- Email: [email protected]
On radiative gauge symmetry breaking in the minimal supersymmetric model
Nuclear Physics B331 (1990) 331-349 North-Holland
O N RADIATIVE GAUGE SYMMETRY BREAKING IN THE MINIMAL SUPERSYMMETRIC MODEL* Giorgio GAMBERINI**, Giovanni RIDOLFI t and Fabio ZWIRNERt*
Department of Physics, UCB, Berkeley, and Lawrence Berkeley Laboratory, Berkeley, USA Received 20 June 1989
Giorgio Gamberini died on September 30, 1989. His two co-authors will deeply miss him as an invaluable friend and collaborator. We present a critical reappraisal of radiative gauge symmetry breaking in the minimal supersymmetric standard model. We show that a naive use of the renormalization group improved tree-level potential can lead to incorrect conclusions. We specify the conditions under which the above method gives reliable results, by performing a comparison with the results obtained from the full one-loop potential. We also point out how the stability constraint and the conditions for the absence of charge- and colour-breaking minima should be applied. Finally, we comment on the uncertainties affecting the model predictions for physical observables, in particular for the top quark mass.
1. Introduction The most conservative solution to the naturalness problem of the standard model is low-energy supersymmetry, which allows one to stay within the framework of a weakly coupled renormalizable gauge theory even when describing particle interactions at energies around and above the electroweak scale. This motivated, in recent years, an intense theoretical activity on supersymmetric extensions of the standard model (for reviews and references see, for example, ref. [1]). The simplest one is the so-called minimal supersymmetric standard model (MSSM), with gauge group SU(3)c × SU(2)L × U(1)v, three generations of quark and lepton chiral superfields ( Q - (U, D ) , U c , D c, L-= (N, E), Ec), two Higgs chiral superfields ( H 1 + H 2o )), and a superpotential of the form ( H ° , H1-), H2 -= ( H 2,
f = huQUCH2 + hoQDCH1 + hELECH1 + I.tH1H2,
(1.1)
where contracted group and generation indices are understood. The above ingredi* This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098, and in part by NSF under grant PHY-8515857. ** Supported by a Fellowship of Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Italy. ~ Supported by a Fellowship of Istituto Nazionale di Fisica Nucleare, Sezione di Genova, Italy. tt On leave from Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Italy. 0550-3213/90/$03.50©Elsevier Science Publishers B.V. (North-Holland)
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ents are enough to specify a globally supersymmetric gauge-invariant lagrangian, *~susy. The complete low-energy effective lagrangian is assumed to be of the general form c~= £¢susy+ £'¢soft, where ~soft is a collection of explicit but soft supersymmetry-breaking terms, including gaugino masses, scalar masses, and scalar interaction terms of the form - (huAuQU~H2 + h.c.) - (hDADQD~H1 + h.c.)
(1.2) - (hEAELE~H1 + h.c.) - (B#H1H 2 + h.c.).
In the MSSM, the origin and the magnitude of the different terms in &°soft are undetermined, but they are usually interpreted as remnants of the spontaneous breaking of local supersymmetry in the underlying fundamental theory (supergravity and eventually superstrings). A very attractive feature of the MSSM is the possibility of "radiative breaking" of the electroweak symmetry [2], through a generalization of the original ColemanWeinberg mechanism [3]. Combined with some plausible assumptions on gauge couplings unification and on the origin of supersymmetry breaking, the requirement of a phenomenologically acceptable electroweak symmetry breaking is extremely powerful, since it rules out large regions of parameter space, and thus considerably enhances the predictiveness of the MSSM. It is therefore very important to understand the approximations and ambiguities involved in the calculations of radiative symmetry breaking. In this paper we present a critical reappraisal of the existing analyses, pointing out their limitations and introducing possible improvements. At the field-theoretical level, we do not introduce any new method: the treatment of ref. [3] is perfectly adequate. On the other hand, we try to clarify its application to the MSSM, since in the literature on the subject [2] one can often find confusing statements. We keep the discussion at the general level, limiting the applications to some illustrative examples. A complete analysis of the full parameter space, including all the theoretical and phenomenological constraints, will be given elsewhere [4]. Our considerations can be extended to the "no-scale" supergravity models [5], where also the scale of supersymmetry breaking is undetermined at the tree-level and can be fixed by radiative corrections, and to non-minimal supersymmetric extensions of the standard model [6]. The structure of the present work is as follows. We conclude this introduction by reviewing the conventional approach to radiative SU(2)× U(1) breaking, which makes use of the renormalization group improved tree-level potential, Vo(Q), evaluated at a renormalization scale Q - mw. In sect. 2 we compare the above method with the minimization of the full one-loop effective potential, VI(Q): we show that Vo(Q) is very sensitive to the choice of the renormalization scale Q, and only for a small and model-dependent range of Q it gives sensible results. Conversely, the results obtained from VI(Q) are remarkably stable for a wide range of Q
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around the electroweak scale, GF 1/2- 300 GeV. In sect. 3 we introduce three representative examples, to illustrate the general considerations of sect. 2 in three qualitatively different situations. We also formulate a practical rule to determine the scale Q at which the use of Vo(Q) is justified. In sect. 4 we re-examine some theoretical constraints on the model parameters, which were originally obtained [2, 7] by requiring that Vo(Q) be bounded from below and that charge and colour be unbroken at the minimum. Taking into account the structure of the one-loop effective potential, we discuss how these constraints should be implemented. The result is that, if correctly applied, they are much less stringent than previously thought, and some of them become completely irrelevant. Finally, in sect. 5 we discuss the uncertainties affecting the model predictions for the physical observables, concentrating on the top quark mass as a particularly interesting example. We review now, to establish language and notation, the conventional approach to radiative S U ( 2 ) × U(1) breaking. To reduce the number of free parameters, it is usually assumed that the running gauge coupling constants (with the U(1) factor properly normalized, gl - 5Vf~ g') unify at a certain super-high scale Mu,
g3(Q )[ Mu = g2 (Q)I MU= gl(Q)[ M~ = gu,
(1.3)
and that the running soft supersymmetry-breaking parameters can be summarized, at the scale Mu, by a universal gaugino mass
M3(Q)IMu = M2(Q)IM,~= MI(Q)[MU= M1/2,
(1.4)
a universal scalar mass m~(Q)IM~ = m2¢(Q)]Mu = m~(Q)IM~ = m2L \(Q~' 11 MU
=mZc(Q)]Mu=rn~(Q)[M =mZ2(Q)]M =rn~o,
(1.5)
and the scalar couplings
Au(Q)[ M. = AD(Q)] M. = AE(Q)] M. = Ao, B(Q)[ Mu = B0"
(1.6) (1.7)
It is also customary to neglect intergenerational mixing and the masses of all quarks and leptons apart from the top quark, so that the only non-negligible terms in the superpotential of eq. (1.1) can be written as
f=ht(UUCH ° - DUCHy)+ ~(H°H ° - HI Hf ),
(1.8)
where now U, D and U c refer only to the third generation. Correspondingly, one
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considers in £,°soft only mass terms and scalar interaction terms of the form -- [ h t A t ( U U C H
0 -
DUCH; ) + h.c.] - [ BIz( H ° H ° - H { H ; ) + h.c.] . (1.9)
To determine the lagrangian £.,9, one has then to specify, besides the known gauge and Yukawa couplings, the six unknown parameters h t(Mu) (related to the top quark mass mr), I~(Mv), Ma/2, mo, A o and B 0. To discuss radiative symmetry breaking, one usually assumes that all the parameters in the MSSM lagrangian are real, and considers the part of the tree-level potential depending on the neutral components of the Higgs fields, Vo= m2]Hl°] 2 + m~lH°] 2 + m ~ ( H ° H ° + h.c.) + 4(g22 + g'2)(]H1°]2- ]H°]2) 2, (1.10) where ml2 _
m 2 H1
+ ~2,
m~ --=m22 + ~2,
rn32= B/I.
(1.11)
The minimization of the tree-level potential of eq. (1.10) is straightforward. In particular, one can always redefine the phases of the Higgs field in such a way that m32 < 0 and the vacuum expectation values (v.e.v.'s) computed from V0 are real and positive. The condition for Vo to be bounded from below is
5 : - rn~ + m 2 - 2]m~] > O,
(1.12)
and the condition for the origin H ° = H ° = 0 to be unstable, so that the minimum of Vo corresponds to non-vanishing v.e.v.'s, is ~
m a2 m 22 - m~ < O.
(1.13)
However, one has to pay attention to the fact that loop corrections to V0 are in general not negligible. The one-loop effective potential V~ has the form
vl = Vo +
vl,
(1.14)
where [3]
1 [
AV1 - 64~2 Str ,///,4 log Q2
2
(1.15)
depends on H1°, H2° through the tree-level squared-mass matrix ./g2. In the above expression for V1, all parameters of the theory are running parameters, functions of the renormalization point Q. They can be evaluated for any Q by solving the
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standard renormalization group equations (RGE), whose form is well known [8] and will not be repeated here. We just remind the reader, for completeness, that the R G E of ref. [8] and the effective potential of eq. (1.14) are written in the Landau gauge (to avoid the appearance of the gauge-fixing parameter) and in the (massindependent) modified minimal subtraction scheme, MS. If one chooses a low renormalization scale Q conveniently, the large logarithmic corrections, proportional to l o g ( M u / Q ) , are reabsorbed in the running parameters of Vo(Q), and AVI(Q) contains only small logarithmic corrections. One therefore neglects AVI(Q) and minimizes Vo(Q), keeping only the sets of boundary conditions for which the v.e.v.'s v° - ( H ° ) , v°-- ( H ° ) give the correct value of the W mass, m w = ( g 2 / v / 2 ) ~ ( v ° ) 2 + ( v ° ) 2 - 81 GeV, and the rest of the particle spectrum and interactions are phenomenologically acceptable.
2. Radiative SU(2) × U(1) breaking
Having introduced the general calculational framework for radiative SU(2) × U(1) breaking, we are now ready to discuss some specific points in more detail. One issue is the validity of the boundary conditions, eqs. (1.3)-(1.7). We first concentrate on the grand unification condition, eq. (1.3). To determine correctly M U, a U =- g ~ / ( 4 ~ ) and the electroweak mixing parameter sin20w, one should take into account threshold effects. First, threshold effects at the unification scale [9] can modify the naive unification condition in eq. (1.3). However, this modification is only required if one performs a two-loop renormalization, as was done in grand-unified models [9], but for one-loop renormalization the boundary conditions in eqs. (1.3)-(1.7) are perfectly adequate. Vice versa, since threshold effects at the unification scale can only be computed in the framework of a more fundamental theory*, in the MSSM one should stick to the one-loop approximation. We therefore assume consistently that eq. (1.3) holds at the one-loop level in the M S scheme. In addition, one has to worry about the thresholds around the electroweak scale. It is well known [10] that mass-independent renormalization schemes do not automatically incorporate threshold effects in the evolution of the running coupling constants. These threshold effects can be introduced, invoking the decoupling theorem [11], by using a step approximation for the fl-functions and suitable matching conditions for the running parameters in the vicinity of each physical threshold. At the one-loop level, one can just change the fi-function coefficients and require that the running parameters are continuous at each physical threshold. A complete treatment, depending on the top quark mass and on the different supersymmetric particle masses, would be very complicated. To simplify drastically the problem, at least as far as the evolution of the dimensionless couplings is concerned, we shall assume that rn t - m w and that * In the case of a supergravity or superstring grand unification, such a theory should also include a mechanism for the spontaneous breaking of local N = 1 supersymmetry.
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the supersymmetric particle thresholds can be summarized by a single parameter Msusy, which in the phenomenologically relevant cases is always larger than m w. Our input parameters will then be [12] t
x,
O/emtmw)
=
1 127.93 + 0 . 3 0 '
(2.1)
and [13] a3(mw)
f~ 11 A+0.012 ~'~0.010 -
(2.2)
For scales Q between m w and Msusy we shall evolve the gauge couplings according to the R G E of the ordinary standard model*, whose one-loop B-function coefficients are given by b ° = - 7, b ° = - 19/6 and b ° = 41/10. Between Msusy and M U we shall evolve the gauge couplings according to the R G E of the MSSM, with b 3 = - 3 , b 2 = 1 and b I = 33/5. To take into account the residual uncertainties, we shall allow Msusy tO vary between m w and 1 TeV (larger values of Msusy would re-introduce the fine-tuning problem that supersymmetry is supposed to cure [15]). The region of Mtj, a U and sin20w(mw) corresponding to the allowed range for O%m(mw), o~3(mw) and Msusy is summarized in fig. 1. Notice how the unification constraint of eq. (1.3) is in remarkable agreement** with the measured value [16] of the electroweak mixing angle! Moving to the remaining boundary conditions, eq. (1.4) is consistent with the unification relations in eq. (1.3), whereas eqs. (1.5)-(1.7) are usually motivated by the assumption that the super-Higgs effect takes place in a " h i d d e n " sector of the theory, which interacts with the "observable" sector only gravitationally and is therefore flavour-blind. Proceeding further in the examination of the standard approach, the effect of intergenerational mixing on the R G E and on the effective potential can be certainly neglected in a first approximation, and in any case its inclusion does not involve any conceptually new problem and any radical modification of the results. A slightly more delicate assumption is the neglect of the flavour-diagonal bottom Yukawa coupling, h b, with respect to the corresponding top Yukawa coupling, h t : this approximation is justified only if (H°)/(H°) << m t / m b , and there might be some marginal region of parameter space which does not satisfy this condition. Even in this case, however, the explicit introduction of h b in the R G E and in the effective potential is straightforward and does not introduce any qualitatively new feature. An extremely important assumption is that all fields, apart from the real neutral components of H a and H 2, have zero v.e.v.'s. The consistency of this assumption should be tested for each set of boundary conditions, and will be discussed in some * The c o r r e s p o n d i n g R G E for the top q u a r k Y u k a w a c o u p l i n g can b e found, for example, in ref. [14]. * * I n fig. l , sin 2 0 w ( m w ) is evaluated in the ~ scheme, a n d should not be confused with sin 2 0 w ~- 1 2 rnw/m ~, w h i c h is the definition used in ref. [16]. However, the difference b e t w e e n the two is small a n d c a l c u l a b l e [12].
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detail in sect. 4. The above assumption allows one to concentrate on the part of the effective potential which depends only on the real neutral components of H 1 and H 2. Notice, however, that whereas V0 involves only the mass parameters of the Higgs sector (m 2, m~, m~), AV1 depends on the masses of all fields in the theory (vector bosons, spin- ~2 fermions and spin-0 bosons). This will make the analysis of radiative s y m m e t r y breaking slightly different from the usual Coleman Weinberg case, where only a single scalar field 0 is involved. We n o w c o m e to the most crucial assumption of all, the discussion of which is the central subject of this paper: that for a convenient choice of the renormalization scale, Q, one can neglect the logarithmic contribution AVI(Q) to the effective potential in the minimization of Vo(Q) and in the determination of the low-energy p h e n o m e n o l o g y . In previous calculations [2,5,6], different choices were made: Q_ = row, Q_ = Gv 1/2, Q = m~, etc. However, it was realized from the very beginning
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(e.g. by Kounnas et al. in ref. [2]) that the minimum of V0(Q), characterized by the v.e.v.'s v ° and v°, can depend strongly on the choice of the scale Q at which to compute. This fact will be illustrated graphically in the examples of sect. 3. At large scales Q, both 5 ~ and ~ , defined in eqs. (1.12)-(1.13), are positive*. Going down with the scale Q, one can identify a certain value Q b , such that ~ < 0 for Q < QbEventually, there will also be a scale Qs such that 5°< 0 for Q < Qs. It can be easily proven that Qs _< Qb must hold. For Q > Qb one has v° = v° = 0, then v ° and v° increase monotonically as long as Q moves from Q b tO Qs, where the tree-level potential becomes unbounded and v° and v° escape to infinity. One sees therefore that, in the finite interval (Qs, Q b ) , which is usually not larger than a few TeV in the cases of interest, the v.e.v.'s computed from the RGE-improved tree-level potential vary from zero to infinity! The correct way of determining the ground state of the theory is to take into account also the field-dependent part AV1 in the effective potential V1 (the latter is independent of Q by definition, at least up t o O ( h 2) corrections). One then expects the v.e.v.'s vl and v2, determined from V1, to depend on the scale Q only via wave-function renormalization effects, given by [8] 0 log v~ 1 0 log Q - 64~ 2 (3g2 + g,2),
01ogv 2 1 g , 2 _ 12ht2) O logQ - 64~ 2 (3g~ +
(2.3)
(2.4)
We can see that, even in the case of a large top Yukawa coupling ( h t - 1) and for a variation of Q of one order of magnitude, the variation of v2 due to wave-function renormalization is about 4%, whilst the corresponding variation of v I is always around 0.5%. Before moving to the examples of sect. 3, we would like to discuss a subtlety in the definition of AVv At a given scale Q, even in the vicinity of the tree-level m i n i m u m (v °, v°), there can be field configurations corresponding to some negative eigenvalues for the mass matrix jg2, which would cause the appearance of an imaginary part in AV1. This problem arises because of the non-convexity of the tree-level potential V0 when gauge symmetry is spontaneously broken. As emphasized in ref. [17], one should not use eqs. (1.14) and (1.15) in the region of field configurations where the tree-level potential V0 is non-convex, since the effective one-loop potential in this region would be the convex envelope of eq. (1.14). In our case, since we are only interested in finding the minimum of the effective potential, this problem may be circumvented in two different ways. * In fact, boundary conditions such that both 5p and ~ are negative at Mtj would destabilize the tree-level potential and induce a minimum of the full effective potential with v.e.v.'s of order M U. Moreover, due to the boundary conditions of eq. (1.5), 5p and 5~ have the same sign at M v.
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The first way consists in using eqs. (1.14)-(1.15) to find the correct one-loop minimum perturbatively, expanding around the tree-level solution (v °, v°). Accordingly, one finds
v~=v°i+Avi
( i : 1,2),
(2.5)
where
02Vo AUi ~ __
t -1
OAF1
(2.6)
I
OUiOVj[v~vo ]
OVj
l)~V O"
Doing so, one never encounters the problem of negative squared masses, since in eq. (2.6) j/g2 is evaluated at the tree-level minimum, and therefore is automatically positive semi-definite. However, we want to stress that the perturbative expansion in eqs. (2.5) and (2.6) can fail, and indeed it does in many relevant cases. The reason is that the one-loop correction AVI(Q) to the potential is not necessarily small, unless one carefully chooses a scale Q which one does not know a priori. Only the sum VI(Q) = Vo(Q)+ AVI(Q) is scale-independent (up to order O(h 2) corrections): it does not make much sense to expand perturbatively around a solution v° which, as we have seen, depends so strongly on the scale at which it is computed, or even does not exist for scales Q < Qs. The second approach, which is the one we have chosen to deal with the examples of sect. 3, is the following. We know that at any acceptable minimum all the eigenvalues of ~ , 2 corresponding to squarks and sleptons must be positive, to avoid the spontaneous breaking of colour, electric charge or lepton number (the squared masses for all the fermions and vector bosons are positive by definition). However, in the Higgs sector we have some zero eigenvalues corresponding to the Goldstone modes, which appear when we compute the squared Higgs mass matrix for the values (v °, v°) which minimize the tree-level potential V0. When we try to find the true minimum (Vl, v2) of the one-loop potential V1, negative eigenvalues for the tree-level squared-masses in the Higgs sector can in general appear. However, since these negative eigenvalues are of order O(h) (they come from a one-loop correction to (v °, v°)), we can safely neglect the corresponding terms in AV1 (which would come out of order O(h3)) without spoiling the one-loop approximation of the complete potential, which, as we mentioned before, is renormalization group invariant up to order O(h 2) corrections. In practice, we can take the absolute value of ~,2 inside the logarithm, or, equivalently, neglect the contribution of the negative mass eigenvatues.
3. Examples In this section, we present some examples to illustrate the previous considerations. We have chosen three representative cases, corresponding to the three qualitatively different situations depicted in figs. 2a-c. They can be easily inter-
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preted in terms of the scales Q b , Qs and Msu~y introduced before*, and are characterized by the three different orderings (a)
M~o~y< Q~ < Qb,
(b)
Q~
(c)
Q~ < Q b < msusy,
(3.1)
The scales Qs and Q b c a n be read off fig. 2 as the values of Q where the tree-level v.e.v.'s (v °, v°), represented by dashed lines, go to infinity and to zero, respectively. A striking feature of all three cases is the good Q-independence of the one-loop v.e.v.'s (v 1, v2), represented by solid lines, specially when compared with the wild behaviour of the tree-level result. The dotted lines give the v.e.v.'s (v~, vp) obtained with the perturbative expansion of eqs. (2.5) and (2.6), and show that such an expansion is good only in case (b), and even there for a rather small range of Q. The fact that v p, v~ ~ ~ for Q ~ Q~ can be explained by noticing that the smaller eigenvalue of the matrix (8 2Vo/8 vi 8 vj)],,=v0 appearing in eq. (2.6) vanishes in that limit. We now discuss the single cases in more detail. Case (a) is an example of SU(2) × U(1) breaking via a genuine Coleman-Weinberg mechanism: Vo(Q) becomes unbounded below a scale Q s - 80 GeV, whereas for Q = Qs it is flat in the direction v 1 = v2. This degeneracy is removed by the one-loop correction AVe, and the potential V1 develops a symmetry-breaking minimum with v I = V2. Since all the mass parameters in AV~ are much smaller than the scale under consideration (Msu,~y - 25 GeV << Q~), v 1 and 02 are essentially determined by the renormalization scale itself (dimensional transmutation). The v.e.v.'s are approximately given by v-QJg, where g is a typical gauge coupling: in fact, ~ 2 ( v ) - Q 2 at the minimum, and for M,o,y ~ 0 it is Jg2(v) - g2v2. In this case we can also see that Q, defined as the scale where (v 1, v2) -- (v °, v°), is very close to Q~. However, the rapid variation of (v °, v°) with the scale makes the tree-level potential approach useless. Case (a) has been thoroughly discussed in the literature [2, 5]: although theoretically appealing, this situation is ruled out by the present experimental bounds on the super-particle masses [18]. In case (b) the scale Q, where the tree-level and one-loop results coincide, is essentially M ~ y itself. In fact, it is sufficient to remember that m 2 << Q 2 ~ im 2 log(m 2 / Q 2) I << Q 2 t o understand that the dominant contributions to J Vl come from the largest field-dependent ~ 2 eigenvalues, typically the squark masses. In our case, it is at Q - - 2 0 0 G e V - Ms~sy that AV1 in eq. (1.15) becomes negligible. * We recall that M~u~v is the typical scale of the supersymmetry-breaking masses, and in practice coincides with an "average" squark mass, since squarks give the most important field-dependent contribution to AV1 in eq. (1.15).
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G. Gamberini et al. / Gaugesymmetry breaking 250
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Fig. 2. Vacuum expectation values computed from the tree-level potential ((v°, v°), dashed lines), the perturbative formulas of eqs. (2.5) and (2.6) ((v p, VzP),dotted lines) and the one-loop effective potential ((vl, v2), solid lines), as functions of the renormalization scale Q. Cases (a), (b) and (c) correspond to the following sets of boundary conditions:
(a)
M 1 / 2 = 10 G e V ,
/l(Mu)
= 2 GeV,
ht(Mu)
(b)
M r / 2 = 100 G e V ,
l~(Mu)
= 20 G e V ,
ht(Mu)=O.165,
(c)
M,/2 = 500 GeV,
t~(Mu)= 100 GeV,
In all cases we have taken
= 0.160,
ht(Mu)= 0.140.
m 0 = A 0 = B 0 = 0, a U = 0 . 0 4 a n d M U = 1.335 × 10 ]6 G e V .
N o t i c e that the prescription of c o m p u t i n g the tree-level m i n i m u m at Q = m w (used in s o m e of the previous analyses) would give a c o m p l e t e l y wrong result: it just happens in this case that at Q - - m w the tree-level potential is u n b o u n d e d from below. On the other hand, minimizing correctly V1, one finds a p h e n o m e n o l o g i c a l l y acceptable breaking of SU(2) × U(1), with the right W and Z masses and supersymmetric particle masses above the current experimental limits. For example, one finds
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a gluino mass rn~ = 255 GeV, squark masses mq between 170 and 275 GeV, and slepton masses m~L = 82 GeV, meR = 55 GeV, m~ -- 43 GeV; the lightest neutralino weighs m~, -- 20 GeV and the lightest chargino m¢, = 50 GeV; the top quark mass is m t - - 8 2 GeV and the lightest neutral Higgs has a mass m H --35 GeV. One can also see from fig. 2b that the ratio v2/v 1 is small enough to make the approximation h b << h t reliable. Finally, case (c) is an example of unbroken SU(2) × U(1), despite the fact that there are scales Q at which Vo(Q) admits symmetry-breaking minima. This case corresponds t o M s u s y - 1500 GeV > Q b - 600 GeV. What happens here is the following. When one evolves the parameters of the theory from Q = Mr; down to lower energies, using the RGE of the supersymmetric theory, one eventually reaches Q = Msusy. For Q < Ms,sy the supersymmetric particles, whose masses are larger than the renormalization scale, are effectively decoupled from the renormalization group evolution of the different parameters, in accordance with the decoupling theorem [11]. If one wanted to keep using the tree-level potential below the scale M~usy, one should use the RGE of the non-supersymmetric theory, which, however, would modify very little the mass parameters appearing in Vo(Q). As a result, these parameters would be frozen at the values assumed for Q = Msosy, where not even Vo(Q) breaks the symmetry, as apparent from fig. 2c. The fake symmetry breaking in Vo(Q), described by the dashed line, is due to the fact that, following the approach of most of the previous studies, we kept using Vo(Q) without including the decoupling of the supersymmetric particles in the R G E evolution. Coming back to our approach, notice that, although we did not explicitly use the "decoupling" in determining the evolution of the running parameters, the inclusion of the one-loop corrections to the potential takes care of it automatically, since VI(Q) never breaks the symmetry, even for Q < Msu~y.
4. Constraints In the discussion of sects. 2 and 3, we have assumed that all scalar fields other than H ° and H2° have vanishing v.e.v.'s. We now consider the constraints that should be satisfied for the consistency of this assumption, with the aim of clarifying some points that have been frequently misunderstood. In the approach which makes use of the renormalization group improved tree-level potential*, Vo(Q), one can formulate a series of consistency conditions. For example, the tree-level squared-mass matrices for all the fields but the Higgs doublet should be positive definite. Moreover, the trilinear scalar couplings in eq. (1.2) * For the discussion of this section, of course, we extend the definitions of Vo(Q) and AVI(Q) given in eqs. (1.10) and (1.15), respectively, to include the dependence on all the scalar fields of the MSSM.
G. Gamberini et al. / Gauge symmetry breaking
343
should satisfy the constraints [7]
< 3(m
+
+ m2),
(4.1)
< 3(m
+
+
(4.2)
A~ < 3(m 2 + m~c + m2),
(4.3)
in order not to break colour or electric charge*. Further conditions to avoid instabilities along charge and colour breaking directions have been formulated, for example [7]
m 2H2 + m 2 > 0 .
(4.4)
The above conditions, eqs. (4.1)-(4.4), are in general necessary conditions (a rigorous determination of sufficient conditions is still lacking), and they strongly constrain the parameter space of the MSSM. We will show below how they should be correctly implemented, by taking into account the effects of the field-dependent part AVI(Q) of the one-loop potential, and in doing so we shall modify, sometimes drastically, some statements which can be found in the literature. As a preliminary exercise before the discussion of eqs. (4.1)-(4.4), we return for a moment to the condition of eq. (1.12), which is necessary for the boundedness of Vo(Q). It is clear from the previous discussion that the condition of eq. (1.12), 5°> 0, cannot be satisfied at all scales, since we know that, for realistic values of the parameters, there always exists a scale, Qs, below which the tree-level potential becomes unbounded from below. In the literature, it is often required that ,9°>/0 for all scales Q between m w and M u. However, this cannot be the correct criterion either: we have presented an explicit example (the one in fig. 2b) in which this criterion would have given a wrong answer. The explanation is clear if one remembers the previous analysis of fig. 2b. When the instability of Vo(Q) is reached, just below the scale Qs, v° and v° go to infinity. However, the correct v.e.v.'s do not really go to infinity, since the one-loop correction AVI(Q) stabilizes V1 and does not allow v 1 and v2 to reach values for which ~ ' 2 ( v ) > > Qs2. Even if one insists on using Vo(Q), but includes the decoupling, v° and v° never run away to infinity, because ~ ' 2 ( v ° ) = Q2 for Q > Q~, and there the evolution of the mass parameters of the tree-level potential must be stopped. We can therefore identify Q as the value of Q for which the largest field-dependent eigenvalues of Jg2[v°(Q)] are of the order of Q 2 . The criterion of eq. (1.12) for the boundedness of Vo(Q) should therefore be interpreted as a criterion for avoiding v.e.v.'s larger than desired, and should be correctly * Actually, the first condition is not applicable in the case of the top quark coupling, but this will not affect the conclusions of the following discussion.
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G. Gamberini et aL / Gauge symmetry breaking
implemented as follows: (i) Check if 5P> 0 at Q = M U. If not, this means that v I and v2 are of order Mr/ and thus phenomenologically unacceptable. (ii) If 50> 0 at M U, evolve down the running parameters until the scale Q reaches Q,~ or M~usy, and stop there the RGE evolution. If Qs has been reached, then u1 ~---U2~---Qs/g. Otherwise, 01 and v2 are obtained by minimizing Vo(Q) at 0 - Msusy. Using the values (vl, v2) obtained in this way, one can then check if a phenomenologically acceptable particle spectrum is obtained. It is important to stress the fact that the one-loop potential VI(Q) takes into account all this procedure automatically, and we do not have to worry about fine tuning the scale Q at which to minimize it, since the result is Q-independent as long as AV t is not too large. Now we come to the problem of the possible existence of unwanted chargea n d / o r colour-breaking minima, along certain directions in the scalar field space. If one of the conditions in eqs. (4.1)-(4.4) is violated, Vo(Q) develops these undesired minima. In principle, to see if these new minima survive the one-loop corrections, one should compute AV 1 as a function of all the scalar fields. This, however, would be a terribly complicated algebraic task. Instead, we can formulate a procedure analogous to that applied to the condition of eq. (1.12), exploiting the fact that the unwanted minima usually have large field v.e.v.'s inversely proportional to a small Yukawa coupling, e.g. 0 - Ai/hi. One should check if Vo(Q) has any unwanted minimum for scales Q between M U and Q. Such a minimum does really occur (and is not washed out by the one-loop corrections), only if there exists a scale Qc (Q < Qc < Mu) such that the largest field-dependent eigenvalues of ,///'2[~b(Qc) ] are of the order of Q2 (a rough estimate of these eigenvalues is given by max{ g2(q~)2, M 2 y )). The above criterion is another way of stating that a minimum of Vo(Q) is trustable only if AVI(Q) is small. The smallness of AV 1 depends on the scale Q, but so does also the existence of a given minimum of V0. If no scale Q~ can be found, where both conditions are simultaneously satisfied, then the would-be minimum does not exist. To illustrate the above considerations, we examine the condition in eq. (4.4), introduced in the last paper of ref. [15]. We reproduce here, for the reader's convenience, the essential points of its derivation. Along the field direction
a2/, (H°) -
hD' ,
a/x (DL,) = (DR,> -- hD ,,
a~/1 + a2/~ (~L,) --
hD'
,
(4.5)
where a is an arbitrary parameter, one can encounter an instability of the tree-level
345
G. Gamberini et al. / Gauge symmetry breaking potential, which now reads
Vo=
ho,] [a2( m2N2
m2D,
m2
.
(4.6)
Then, if (m22 + m2j) < 0, for a large enough V0 assumes arbitrarily large negative values. This is how the constraint of eq. (4.4) was derived. It was also claimed that it should be satisfied for any Q between m w and M v. If this were true, one could set a strong upper limit on the top quark mass in the restricted class of models where the only seed of supersymmetry breaking at M v is a universal gaugino mass. First of all we notice that, if we include one-loop corrections to the potential, the p a r a m e t e r a can no longer be arbitrarily large, but is constrained by the requirements
'lgz(O°)l ~< Q, Ig3(DL,)I = ]g3(D,,)[ ~ Q,
(4.7)
Ig2<~L,)l ~< Q. Using eq. (4.5), we can summarize the constraint of eq. (4.7) as follows
a ~ min
, /xg2
.
(4.8)
/xg3
Secondly, for realistic values of the parameters the condition in eq. (4.4) can only be violated at scales Q around the electroweak scale. As a consequence, eq. (4.8) tells us that a cannot be very large and that the value of the potential in eq. (4.6) can hardly become negative. We have checked that, under the restriction of eq. (4.8) and varying the values of the boundary condition at M v in a considerable range, the potential in eq. (4.6) never becomes negative, and the constraint of eq. (4.4) is ineffective. The constraints of eqs. (4.1)-(4.3) can be discussed in a similar fashion. Without presenting a detailed analysis, we just want to stress that, since the unwanted charge- and colour-breaking minima would be of order q~- A i / h i, eqs. (4.1)-(4.3) need not be satisfied for all scales between M u and m w, but only at Q - g4 - gAi/hi.
5. Predictions for the physical masses In this last section we discuss the predictions for the physical masses of the MSSM, paying particular attention to the W mass (which sets the electroweak scale and is used as a constraint) and to the top quark mass (which is a hot experimental issue at the time of this writing).
G. Gamberini et aL / Gauge symmetry breaking
346
T o begin with, we discuss the relation between the running masses in the MS scheme, m w ( Q ) and mr(Q), and the corresponding physical masses mw and mtThe running of v I and v2 with the scale, although negligible in determining the ground state through the minimization of the one-loop potential, may become important when dealing with a computation of physical masses. To calculate the physical mass of a particle in the one-loop approximation, starting from the running mass and coupling constant parameters of the low-energy effective theory, one should in principle compute the one-loop propagator for that particle, using the same renormalization prescription which has been used in deriving the running parameters: the physical mass then is the position of the pole of that propagator. However, for practical purposes, and in the MS scheme, a good approximation [19] is achieved if we define the physical masses N, in terms of the running masses m(Q), by the formula ~ --- m(Q = ~). For example, we have m2(Q)
= 1 2
(5.1)
+ 4(o)],
and mt(Q)
=ht(Q)vz(Q),
(5.2)
where by v i (Q) we mean the minimum of the one-loop potential V1 calculated at the scale Q (its Q-dependence is only through wave-function renormalization, as discussed in sect. 2). Therefore we have
-
m2~2
g2 mw){V (0) 2 ~,
O[o~(Q)+vZ(Q)] + v2(0) +
mt = ht(mt)[v2(0) + 0- log Q
0 log Q
0 l°g
~} 0 l°g
'
(5.3)
"
These are the formulae which should be used, in the absence of a complete one-loop computation, for the comparison with the experimental values (eq. (5.3)) and limits (eq. (5.4)). We now work out an application of eqs. (5.3)-(5.4), in order to convince the reader that the considerations of the present paper have not onl 7 academical, but also practical relevance. We fix a set of boundary conditions at the unification scale, for example* aU= 0.0394, M u = 1.24 × 1016 GeV and
M1/2=
200 GeV,
* These values of a t M~u~y = 375 GeV.
/~(M v) = 40 GeV,
m o = A o = Bo = 0.
(5.5)
and of M U are o b t a i n e d from the central values in eqs. (2.1) and (2.2) with
G. Gamberini et al. / Gauge symmetry breaking
347
The requirement of a correct symmetry breaking, with Nw = 81 GeV, allows then for a unique value of the top quark mass N t. We compare the numerical predictions for N t obtained in the conventional approach and in the approach outlined in this paper. In the first case, one uses supersymmetric RGE to evolve all parameters from M U to m w, and minimizes Vo(Q) at Q=m w. Then one equates mw with
gz(mw)~(v°2(mw) + v°2(mw))/2,
and mt with Iht(mw)v°(mw)[. In this way one obtains m r - - 6 9 GeV for the given boundary conditions. In the second case ( M s u s y - 3 7 5 GeV), one uses supersymmetric RGE between M U and Msu~y, and ordinary R G E below Msu~y. One then minimizes the full one-loop potential at a scale Q - M~u~y, and uses eqs. (5.3) and (5.4) for the computation of Nw and mr. Doing so, one obtains that ~i w -- 81 GeV implies, for the same boundary conditions as above, m t ~-- 81 GeV. Therefore, the difference between the two methods gives a significant discrepancy in the predictions for the top quark mass. If one day some fundamental theory (superstrings?) will determine for us the boundary conditions at Mu, one should pay attention in determining its low-energy predictions! More realistically, one should be careful in interpreting the present experimental limits as constraints in the space of boundary conditions. As a second application of eqs. (5.3) and (5.4), and of the results presented in fig. 1, we test how the prediction for Nt is affected by the uncertainties in the knowledge of aem(mw) and of c%(mw), eqs. (2.1) and (2.2). For example, we fix the boundary conditions as in eq. (5.5), which implies that M~u~y- 375 GeV, and we vary aem(mw) and % ( m w ) in the interval allowed by eqs. (2.1) and (2.2). Correspondingly, we span a certain interval of (c~, My) values: in turn, they lead to different predictions for the value of mt which induces the correct radiative breaking of the electroweak symmetry 78 GeV ~< ~ t ~< 84 GeV.
(5.6)
This is another uncertainty which should be kept in mind when examining the low-energy consequences of a given set of boundary conditions. This kind of considerations can be extended to all the low-energy predictions of the MSSM: a complete treatment, however, is outside the scope of the present paper and will be given elsewhere. The authors would like to thank R. Barbieri, C.M. Becchi, S. Coleman, J. Ellis and M.K. Gaillard for discussions and suggestions, and in particular Massimo Porrati for exhausting but enlightening discussions. References
[1] H.-P. Nilles, Phys. Rep. 110 (1984) 1; H.E. Haber and G.L. Kane, Phys. Rep. 117 (1985) 75; A.B. Lahanas and D.V. Nanopoulos, Phys. Rep. 145 (1987) 1
348
G. Gamberini et al. / Gauge symmetry breaking
[2] L.E. Ib~raez and G.G. Ross, Phys. Lett. Bll0 (1982) 215; K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, Prog. Theor. Phys. 68 (1982) 927; 71 (1984) 413; L. Alvarez-Gaum~, M. Claudson and M.B. Wise, Nucl. Phys. B207 (1982) 96; J. Ellis, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B121 (1983) 123; L.E. Ib~raez, Nucl. Phys. B218 (1983) 514; L. Alvarez-Gaum6, J. Polchinski and M.B. Wise, Nucl. Phys. B221 (1983) 495; J. Ellis, J.S. Hagelin, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B125 (1983) 275; L.E. Ib~gaez and C. Lopez, Phys. Lett. B126 (1983) 54; Nucl. Phys. B233 (1984) 511; C. Kounnas, A.B. Lahanas, D.V. Nanopoulos and M. Quir6s, Nucl. Phys. B236 (1984) 438; U. Ellwanger, Nucl. Phys. B238 (1984) 665; L.E. Ib~raez, C. Lopez and C. Mufioz, Nucl. Phys. B256 (1985) 218; A. Bouquet, J. Kaplan and C.A. Savoy, Nucl. Phys. B262 (1985) 299; H.J. Kappen, Phys. Rev. D35 (1987) 3923; M. Drees, Phys. Rev. D38 (1988) 718; H.J. Kappen, Phys. Rev. D38 (1988) 721; B. Gato, Z. Phys. C37 (1988) 407; G.F. Giudice and G. Ridolfi, Z. Phys. C41 (1988) 447; S.P. Li and H. Navelet, Nucl. Phys. B319 (1989) 239 [3] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888; S. Weinberg, Phys. Rev. D7 (1973) 2887 [4] J. Ellis and F. Zwirner, preprint U C B / P T H / 8 9 / 1 5 , LBL-27310 (1989) [5] J. Ellis, A.B. Lahanas, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B134 (1984) 429; J. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B241 (1984) 406; B247 (1984) 373 [6] J.-P. Derendinger and C.A. Savoy, Nucl. Phys. B237 (1984) 307; J.-P. Derendinger, L.E. Ib~i~aezand H.-P. Nilles, Nucl. Phys. B267 (1986) 365; P. Binetruy, S. Dawson, I. Hinchliffe and M. Sher, Nucl. Phys. B273 (1986) 501; J. Ellis, K. Enqvist, D.V. Nanopoulos and F. Zwirner, Nucl. Phys. B276 (1986) 14; L.E. Ib~glez and J. Mas, Nucl. Phys. B286 (1987) 107; G. Costa, F. Feruglio, F. Gabbiani and F. Zwirner, Nucl. Phys. B286 (1987) 325; J. Ellis, J.S. Hagelin, S. Kelley and D.V. Nanopoulos, Nucl. Phys. B311 (1989) 1; J. Ellis, J.F. Gunion, H.E. Haber, L. Roszkowski and F. Zwirner, Phys. Rev. D39 (1989) 844 [7] J.M. Fr~re, D.R.T. Jones and S. Raby, Nucl. Phys. B222 (1983) 11; M. Claudson, L.J. Hall and I. Hinchliffe, Nucl. Phys. B228 (1983) 501; J.-P. Derendinger and C.A. Savoy, Nucl. Phys. B237 (1984) 307; M. Drees, M. Gliick and K. Grassie, Phys. Lett. B157 (1985) 164; J.F. Gunion, H.E. Haber and M. Sher, Nucl. Phys. B306 (1988) 1; H. Komatsu, Phys. Lett. B215 (1988) 323 [8] R. Barbieri, S. Ferrara, L. Maiani, F. Palumbo and C.A. Savoy, Phys. Lett. Bl15 (1982) 212; M.B. Einhorn and D.R.T. Jones, Nucl. Phys. B196 (1982) 475; J.-P. Derendinger and C.A. Savoy, Nucl. Phys. B237 (1984) 307; B. Gato, J. Leon, J. P~rez-Mercader and M. Quir6s, Nucl. Phys. B253 (1985) 285; N.K. Falck, Z. Phys. C30 (1986) 247 [9] S. Weinberg, Phys. Lett. B91 (1980) 51; L.J. Hall, Nucl. Phys. B178 (1981) 75 [10] D.A. Ross, Nucl. Phys. B140 (1978) 1; T. Goldman and D.A. Ross, Nucl. Phys. B171 (1980) 273; P. Bin6truy and T. Schiicker, Nucl. Phys. B178 (1981) 293; B178 (1981) 307; I. Antoniadis, C. Kounnas and C. Roiesnel, Nucl. Phys. B198 (1982) 317; W. Bernreuther, Ann. Phys. (N.Y.) 151 (1983) 127 [11] K. Symanzik, Commun. Math. Phys. 34 (1973) 7; T. Appelquist and J. Carazzone, Phys. Rev. D l l (1975) 2856 [12] W.J. Marciano, in Proc. Aspen Winter Physics Conf. 1985, Ann. N.Y. Acad. Sci. 461 (1986) 367
G. Gamberini et al. / Gauge symmet~ breaking
349
[13] R.M. Barnett, I. Hinchliffe and W.J. Stirling, in Review of particle properties, Phys. Lett. B204 (1988) 96 [14] C.T. Hill, Phys. Rev. D24 (1981) 691; M.E. Machacek and M.T. Vaughn, Phys. Lett. B103 (1981) 427 [15] J. Ellis, K. Enqvist, D.V. Nanopoulos and F. Zwirner, Mod. Phys. Lett. A1 (1986) 57; R. Barbieri and G.F. Giudice, Nucl. Phys. B306 (1988) 63 [16] U. Amaldi, A. B6hm, L.S. Durkin, P. Langacker, A.K. Mann, W.J. Marciano and A. Sirlin, Phys. Rev. D36 (1987) 1385; G. Costa, J. Ellis, G.L. Fogli, D.V. Nanopoulos and F. Zwirner, Nucl. Phys. B297 (1988) 244 [17] Y. Fujimoto, L. O'Raifeartaigh and G. Parravicini, Nucl. Phys. B212 (1983) 268; A. Dannenberg, Phys. Lett. B202 (1988) 110 [18] C. Albajar et al. (UA1 Collaboration), Phys. Lett. B198 (1987) 261 [19] J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77
O N RADIATIVE GAUGE SYMMETRY BREAKING IN THE MINIMAL SUPERSYMMETRIC MODEL* Giorgio GAMBERINI**, Giovanni RIDOLFI t and Fabio ZWIRNERt*
Department of Physics, UCB, Berkeley, and Lawrence Berkeley Laboratory, Berkeley, USA Received 20 June 1989
Giorgio Gamberini died on September 30, 1989. His two co-authors will deeply miss him as an invaluable friend and collaborator. We present a critical reappraisal of radiative gauge symmetry breaking in the minimal supersymmetric standard model. We show that a naive use of the renormalization group improved tree-level potential can lead to incorrect conclusions. We specify the conditions under which the above method gives reliable results, by performing a comparison with the results obtained from the full one-loop potential. We also point out how the stability constraint and the conditions for the absence of charge- and colour-breaking minima should be applied. Finally, we comment on the uncertainties affecting the model predictions for physical observables, in particular for the top quark mass.
1. Introduction The most conservative solution to the naturalness problem of the standard model is low-energy supersymmetry, which allows one to stay within the framework of a weakly coupled renormalizable gauge theory even when describing particle interactions at energies around and above the electroweak scale. This motivated, in recent years, an intense theoretical activity on supersymmetric extensions of the standard model (for reviews and references see, for example, ref. [1]). The simplest one is the so-called minimal supersymmetric standard model (MSSM), with gauge group SU(3)c × SU(2)L × U(1)v, three generations of quark and lepton chiral superfields ( Q - (U, D ) , U c , D c, L-= (N, E), Ec), two Higgs chiral superfields ( H 1 + H 2o )), and a superpotential of the form ( H ° , H1-), H2 -= ( H 2,
f = huQUCH2 + hoQDCH1 + hELECH1 + I.tH1H2,
(1.1)
where contracted group and generation indices are understood. The above ingredi* This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098, and in part by NSF under grant PHY-8515857. ** Supported by a Fellowship of Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Italy. ~ Supported by a Fellowship of Istituto Nazionale di Fisica Nucleare, Sezione di Genova, Italy. tt On leave from Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Italy. 0550-3213/90/$03.50©Elsevier Science Publishers B.V. (North-Holland)
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G. Gamberini et al. / Gauge symmetry breaking
ents are enough to specify a globally supersymmetric gauge-invariant lagrangian, *~susy. The complete low-energy effective lagrangian is assumed to be of the general form c~= £¢susy+ £'¢soft, where ~soft is a collection of explicit but soft supersymmetry-breaking terms, including gaugino masses, scalar masses, and scalar interaction terms of the form - (huAuQU~H2 + h.c.) - (hDADQD~H1 + h.c.)
(1.2) - (hEAELE~H1 + h.c.) - (B#H1H 2 + h.c.).
In the MSSM, the origin and the magnitude of the different terms in &°soft are undetermined, but they are usually interpreted as remnants of the spontaneous breaking of local supersymmetry in the underlying fundamental theory (supergravity and eventually superstrings). A very attractive feature of the MSSM is the possibility of "radiative breaking" of the electroweak symmetry [2], through a generalization of the original ColemanWeinberg mechanism [3]. Combined with some plausible assumptions on gauge couplings unification and on the origin of supersymmetry breaking, the requirement of a phenomenologically acceptable electroweak symmetry breaking is extremely powerful, since it rules out large regions of parameter space, and thus considerably enhances the predictiveness of the MSSM. It is therefore very important to understand the approximations and ambiguities involved in the calculations of radiative symmetry breaking. In this paper we present a critical reappraisal of the existing analyses, pointing out their limitations and introducing possible improvements. At the field-theoretical level, we do not introduce any new method: the treatment of ref. [3] is perfectly adequate. On the other hand, we try to clarify its application to the MSSM, since in the literature on the subject [2] one can often find confusing statements. We keep the discussion at the general level, limiting the applications to some illustrative examples. A complete analysis of the full parameter space, including all the theoretical and phenomenological constraints, will be given elsewhere [4]. Our considerations can be extended to the "no-scale" supergravity models [5], where also the scale of supersymmetry breaking is undetermined at the tree-level and can be fixed by radiative corrections, and to non-minimal supersymmetric extensions of the standard model [6]. The structure of the present work is as follows. We conclude this introduction by reviewing the conventional approach to radiative SU(2)× U(1) breaking, which makes use of the renormalization group improved tree-level potential, Vo(Q), evaluated at a renormalization scale Q - mw. In sect. 2 we compare the above method with the minimization of the full one-loop effective potential, VI(Q): we show that Vo(Q) is very sensitive to the choice of the renormalization scale Q, and only for a small and model-dependent range of Q it gives sensible results. Conversely, the results obtained from VI(Q) are remarkably stable for a wide range of Q
G. Gamberini et a L / Gauge symmeto~ breaking
333
around the electroweak scale, GF 1/2- 300 GeV. In sect. 3 we introduce three representative examples, to illustrate the general considerations of sect. 2 in three qualitatively different situations. We also formulate a practical rule to determine the scale Q at which the use of Vo(Q) is justified. In sect. 4 we re-examine some theoretical constraints on the model parameters, which were originally obtained [2, 7] by requiring that Vo(Q) be bounded from below and that charge and colour be unbroken at the minimum. Taking into account the structure of the one-loop effective potential, we discuss how these constraints should be implemented. The result is that, if correctly applied, they are much less stringent than previously thought, and some of them become completely irrelevant. Finally, in sect. 5 we discuss the uncertainties affecting the model predictions for the physical observables, concentrating on the top quark mass as a particularly interesting example. We review now, to establish language and notation, the conventional approach to radiative S U ( 2 ) × U(1) breaking. To reduce the number of free parameters, it is usually assumed that the running gauge coupling constants (with the U(1) factor properly normalized, gl - 5Vf~ g') unify at a certain super-high scale Mu,
g3(Q )[ Mu = g2 (Q)I MU= gl(Q)[ M~ = gu,
(1.3)
and that the running soft supersymmetry-breaking parameters can be summarized, at the scale Mu, by a universal gaugino mass
M3(Q)IMu = M2(Q)IM,~= MI(Q)[MU= M1/2,
(1.4)
a universal scalar mass m~(Q)IM~ = m2¢(Q)]Mu = m~(Q)IM~ = m2L \(Q~' 11 MU
=mZc(Q)]Mu=rn~(Q)[M =mZ2(Q)]M =rn~o,
(1.5)
and the scalar couplings
Au(Q)[ M. = AD(Q)] M. = AE(Q)] M. = Ao, B(Q)[ Mu = B0"
(1.6) (1.7)
It is also customary to neglect intergenerational mixing and the masses of all quarks and leptons apart from the top quark, so that the only non-negligible terms in the superpotential of eq. (1.1) can be written as
f=ht(UUCH ° - DUCHy)+ ~(H°H ° - HI Hf ),
(1.8)
where now U, D and U c refer only to the third generation. Correspondingly, one
G. Gamberini et al. / Gauge symmetry breaking
334
considers in £,°soft only mass terms and scalar interaction terms of the form -- [ h t A t ( U U C H
0 -
DUCH; ) + h.c.] - [ BIz( H ° H ° - H { H ; ) + h.c.] . (1.9)
To determine the lagrangian £.,9, one has then to specify, besides the known gauge and Yukawa couplings, the six unknown parameters h t(Mu) (related to the top quark mass mr), I~(Mv), Ma/2, mo, A o and B 0. To discuss radiative symmetry breaking, one usually assumes that all the parameters in the MSSM lagrangian are real, and considers the part of the tree-level potential depending on the neutral components of the Higgs fields, Vo= m2]Hl°] 2 + m~lH°] 2 + m ~ ( H ° H ° + h.c.) + 4(g22 + g'2)(]H1°]2- ]H°]2) 2, (1.10) where ml2 _
m 2 H1
+ ~2,
m~ --=m22 + ~2,
rn32= B/I.
(1.11)
The minimization of the tree-level potential of eq. (1.10) is straightforward. In particular, one can always redefine the phases of the Higgs field in such a way that m32 < 0 and the vacuum expectation values (v.e.v.'s) computed from V0 are real and positive. The condition for Vo to be bounded from below is
5 : - rn~ + m 2 - 2]m~] > O,
(1.12)
and the condition for the origin H ° = H ° = 0 to be unstable, so that the minimum of Vo corresponds to non-vanishing v.e.v.'s, is ~
m a2 m 22 - m~ < O.
(1.13)
However, one has to pay attention to the fact that loop corrections to V0 are in general not negligible. The one-loop effective potential V~ has the form
vl = Vo +
vl,
(1.14)
where [3]
1 [
AV1 - 64~2 Str ,///,4 log Q2
2
(1.15)
depends on H1°, H2° through the tree-level squared-mass matrix ./g2. In the above expression for V1, all parameters of the theory are running parameters, functions of the renormalization point Q. They can be evaluated for any Q by solving the
G. Gamberini et al. / Gaugesymmet~ breaking
335
standard renormalization group equations (RGE), whose form is well known [8] and will not be repeated here. We just remind the reader, for completeness, that the R G E of ref. [8] and the effective potential of eq. (1.14) are written in the Landau gauge (to avoid the appearance of the gauge-fixing parameter) and in the (massindependent) modified minimal subtraction scheme, MS. If one chooses a low renormalization scale Q conveniently, the large logarithmic corrections, proportional to l o g ( M u / Q ) , are reabsorbed in the running parameters of Vo(Q), and AVI(Q) contains only small logarithmic corrections. One therefore neglects AVI(Q) and minimizes Vo(Q), keeping only the sets of boundary conditions for which the v.e.v.'s v° - ( H ° ) , v°-- ( H ° ) give the correct value of the W mass, m w = ( g 2 / v / 2 ) ~ ( v ° ) 2 + ( v ° ) 2 - 81 GeV, and the rest of the particle spectrum and interactions are phenomenologically acceptable.
2. Radiative SU(2) × U(1) breaking
Having introduced the general calculational framework for radiative SU(2) × U(1) breaking, we are now ready to discuss some specific points in more detail. One issue is the validity of the boundary conditions, eqs. (1.3)-(1.7). We first concentrate on the grand unification condition, eq. (1.3). To determine correctly M U, a U =- g ~ / ( 4 ~ ) and the electroweak mixing parameter sin20w, one should take into account threshold effects. First, threshold effects at the unification scale [9] can modify the naive unification condition in eq. (1.3). However, this modification is only required if one performs a two-loop renormalization, as was done in grand-unified models [9], but for one-loop renormalization the boundary conditions in eqs. (1.3)-(1.7) are perfectly adequate. Vice versa, since threshold effects at the unification scale can only be computed in the framework of a more fundamental theory*, in the MSSM one should stick to the one-loop approximation. We therefore assume consistently that eq. (1.3) holds at the one-loop level in the M S scheme. In addition, one has to worry about the thresholds around the electroweak scale. It is well known [10] that mass-independent renormalization schemes do not automatically incorporate threshold effects in the evolution of the running coupling constants. These threshold effects can be introduced, invoking the decoupling theorem [11], by using a step approximation for the fl-functions and suitable matching conditions for the running parameters in the vicinity of each physical threshold. At the one-loop level, one can just change the fi-function coefficients and require that the running parameters are continuous at each physical threshold. A complete treatment, depending on the top quark mass and on the different supersymmetric particle masses, would be very complicated. To simplify drastically the problem, at least as far as the evolution of the dimensionless couplings is concerned, we shall assume that rn t - m w and that * In the case of a supergravity or superstring grand unification, such a theory should also include a mechanism for the spontaneous breaking of local N = 1 supersymmetry.
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the supersymmetric particle thresholds can be summarized by a single parameter Msusy, which in the phenomenologically relevant cases is always larger than m w. Our input parameters will then be [12] t
x,
O/emtmw)
=
1 127.93 + 0 . 3 0 '
(2.1)
and [13] a3(mw)
f~ 11 A+0.012 ~'~0.010 -
(2.2)
For scales Q between m w and Msusy we shall evolve the gauge couplings according to the R G E of the ordinary standard model*, whose one-loop B-function coefficients are given by b ° = - 7, b ° = - 19/6 and b ° = 41/10. Between Msusy and M U we shall evolve the gauge couplings according to the R G E of the MSSM, with b 3 = - 3 , b 2 = 1 and b I = 33/5. To take into account the residual uncertainties, we shall allow Msusy tO vary between m w and 1 TeV (larger values of Msusy would re-introduce the fine-tuning problem that supersymmetry is supposed to cure [15]). The region of Mtj, a U and sin20w(mw) corresponding to the allowed range for O%m(mw), o~3(mw) and Msusy is summarized in fig. 1. Notice how the unification constraint of eq. (1.3) is in remarkable agreement** with the measured value [16] of the electroweak mixing angle! Moving to the remaining boundary conditions, eq. (1.4) is consistent with the unification relations in eq. (1.3), whereas eqs. (1.5)-(1.7) are usually motivated by the assumption that the super-Higgs effect takes place in a " h i d d e n " sector of the theory, which interacts with the "observable" sector only gravitationally and is therefore flavour-blind. Proceeding further in the examination of the standard approach, the effect of intergenerational mixing on the R G E and on the effective potential can be certainly neglected in a first approximation, and in any case its inclusion does not involve any conceptually new problem and any radical modification of the results. A slightly more delicate assumption is the neglect of the flavour-diagonal bottom Yukawa coupling, h b, with respect to the corresponding top Yukawa coupling, h t : this approximation is justified only if (H°)/(H°) << m t / m b , and there might be some marginal region of parameter space which does not satisfy this condition. Even in this case, however, the explicit introduction of h b in the R G E and in the effective potential is straightforward and does not introduce any qualitatively new feature. An extremely important assumption is that all fields, apart from the real neutral components of H a and H 2, have zero v.e.v.'s. The consistency of this assumption should be tested for each set of boundary conditions, and will be discussed in some * The c o r r e s p o n d i n g R G E for the top q u a r k Y u k a w a c o u p l i n g can b e found, for example, in ref. [14]. * * I n fig. l , sin 2 0 w ( m w ) is evaluated in the ~ scheme, a n d should not be confused with sin 2 0 w ~- 1 2 rnw/m ~, w h i c h is the definition used in ref. [16]. However, the difference b e t w e e n the two is small a n d c a l c u l a b l e [12].
337
G. Gamberini et al. / Gaugesymmetry breaking 0.240
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Fig. 1. Allowed regions in the planes: (a) (a{1,sinZOw); (b) (Mu,sin20w); (c) (au a, Mu). They correspond to the following allowed intervals for the input parameters: 127.63 < a[ml(mw)~< 128.23, 0.104~
detail in sect. 4. The above assumption allows one to concentrate on the part of the effective potential which depends only on the real neutral components of H 1 and H 2. Notice, however, that whereas V0 involves only the mass parameters of the Higgs sector (m 2, m~, m~), AV1 depends on the masses of all fields in the theory (vector bosons, spin- ~2 fermions and spin-0 bosons). This will make the analysis of radiative s y m m e t r y breaking slightly different from the usual Coleman Weinberg case, where only a single scalar field 0 is involved. We n o w c o m e to the most crucial assumption of all, the discussion of which is the central subject of this paper: that for a convenient choice of the renormalization scale, Q, one can neglect the logarithmic contribution AVI(Q) to the effective potential in the minimization of Vo(Q) and in the determination of the low-energy p h e n o m e n o l o g y . In previous calculations [2,5,6], different choices were made: Q_ = row, Q_ = Gv 1/2, Q = m~, etc. However, it was realized from the very beginning
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(e.g. by Kounnas et al. in ref. [2]) that the minimum of V0(Q), characterized by the v.e.v.'s v ° and v°, can depend strongly on the choice of the scale Q at which to compute. This fact will be illustrated graphically in the examples of sect. 3. At large scales Q, both 5 ~ and ~ , defined in eqs. (1.12)-(1.13), are positive*. Going down with the scale Q, one can identify a certain value Q b , such that ~ < 0 for Q < QbEventually, there will also be a scale Qs such that 5°< 0 for Q < Qs. It can be easily proven that Qs _< Qb must hold. For Q > Qb one has v° = v° = 0, then v ° and v° increase monotonically as long as Q moves from Q b tO Qs, where the tree-level potential becomes unbounded and v° and v° escape to infinity. One sees therefore that, in the finite interval (Qs, Q b ) , which is usually not larger than a few TeV in the cases of interest, the v.e.v.'s computed from the RGE-improved tree-level potential vary from zero to infinity! The correct way of determining the ground state of the theory is to take into account also the field-dependent part AV1 in the effective potential V1 (the latter is independent of Q by definition, at least up t o O ( h 2) corrections). One then expects the v.e.v.'s vl and v2, determined from V1, to depend on the scale Q only via wave-function renormalization effects, given by [8] 0 log v~ 1 0 log Q - 64~ 2 (3g2 + g,2),
01ogv 2 1 g , 2 _ 12ht2) O logQ - 64~ 2 (3g~ +
(2.3)
(2.4)
We can see that, even in the case of a large top Yukawa coupling ( h t - 1) and for a variation of Q of one order of magnitude, the variation of v2 due to wave-function renormalization is about 4%, whilst the corresponding variation of v I is always around 0.5%. Before moving to the examples of sect. 3, we would like to discuss a subtlety in the definition of AVv At a given scale Q, even in the vicinity of the tree-level m i n i m u m (v °, v°), there can be field configurations corresponding to some negative eigenvalues for the mass matrix jg2, which would cause the appearance of an imaginary part in AV1. This problem arises because of the non-convexity of the tree-level potential V0 when gauge symmetry is spontaneously broken. As emphasized in ref. [17], one should not use eqs. (1.14) and (1.15) in the region of field configurations where the tree-level potential V0 is non-convex, since the effective one-loop potential in this region would be the convex envelope of eq. (1.14). In our case, since we are only interested in finding the minimum of the effective potential, this problem may be circumvented in two different ways. * In fact, boundary conditions such that both 5p and ~ are negative at Mtj would destabilize the tree-level potential and induce a minimum of the full effective potential with v.e.v.'s of order M U. Moreover, due to the boundary conditions of eq. (1.5), 5p and 5~ have the same sign at M v.
G. Gamberini et al. / Gaugesymmetry breaking
339
The first way consists in using eqs. (1.14)-(1.15) to find the correct one-loop minimum perturbatively, expanding around the tree-level solution (v °, v°). Accordingly, one finds
v~=v°i+Avi
( i : 1,2),
(2.5)
where
02Vo AUi ~ __
t -1
OAF1
(2.6)
I
OUiOVj[v~vo ]
OVj
l)~V O"
Doing so, one never encounters the problem of negative squared masses, since in eq. (2.6) j/g2 is evaluated at the tree-level minimum, and therefore is automatically positive semi-definite. However, we want to stress that the perturbative expansion in eqs. (2.5) and (2.6) can fail, and indeed it does in many relevant cases. The reason is that the one-loop correction AVI(Q) to the potential is not necessarily small, unless one carefully chooses a scale Q which one does not know a priori. Only the sum VI(Q) = Vo(Q)+ AVI(Q) is scale-independent (up to order O(h 2) corrections): it does not make much sense to expand perturbatively around a solution v° which, as we have seen, depends so strongly on the scale at which it is computed, or even does not exist for scales Q < Qs. The second approach, which is the one we have chosen to deal with the examples of sect. 3, is the following. We know that at any acceptable minimum all the eigenvalues of ~ , 2 corresponding to squarks and sleptons must be positive, to avoid the spontaneous breaking of colour, electric charge or lepton number (the squared masses for all the fermions and vector bosons are positive by definition). However, in the Higgs sector we have some zero eigenvalues corresponding to the Goldstone modes, which appear when we compute the squared Higgs mass matrix for the values (v °, v°) which minimize the tree-level potential V0. When we try to find the true minimum (Vl, v2) of the one-loop potential V1, negative eigenvalues for the tree-level squared-masses in the Higgs sector can in general appear. However, since these negative eigenvalues are of order O(h) (they come from a one-loop correction to (v °, v°)), we can safely neglect the corresponding terms in AV1 (which would come out of order O(h3)) without spoiling the one-loop approximation of the complete potential, which, as we mentioned before, is renormalization group invariant up to order O(h 2) corrections. In practice, we can take the absolute value of ~,2 inside the logarithm, or, equivalently, neglect the contribution of the negative mass eigenvatues.
3. Examples In this section, we present some examples to illustrate the previous considerations. We have chosen three representative cases, corresponding to the three qualitatively different situations depicted in figs. 2a-c. They can be easily inter-
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preted in terms of the scales Q b , Qs and Msu~y introduced before*, and are characterized by the three different orderings (a)
M~o~y< Q~ < Qb,
(b)
Q~
(c)
Q~ < Q b < msusy,
(3.1)
The scales Qs and Q b c a n be read off fig. 2 as the values of Q where the tree-level v.e.v.'s (v °, v°), represented by dashed lines, go to infinity and to zero, respectively. A striking feature of all three cases is the good Q-independence of the one-loop v.e.v.'s (v 1, v2), represented by solid lines, specially when compared with the wild behaviour of the tree-level result. The dotted lines give the v.e.v.'s (v~, vp) obtained with the perturbative expansion of eqs. (2.5) and (2.6), and show that such an expansion is good only in case (b), and even there for a rather small range of Q. The fact that v p, v~ ~ ~ for Q ~ Q~ can be explained by noticing that the smaller eigenvalue of the matrix (8 2Vo/8 vi 8 vj)],,=v0 appearing in eq. (2.6) vanishes in that limit. We now discuss the single cases in more detail. Case (a) is an example of SU(2) × U(1) breaking via a genuine Coleman-Weinberg mechanism: Vo(Q) becomes unbounded below a scale Q s - 80 GeV, whereas for Q = Qs it is flat in the direction v 1 = v2. This degeneracy is removed by the one-loop correction AVe, and the potential V1 develops a symmetry-breaking minimum with v I = V2. Since all the mass parameters in AV~ are much smaller than the scale under consideration (Msu,~y - 25 GeV << Q~), v 1 and 02 are essentially determined by the renormalization scale itself (dimensional transmutation). The v.e.v.'s are approximately given by v-QJg, where g is a typical gauge coupling: in fact, ~ 2 ( v ) - Q 2 at the minimum, and for M,o,y ~ 0 it is Jg2(v) - g2v2. In this case we can also see that Q, defined as the scale where (v 1, v2) -- (v °, v°), is very close to Q~. However, the rapid variation of (v °, v°) with the scale makes the tree-level potential approach useless. Case (a) has been thoroughly discussed in the literature [2, 5]: although theoretically appealing, this situation is ruled out by the present experimental bounds on the super-particle masses [18]. In case (b) the scale Q, where the tree-level and one-loop results coincide, is essentially M ~ y itself. In fact, it is sufficient to remember that m 2 << Q 2 ~ im 2 log(m 2 / Q 2) I << Q 2 t o understand that the dominant contributions to J Vl come from the largest field-dependent ~ 2 eigenvalues, typically the squark masses. In our case, it is at Q - - 2 0 0 G e V - Ms~sy that AV1 in eq. (1.15) becomes negligible. * We recall that M~u~v is the typical scale of the supersymmetry-breaking masses, and in practice coincides with an "average" squark mass, since squarks give the most important field-dependent contribution to AV1 in eq. (1.15).
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G. Gamberini et al. / Gaugesymmetry breaking 250
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50
100
500
1000
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50
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100
i
i
i
i
500
1000
5OOO
Q (GeV)
q (c~v)
Fig. 2. Vacuum expectation values computed from the tree-level potential ((v°, v°), dashed lines), the perturbative formulas of eqs. (2.5) and (2.6) ((v p, VzP),dotted lines) and the one-loop effective potential ((vl, v2), solid lines), as functions of the renormalization scale Q. Cases (a), (b) and (c) correspond to the following sets of boundary conditions:
(a)
M 1 / 2 = 10 G e V ,
/l(Mu)
= 2 GeV,
ht(Mu)
(b)
M r / 2 = 100 G e V ,
l~(Mu)
= 20 G e V ,
ht(Mu)=O.165,
(c)
M,/2 = 500 GeV,
t~(Mu)= 100 GeV,
In all cases we have taken
= 0.160,
ht(Mu)= 0.140.
m 0 = A 0 = B 0 = 0, a U = 0 . 0 4 a n d M U = 1.335 × 10 ]6 G e V .
N o t i c e that the prescription of c o m p u t i n g the tree-level m i n i m u m at Q = m w (used in s o m e of the previous analyses) would give a c o m p l e t e l y wrong result: it just happens in this case that at Q - - m w the tree-level potential is u n b o u n d e d from below. On the other hand, minimizing correctly V1, one finds a p h e n o m e n o l o g i c a l l y acceptable breaking of SU(2) × U(1), with the right W and Z masses and supersymmetric particle masses above the current experimental limits. For example, one finds
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a gluino mass rn~ = 255 GeV, squark masses mq between 170 and 275 GeV, and slepton masses m~L = 82 GeV, meR = 55 GeV, m~ -- 43 GeV; the lightest neutralino weighs m~, -- 20 GeV and the lightest chargino m¢, = 50 GeV; the top quark mass is m t - - 8 2 GeV and the lightest neutral Higgs has a mass m H --35 GeV. One can also see from fig. 2b that the ratio v2/v 1 is small enough to make the approximation h b << h t reliable. Finally, case (c) is an example of unbroken SU(2) × U(1), despite the fact that there are scales Q at which Vo(Q) admits symmetry-breaking minima. This case corresponds t o M s u s y - 1500 GeV > Q b - 600 GeV. What happens here is the following. When one evolves the parameters of the theory from Q = Mr; down to lower energies, using the RGE of the supersymmetric theory, one eventually reaches Q = Msusy. For Q < Ms,sy the supersymmetric particles, whose masses are larger than the renormalization scale, are effectively decoupled from the renormalization group evolution of the different parameters, in accordance with the decoupling theorem [11]. If one wanted to keep using the tree-level potential below the scale M~usy, one should use the RGE of the non-supersymmetric theory, which, however, would modify very little the mass parameters appearing in Vo(Q). As a result, these parameters would be frozen at the values assumed for Q = Msosy, where not even Vo(Q) breaks the symmetry, as apparent from fig. 2c. The fake symmetry breaking in Vo(Q), described by the dashed line, is due to the fact that, following the approach of most of the previous studies, we kept using Vo(Q) without including the decoupling of the supersymmetric particles in the R G E evolution. Coming back to our approach, notice that, although we did not explicitly use the "decoupling" in determining the evolution of the running parameters, the inclusion of the one-loop corrections to the potential takes care of it automatically, since VI(Q) never breaks the symmetry, even for Q < Msu~y.
4. Constraints In the discussion of sects. 2 and 3, we have assumed that all scalar fields other than H ° and H2° have vanishing v.e.v.'s. We now consider the constraints that should be satisfied for the consistency of this assumption, with the aim of clarifying some points that have been frequently misunderstood. In the approach which makes use of the renormalization group improved tree-level potential*, Vo(Q), one can formulate a series of consistency conditions. For example, the tree-level squared-mass matrices for all the fields but the Higgs doublet should be positive definite. Moreover, the trilinear scalar couplings in eq. (1.2) * For the discussion of this section, of course, we extend the definitions of Vo(Q) and AVI(Q) given in eqs. (1.10) and (1.15), respectively, to include the dependence on all the scalar fields of the MSSM.
G. Gamberini et al. / Gauge symmetry breaking
343
should satisfy the constraints [7]
< 3(m
+
+ m2),
(4.1)
< 3(m
+
+
(4.2)
A~ < 3(m 2 + m~c + m2),
(4.3)
in order not to break colour or electric charge*. Further conditions to avoid instabilities along charge and colour breaking directions have been formulated, for example [7]
m 2H2 + m 2 > 0 .
(4.4)
The above conditions, eqs. (4.1)-(4.4), are in general necessary conditions (a rigorous determination of sufficient conditions is still lacking), and they strongly constrain the parameter space of the MSSM. We will show below how they should be correctly implemented, by taking into account the effects of the field-dependent part AVI(Q) of the one-loop potential, and in doing so we shall modify, sometimes drastically, some statements which can be found in the literature. As a preliminary exercise before the discussion of eqs. (4.1)-(4.4), we return for a moment to the condition of eq. (1.12), which is necessary for the boundedness of Vo(Q). It is clear from the previous discussion that the condition of eq. (1.12), 5°> 0, cannot be satisfied at all scales, since we know that, for realistic values of the parameters, there always exists a scale, Qs, below which the tree-level potential becomes unbounded from below. In the literature, it is often required that ,9°>/0 for all scales Q between m w and M u. However, this cannot be the correct criterion either: we have presented an explicit example (the one in fig. 2b) in which this criterion would have given a wrong answer. The explanation is clear if one remembers the previous analysis of fig. 2b. When the instability of Vo(Q) is reached, just below the scale Qs, v° and v° go to infinity. However, the correct v.e.v.'s do not really go to infinity, since the one-loop correction AVI(Q) stabilizes V1 and does not allow v 1 and v2 to reach values for which ~ ' 2 ( v ) > > Qs2. Even if one insists on using Vo(Q), but includes the decoupling, v° and v° never run away to infinity, because ~ ' 2 ( v ° ) = Q2 for Q > Q~, and there the evolution of the mass parameters of the tree-level potential must be stopped. We can therefore identify Q as the value of Q for which the largest field-dependent eigenvalues of Jg2[v°(Q)] are of the order of Q 2 . The criterion of eq. (1.12) for the boundedness of Vo(Q) should therefore be interpreted as a criterion for avoiding v.e.v.'s larger than desired, and should be correctly * Actually, the first condition is not applicable in the case of the top quark coupling, but this will not affect the conclusions of the following discussion.
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implemented as follows: (i) Check if 5P> 0 at Q = M U. If not, this means that v I and v2 are of order Mr/ and thus phenomenologically unacceptable. (ii) If 50> 0 at M U, evolve down the running parameters until the scale Q reaches Q,~ or M~usy, and stop there the RGE evolution. If Qs has been reached, then u1 ~---U2~---Qs/g. Otherwise, 01 and v2 are obtained by minimizing Vo(Q) at 0 - Msusy. Using the values (vl, v2) obtained in this way, one can then check if a phenomenologically acceptable particle spectrum is obtained. It is important to stress the fact that the one-loop potential VI(Q) takes into account all this procedure automatically, and we do not have to worry about fine tuning the scale Q at which to minimize it, since the result is Q-independent as long as AV t is not too large. Now we come to the problem of the possible existence of unwanted chargea n d / o r colour-breaking minima, along certain directions in the scalar field space. If one of the conditions in eqs. (4.1)-(4.4) is violated, Vo(Q) develops these undesired minima. In principle, to see if these new minima survive the one-loop corrections, one should compute AV 1 as a function of all the scalar fields. This, however, would be a terribly complicated algebraic task. Instead, we can formulate a procedure analogous to that applied to the condition of eq. (1.12), exploiting the fact that the unwanted minima usually have large field v.e.v.'s inversely proportional to a small Yukawa coupling, e.g. 0 - Ai/hi. One should check if Vo(Q) has any unwanted minimum for scales Q between M U and Q. Such a minimum does really occur (and is not washed out by the one-loop corrections), only if there exists a scale Qc (Q < Qc < Mu) such that the largest field-dependent eigenvalues of ,///'2[~b(Qc) ] are of the order of Q2 (a rough estimate of these eigenvalues is given by max{ g2(q~)2, M 2 y )). The above criterion is another way of stating that a minimum of Vo(Q) is trustable only if AVI(Q) is small. The smallness of AV 1 depends on the scale Q, but so does also the existence of a given minimum of V0. If no scale Q~ can be found, where both conditions are simultaneously satisfied, then the would-be minimum does not exist. To illustrate the above considerations, we examine the condition in eq. (4.4), introduced in the last paper of ref. [15]. We reproduce here, for the reader's convenience, the essential points of its derivation. Along the field direction
a2/, (H°) -
hD' ,
a/x (DL,) = (DR,> -- hD ,,
a~/1 + a2/~ (~L,) --
hD'
,
(4.5)
where a is an arbitrary parameter, one can encounter an instability of the tree-level
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G. Gamberini et al. / Gauge symmetry breaking potential, which now reads
Vo=
ho,] [a2( m2N2
m2D,
m2
.
(4.6)
Then, if (m22 + m2j) < 0, for a large enough V0 assumes arbitrarily large negative values. This is how the constraint of eq. (4.4) was derived. It was also claimed that it should be satisfied for any Q between m w and M v. If this were true, one could set a strong upper limit on the top quark mass in the restricted class of models where the only seed of supersymmetry breaking at M v is a universal gaugino mass. First of all we notice that, if we include one-loop corrections to the potential, the p a r a m e t e r a can no longer be arbitrarily large, but is constrained by the requirements
'lgz(O°)l ~< Q, Ig3(DL,)I = ]g3(D,,)[ ~ Q,
(4.7)
Ig2<~L,)l ~< Q. Using eq. (4.5), we can summarize the constraint of eq. (4.7) as follows
a ~ min
, /xg2
.
(4.8)
/xg3
Secondly, for realistic values of the parameters the condition in eq. (4.4) can only be violated at scales Q around the electroweak scale. As a consequence, eq. (4.8) tells us that a cannot be very large and that the value of the potential in eq. (4.6) can hardly become negative. We have checked that, under the restriction of eq. (4.8) and varying the values of the boundary condition at M v in a considerable range, the potential in eq. (4.6) never becomes negative, and the constraint of eq. (4.4) is ineffective. The constraints of eqs. (4.1)-(4.3) can be discussed in a similar fashion. Without presenting a detailed analysis, we just want to stress that, since the unwanted charge- and colour-breaking minima would be of order q~- A i / h i, eqs. (4.1)-(4.3) need not be satisfied for all scales between M u and m w, but only at Q - g4 - gAi/hi.
5. Predictions for the physical masses In this last section we discuss the predictions for the physical masses of the MSSM, paying particular attention to the W mass (which sets the electroweak scale and is used as a constraint) and to the top quark mass (which is a hot experimental issue at the time of this writing).
G. Gamberini et aL / Gauge symmetry breaking
346
T o begin with, we discuss the relation between the running masses in the MS scheme, m w ( Q ) and mr(Q), and the corresponding physical masses mw and mtThe running of v I and v2 with the scale, although negligible in determining the ground state through the minimization of the one-loop potential, may become important when dealing with a computation of physical masses. To calculate the physical mass of a particle in the one-loop approximation, starting from the running mass and coupling constant parameters of the low-energy effective theory, one should in principle compute the one-loop propagator for that particle, using the same renormalization prescription which has been used in deriving the running parameters: the physical mass then is the position of the pole of that propagator. However, for practical purposes, and in the MS scheme, a good approximation [19] is achieved if we define the physical masses N, in terms of the running masses m(Q), by the formula ~ --- m(Q = ~). For example, we have m2(Q)
= 1 2
(5.1)
+ 4(o)],
and mt(Q)
=ht(Q)vz(Q),
(5.2)
where by v i (Q) we mean the minimum of the one-loop potential V1 calculated at the scale Q (its Q-dependence is only through wave-function renormalization, as discussed in sect. 2). Therefore we have
-
m2~2
g2 mw){V (0) 2 ~,
O[o~(Q)+vZ(Q)] + v2(0) +
mt = ht(mt)[v2(0) + 0- log Q
0 log Q
0 l°g
~} 0 l°g
'
(5.3)
"
These are the formulae which should be used, in the absence of a complete one-loop computation, for the comparison with the experimental values (eq. (5.3)) and limits (eq. (5.4)). We now work out an application of eqs. (5.3)-(5.4), in order to convince the reader that the considerations of the present paper have not onl 7 academical, but also practical relevance. We fix a set of boundary conditions at the unification scale, for example* aU= 0.0394, M u = 1.24 × 1016 GeV and
M1/2=
200 GeV,
* These values of a t M~u~y = 375 GeV.
/~(M v) = 40 GeV,
m o = A o = Bo = 0.
(5.5)
and of M U are o b t a i n e d from the central values in eqs. (2.1) and (2.2) with
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347
The requirement of a correct symmetry breaking, with Nw = 81 GeV, allows then for a unique value of the top quark mass N t. We compare the numerical predictions for N t obtained in the conventional approach and in the approach outlined in this paper. In the first case, one uses supersymmetric RGE to evolve all parameters from M U to m w, and minimizes Vo(Q) at Q=m w. Then one equates mw with
gz(mw)~(v°2(mw) + v°2(mw))/2,
and mt with Iht(mw)v°(mw)[. In this way one obtains m r - - 6 9 GeV for the given boundary conditions. In the second case ( M s u s y - 3 7 5 GeV), one uses supersymmetric RGE between M U and Msu~y, and ordinary R G E below Msu~y. One then minimizes the full one-loop potential at a scale Q - M~u~y, and uses eqs. (5.3) and (5.4) for the computation of Nw and mr. Doing so, one obtains that ~i w -- 81 GeV implies, for the same boundary conditions as above, m t ~-- 81 GeV. Therefore, the difference between the two methods gives a significant discrepancy in the predictions for the top quark mass. If one day some fundamental theory (superstrings?) will determine for us the boundary conditions at Mu, one should pay attention in determining its low-energy predictions! More realistically, one should be careful in interpreting the present experimental limits as constraints in the space of boundary conditions. As a second application of eqs. (5.3) and (5.4), and of the results presented in fig. 1, we test how the prediction for Nt is affected by the uncertainties in the knowledge of aem(mw) and of c%(mw), eqs. (2.1) and (2.2). For example, we fix the boundary conditions as in eq. (5.5), which implies that M~u~y- 375 GeV, and we vary aem(mw) and % ( m w ) in the interval allowed by eqs. (2.1) and (2.2). Correspondingly, we span a certain interval of (c~, My) values: in turn, they lead to different predictions for the value of mt which induces the correct radiative breaking of the electroweak symmetry 78 GeV ~< ~ t ~< 84 GeV.
(5.6)
This is another uncertainty which should be kept in mind when examining the low-energy consequences of a given set of boundary conditions. This kind of considerations can be extended to all the low-energy predictions of the MSSM: a complete treatment, however, is outside the scope of the present paper and will be given elsewhere. The authors would like to thank R. Barbieri, C.M. Becchi, S. Coleman, J. Ellis and M.K. Gaillard for discussions and suggestions, and in particular Massimo Porrati for exhausting but enlightening discussions. References
[1] H.-P. Nilles, Phys. Rep. 110 (1984) 1; H.E. Haber and G.L. Kane, Phys. Rep. 117 (1985) 75; A.B. Lahanas and D.V. Nanopoulos, Phys. Rep. 145 (1987) 1
348
G. Gamberini et al. / Gauge symmetry breaking
[2] L.E. Ib~raez and G.G. Ross, Phys. Lett. Bll0 (1982) 215; K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, Prog. Theor. Phys. 68 (1982) 927; 71 (1984) 413; L. Alvarez-Gaum~, M. Claudson and M.B. Wise, Nucl. Phys. B207 (1982) 96; J. Ellis, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B121 (1983) 123; L.E. Ib~raez, Nucl. Phys. B218 (1983) 514; L. Alvarez-Gaum6, J. Polchinski and M.B. Wise, Nucl. Phys. B221 (1983) 495; J. Ellis, J.S. Hagelin, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B125 (1983) 275; L.E. Ib~gaez and C. Lopez, Phys. Lett. B126 (1983) 54; Nucl. Phys. B233 (1984) 511; C. Kounnas, A.B. Lahanas, D.V. Nanopoulos and M. Quir6s, Nucl. Phys. B236 (1984) 438; U. Ellwanger, Nucl. Phys. B238 (1984) 665; L.E. Ib~raez, C. Lopez and C. Mufioz, Nucl. Phys. B256 (1985) 218; A. Bouquet, J. Kaplan and C.A. Savoy, Nucl. Phys. B262 (1985) 299; H.J. Kappen, Phys. Rev. D35 (1987) 3923; M. Drees, Phys. Rev. D38 (1988) 718; H.J. Kappen, Phys. Rev. D38 (1988) 721; B. Gato, Z. Phys. C37 (1988) 407; G.F. Giudice and G. Ridolfi, Z. Phys. C41 (1988) 447; S.P. Li and H. Navelet, Nucl. Phys. B319 (1989) 239 [3] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888; S. Weinberg, Phys. Rev. D7 (1973) 2887 [4] J. Ellis and F. Zwirner, preprint U C B / P T H / 8 9 / 1 5 , LBL-27310 (1989) [5] J. Ellis, A.B. Lahanas, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B134 (1984) 429; J. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B241 (1984) 406; B247 (1984) 373 [6] J.-P. Derendinger and C.A. Savoy, Nucl. Phys. B237 (1984) 307; J.-P. Derendinger, L.E. Ib~i~aezand H.-P. Nilles, Nucl. Phys. B267 (1986) 365; P. Binetruy, S. Dawson, I. Hinchliffe and M. Sher, Nucl. Phys. B273 (1986) 501; J. Ellis, K. Enqvist, D.V. Nanopoulos and F. Zwirner, Nucl. Phys. B276 (1986) 14; L.E. Ib~glez and J. Mas, Nucl. Phys. B286 (1987) 107; G. Costa, F. Feruglio, F. Gabbiani and F. Zwirner, Nucl. Phys. B286 (1987) 325; J. Ellis, J.S. Hagelin, S. Kelley and D.V. Nanopoulos, Nucl. Phys. B311 (1989) 1; J. Ellis, J.F. Gunion, H.E. Haber, L. Roszkowski and F. Zwirner, Phys. Rev. D39 (1989) 844 [7] J.M. Fr~re, D.R.T. Jones and S. Raby, Nucl. Phys. B222 (1983) 11; M. Claudson, L.J. Hall and I. Hinchliffe, Nucl. Phys. B228 (1983) 501; J.-P. Derendinger and C.A. Savoy, Nucl. Phys. B237 (1984) 307; M. Drees, M. Gliick and K. Grassie, Phys. Lett. B157 (1985) 164; J.F. Gunion, H.E. Haber and M. Sher, Nucl. Phys. B306 (1988) 1; H. Komatsu, Phys. Lett. B215 (1988) 323 [8] R. Barbieri, S. Ferrara, L. Maiani, F. Palumbo and C.A. Savoy, Phys. Lett. Bl15 (1982) 212; M.B. Einhorn and D.R.T. Jones, Nucl. Phys. B196 (1982) 475; J.-P. Derendinger and C.A. Savoy, Nucl. Phys. B237 (1984) 307; B. Gato, J. Leon, J. P~rez-Mercader and M. Quir6s, Nucl. Phys. B253 (1985) 285; N.K. Falck, Z. Phys. C30 (1986) 247 [9] S. Weinberg, Phys. Lett. B91 (1980) 51; L.J. Hall, Nucl. Phys. B178 (1981) 75 [10] D.A. Ross, Nucl. Phys. B140 (1978) 1; T. Goldman and D.A. Ross, Nucl. Phys. B171 (1980) 273; P. Bin6truy and T. Schiicker, Nucl. Phys. B178 (1981) 293; B178 (1981) 307; I. Antoniadis, C. Kounnas and C. Roiesnel, Nucl. Phys. B198 (1982) 317; W. Bernreuther, Ann. Phys. (N.Y.) 151 (1983) 127 [11] K. Symanzik, Commun. Math. Phys. 34 (1973) 7; T. Appelquist and J. Carazzone, Phys. Rev. D l l (1975) 2856 [12] W.J. Marciano, in Proc. Aspen Winter Physics Conf. 1985, Ann. N.Y. Acad. Sci. 461 (1986) 367
G. Gamberini et al. / Gauge symmet~ breaking
349
[13] R.M. Barnett, I. Hinchliffe and W.J. Stirling, in Review of particle properties, Phys. Lett. B204 (1988) 96 [14] C.T. Hill, Phys. Rev. D24 (1981) 691; M.E. Machacek and M.T. Vaughn, Phys. Lett. B103 (1981) 427 [15] J. Ellis, K. Enqvist, D.V. Nanopoulos and F. Zwirner, Mod. Phys. Lett. A1 (1986) 57; R. Barbieri and G.F. Giudice, Nucl. Phys. B306 (1988) 63 [16] U. Amaldi, A. B6hm, L.S. Durkin, P. Langacker, A.K. Mann, W.J. Marciano and A. Sirlin, Phys. Rev. D36 (1987) 1385; G. Costa, J. Ellis, G.L. Fogli, D.V. Nanopoulos and F. Zwirner, Nucl. Phys. B297 (1988) 244 [17] Y. Fujimoto, L. O'Raifeartaigh and G. Parravicini, Nucl. Phys. B212 (1983) 268; A. Dannenberg, Phys. Lett. B202 (1988) 110 [18] C. Albajar et al. (UA1 Collaboration), Phys. Lett. B198 (1987) 261 [19] J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77