The gauge hierarchy problem, technicolor, supersymmetry, and all that

PHYSICS REPORTS (Review Section of Physics Letters) 104, Nos. 2-4 (1984) 181-193. North-Holland, Amsterdam

The Gauge Hierarchy Problem, Technicolor, Supersymmetry, and all that Leonard SUSSKIND*

1. The gauge hierarchy problem The standard SU(3) x SU(2) x U(1) theory of strong, electromagnetic and weak interactions appears to correctly describe all physics down to the smallest distance scales yet probed. Most probably the Z °, W -+ and top quark will be discovered. Should this occur we will be left with the following situation: a highly accurate phenomenology of all known physics (excepting gravity) exists but no explanation is in sight for a variety of fundamental questions such as the origin of generations, the values of fundamental constants, the exact equality of the electric charge of the proton and the positron, and the role of gravitation. Clearly the standard theory must give way to some more complete theory at some small distance scale. There are well known reasons to think that such a complete fundamental synthesis can only take place somewhere between 1014 and 1019 GeV. What if anything lies between the weak scale -102 GeV and the scale of unification? The simplest answer is nothing beyond the Z, W top quark and Higgs. Nothing new will be encountered below 11) 14 GeV. It is obviously of the utmost importance to know whether there are any indications, one way or another, for this "Boring Desert" hypothesis. These lectures are about one such negative indication, namely the gauge hierarchy problem (GHP) or the problem of naturalness of the two widely separated scales, represented by the expectation value of the Higgs field (~b)-250 GeV and the unification or Planck scale. The GHP is a consequence of several assumptions which I will now list. (1) The standard SU(3) x SU(2) x U(1) theory including quark, lepton, photon, Z °, W -~ and one or more Higgs scalar fields correctly describes nature up to energies - M very much larger than the weak scale -250 GeV. (2) New physics of an unspecified nature occur at scale - M possibly including SU(5), gravity, etc. (3) The behavior of the world at ordinary energies is not exceedingly sensitive to the values of fundamental parameters. In particular the very existence of a low energy world, characterized by the Higgs expectation value -250 GeV should not require the fundamental parameters of the microscopic world at scale ~ M to be "Fine Tuned". This last assumption, usually called "Naturalness" is less familiar than the others and requires some explanation. To understand its meaning we must recall that a phenomenological field theory such as the standard model must be cut off at small distances to be defined. The formal path integral is meaningless until a prescription is given for regularizing its short distance behavior. This prescription specifies a momentum k as a cutoff and a set of parameters (coupling constant g and mass parameters #) which will generally depend on k. The cutoff theory is only useful for momenta and energies less than k. * Note from the Editor: Dr. Susskind's contribution to the Solvay Conference covered material which he had already presented to a large extent at the SLAC Summer Institute. Whilst he did not deem it appropriate to write a new text, we are happy, for the sake of completeness, to reproduce here his written contribution to the SLAC Summer Institute. We are much indebted to the organizers G. Feldman, F. Gilman and D. Leith for their very kind cooperation.

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182

What about the effects of quantum fluctuations involving wavelengths smaller than k -1. The central point of renormalization theory is that these effects can be lumped into the values of the constants g and/z (apart from very small effects which vanish like inverse powers of k). A change in the cutoff from k to k' can be compensated by a change in g and tz. Accordingly we consider (g, # ) to be functions [(g(k), # (k)]. The important point that I want to emphasize is that the phenomenological parameters g, /z are dynamical objects which depend in a complicated way on all the physics at length scales smaller than k -1. Let us now apply this idea to some hypothetical unified theory. I will begin by supposing that the theory contains no light fields of mass < M other than those of the standard model. In this case if the cutoff k is less than M the physics of low energies (
(1)

These parameters summarize all the unknown physics at momenta >k. In particular they depend non-trivially on the structure of the unified theory at distances shorter than M -1. The mass of every ordinary particle in the standard theory is proportional to the vacuum expectation value (VEV) of (¢). For example quark and lepton masses have the form mq ~ gv(¢) •

(2)

The Z and W masses are proportional to

g20)

(3)

and the Higgs mass to

x/TO).

(4)

The VEV (¢) is not a new parameter but can be calculated from A(k), /z(k), g(k) and gv(k). For example suppose we begin by ignoring all quantum fluctuations and calculate (¢) by minimizing V(~0). This gives (¢) =/., (k)/2"X/A--~(k).

(5)

This however is not a good approximation if the cutoff k is large ( - M ) . It is spoiled by quantum fluctuations of all modes down to wavelengths ~ k -1. A much better approximation is to use a low energy cutoff k' say of order 1 TeV. For most purposes we may write k' = 0 (since k ' ~ k ) , ( ¢ ) ~ ~ (0)/2"X/~-6-).

(6)

Thus (~0) may be well approximated by computing the "renormalized" mass and coupling ~(0), h(0). For example consider /x(0) which is calculated by Feynman graphs. Each graph in fig. 1 is cut off

L. Susskind, The gauge hierarchy problem, technicolor, supersymmetry, and all that

....

1

183

I

_~__/_ . . . . +

+

-- ~)--

-

+...

Fig. 1.

dependent and proportional to k 2. Thus/z2(0) has the form

2(o) = u2(k )+ k:(C,a + C2g: +. . .).

(7)

Now the point is that we believe that the values of (g(k) . . . . . /z(k)) are most closely related to the fundamental underlying unified theory when k is as large as possible, say of order M. Then eq. (7) becomes

(8)

#2(0) = / z 2 ( M ) + M2(C1 , ~ ( M ) + . . . ) .

How big is #2(0). From eq. (6) we see that/z(0) should be -(~0) ~ - ( 0 ) and for typical V'-~-- 1, #(0) is of order 102 GeV. Suppose the scale M - 1015GeV. Then we may write eq. (8) as /z2(0)/M 2 = 10 -26 = tza(M)/M 2 + (C1/~

Jr-'..).

(9)

This is a very unreasonable situation. It requires the dimensionless parameter #E(M)/M2 to cancel the complicated series (C1A + • • ") to 26 decimal places in order that the world of ordinary particles be as light as it is. But lzZ(M)/M 2 only knows about physics at small distances. To obtain such cancellation would require a miraculous conspiracy among the parameters of the unified theory. In other words, the existence of a scale - 1 0 2 GeV in the standard theory with a second more fundamental scale at 1015 GeV is unnatural. This is called the GHP.

Possible solutions to the GHP (1) Forget it for now. Some future theory will explain the fine tuning of/z(M). The boring desert exists just like in the usual SU(5) model. No new discoveries above 250 GeV until 1015GeV. (2) Technicolor. It is possible for a low energy world to emerge naturally. A familiar example is QCD with massless quarks and gluons. In this case chirai symmetry and gauge invariance insure the absence of any perturbative renormalizations of quark and gluon masses. The coupling constant g(k) is renormalized and therefore depends on the cutoff scale k. The correct dependence is given by the renormalization group

dg(k)/d log k = fl(g(k)) = -rio g(k) 3 + . . . .

(lo)

Thus if g is chosen fairly small at k - M then at a much smaller momentum scale g will become large.

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184

This smaller scale is called AOCDand satisfies 1

AOCD-- M e x p ( 2 / 3 0 g ~ ( M ) ) .

(11)

Evidently AOcD/M can be very small (-10 15) with no fine adjustment of parameters. In fact we need not even introduce a very small parameter to get AocD/M very small. In this case it is sufficient to make g ( M ) - 0.3 to get AOcD/M -- 10-is. This brings us to a distinction between degrees of naturalness. I will consider a theory to be "seminatural" if it requires very small but not finely adjusted parameters. In these theories the smallness of the Higgs expectation value is stable but unexplained. On the other hand, theories like QCD I will call completely natural. In these theories all very small mass scales are exponentially small in terms of fundamental parameters such as g. Technicolor is an attempt to build a completely natural theory by mimicking QCD. The important observation leading to T.C. is that the real role of the fundamental scalar ¢ is to spontaneously break the chiral gauge symmetry SU(2) x U(1). Now in QCD chiral symmetries are also broken, not by vacuum expectations of fundamental scalars but by quark bilinears (01qql0) - h

(12)

In fact, if no Higgs fields were introduced into the standard SU(3) x SU(2) x U(1) theory, the ordinary QCD chiral breaking would break SU(2) x U(1) and give mass to W ± and Z. The pattern of masses Mw = cos 0w Mz,

m~, = 0

(13)

would be the same as in the conventional theory because the chiral multiplet containing (qq) is isomorphic to the Higgs multiplet. In fact the massless pion multiplet would mix with the gauge fields and become the longitudinal Z °, W ~. However, because the strength of the condensate (qq) is of order (AocD)3 the resulting masses of Z, W -+ are mw ~ ½g2AocD

(14)

(g2 SU(2) coupling). The correct formula replaces (~) by the pion decay constant F=. This leads to masses -2000 times too small. However suppose that a second QCD-like gauge sector (technicolor) exists. The gauge group does not have to be SU(3) but should become strong at a mass scale -2000 Aoco. If this sector contains fermions (F) which carry T.C. instead of color but which form conventional weak doublets then condensates =

will induce SU(2) x U(1) breaking. The Z and W will get masses of the correct magnitude and no very small or finely adjusted parameters are needed. The main prediction of this kind of theory is the existence of a rich spectrum of "technihadrons" with masses, widths and splittings -2000 times their hadron's counterparts.

185

L. Susskind, The gauge hierarchyproblem, technicolor, supersymmetry, and all that

The technicolor idea is a very elegant solution to the GHP but unfortunately runs into grave difficulties. As I have described it T.C. can explain the weak scale and masses of Z, W. What it fails to do is to provide a mechanism for ordinary fermion masses. In the conventional theory these arise from the Yukawa couplings of fermions (say quarks) to Higgs. Schematically we write gvqq¢ where gv is a dimensionless coupling. When ¢ gets a vacuum value a fermion mass term gT(¢)qq results. In T.C. theory ¢ is replaced by the fermion bilinear ~'F. To get masses for ordinary quarks an effective coupling Gqq~'F must exist with fermion coupling constant G (with units of m-2). The light fermion mass is then G(FF). Where can the required 4 fermion couplings come from? Ordinary SU(3) × SU(2) × U(1) interactions do not induce couplings of the right kind. Thus we are forced to introduce yet further physics. Several possibilities have been suggested. The first possibility called extended T.C. introduced new gauge degrees of freedom (E) which couple ordinary fermions to technifermions as in fig. 2. The required 4-Fermi couplings are generated by fig. 3. The coupling G is given by G-

g E2 / M E2

(15)

where gE is the coupling of the E-bosons. The light fermions then have mass mf

M2(FF)-g2~=ATC

(16)

E /

Another popular idea is that quarks and leptons and technifermions are massless composites of some hypothetical preons bound by yet a third strong interaction called metacolor. The 4 Fermi couplings FFqq, FFt~t~would be due to exchange of preons as in fig. 4. F

F

/~f Fig. 2.

f Fig. 3.

f Fig. 4.

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186

q Fig. 5.

Fig. 6.

Unfortunately these ideas do not seem to be consistent with the observed suppression of neutral strangeness changing currents. In other words there is no known techni-Gim mechanism. For example in the ETC theory the existence of E-bosons coupling q to F through the vertex in fig. 5 would imply gauge coupling of q to q and F to F to close the non-Abelian gauge algebra. Thus we will inevitably find additional 4-Fermi couplings qqftq from fig. 6. These couplings will generally include neutral strangeness changing interactions with strength GNSC ~

2

2 .

gEIME,

(17)

of course if gEIME 2 2 is small enough there is no danger. However, from eq. (16) we see that

g~/M~~

mq/A 3TC.

(18)

In other words g z2/ M z2 must be large enough to do the job it was designed to do, i.e., provide quark masses. When typical quark masses are used in eq. (18) it is found that the resulting magnitude of GNsc is too large by many orders of magnitude to be consistent with known bounds on GNso The worst disaster is the K L - K s mass difference which bounds GNsc to be 10 6 times smaller than mq/A3o Similar considerations apply to preon models where preon exchange will generate unacceptable NSC currents if they are made large enough to account for quark masses. Perhaps somebody will discover a techni-Gim mechanism which can operate in a class of technicolor models. However, without such a mechanism I am forced to conclude that T.C. is very unlikely. (3) This brings us to the third possible solution. Consider the electron mass in QED. The electron mass (like the Higgs mass) gets renormalized. Offhand, if we did not know better we would expect ~m

~ o~k

where a is the fine structure constant and k the cutoff. This however, is incorrect. The reason is that if the bare mass of the electron is zero the theory has a symmetry (chiral symmetry) which prevents a mass counter-term from occurring. Therefore if too, the bare mass, vanishes so does 6m. Accordingly ~rn is proportional to m0 and the correct formula is

3l~2 ~ am 2olog(k2/m 2).

(19)

Thus requiring ~m < m (naturalness) allows k to be enormous. This is many times bigger than the Planck mass. Now suppose we could find a symmetry which was exact only if ~2 (Higgs mass) were zero. In the same way we would argue that the quadratic divergences in the renormalization of # 2 would cancel and

L. Susskind, The gauge hierarchy problem, technicolor, supersymmetry, and all that

187

8/.t 2 ~ am 2 log(k2/m 2). As long as log(k2/m 2) is not too big the stability of a very small ~2/k2 or #2/rn2 would be insured. This kind of theory does not explain the smallness of/.t 2 or rn 2, but it does not require fine tuning of too. It is seminatural. A general definition of a seminatural theory is that a symmetry exists in the limit ~o-~0 which insures the vanishing of 8#. Small breaking characterized by dimensionless parameters of order i.t~/rn 2 will then allow t.t2/m 2 to have a nonvanishing small value which is stable with respect to radiative corrections. Are there known symmetries which can protect massless scalar bosons from acquiring mass? Until the discovery of supersymmetry in the early 1970's the answer was no. Today we know that a combination of super and chiral symmetries can do the job. The rest of these lectures are about the supersymmetry solution to the GHP.

2. Supersymmetry Supersymmetry transforms bosons into fermions and fermions into bosons. If the symmetry is unbroken it insures the equality of fermion and boson masses even after quantum radiative corrections. Now suppose that when a multiplet including a scalar a and a fermion ~b is massless, there is a chiral symmetry under which

~-+ ei~,,o~0. The fermion mass will then remain zero to all orders in the coupling. But because supersymmetry is operative the boson mass will also remain zero. Let us begin with a scalar complex boson field A and a 2-component left-handed spinor ~b,~(a = l, 2) and a second boson field F. The field F will not be an independent quantum field but will be eliminated. It is called an auxiliary field and is useful in doing the bookkeeping of supersymmetry. The three fields -- (A, ~O,F) form a so-called chiral superfield. A supersymmetry transformation mixes them. An infinitesimal supertransformation is parametrized by a complex infinitesimal spinor parameter e,~. The rule is 8A = ie~4/~

~b = - i e F + 2go%, cg,.A

(20)

8F = 2i&r,. ,9,.~. Several points are unusual about this transformation law. First of all if we consider e to be an ordinary (commuting) number, then 8A is a fermion field and 3~b is a boson. We could work with such a convention but it is far more efficient to assume e is an element of an anticommuting algebra, i.e., that it is fermionic =

=

o.

188

Higher Energy Physics

The second odd fact concerns dimension counting. If ~ and ~0 are canonical fields then they have dimensions m and m 3/2 respectively. This means that e has dimensions of m -~/2. The dimension of F is then forced to be m 2 or else eq. (20) cannot be consistent. Thus F cannot be a canonical field. To illustrate how supersymmetry works we consider a simple free field example. The Lagrangian for A is

O,.,A* O,,A

(21)

and for 0 q~0.

(22)

For F the only term in L is F*F. Note that a free kinetic term for F is dimensionally inconsistent. Now it is an easy exercise to show that gL = 0 under the supersymmetry transformation (20) the field F does not propagate. Its equation of motion is

OL/OF* = F = 0.

(23)

To add interactions to L, we must know how to multiply superfields. Consider two superfields 4)1 and 4)2

4)1 = (A,, ~Ot,Ft) 4)2 = (A2, ~02,F2). Their product 4)~4)2 is a superfield defined by Or@2 = (A,A2, t,O2A,++ OtA2, A~F-, + A2F~ + 2Or" 02).

(24)

You can check that if 4)~ and 4)2 transform according to (20) so does 4)~4)2. Next we see that under (20) the F-term of a superfield gets changed by a pure divergence under a supertransformation. Thus we may put it into L without destroying the SS of the action; for example we can put

½m (4)4))v = ( A F +ftb),,

(25)

in L. This term gives mass m to 0 and couples A to F. However, since derivatives of F do not appear in L, it may be eliminated. The equation of motion for F gives

F = -mA*.

(26)

Using F = - mA* in L then gives m 2 A * A in L so that A is also a massive field. We can also add 4)3 to the Lagrangian without spoiling either supersymmetry or renormalizability

g4)3 ~ g(AZF + i0OA).

(27)

L. Susskind, The gauge hierarchy problem, technicolor, supersymmetry, and all that

189

This term evidently contains a Yukawa coupling O~A between the scalar and spinor components of 4. T h e A 2 F term can be understood in one of two ways. Since derivatives of F do not appear in L~ we

may eliminate it. For example if no ¢2 term is present then F appears in the combination (28)

FF* + g A 2 F + g A * 2 F * .

The equation of motion for F* is F + gA .2 = 0 or

F

=

- g A .2

.

Plugging this back into (28) gives -g(A*A) 2

in the scalar field potential. Two things should be noted. First although we began with qb3 in the "superpotential" the result in the ordinary scalar Lagrangian was not A 3 but the renormalizable A 4 type coupling. Any higher powers of 45 in the superpotential inevitably lead to nonrenormalizable theories. The second noteworthy feature is that the Yukawa coupling g and quartic coupling g2 are linked. This link cannot be broken without destroying supersymmetry. Conversely supersymmetry insures that the renormalization of the Yukawa and quartic couplings preserve the connection between them. If in addition the supermultiplet qb was given a mass m, then the term would also yield a cubic term g m A * A 2 + c.c.

Again SUSY insures that the connection between the Yukawa coupling, mass and cubic coupling is unrenormalized. The second way to understand the auxiliary field F is to treat it like a quantum field without a kinetic term. The propagator for F is not 1/k z but instead it is just 1. The F field must be included in internal lines of graphs but since it does not have poles in its propagator it never appears externally on shell. Thus consider the coupling gAZF. This can be interpreted as vertex shown in fig. 7. By exchanging F we generate diagrams with 4 external A-lines with amplitude g2. This is of course just the quartic coupling induced by solving for F and eliminating it. Individual Feynman graphs can be combined into supersymmetric combinations according to a simple rule. Do not eliminate the auxiliary fields. Draw graphs which would occur in ordinary qb3 A /

A\ \ 9

/A \ ~)

---- g

/

F

/

/A

\

A

Fig. 7.

/ \

Fig. 8.

190

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theory. Then for each graph allow each internal line to be an A, ~ or F in all combinations. The vertices which are non-zero are:

}__.~_A

a)

/,

:

A ~p,

=

AAF

A\ 'k 4l

b)

F

A// Fig. 9.

(If in addition a mass term 4 2 is present then the rules must be slightly m o d i f i e d - w e leave it as an exercise to the reader.) Thus consider the vacuum supergraph in fig. 10.

~b Fig. 10.

According to our rule this includes the kinds of graphs in fig. 11. A

{al

(b)

Fig. 11.

The graphs (b) are really of the form as in fig. 12.

\ ¢

/ I

Fig. 12.

It is such supersymmetric combinations which satisfy the constraints of supersymmetry. For example, if we consider boson and fermion mass renormalization we can begin with a supergraph as in fig. 13.

Fig. 13,

Fig. 14,

L. Susskind, The gauge hierarchy problem, technicolor, supersymmetry, and all that

_m_

/ _ _ - - - - - -

/

I

"l-

.

.

191

C>

.

.

.

.

-653 ....

\ I

Fig. 15.

For external fermion lines this gives fig. 14, while for external bosons it is fig. 15. These contributions to fermion and boson masses are exactly equal. Similar considerations apply to renormalization of the coupling constants to find the combinations of graphs which preserve the connections between the quartic and Yukawa couplings. That scalar and spinor masses and couplings renormalize in exactly the same way is remarkable in a quantum field theory but can be understood as a direct consequence of supersymmetry. A far more remarkable fact for which I know no obvious explanation is that these couplings do not renormalize at all. This is a strange result which can be proved graph by graph using the supergraph formalism of Grisaru, Rocek, and Siegel but which has no general explanation. The general statement is that no coupling of the form ~ n + C.C.

is ever generated in the effective action by loop diagrams. Before concluding this survey of SUSY I will briefly describe another kind of supermultiplet which includes gauge bosons. This is the so-called real superfield containing components R=(V~,A,D)

where V~, is a vector field, A a real (majorana) spinor and D a scalar auxiliary field. The three components must have identical gauge quantum numbers (electric and weak charge, color, etc.) since V, will be gauge bosons such as photons, W-, Z °, gluons and so forth. The so-called gauginos A must consist of neutral colorless photinos, zinos, charged colorless winos and an electrically neutral color octet of gluinos. If the gauge symmetry of R is non-Abelian, R couples to itself through cubic and quartic couplings. The vertices include V V V Yang Mills vertices, AAV, vertices VDD. In addition, R couples to matter fields ~0 with couplings which include all the usual gauge coupling as well as A0A and DAA* coupling. In particular the A A * D couplings generate new quartic A A * A A * couplings when the auxiliary field D is eliminated.

3. Breaking SUSY Supersymmetry requires that particles come in multiplets forming chiral (AO) or real (VA) supermultiplets. This would require partners for all the usual particles. For example, quarks and leptons should have scalar partners, squarks and sleptons; Higgs bosons should be accompanied by fermionic higgsinos and gauge bosons by photinos, zinos, winos, and gluinos. Since no such particles exist at low energy we must assume SUSY is broken, and that the partners are massive enough to escape detection.

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As with ordinary symmetries supersymmetry can be broken either explicitly or spontaneously. We shall begin by considering explicit symmetry breaking. The context of the discussion will be a minimal supersymmetric extension of the standard SU(3) x SU(2) x U(1) + quarks + leptons + Higgs. Let us begin by enumerating the particle content of such a theory. (1) Gauge fields. The usual massless (before Higgs) gauge fields for photons, Z's, W's and gluons are accompanied by majorana photinos, zinos, winos and gluinos. The photino and zino are electrically neutral and colorless. A linear combination of photino and zino form the neutral partner of the charged winos. Together they form an SU(2)w~ak triplet. Gluinos are electroweak singlets which form an octet under color. Since the Lagrangian mass term for gauge bosons (pre Higgs) vanishes, supersymmetry would require massless gauginos. Thus bare masses for gauginos explicitly breaks SUSY. (2) Quarks and leptons are accompanied by scalar and pseudoscalar particles. The left (right) handed fermion SU(2)w doublets (singlets) are accompanied by complex zero spin partners to form chiral supermultiplets. The "left- and right-handed" spin zero particles can be added (subtracted) to form scalar (pseudoscalar) partners. In the absence of supersymmetry breaking these are degenerate with their fermionic partners. Furthermore if the Yukawa couplings of quarks and leptons to Higgs vanish then they must all be massless by virtue of chiral symmetry. In this limit any mass for scalar quarks (squarks) and leptons (sleptons) explicitly breaks SUSY. (3) Higgs doublets are accompanied by fermions. A simple Higgs doublet has a Weyl partner, one component of which is charged and the other neutral. I will now explain why two Higgs doublets are required. In the standard theory the Higgs field H can be used to give mass to both up and down quarks. This is because H and its complex conjugate form SU(2) doublets. Thus Yukawa couplings of the form

gDOLHD*R+ guQLH* U*R

(29)

are allowed. In a supersymmetric theory these couplings arise from cubic terms in the superpotential formed from the chiral supermultiplets of the same chirality. Now if H is an 1.h. chiral supermultiplet then H* is r.h. since OK, D~ and U~ are 1.h. The coupling OLHO~. is allowed but QLH*U*Ris not. The remedy for this situation is to double the Higgs content by introducing a second Higgs doublet H which transforms like/q** under the electroweak group but which is l.h. Eq. (29) is replaced by

gDQLHD *R+ guQLIZIU*R

(30)

in the superpotential. The supersymmetric part of the Lagrangian contains all the usual interactions generalized to be supersymmetric. This means gauge couplings and Yukawa interactions. No bare masses are allowed for gauge, quark, or lepton supermultiplets. SUSY does however, allow the Higgs mass term

which produces equal (positive) masses for all Higgs bosons. In order to produce spontaneous symmetry breaking in the electroweak sector, the Higgs mass matrix must have some negative eigenvalues. This can only arise from supersymmetry breaking. The breaking of supersymmetry should satisfy the criterion that it not lead to quadratic divergences

L. Susskind, The gauge hierarchy problem, technicolor, supersymmetrv, and all that

193

in the scalar masses and thereby render the theory unnatural. This constrains the breaking term in the Lagrangian to the so-called soft terms. Generally these include scalar and fermion masses as well as cubic scalar interactions. However, the constraints of SU(3) × SU(2) × U(1) limit the breakings to: (1) Scalar quark and lepton masses (2) Gaugino masses (3) Higgs and higgsino masses (4) Trilinear squark operators. Each of these operators has a coupling constant with dimensions of a mass or mass squared. Accordingly the radiative contribution to the scalar masses arising from soft breaking will generally have a coefficient of mass squared. This means that it is no worse than logarithmically dependent on the cutoff mass. The predictions of supersymmetry include the existence of a spectrum of superpartners. One of these superpartners must be stable. To see this let us define a quantum number R R = (- 1) F÷L÷3B where F = fermion number, L = lepton number, and B = baryon number. This quantum number is 1 for all ordinary particles and - 1 for their partners. Thus, to the extent that lepton and baryon numbers are conserved, R is a good quantum number. Accordingly, the lightest odd-R particle must be stable or almost SO.

Broken supersymmetry does not predict the masses of the superpartners. However, if supersymmetry is to solve the GHP, then they must be no heavier than a few TeV and possibly much lighter. If and when these particles are discovered, they will be unmistakable because supersymmetry predicts their couplings. For example, suppose a scalar electron is discovered which can decay into a photino. Supersymmetry tells us that the coupling constant for this process is the electric charge e.