-zero in the value of a parameter is not actually a small change at all, since it involves an infinite percentage change i.e. 8 X/X >> 1. Thus when we talk about stability against variation of parameters we are referring to the non-zero parameters of the theory. The part within the square brackets is the most general sup ersymmetric O (10) invariant superpotential that can be written using so, Z, X1, ×2 and is reponsible for breaking O(10) to SU(3) X SU(2) X U(1). The terms involving Y'I (within parentheses)are there just to give a mass to the colour triplet parts of the higgs H and H ' while having the doublets massless [ 11 ]. H contains the ordinary Weinberg-Salam higgs fields and is responsible for the weak breaking: SU(2) X U(1) 50

24 February 1983

-+ U(1). H ' has to be included because the antisymmetry of E1 does not allow a coupling HN1H. While we have achieved the right superheavy gauge symmetry breaking, the lagrangian as it stands is phenomenologically unacceptable because supersymmetry is still unbroken. As a consequence we have a host of unwanted light particles, namely, the superpartners of the usual gauge and matter particles. To break supersymmetry softly, to give masses to these particles and to break SU(2) X U(1) we add some explicit nonsupersymmetric terms of dimension two [ 12]. Explicit supersymmetry breaking has been often criticised as being arbitrary and unnatural. However, one should in fact expect to see such terms amongst the low-energy remnants of a spontaneously broken supergravity theory [13] unifying gravity with the other three interactions. Once supersymmetry is broken explicitly by a term of dimension two we are forced to add all the dimension two [ 12,14] terms which are O (10) invariant. The scale of supersymmetry breaking is estimated later in this letter to be O(10 6 GeV) from phenomenological requirements. We proceed to the analysis of the minimum of the potential. We first consider the supersymmetric case, and then the effects of the non-supersymmetric terms. Setting the auxiliary fields equal to zero gives [matter fields have been put equal to zero in eqs. (1)-(6)]: D/i" -- 0 ' F , = 0 : Mlso+ XI(¢2 - ~ tr ~02) + X2(E2 - ~ tr Z 2 ) + X3(~2 _ 1 tr E 2) = 0 ,

(1)

a i'b F 2 = 0 : M 2 Ei/ + X2(ESO + SON)i~ + g-X1 (o J)a X2b = 0

(2) Fzc I =0: M3Y,~J+)t3(Zlso+~o~I) ij

+ ~ f ( H i H 'j - H'iH j) = 0 ,

(3)

Fx, = 0 : r e X 2 a+g~abX2b = 0 ,

(4)

F H=O:

(5)

fElH'+2~/=0,

F H,=O: f H ~ 1 + 2 p ' H ' = 0 .

(6)

SettingD = 0 forces Xl = X2 and [A x , B x ] = 0 = [A,, Be] and A H C~BH,A H, OCBH,, where A and B are the real and imaginary parts respectively of the scalar components of the superfield.

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The following parametrization can be shown to have a non-trivial solution for eqs. (1)-(6) [we use the conventions of ref. [15] for the generators of O(10)] tp = o d i a g ( 3 , 3 , 3 , 3 , - 2 , - 2 , - 2 , - 2 , - 2 - 2 ) , = xT 2 = x(o, 0,0,...0,1)

H T = (0,0 .... 0),

H 'T = (0,0...0),

24 February 1983

ticles get a mass due to loop corrections. On substituting the VEV for Z t in the term H Z 1 H ' it is easy to see that the colour triplets of H and H ' become superheavy while the Weinberg-Salam doublets are unaffected [11]. There are two other possible solutions the completely symmetric one and the one with symmetry group SU(4) × U(1). By adding tr ~02 to the lagrangian we pick this required solution. Now we have to study the effects of the supersymmetry breaking terms which are summarized here:

a

f d4t~[U1

a -a -a

+ U5X1X-1 + U6X2~-2] +

~=O

+ U 3 tr E~ + U 4 H ' / ~ '

fdao(U7MaMa + U3HH)

b -b

+ f d20(X1 M2 + X2Y,2) + ....

-b' -b 0 1 G1 = O1

tr E I ~ 1 + U 2 tr ~

-1 1 1

-1 -1 x, o, o 1 and o are all of O(1016 Gev). E, ¢, Xt, X2 as is clear from the above parametrization leave a residual symmetry which is SU(3) X SU(2) X U(1). This symmetry breaking occurs at 1016 GeV. The next scale is the weak interaction scale where SU(2) X U(1) is broken to U(1) by H. This is done when supersymmetry is broken explicitly. Y~I does not break the symmetry any further. In fact the parametrization chosen for Z 1 is not unique. It has a little group SO(4) X U(3). But, being coupled only to ~p,H and H ' eqs. (1) and (3) are invariant under a larger group SO(4) X SO(6), i.e. any transformation by the elements of [SO(6) SU(4)]/U(3) changes the parametrization but leaves the equations satisfied. Thus at the tree level there are massless particles - pseudo goldstones. They are a 3 + 3- of scalars and their fermionic partners. When supersymmetry is broken, as will be seen, these par-

Here U stands for/a202~ 2 a n d X stands for//202. It is argued below that the supersymmetry breaking scale is O(106) GeV, in particular X1, X2 have to be of that order of magnitude. We add f U8HH ~ -I~ZH2 to break SU(2) X U(1) -* U(1) giving HT a VEV (h0, 0,0,h 3,0 .... 0). Therefore ~t .~ 0 (102 GeV). Similarly, U 1 has to be negative in order that SU(3) X SU(2) × U(1) be the minimum and U 7 is positive to give the scalar partners of the quarks and leptons a positive mass squared, and can be as large as 106 GeV. We have to ensure that the supersymmetry breaking terms do not change h 0 and h 3 by large amounts. These terms affect h0, h 3 indirectly through changes in VEVs of ~0, Z 1 , Z, X. However only those fields which are not superheavy can have their VEVs changed significantly [for a field ff of massM, the change 5~b due to the addition of/12~k 2 to the potential is O(/a2/M)] - thus we have only to look at the pseudo-goldstones in Z 1. Since these are contained in Z~/ with i,j = 4 - 9 , they do not affect the equation for h0, h 3. Note that dimension three terms have to be excluded because they would change the minimum drastically. The spectrum of the theory looks as follows: At the tree level we have O(10)/SU(3) X SU(2) × U(1) gauge fields and their fermionic partners, colour triplet higgs and their fermionic partners, ~0, Z, Y'I (except for the pseudo goldstone bosons), Xl, X2 all with masses O(1016 GeV). The SU(2) X U(1)/U(1) gauge fields and their fermionic partners have masses O (102 GeV) as required by low-energy phenomenology. There are also some scalar doublets which have 51

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masses o f this order coming from H ' . Finally the scalar partners of the quarks and leptons also have masses of this order coming from explicit mass terms. All this has been achieved without any fine tuning of parameters. As mentioned before the supersymmetry breaking terms of dimension two break SU(2) X U(1) give masses to the scalar partners of the quarks and leptons and also pick out the right vacuum. However the fermionic partners of the massless gauge bosons - the gluinos and the photino as well as the pseudo goldstones and their fermionic partners are all massless at the tree level (since explicit supersymmetry breaking fermion mass terms have dimension three and are excluded from the lagrangian). We have to show that loop corrections do indeed give them masses which are realistic. Fig. la shows how the fermion partners of the pseudo goldstone boson get masses. Fig. lb is the corresponding diagram for the boson. Here X = /d202 and U =/a2~202 . For details of calculations involving explicit supersymmetry breaking terms see ref. [12]. The graph in fig. la is estimated to be (ot2/16n2)la4/ rn 3, where ~uis the scale of supersymmetry breaking, a s ~ 0.I, m is the mass o f the particles running around and has to be O(/a) for this approximation to be valid.

I~

co )

I

Y~

YI

3

3

, I I

~

x

24 February 1983

Thus we need g ~ m ~ O(106) GeV for the mass of the colour triplet pseudo goldstone particles to be O(100) GeV. Therefore we conclude that the scale of supersym. metry breaking is 106 GeV which is higher than the weak interaction scale. This introduces a certain amount of tuning, due to radiative corrections, to the tune of three decimal places. However, as we shall see later, this much of fine tuning is required anyway for other reasons. We also find it necessary to have some colour triplet scalars wkich are not superheavy and belong to a real representation of O(10) in order to be able to write a supersymmetry breaking mass term f d20 xM 2 which is required for the graph. The ordinary quark scalars belong to a 16 which is not real and the colour triplet in the 10 of higgs is necessarily superheavy. So we have to introduce a multiplet - say a 10 - having a mass of O(106 Gev). We note in passing that the SU(3) gauge coupling is still asymptotically free. Below the grand unification scale this model as an SU(3) × SU(2) X U(1) theory is similar to many supersymmetric grand unified models already existing in the literature [ 16]. For this value of the supersymmetry breaking scale the gluino mass which comes from fig. 2 is O(104) GeV which is compatible with experiment. (Note that the gluino gets a mass at linear order in the breaking of supersymmetry in contrast to the O'Raifeartaigh models in which the leading one loop contribution is cubic in supersymmetry breaking [ 17] .) sin20 is ~0.23. It should be pointed out that it is possible to construct models with only one 45, but such models seem to require an intermediate scale where SU(2)R is broken. This results in too high a value for sin20. Proton life times are O(1031 y) as in any minimal supersymmetric GUT [ 18]. Finally as in the models of ref. [ 16] to ensure the operation of the "super GIM" mechanism and obtain K 0 - K"0 mixing of the

fd48 U~l~ I

I

Ix 3

3

3

3

~, fd4e x(52 D" V)(D~.v)

I

U

Fig. 1. (a) Graph which gives a mass to the fermionic partners of the pseudo goldstone bosons, m-3 f d4O xx(DaE 1)(Da ~ 1) (#4/m3) ~a~0a. Here E1 and ~ stand for the 3 +3 piece of the 45. (b) The finite part of this graph gives a mass to the pseudo goldstone bosons A and B which are a 3 + 3. f daouEI~-,I ~ u2(A 2 +B2). 52

Fig. 2. Graph which gives a mass to the gluino, m -1 f d40 X tr(/)2D°~V)(D~V) ~ (~21m) tr ~aX~.

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right magnitude the mass splittings Am 2 of the scalar partners of the quarks and leptons must be ~ 1 0 - 3 times their average (mass) 2 [ 19]. This introduces, as mentioned before, a three decimal place tuning of parameters. I would like to thank T. Tomaras for many useful discussions and also for a careful reading of the manuscript. I am also grateful to M. Grisaru and M. Rocek for some discussions and J. Schwarz for encouragement.

References [1] E. Gildener, Phys. Rev. D14 (1976) 1667; S. Weinberg, Harvard preprint (1976). [2] A. Buras, J. Ellis, M.K. GaiUardand D.V. Nanopoulos, Nucl. Phys. B135 (1978) 66. [31 S. Weinberg,Phys. Rev. D13 (1976) 974; D19 (1976) 1277; L. Susskind, Phys. Rev. D20 (1979) 2619; M. Veltman, Acta Phys. Polon. B12 (1981) 437. [4] E. Eichten and K. Lane, Phys. Lett. 90B (1980) 125; S. Dimopoulos and S. Raby, Nucl. Phys. B192 (1981) 353; M. Dine, W. Fischler and M. Srednicki, Nucl. Phys. B189 (1981) 575. [5] S. Weinberg,Phys. Lett. 82B (1979) 387; E. Witten, Phys. Lett. 105B (1981) 267. [6] S. Dimopoulos and H. Georgi, Harvard University preprint, HUTP-82/A046.

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[7] A. Masiero et al., CERN preptint TH3298; B. Grinstein, Harvard University preprint, HUTP-82/A014. [8] J. Wess and B. Zumino, Nucl. Phys. BT0 (1974) 39; B78 (1974) 1; A. Salam and J. Strathdee, Nucl. Phys. B76 (1974) 477; B80 (1974) 499; Y.A. Gol'fand and E.P. Liktman, JETP Lett. 13 (1971) 323; D.V. Volkov and V.P. Akulov, Phys. Lett. 46B (1973) 109; P. Fayet and S. Ferrera, Phys. Rep. 32 (1977) 249. [9] F. Bucella et al., CERN preprint no. TH3212. [10] M.T. Grisaru, M. Rocek and W. Siegel, Nucl. Phys. B159 (1979) 429. [11] S. Dimopoulos and F. Wilczek, Institute for Theoretical Physics at Santa Barbara preprint (UMHE 81-71) 1981. [ 12] L. GirardeUo and M. Grisaru, Nucl. Phys. B194 (1982) 65. [13] J. Scherk and J. Schwarz, Nucl. Phys. B153 (1979) 61; Phys. Lett. 82B (1979) 60. [14] J. Iliopoulos and B. Zumino, Nucl. Plays. B76 (1974) 510. [15] H. Georgi and D.V. Nanopoulos, Nucl. Phys. B155 (1979) 52. [16] S. Dimopoulos and H. Georgi, Nucl. Phys. B193 (1981) 150; N. Sakai, Z. Phys. C l l (1981) 153; R.K. Kaul, Tata Institute preprint TIFR/TH/81-32 (1981). [ 17] J. Polchinski, SLAC preptint SLAC-PUB-2931; J. Polchinski and L. Susskind, SLAC preprint SLAC-PUB2924. [18] J. Ellis, D.V. Nanopoulos and S. Rudaz, CERN preptint TH3184. [19] J. Ellis and D.V. Nanopoulos, CERN preprint TH3216 (1981).

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