New limits on the SUSY Higgs boson mass1

7 January 1999

Physics Letters B 445 Ž1999. 331–336

New limits on the SUSY Higgs boson mass Konstantin T. Matchev a , Damien M. Pierce

1

b

a

b

Theory Group, Fermi National Laboratory, BataÕia, IL 60510, USA Stanford Linear Accelerator Center, Stanford UniÕersity, Stanford, CA 94309, USA Received 22 July 1998; revised 15 November 1998 Editor: H. Georgi

Abstract We present new upper limits on the light Higgs boson mass m h in supersymmetric models. We consider two gravity-mediated models Žwith and without universal scalar masses. and two gauge-mediated models Žwith a 5 q 5 or 10 q 10 messenger sector.. We impose standard phenomenological constraints, as well as SUŽ5. Yukawa coupling unification. Requiring that the bottom and tau Yukawa couplings meet at the unification scale to within 15%, we find the upper limit m h - 114 GeV in the universal supergravity model. This reverts to the usual upper bound of 130 GeV with a particular nonuniversality in the scalar spectrum. In the 5 q 5 gauge-mediated model we find m h - 97 GeV for small tan b and m h , 116 GeV for large tan b , and in the 10 q 10 model we find m h - 94 GeV. We discuss the implications for upcoming searches at LEP-II and the Tevatron. q 1999 Elsevier Science B.V. All rights reserved. PACS: 11.30.Pb; 12.10.Kt; 14.80.Cp

If weak-scale supersymmetry ŽSUSY. exists it may be a challenge to discover. The superpartners may all be so heavy that they do not appreciably affect any low energy observables, and are below threshold for production at LEP-II and the Tevatron. In that case they will go undiscovered until the LHC turns on in 2005. However, one of the most robust and enticing hallmarks of supersymmetric models is the prediction of a light Higgs boson. At tree level, m h F MZ , but it receives large radiative corrections from top and stop loops w1x. Naturalness suggests that the third generation squarks should not be too 1 K.T.M. ŽD.M.P.. is supported by Department of Energy contract DE-AC02-76CH03000 ŽDE–AC03–76SF00515.. D.M.P. contact information – address: SLAC, MS 81, P.O. Box 4349, Stanford, CA 94309, USA. E-mail: [email protected]

heavy. Hence, we only consider squark masses below 1 TeV. Then, with a top quark mass of 175 GeV, the upper limit on m h is about 130 GeV Žincluding recent two-loop corrections w2,3x.. The largest possible value is obtained with heavy stops and large squark mixing. Supersymmetry goes hand in hand with grand unification. Grand unified theory ŽGUT. models predict that the gauge couplings unify at the GUT scale. In the MSSM, with superpartners below the TeV scale, this prediction is nearly satisfied w4x. The couple of percent discrepancy in gauge coupling unification finds a ready explanation in GUTs, from GUT threshold effects. In typical GUT models the bottom and tau Yukawa couplings Ž l b and lt . are predicted to unify as well. At leading order this only happens for either very small ŽQ 2. or rather large

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 4 7 4 - 9

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K.T. MatcheÕ, D.M. Piercer Physics Letters B 445 (1999) 331–336

Ž; m trm b . values of tan b Žtan b is the ratio of expectation values of the two Higgs doublets.. We take the muon decay constant, the Z-boson mass, the fermion masses, tan b , and the strong and electromagnetic couplings as inputs to determine l b and lt at the GUT scale Žthe scale where the UŽ1. and SUŽ2. gauge couplings meet.. We define the Yukawa coupling mismatch at the GUT scale to be ´ b ' Ž l b y lt .rlt , and, allowing for variations in the input parameters, GUT scale threshold corrections, and corrections from gravity-induced higherdimensional Planck-suppressed operators, we conservatively expect ´ b to be less than 15% in magnitude. At next-to-leading order bottom-tau unification is sensitive to the supersymmetric spectrum through radiative corrections. The corrections to l b are enhanced at large tan b and can be quite large w5–7x. They broaden the region at large tan b where exact unification is possible to 15 Q tan b Q 50. The branching ratio BŽ B ™ X sg . also receives large tan b enhanced corrections. The requirements of Yukawa unification and compliance with the BŽ B ™ X sg . measurement tend to conflict with each other. Depending on the model, imposing both constraints can single out a very particular parameter space, resulting in predictions for the superpartner and Higgs boson masses. In this letter we examine the Higgs boson mass predictions in four supersymmetric models – two gravity-mediated models Žwith and without universal scalar masses., and two gauge-mediated models Žwith a 5 q 5 or 10 q 10 messenger sector.. Yukawa coupling unification together with the b ™ sg constraint has been previously discussed within the context of the gravity-mediated models in Refs. w6,8x, but no conclusions about the light Higgs boson mass were drawn. Ref. w9x uses fine-tuning criteria in addition to the BŽ B ™ X sg . constraint to derive some limits on the light Higgs boson mass. In each model we randomly pick points in the supersymmetric parameter space Žthe parameter spaces are discussed below.. Rather than fix all the input parameters, we determine the Z-boson mass, the top-quark mass, and the electromagnetic and strong couplings at the Z-scale, in a global fit to precision data. We construct a x 2 function and minimize it with respect to the four standard model inputs. The x 2 function contains 30 electroweak precision observables, and BŽ B ™ X sg .. The list of

observables, the measurements we use, and further details are given in Ref. w10x. We fix the remaining inputs Gm s 1.16639 = 10y5 GeVy2 , mt Žpole. s 1.777 GeV, and m b Žpole. s 4.9 GeV. In the MSSM, the most volatile of the 30 observables is the b ™ sg observable. While we could exclude regions of parameter space by employing the full 30-observable x 2 , for the purposes considered in this letter it suffices to consider the b ™ sg observable alone as an additional constraint. This serves to focus, simplify and clarify the results and discussion. We impose a number of phenomenological constraints at each point in parameter space. We require radiative electroweak symmetry breaking and determine the CP-odd Higgs boson mass m A and the absolute value of the Higgsino mass parameter m to full one-loop order w7x. In the models we consider Žexcept the non-universal model. very large values of tan b Že.g. tan b R 60. are excluded since m2A is found to be negative 2 . We also require that all Yukawa couplings remain perturbative up to the GUT scale. Small values of tan b Ži.e. tan b Q 1.2. are excluded by this constraint, due to the top-quark Yukawa coupling Landau pole. Finally, we require that all the superpartner and Higgs boson masses are above the bounds set by direct particle searches. We calculate the gauge and Yukawa couplings using the full one-loop threshold corrections w7x and two-loop renormalization group equations w11x ŽRGEs.. The parameter dependence of the l b threshold corrections can be understood from the simplified approximation

dl b lb

1 ,y 16p

2

ž

8 3

g 32 m g˜ q l2t A t

m tan b

/

m 2q˜

.

Ž 1.

The first Žsecond. term is the gluino-sbottom Žchargino-stop. loop contribution. m q˜ is an average Žstop or sbottom. squark mass, m g˜ is the gluino mass, A t is the stop-stop-Higgs trilinear coupling and g 3 Ž l t . is the strong Žtop Yukawa. coupling. In a leading-order analysis, where the corrections Ž1. are neglected, l b and lt unify well below the GUT scale for intermediate values of tan b . With m ) 0

2

In those cases there are solutions with m2A ) 0, but tan b <1 and m b 4 m t .

K.T. MatcheÕ, D.M. Piercer Physics Letters B 445 (1999) 331–336

the corrections Ž1. make this situation worse, so that with tan b ) 2 ´ b falls in the range y20 to y60%. This discrepancy is larger than can be accounted for in realistic GUT models w12,13x. Also, variations in the input parameters D m t s "3 GeV, D m b s "0.15 GeV and D a s s "0.003 result in D ´ b s "1%, "3%, and .3%, respectively. With m - 0 the threshold corrections Ž1. help Yukawa unification by increasing l b at the weak scale, thus delaying its unification with lt to higher scales. Our conservative requirement < ´ b < - 15% restricts us to either tan b - 2 with m of either sign, or tan b R 5 and m - 0. The allowed values of A t play a central role in our discussion. Each model allows for a different range of values of A t , with corresponding implications. We start by discussing the results in the gravity-mediated model with universal soft parameters ŽmSUGRA.. In this model three inputs are specified at the GUT scale. They are a universal scalar mass M0 , a universal gaugino mass M1r2 , and a universal trilinear scalar coupling A 0 . The remaining two inputs are tan b and the sign of m. Because of the large top Yukawa coupling, the value of A t at the weak scale exhibits a quasi-fixed point behavior. Hence, the sensitivity to its high scale boundary condition is reduced. The quasi-fixed point behavior is illustrated in Fig. 1Ža., where we plot the dimensionless parameter a t ' A trM q˜ as a function of the renormalization scale Q in the 2 mSUGRA model Ž M q˜ ' M02 q 4 M1r2 is approximately equal to the first or second generation squark mass.. We see that A t tends to be negative at the weak scale. In that case the stop-chargino contribution in Eq. Ž1. partially cancels the sbottom-gluino contribution. At intermediate values of tan b Ž10 Q tan b Q 20. Yukawa unification requires that the correction Ž1. be maximized. This happens when A t ) 0, so that the stop-chargino contribution adds constructively to the sbottom-gluino contribution. Hence, at intermediate tan b large and positive values of a 0 ' A 0rM q˜ are necessary to achieve small < ´ b < in the mSUGRA model. This is illustrated in Fig. 1Žb., where we show the results of a scan over the mSUGRA parameter space. The striking correlation between a0 and tan b is emphasized by requiring < ´ b < - 5% in this figure. At large tan b the tan b

(

333

Fig. 1. Ža. Renormalization group trajectories of a t for M0 s 500 GeV, M1r 2 s 200 GeV, tan b s 20 and m - 0; Žb. The stop mixing parameter a 0 versus tan b with < ´ b < - 5%; Žc. The branching ratio BŽ B ™ X sg . versus a 0 , with the additional constraints < ´ b < -15% and m h )100 GeV. The horizontal line indicates the upper bound 4.2=10y4 . Žd. The light Higgs boson mass as a function of a0 , scanned over the mSUGRA parameter space with no additional constraints.

enhancement in Eq. Ž1. is by itself enough for successful Yukawa unification. In fact, at some points cancellation between the two terms in Ž1. is necessary, so small or negative values of a 0 are preferred. We also see from Fig. 1Žb. that in the small tan b region large positive A 0 is required, signifying that the corrections in Eq. Ž1. are relevant. The points in this region have values of the Higgs boson mass below 100 GeV. In 1995 the CLEO collaboration reported the measurement of the b ™ sg rate w14x. The upper bound on the rate is relevant for our analysis. CLEO found the upper bound BŽ B ™ X sg . - 4.2 = 10y4 at 95% C.L. In Ref. w15x it has been pointed out that in the extraction of the total branching ratio from the data the CLEO collaboration used an inappropriate model for the photon spectrum below 2.1 GeV. Using an appropriate model, Ref. w15x finds the 90% C.L. upper bound on the rate varies from 4.6 = 10y4 to 3.6 = 10y4 as m b varies from 4.65 to 4.95 GeV. This year the CLEO and ALEPH collaborations have reported new measurements of BŽ B ™ X sg . w16x. We use CLEO’s original bound, BŽ B ™ X sg . - 4.2 = 10y4 , as a constraint. The upper bound on BŽ B ™ X sg . imposes strong constraints on supersymmetric models. If m - 0 the

K.T. MatcheÕ, D.M. Piercer Physics Letters B 445 (1999) 331–336

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chargino-stop and Higgs boson contributions to the b ™ sg amplitude add constructively to the SM amplitude. Due to the tan b enhancement of the chargino loop contribution, very large total amplitudes can result, leading to predictions for BŽ B ™ X sg . well above the upper bound. As a result, significant regions of the SUSY parameter space are excluded. We can identify those by considering the following approximate formula for the leading supersymmetric corrections to the O 7 operator coefficient. With m 0 and tan b large, we have

d C 7 Ž MW . , y

3tan b 16p 2 < m <

ž

MW2 M2

y

1 A t m2t 2

m2t˜

/

,

Ž 2.

where the first Žsecond. term is the contribution from the t˜L y x˜q Ž t˜y h˜ q . loop Žwe work to first order in stop and chargino mixing.. MW Ž m t . is the W-boson Žtop-quark. mass, M2 is the SUŽ2. gaugino soft mass, and m t˜ is the average stop mass. If A t ) 0 there is destructive interference between the two terms, and the supersymmetric contribution to the b ™ sg amplitude is reduced. In Fig. 1Žc. we show the full one-loop prediction for BŽ B ™ X sg . in the mSUGRA model, subject to the b y t unification constraint < ´ b < - 15%. As expected, the rate is suppressed for large and positive values of a 0 . We see from Fig. 1Žc. that in the mSUGRA model with tan b ) 2, reasonably small < ´ b < and BŽ B ™ X sg . can occur only for relatively large and positive a0 Ž a 0 ) 1.1.. Because of the focusing towards negative values, the resulting values of a t at the squark mass scale are rather small Žy0.4 - a t 0.7.. Hence, top squark mixing is suppressed and the corrections to the Higgs boson mass are minimized. The scatter plot in Fig. 1Žd. shows m h versus a0 in the mSUGRA model. The Higgs boson mass is maximal at a0 s y1.7 and decreases with increasing a0 . In Fig. 2Ža. we show the scatter plot of m h versus ´ b in the mSUGRA model. We have imposed the b ™ sg constraint in Fig. 2. The vertical lines indicate the region < ´ b < - 15%. We see that the Higgs boson mass is below 114 GeV in this region. The mSUGRA model suffers from the rather ad hoc assumption of scalar mass unification. While we see no compelling justification for this boundary condition, if it did apply it would naturally hold at the Planck scale. The effects of running between the

Fig. 2. Scatter plots of m h versus ´ b in four supersymmetric models: SUGRA with Ža. universal or Žb. nonuniversal scalar masses; and minimal gauge mediation with a Žc. 5q5 or Žd. 10q10 messenger sector. In each case, we require 1.0=10y4 BŽ B ™ X sg . - 4.2=10y4 . The vertical lines on the plots delineate the region < ´ b < -15%.

Planck scale and the GUT scale can be significant w17x. Regardless of the boundary condition at the Planck scale, a GUT symmetry will ensure that GUT multiplets remain degenerate above the GUT scale. In SUŽ5., the parameter space at the GUT scale in general includes the soft mass parameters MH 1, MH 2 , M5 and M10 , corresponding to the 5 and 5 representations of Higgs fields, and the 5 and 10 representations of sfermion fields, respectively. ŽWe impose generation independence of the M5 and M10 masses.. Because of the larger parameter space of the nonuniversal SUŽ5. model, the upper limit on the Higgs boson mass, including the BŽ B ™ X sg . and approximate bottom-tau unification constraints, reverts to the general upper limit of 125 GeV Žsee Fig. 2Žb... Two-loop threshold corrections not taken into account in Fig. 2Žb. increase this limit to 130 GeV w3x. We can contrast the situation here with the standard mSUGRA case. Large splitting between MH 2 and M10 can lead to much larger values of m and m A for a given tan b . The larger value of m gives larger corrections to the bottom Yukawa coupling, making bottom-tau unification possible for smaller values of tan b . The smaller values of tan b , with larger m and m A , lead to reductions in the

K.T. MatcheÕ, D.M. Piercer Physics Letters B 445 (1999) 331–336

supersymmetric contributions to the b ™ sg amplitude. This, in turn, allows compliance with the BŽ B ™ X sg . upper bound with large and negative values of A 0 . Such A 0 values result in large squark mixing contributions to the Higgs boson mass Žsee Fig. 1Žd... The points with the largest m h have M10 , 1 TeV, M1r2 , 400 " 100 GeV, tan b , 14 " 3, A 0 in the range y1 to y2 TeV and, typically, MH 2 Q 100 GeV. Gauge mediation w18x is an attractive alternative to gravity-mediated supersymmetry breaking. One of the nice features of the minimal gauge mediation models we consider is the automatic scalar mass degeneracy w19x. All sfermions with identical quantum numbers have the same mass at the messenger scale. This provides a natural solution to the supersymmetric flavor problem. In order to preserve gauge coupling unification we consider two models with full SUŽ5. messenger sector representations, the 5 q 5 and 10 q 10 models. We assume a minimal Higgs sector, where the mechanism which gives rise to the B and m terms does not give additional contributions to the scalar masses. In the minimal models w19x the interactions between the dynamical supersymmetry breaking sector and a standard model singlet give rise to a vev in its scalar and F components. The coupling of the singlet to the messenger fields results in supersymmetry breaking and conserving messenger masses. To determine the effective theory below the messenger mass scale, the messenger fields are integrated out. The MSSM superpartners then receive masses proportional to L s FrS, where F Ž S . is the singlet F-term Žscalar. vev. At this order, there is no A-term generated. Hence, we set A 0 s 0 at the messenger scale. The messenger scale determines the amount of running of the soft parameters. The smallest allowed value of the messenger scale is 3 L. Since we do not want the gravity-mediated contributions to the scalar masses to spoil the solution to the supersymmetric flavor problem, we suppress the gravity-mediated contributions by requiring the messenger scale to be below MGU Tr10.

3 To be precise, we require M )1.03 L. Setting M s L would result in a phenomenologically excluded massless messenger squark.

335

Because A 0 s 0, we find the maximal Higgs boson mass in the gauge-mediated models is about 117 GeV. This bound arises solely from perturbative Yukawa coupling and electroweak symmetry breaking requirements, and a 1 TeV naturalness bound on the squark masses. The bound is not appreciably affected by the two-loop threshold corrections w3x. As in the gravity-mediated models we just considered, we perform a full next-to-leading order analysis. Hence we include effects in the one-loop RG evolution of the gauge couplings due to the splitting of the messenger multiplets, as well as messenger contributions to the two-loop gauge and Yukawa coupling RGE’s w13x. In the mSUGRA model we found that the BŽ B ™ X sg . and bottom-tau unification constraints required a 0 ) 1.1 at intermediate to large tan b . Since the gauge-mediated models have a0 s 0, one would expect that these models would not be compatible with the constraints at intermediate to large tan b if the spectrum did not significantly differ from the mSUGRA model. However, it is well known that the spectra in gauge- and gravity-mediated models can be quite different w20x. For example, in the 5 q 5 gauge-mediated model, the scalar masses and the m-term are significantly heavier for a given gaugino mass than in the mSUGRA model. Just as in the nonuniversal model, the larger m allows for bottom-tau unification with smaller values of tan b , and the reduced tan b and larger m and m A suppress the supersymmetric contribution to the b ™ sg amplitude. Hence, in the 5 q 5model there is a small amount of parameter space at intermediate tan b where the constraints are satisfied, even though A t - 0. In this region we find the prediction m h , 116 GeV. In the small tan b region m h - 97 GeV. The results are shown in Fig. 2Žc.. For a given gaugino mass, larger messenger sector representations result in lighter scalar masses. Hence, the 10 q 10 model has relatively lighter scalars than the 5 q 5 model. The lighter scalars make Yukawa unification more difficult, and readily result in too large values of BŽ B ™ X sg . at intermediate to large tan b . As can be seen in Fig. 2Žd., the two constraints taken together exclude the 10 q 10 model outright for intermediate to large values of tan b . The only allowed points with < ´ b < - 15% correspond to values of tan b - 2. In this region m h is less than 94 GeV.

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K.T. MatcheÕ, D.M. Piercer Physics Letters B 445 (1999) 331–336

Our results are particularly interesting in light of the upcoming Higgs boson searches at LEP and the Tevatron. LEP-II should be able to either discover or rule out a light Higgs boson up to about 105 GeV. If LEP finds a Higgs boson heaÕier than 96 Ž94. GeV, the minimal 5 q 5 Ž10 q 10. gauge-mediated model will require some modification in order to be compatible with bottom-tau unification. If, on the other hand, LEP does not find a light Higgs boson, the Yukawa unification criterion excludes the minimal 10 q 10 gauge-mediated model. Furthermore, if bottom-tau unification is taken seriously, upcoming runs at the Tevatron stand a chance to explore both the mSUGRA and minimal 5 q 5 gauge-mediated models. The Tevatron reach in m h as a function of its total integrated luminosity is currently under active investigation and no definite conclusions can be made at this point, but the upper limits of 114 and 116 GeV, correspondingly, can serve as important benchmarks in the design of an extended Run 2. Finally, if the Tevatron can place a limit on the Higgs boson mass above 116 GeV, this would point towards a particular nonunified scenario in the gravity-mediated models, and exclude minimal gauge-mediation altogether.

w3x w4x

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w6x w7x w8x

w9x w10x w11x

w12x w13x w14x w15x w16x

Acknowledgements w17x

KTM wishes to thank the SLAC theory group for its hospitality during a recent visit.

w18x

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