the minimal u (1)r-symmetric model revisited

Physics Letters B 286 (1992) 299-306 North-HoUand PHYSICS LETTERS B

The minimal U ( 1 )R-symmetric model revisited

Lisa Randal l ~ and Nur ia Rius Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 20 February 1992; revised manuscript received 12 May 1992

U ( 1 )R-symmetric models of low energy supersymmetry, in which R-parity is extended to a continuous symmetry, make definitive experimental predictions in contrast to the standard supersymmetric model. We show that, contrary to previous claims, the minimal model is phenomenologically viable because large radiative corrections to the Higgs mass can invalidate the Higgs mass constraint on the parameters of the model. We incorporate these corrections and demonstrate that, unlike the standard supersymmetric light Higgs boson, which may be much heavier than the Z if the top quark mass is large, the light Higgs of our model will very likely be accessible to LEP II. We also show that the sleptons should be within reach.

1. Introduction

In this letter, we reconsider the minimal U ( 1 ) R -

symmetric model (MR) [ 1 ] in which R-parity is extended to a continuous U ( 1 ) symmetry [ 2-4 ]. This model is worthy of further investigation because it has several advantages over the standard supersymmetric model. Perhaps the most significant experimental feature of this model is the restrictive gaugino spectrum. Unlike the standard supersymmetric model, which is difficult to pin down because of the extensive parameter space, the MR model is very constrained. This gives rise to definitive experimental tests of the model, including searches for wino pair and zino-photino production at LEP II. Further- more, the minimal U ( 1 )R-symmetric model has several theoretically appealing features. In the MR model, the fine tunings that are necessary in the standard supersymmetric model in order to get a sufficiently small neutron electric dipole moment and to get a/z Higgs mass mixing parameter of order of the electroweak scale are absent. Recall that in the standard supersymmetric model with a supergravity hid- den sector, the operator HIH2 appears in the superpotential, so its coefficient will only naturally be of order of the supersymmetry breaking scale with cer- tain structures of the underlying supergravity theory

On leave from Harvard Society of Fellows.

(see for example ref. [5] ). Recall also that the parameter A which appears in the supersymmetry breaking potential multiplying three scalar operators (as determined by the superpotential) can be com- plex, allowing for a neutron electric dipole moment at one loop [ 6 ]. In the MR model, the # parameter is naturally a supersymmetry breaking parameter whose magnitude is similar to other supersymmetry breaking parameters, and the A terms which could have CP violating phases are not present.

However, in the original paper [ 1 ], it was claimed that the minimal U(1 )R-symmetric model, though satisfying all the above properties, was excluded at the 2a level by data from LEP. The reason was that the combination of invisible Z width and Higgs mass measurements conspired to eliminate the entire region of parameter space. The bound on the Z width bounds tan fl from above, while the tree level Higgs mass constraint bounds tan fl from below. However, the bound based on a tree level calculation of the Higgs boson mass is incorrect. It has since been shown by several authors [ 7 ] that the tree level prediction for the light Higgs can be very inaccurate because of the potentially large one-loop radiative correction from top quark and top squark loops. The sum of these corrections, which grows as the fourth power of the top quark mass, can be very large. This correction can raise the light Higgs mass by tens of GeV. These authors showed that the corrections could be so large

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved. 299

Volume 286, number 3,4 PHYSICS LETTERS B 30 July 1992

that the Higgs of the standard supersymmetric model would escape the range of LEP II. In this paper, we investigate the consequences of this large radiative correction for the Higgs boson of the MR model. We show that with the one-loop correction to the Higgs mass incorporated, the Higgs mass constraint is extremely weak. Any region of mA-tanfl parameter space which is permitted without the Higgs mass constraint can be accommodated with a sufficiently heavy top quark. We reinterpret the Higgs mass constraint as a constraint on the minimum top quark mass required to permit a point in the mA-tan fl parameter space.

It is also of interest that the range of radiative corrections to the Higgs mass of the MR model is smaller than that of the standard supersymmetric model. This is because the photino, which in this model is massless at tree level, has a one-loop mass which decreases as the top squarks become more degenerate. This mass must satisfy the cosmological constraint on its present mass density. This means that in general, the radiative correction to the Higgs mass generated by the 7L and i'R loops is smaller than the maximum possible for the standard SUSY Higgs, which is achieved with degenerate left and right handed top squarks (assuming a maximum value for the supersymmetric particle masses). Because of the reduced radiative correction to the Higgs mass, we find that the Higgs of the MR model should be sufficiently light to be accessible to LEP II, even with radiative corrections incorporated. We show that a further consequence of the photino cosmological constraint is that in the fa- vored region of parameter space, tan fl> 1, the slepton mass is less than 100 GeV, so it should also be accessible to LEP II. Combined with the tests on the gaugino spectrum described in ref. [ 1 ], the MR model can be extremely well tested at LEP II.

We briefly review the model below and the constraints from LEP. We then work out the constraint imposed by the Higgs mass with one-loop radiative corrections incorporated. We derive the maximum Higgs boson mass over all the allowed parameter space with the constraint on the mass density in photinos incorporated. We also determine the minimum top quark mass to allow a point in parameter space with the simultaneous Higgs mass and photino constraints. Finally, we find the maximum slepton mass

over our allowed region of parameter space.

2. Review of the model and its gauge spectrum

The model is defined to be the simplest supersymmetric extension of the standard model with a continuous U ( 1 ) R symmetry. We define the continuous R symmetry by giving the coordinate of superspace, 0, charge + 1, all matter superfields charge + 1, and all Higgs superfields charge 0. Expansions of the superfields in terms of the component fields then show that all ordinary particles are R neutral while all superpartners carry nonzero R charge. One consequence of imposing U( 1 )R is that there are no Ma- jorana gaugino masses. This is not a problem for electroweak gauginos, which can acquire masses at SU (2) × U ( 1 ) breaking, but it is a problem for the gluinos. The model therefore contains an additional field to give a gluino Dirac mass. This field appears only in the soft supersymmetry breaking sector and is irrelevant to the rest of this paper so we omit it here. Details can be found in ref. [ 1 ].

The most general lagrangian consistent with the above assumptions is one described by the superpotential

f = U"2L~QH2 + D"2DQH, + E '2ELH, , ( I )

where each term has R = 2 and the quark and lepton superfields Q, U c, D ~, L, E c have the usual SU(3) × SU(2) × U ( 1 ) gauge interactions.

The most general soft supersymmetry breaking potential consistent with our symmetries and a GIM- like mechanism (so that flavor changing neutral cur- rents are naturally suppressed) is

~sof t 2 * = m m H ~ H l +m2~2H*H2 (2)

+ m Z L * E + m2:~Ec*~ ,

+ B H j H 2 + r~oO~+ h.c.,

where we have neglected small Yukawa suppressed corrections to superpartner masses. Notice that we do not assume that all superpartners have the same mass, although it is approximately true for superpartners with the same quantum numbers.

Because there is no H~H2 mass term in the super-

300


potential and there are no Majorana gaugino masses in the supersymmetry breaking potential, the gaugino mass matrix takes a very simple form. The only mass terms are the Dirac masses coupling the partners of the electroweak gauge bosons to the fermionic partners of the Higgs, which arise due to the vacuum expectation values of Ht and HE. Defining ~ as the fermionic partner of the W and ~ as the fermionic partner of the B one obtains the mass terms

, . ~ m = - m _ ~ - H + - m + ~ + _ ~ - - m z Y . B z , (3)

where

m_ = ~ / 2 m w sin fl, (4)

m+ =x /2mwcos fl, (5)

~= - s i n 0w/~+eos 0w~ 3 , (6)

/~z = cos fl/~° - sin fl/~ ° , (7)

where tan fl= Vz/Vt. Notice that at tree level, the zino is degenerate with the Z gauge boson, while the photino, and/Te, the Higgs field orthogonal t o / l z , are massless. Also notice that there is always a charged wino state with mass less than that of the W. This restricted gauge spectrum can have interesting experimental consequences, as discussed in the previous paper [ 1 ].

At one loop, the photino and/7~ get a Dirac mass. The diagram is shown in fig. 1. The result of calculat- ing the photino mass is [ 1 ]

2

rn~= 1.3 cot 20

] m2 2 2 m21 X 2- - - 2 1 n ~ m~. z----Z2... 2 In - - . (8)

mrL-m~ m, rn-&-m, m~ [

Notice that in this model, the invisible Z width is larger than in the standard model because the Z can decay into the Higgsino which is the Dirac partner of the photino. However, because the Z branching frac-

~{{c)

7 N / X

! \

I

t ~'t c) (t) A

Fig. 1. One-loop contribution to the photino mass.

tion into photino and Higgsino is suppressed relative to a single standard model neutrino by a factor cos 22fl, this is consistent with constraints on the invisible Z width so long as tan fl is very close to 1. The present constraint on the Z width [ 8 ] AFz/F~ < O. 11 at 2a translates into the constraint that ]cos2fll < 0.33. This constrains tan fl to lie between 0.71 and 1.41. We assume this constraint throughout the re- mainder of this paper.

3. The Higgs mass at one loop

One can also derive relations among the masses of the scalar, pseudoscalar, and charged Higgs fields. These relations are identical to those of standard supersymmetric models [ 9 ].

The tree level mass matrix for the two-scalar Higgs particles is

m 2 c o s f l + m 2 sin 2fl

--sin fl cos fl( m 2 + m2A) - sin fl cos fl(M2z + m~) M 2sin 2 f l + m 2 cos z fl ]

(9)

Here mA is the tree-level expression for the CP-odd neutral Higgs mass. At one loop, these relations are corrected by top quark and squark loops. If the re- normalization point is chosen to maintain the tree level vacuum expectation value of the Higgs field, one derives the additional oneqoop correction to the (2,2) element of the H iggs mass matrix [ 7 ]:

3g 2 rn 4 (m~Lm~R~ 16~r2M~. sin2 fl log \ - ~ - ~ 4 ] •

(10)

Notice in our model, the correction is extremely simple. This is because the A parameter terms in the supersymmetry breaking potential and the H~H2 term in the superpotential do not appear. We have neglected the b squark contribution which in our range offl is negligible.

The physical Higgs fields are the eigenvectors of this matrix. In general, the expression for the mass is complicated; however in the limit where mA is much larger than the other masses the Higgs mass takes a simple form. In the large mA limit, the light Higgs mass is

301


m 2 = M ~ cos 2 2fl

3g2 rn 4 log ( rn~L rn2R "~ + 16zt2M 2 k , ~ , ] " ( 1 1 )

Notice in this limit we get the sum of the maximum tree-level mass and a fl-independent one-loop contribution. In general, the Higgs mass increases with mA because o f the large tree level contribution but becomes rnA-independent at large mA.

In figs. 2a and 2b, we plot the min imum value o f the top quark mass that generates a sufficiently large Higgs mass as a function of tan ft. Each curve is a con- tour o f fixed mA. For the value o f tan fl on the y axis, we have solved for the min imum top quark mass which yields a Higgs mass bigger than 44 GeV, the current published Higgs mass bound from LEP [ 10 ]. (Although this applies to a standard model Higgs, this is a fairly accurate bound for tan fl~ l . ) In the first plot, we have incorporated radiative corrections from degenerate left and right handed top squarks of mass 1 TeV. We allow tan fl to vary in the range allowed by the Z width bound and show the constraint for mA taking the values 50, 100, 150, 300 and 500 GeV. The region to the left of the curves is excluded. We see that as expected from eq. (11 ) the mA dependence becomes very weak for large m , and the curves become symmetric about tan f l= 1. We see that with the large radiative correction, the Higgs mass constraint is readily satisfied. Even top quarks o f mass lower than 100 GeV can provide a sufficiently large radiative correction if cos 2fl is close to its maximum value. Moreover, for a top quark of mass greater than about 140GeV, all points with 0 . 7 < t a n f l < 1.4 and m4> 50 GeV are permitted. In fig. 2b, we assume the top squarks are not degenerate; we take one to have a mass o f 1 TeV and the other to be degenerate with the top quark. We will see that in the MR model, the top squarks are constrained to be nondegenerate so that the true constraint curves lie between those of plots 2a and 2b. The values of mA are the same as in plot 2a. Here the parameter space is somewhat more restricted; only top quark masses greater than 118 GeV can provide a sufficiently large radiative correction and only a top quark mass greater than about 170 GeV allows the entire region of tan fl-mA. Nevertheless, it is clear that even with a large radiative correction

tg(fl)

-..:. ,, / 1.3 '.i.. ~ ',,

1.1 - , , i I

~!;

0.9 .:~7

0, (a)

100 110 120 130 140 150 160 170

m,(Oev) 180

tg(~) 1.4 •

1.3 ""i '. %\

1.2 ""i '

IA = i ;

09 .fi/

0.8

100 110 120 130 140 150 160 170 180

m,(OeV)

Fig. 2. Minimum value of the top quark mass which yields a Higgs mass bigger than the current bound from LEP (44GeV), as a function of tan ft. The different curves correspond to m A taking the values 50, 100, 150, 300 and 500 GeV, the stronger constraint for the lighter rnA. The region to the left of the curves is excluded. (a) mF L =mr~ = 1 TeV. (b) m/L = 1 TeV; mr~ =m, .

from only one of the top squarks the Higgs mass constraint is not very restrictive; even if the top quark mass turns out to be quite low, a large region of parameter space survives.

302


4. Bounding the Higgs mass

Having shown that the model is viable, we now ask the question of whether the MR model is testable in the Higgs sector, once the radiative contribution to the Higgs mass has been incorporated. The question is whether the Higgs mass can be so large as to elude LEP II. In this section, we study the maximum allowed Higgs mass at any point in parameter space.

Were it not for the cosmological constraint on the mass of the photino, the Higgs mass of our model could vary over the same range as that of the standard supersymmetric model. With the constraint incorporated, we are forced to keep the mass of the left and right handed top squarks nondegenerate. The point is that the photino annihilation cross section grows as the square of the photino mass; too light photinos have too small annihilation cross section and therefore, by the Lee-Weinberg analysis [ 11 ], a large residual mass density today. The photino mass is given by the formula in eq. (8). The min imum allowed mass is determined from the annihilation cross section through s-channel Z exchange and t-channel squark and slepton exchange. We maximize these by taking degenerate squarks of mass 105 GeV and degenerate sleptons of mass 65 GeV (using limits from the Particle Data Book [ 12 ] ). The constraint on the photino mass is then

Ff cos2f l ~2 1 _ _4~_1/2

m~=L~-IT8~--~) + (6 GeV)2 ~ q ? ( ~ " ) J

X (~Qphotino h 2 ) - 1 / 2 , ( 12 )

where i runs over all squarks and sleptons o f charge q, and mass mi such that the corresponding quarks and leptons are accessible in photino annihilation. It is readily seen that the squark contribution is negligible ~l.

In fig. 3, we show the maximum possible Higgs mass as a function of top quark mass when we fix the maximum top squark mass to be 1 TeV and choose the mass of the second top squark to be as large as possible, consistent with the photino mass constraint. Here mA has been taken to be large ( 500 GeV)

#1 Even with the weaker squark mass bound of about 85 GeV which allows for more general squark decay final state, the squark contribution is negligible.

m.(Gev)

100

90

+ !!!!!!!!!:!!. ,0 y!.

f;. 60 ;;... 2; j ' " ., ,

50

4O

30

100 110 120 130 140 150 160 170 180 m,(OeV)

Fig. 3. Maximum possible Higgs mass in the MR model, as a function of the top quark mass. Here, mA=500GeV and tan fl=0.71. One top squark mass is 1 TeV and the second top squark is as heavy as possible, consistent with the photino mass constraint, for different values of ~photl,oh2:1 (solid), 0.5 (dashed), 0.333 (dash-dotted) and 0.25 (dotted).

and tan fl small in order to maximize the Higgs mass. The solid curve represents the constraint ~2photi,oh ~ = 1. We see that over the entire region of parameter space, the Higgs is always lighter than 90 GeV. This is a very important result in that it shows that the Higgs of this model can be found if LEP II succeeds in probing all values of the Higgs mass up to Mz. The remaining curves show the maximum Higgs mass assuming more stringent values of~C2photinoh2: 0.5, 0.333, 0.25. We see that very large values of top quark mass are required to generate a sufficiently heavy photino to get adequate annihilation to reduce the energy density in photinos to these levels. For example, if one believes .Qphotinoh2=0.25, we see that the top quark mass must exceed 155 GeV. Notice that the curves show only that ff2photinoh 2 must be near 1 for a light top but not necessarily for a light Higgs; we have shown in these curves only the maximum Higgs mass consistent with the parameters. It is clear from these curves that one expects the Higgs to be accessible to LEP II in the MR model.

I f we allow mA and tan fl to vary, we find the Higgs mass can take any value between the current lower bound of 44 GeV and the maximum value given in

303


the curves o f fig. 3, for any value of top quark mass. In the above plots, we have assumed the maximum top squark mass is 1 TeV. The maximum reasonable value has been formalized for the standard supersymmetric model by Barbieri and Giudice [ 13 ]; although the top squark mass is in principle an independent parameter in our model, which is not derived from an underlying supergravity theory, this seems like a reasonable value if all supersymmetry breaking parameters are approximately equal. The maximum possible Higgs mass of course depends on the top squark mass. For example, if we allow the top squark mass to be as large as 1.5 TeV, the Higgs mass would be about 10 GeV larger.

Finally, we now give the min imum allowed top quark mass for the various points in tan fl-mA parameter space with both the Higgs mass and photino mass bounds taken into account. In general, the two curves of the constraint versus top quark mass inter- sect at the min imum allowed value of top mass. We give this plot in fig. 4a, with parameters chosen as in fig. 2. Notice that for large tan fl and mA, the photino mass constraint takes over. We see that the model is excluded if the top quark mass is discovered to be smaller than l l0GeV. As expected, the true constraint lies between the regions specified by curves 2a and 2b.

Given the rapidly improving Higgs mass bound from LEP, we think it is worthwhile to also show the true bound on top quark mass for each point in parameter space if LEP does as well as it can and gives a lower bound on the Higgs mass of 65 GeV ~2 (clearly any intermediate Higgs value corresponds to curves between those of figs. 4a and 4b. ). We see that if this mass limit is attained and the top quark is discovered to be lighter than 145 GeV, the model could be excluded before LEP II begins operation. Otherwise, we expect the model to remain viable.

5. Superpartner masses

tg(t~) 1.4 " w ~ ~ I f ~ ~ ~ I , , , , I , ~ , r pf

(a) 1.3

1.2

1,1

1

0.9 1

0.8

0.7 '" I 0 0 t l 0 120 130 140 150 160 170 180

m,(GeV)

tg(¢) 1.4 - ,

~. i

(b) :: i 1.3 ! .

1.2 i

1.1 !. ~ I /

-. i ! -- ..' ,,

:. j /

0.9 / i : : / / / / 4

0,8 "':" " """

0.7 , , ~ 1 . . . . I , , , , I , , , , I t , - ' ~ 100 110 120 130 140 150 160 170 180

m,(GeV)

Fig. 4. Min imum allowed top quark mass with both the photino mass constraint (#pho,i,ohZ= 1 ) and the Higgs mass constraint incorporated. The values ofmA are as in fig. 2. (a) Current bound on the Higgs mass from LEP (44 GeV). (b) Best bound on the Higgs mass attainable at LEP (65 GeV).

Finally, we analyse the bound on the slepton mass due to the cosmological constraint on the mass of the photino, given by eq. ( 12 ). Notice that a slepton contribution to the cross section is always necessary in

~2 We thank Mike Barnett for private communication.

a n ~"2photinoh 2 ~< 1 universe because the Z annihilation cross section alone is inadequate. Given the squark mass constraint, the squark exchange contribution is negligible, so the only way to satisfy the photino mass constraint is with a sufficiently light slepton. By the

304


assumption of the MR model, the sleptons o f a given handedness are degenerate. To generate the most conservative bound on the slepton mass, we maximize the slepton contribution to the annihilation cross section by assuming that both handedness of slepton are degenerate when deriving this bound.

In fig. 5 we show the maximum slepton mass as a function of tan fl for different values of the top mass, assuming that all the sleptons are degenerate. Again, we fix the mass of one of the top squarks to be 1 TeV and we choose for the second stop the mass which gives the heaviest photino from eq. (8), provided the experimental constraint on the Higgs mass is satisfied. ma has been taken to be 500GeV and f2pholi~oh 2 = 1. We get a bound on the slepton mass for any value of m, and tan fl, and moreover in the region of larger tan fl, tan fl> 0.78, the sleptons should be light enough to be accessible to LEP II.

In fig. 5, the second solid curve gives the bound on the slepton mass which is required to generate ~r2photinoh2=0.25. We see the constraint is signifi- cantly stronger; the maximum slepton mass is 73 GeV, always accessible to LEP II.

Notice that except for low values of the top quark

m i ( G e V . . . . I . . . . L ' 7 ~ r ~ ~ ' ' ; ' ] : : ' :

120

110 i ~

tO~

80 .

60

50

40 ~ ~ I , ~ ~ ~ I L L ~ , I L L L I I L ~ ~ m [ ~ , , ~

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

t~(~)

Fig. 5. Maximum slepton mass as a function of tan fl, assuming that all the sleptons are degenerate. In the three upper curves g2photinoh2 = 1 and mr= 180 GeV (solid), 160 GeV (dash-dotted) and 140 GeV (dashed). The second solid curve gives the bound for -Qphotinoh 2 = 0.25 and m, = 180 GeV.

mass, the curves are determined entirely from the energy density constraint, and not the Higgs mass. This can be seen from fig. 2a, where it is clear that for sufficiently heavy top quark, the Higgs mass constraint is satisfied for any point in tan fl-mA parameter space which is not otherwise excluded.

6. Conclusion

We see that contrary to previous reports, the minimal U ( 1 ) R symmetric model remains a very interesting alternative candidate to the standard supersymmetric model. The phenomenological consequences are extremely interesting. The Higgs might be light enough to be discovered at LEP; it should lie within reach of LEP II. The slepton is also very likely to be accessible to LEP II. A further consequence of the light slepton is the zino photino production cross section is large. At LEP II, one can hope to see on the order of 1000 events of this type. Furthermore, LEP II should be sensitive to wino pair production, and the Tevatron can search for W decay to wino photino. Of course many of these signatures apply not just to the minimal model, but to all U( 1 )R symmetric models. However, because of its simplicity, the absence of the lz parameter problem, and the definite prediction for the range of Higgs mass, the minimal model remains the most interesting of these models. Whether it is correct however remains to be determined by experiment. We stress that these tests are definitive because of the very restricted parameter space of the model.

Acknowledgement

We thank Lawrence Hall for a useful discussion at the inception for this work. We also thank Mike Barnett for updates on experimental constraints and Bob Cahn for comments on the manuscript. This work is supported in part by funds provided by the US Department of Energy (DOE) under contract #DE-AC02-76ERO3069 and by CICYT (Spain) under Grant No. AEN90-0040. N.R. is indebted to the MEC (Spain) for a Fulbright scholarship.

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the minimal u (1)r-symmetric model revisited

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