VOLUME 78, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 10 FEBRUARY 1997

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Gauge and Yukawa Coupling Unification in Models with Gauge-MediatedSupersymmetry Breaking

Jonathan A. Bagger,1 Konstantin T. Matchev,1 Damien M. Pierce,2 and Ren-Jie Zhang1

1Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21212Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309

(Received 5 November 1996)

We examine gauge and Yukawa coupling unification in models with gauge-mediated supersymbreaking. We work consistently to two-loop order, and include all weak, messenger, and unificscale threshold corrections. We find that successful unification requires unification-scale threcorrections that are in conflict with the minimal SU(5) model, but are consistent with the modmissing-doublet SU(5) model for small tanb, and large tanb with m . 0. [S0031-9007(97)02325-9]

PACS numbers: 12.60.Jv, 11.30.Qc, 12.10.Kt

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The apparent unification of the gauge couplings inminimal supersymmetric standard model (MSSM) [1] hsparked much interest in supersymmetric extensions tostandard model. In their present form, most phenomelogically viable models have two sectors: a hidden secin which supersymmetry is broken, and a visible secwhich contains the standard-model particles and theirpersymmetric partners. Supersymmetry breaking is tramitted to the visible sector by gravitational interactio(as in supergravity-inspired models) or by standard-mogauge interactions (as in models with gauge-mediatednamical supersymmetry breaking).

Models with gauge-mediated supersymmetry breakare usually constructed to preserve gauge-coupling ucation to one-loop order. In this Letter we will report oncloser look at unification in gauge-mediated models.will present the results of a complete two-loop analyfor gauge and Yukawa coupling unification. Our comptation takes all one-loop thresholds into account, includthose at the weak, messenger, and unification scales.thresholds include finite terms which turn out to be veimportant for our precision analysis.

We will present our results in terms of the modeindependent unification-scale threshold correctionseg andeb [2]. These parameters describe conditions that mbe satisfied by any viable unification model. We willustrate the range of these parameters for the minim[3] and (modified) missing-doublet [4,5] SU(5) modeWe will see that present precision measurements excthe minimal model, but are consistent with gauge aYukawa unification in the modified missing-doublet cas

In the simplest models of gauge-mediated supersymtry breaking [6], the messenger sector contains a set oftorlike fields which couple only to a standard-model singspurion through trilinear terms in the superpotential. Tvectorlike messenger fields are chosen to transform5 1 5 or 10 1 10 representations of SU(5). Requiring thgauge couplings to remain perturbative restricts attento at most four5 1 5 or one10 1 10 plus one5 1 5 pairof fields. (An additional5 1 5 pair can be accommodateif the messenger particles are sufficiently heavy.)

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We assume that the lowest (S) and highest (F) com-ponents of the spurion acquire vacuum expectationues (VEVs) through their interactions with the hiddsector. These interactions remove the mass degeneof the messenger superfields and transmit supersymtry breaking from the hidden to the visible sector throuloop diagrams which contain spurion insertions. At tmessenger scale, gaugino and soft scalar masses aduced by one-loop and two-loop diagrams, respectivThe flavor-blind nature of the gauge interactions ensuthat flavor-changing neutral currents are suppressed.this order, the soft supersymmetry-breakingA parameteris not generated.

The supersymmetric Higgs mass parameterm and thesoft supersymmetry-breakingB parameter violate a PecceQuinn symmetry and cannot be generated by standmodel gauge interactions. We will assume that theygenerated by some minimal mechanism. The region whB ­ 0 is theoretically appealing [7] because it gives rto a large ratio of VEVs (tanb) without fine tuning. Inthis region, all CP-violating phases are generated oradiatively, so CP violation is naturally small.

Our approach is as follows. We start with the Ferconstant, GF , the electromagnetic coupling,aem, theZ-boson mass,MZ , the MS strong coupling constanassMZd, and the top-, bottom-quark, and tau-lepton pmasses,mt , mb , and mt (for details, see [8]). We theassume a supersymmetric spectrum and use the fullloop corrections to calculate theDR couplingsg1, g2,g3, lt, lb , and lt for a given value of tanb. We runthese couplings to the messenger scale,M, using the two-loop MSSM renormalization group equations. AtM wefix the gaugino and soft scalar masses [9]. We thenthe soft parameters back to the squark mass scale, wwe impose electroweak symmetry breaking and calcuthe supersymmetric spectrum. We iterate the procedseveral times to achieve a consistent solution.

Our calculations of the one-loop threshold correctioinclude the finite and logarithmic terms. The finite corretions, which are often neglected in the literature, allowprecise determination of the gauge couplingsg1 andg2 at

© 1997 The American Physical Society

VOLUME 78, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 10 FEBRUARY 1997

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the scaleMZ [2,10]. The finite corrections to the bottomand tau Yukawa couplings also play an important roleour analysis.

Once we determine the gauge and Yukawa couplingsthe messenger scale, we extrapolate them to the unificascale,MGUT, which we define to be the scale whereg1

andg2 meet. We use the usual two-loop beta functionscompute the evolution of the gauge and Yukawa couplingWe also include the messenger contributions, those lisin Ref. [11,14], and [15]

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The sum overf runs overall messenger multiplets,n5

andn10 are the number of5 1 5 and10 1 10 messengerfields, and

Dif ­

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375, f ­ d, ,, q, u, e .

(3)

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3775, a ­ t, b, t . (4)

At MGUT we set the messenger Yukawas to a commvalue, ym. We run the messenger Yukawas back to thmessenger scale according to their one-loop evolutiequations,

mdyf

dm­

yf

16p2

√2y2

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i­1

g2i Cis fd

!, (5)

where T ­ n5s3y2d 1 2y2

,d 1 n10s6y2q 1 3y2

u 1 y2e d and

the Ci ’s are the quadratic Casimirs,3Y2y5, 3y4, and4y3 for fundamental representations. (The messengYukawa evolution equations can receive additionmodel-dependent contributions from the hidden-secparticles. The extra terms do not affect the messenmass splittings, so we can ignore them in our analysNote that the one-loop equations suffice becausemessenger-sector Yukawas enter our calculation othrough the messenger threshold corrections.)

From the set ofyf sMd, we determine the messengerparticle mass spectrum and compute the messenger-sthreshold corrections to the gauge couplings,

Da21i sMd ­

Xf

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ln

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∂∏, (6)

whereL ­ FyS is the supersymmetry-breaking scale anMf is the messenger fermion mass. The second term

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the brackets is small forLyMf ø 1, in which case thereis a near degeneracy among the masses in the vectosupermultiplets. Note that there are no messenger-sYukawa thresholds to this order.

We iterate this procedure to find a consistent solutin the region betweenM and MGUT. At MGUT wedefine the threshold corrections for the gauge and Yukcouplings,eg andeb , as follows,

g3sMGUTd ­ g1sMGUTd s1 1 egd ,

lbsMGUTd ­ ltsMGUTd s1 1 ebd .(7)

The parameterseg and eb describe the unification-scathreshold corrections that are necessary to achieve ucation in any particular model. In what follows, we windicate the allowed ranges ofeg and eb for two of thesimplest unification models, the minimal and the modifimissing-doublet SU(5) models.

In the minimal SU(5) model, the unification-scagauge threshold correction is [2,16]

eg ­3g2

GUT

40p2ln

µMH

MGUT

∂, (8)

whereMH is the mass of the color-triplet Higgs multiplthat mediates nucleon decay. Generally,MH is boundedfrom below by the proton decay limits [17], which impMH * MGUT , soeg * 0.

The missing-doublet model is an alternative SUtheory in which the heavy color-triplet Higgs particlare split naturally from the light Higgs doublets [4This requires large SU(5) representations, such as th75and 50 1 50, so the SU(5) couplingg5 diverges belowthe Planck scale. The modified missing-doublet (MMmodel solves this problem forn5 # 1 by lifting themass of the50 1 50 to the Planck scale and suppressthe nucleon decay rate through an extra Peccei-Qsymmetry [5]. In this way the modified missing-doubmodel can accommodate two color-triplet Higgs particwith masses between1013 1015 GeV.

In the modified missing-doublet model, the unificatioscale gauge threshold can be written as [2,5,18]

eg ­3g2

GUT

40p2

Ωln

µMeff

H

MGUT

∂2 9.72

æ, (9)

where MeffH is the effective mass that enters the pro

decay amplitude, so the previous lower bounds onMH

apply here as well. In the MMD case, the effective mis also bounded from above,Meff

H & 1020 GeV [5].The Yukawa threshold in minimal SU(5) can be writt

as follows [2,17]:

eb ­1

16p2

Ω4g2

GUT

∑ln

µMV

MGUT

∂2

12

∏2 l2

t sMGUTd∑

ln

µMH

MGUT

∂2

12

∏æ, (10)

where MV is the mass of a superheavy SU(5) gauboson. For the minimal SU(5) model, the most string

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lower limit on MV comes from requiring that the5 1 5Higgs coupling remain perturbative to the Planck scaThis impliesMV * 0.5MH [17]. We take the upper limiton MV to be the Planck scale,MV # 1019 GeV.

For the modified missing-doublet model, the Yukawthreshold has the same form as Eq. (10), with the cotriplet Higgs mass,MH , replaced by the effective massMeff

H . In this case, the lower limit onMV comes fromproton decay experiments, which implyMV ygGUT *

3.8 3 1015 GeV [18]. As before, we imposeMV #

1019 GeV. Hence, both models have the same upper lion eb, but the lower limit in the MMD model is lower, byvirtue of the fact thatMV can be smaller andMH larger.

In what follows, we present our results for gaugmediated models. In particular, we calculateeg, eb ,aGUT , and MGUT as functions of the input parameterwhich we take to be tanb, the numbersn5 and n10, thesupersymmetry-breaking scaleL, the messenger scaleM,and the messenger Yukawa at the unification scale,ym.To examine bottom-tau unification, we fix the sign ofm

to be positive.We find the range ofaGUT and MGUT by scan-

ning over the parameter space, withmt ­ 175 GeV,mb ­ 4.9 GeV, L # 300 GeV, 1.03 # MyL # 104,0.03 # ym # 3.0, and tanb in the allowed range.For the casen5 ­ 1, we determineaGUT . s0.0440.054d and MGUT . s1.5 5.0d 3 1016 GeV. For n5 ­n10 ­ 1, we find aGUT . s0.062 0.28d and MGUT .s1.2 7.0d 3 1016 GeV.

In Fig. 1 we plot eg and eb for n5 ­ 1, MyL ­ 2,mb ­ 4.9 GeV, andym ­ 1, versusL and tanb, respec-tively. In (a) we choose tanb ­ 20, while in (b) wetakeL ­ 100 TeV. In each case the short-dashed (londashed) lines correspond toassMZ d ­ 0.124 s0.112d.The black bands correspond toassMZd ­ 0.118 with mt

varying from 170 to 180 GeV. The uncertainty ineb fromvarying mb ­ 4.9 6 0.3 GeV is almost the same as thafrom changingassMZd ­ 0.118 6 0.006.

In Fig. 1(a) we also show the allowed values foreg

in the minimal and modified missing-doublet SU(5) moels. The region of allowedeg in the modified missing-doublet model almost completely overlaps the region wassMZ d ­ 0.118 6 0.006. In contrast, we see that minimal SU(5) is inconsistent withassMZ d by more than2s.

For n5 ­ 1 we find that the messenger-sector corretions decreaseeg. The change is induced by the messeger thresholds and the differences in the two-loop gaucoupling evolution. Both of these effects are of appromately equal importance.

From Fig. 1(a) we see that raising the supersymmebreaking scaleL decreases the size of the gauge-couplunification-scale threshold. This is because the superpner masses scale withL, and larger masses decrease tsize of the required thresholds [2,10].

Figure 1(b) illustrates the well-known fact that bottomtau unification can be achieved only for very small (&1.8)or rather large (*35) tanb. (Very large values of tanb

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FIG. 1. The unification-scale threshold corrections win5 ­ 1, m . 0, MyL ­ 2, and ym ­ 1. (a) The gaugecoupling unification-scale threshold correctioneg versus L,for tanb ­ 20, and assMZd ­ 0.118 (black band), 0.124(short-dashed), and 0.112 (long-dashed). (b) The Yukacoupling unification-scale threshold correction,eb , versustanb, for L ­ 100 TeV and the same values forassMZd as in(a). In each case, the black band is obtained by varyingtop mass from 170 to 180 GeV. The shaded regions indicthe allowed range for (a)eg and (b) eb in the minimal andmodified missing-doublet SU(5) models.

are excluded by the requirement of proper electrowesymmetry breaking.) Figure 1(b) also shows the allowbands foreb in the minimal and modified missing-doubleSU(5) models.

As above, we can compare this plot to the case withmessengers. There, one typically finds that the bottand tau Yukawa couplings meet much earlier thanscale MGUT , so a rather large and negative threshocorrection eb is required. For the case at hand, thextra messenger multiplets change the Yukawa evolutequations at two loops. More importantly, however, thalso increase the gauge couplings, which feed intoYukawa evolution equations and cause the bottom andcouplings to meet even earlier. This makeseb even morenegative.

Fortunately, at large tanb there are significantfinitethreshold corrections to the bottom (and to a smallertent, tau) Yukawa couplings [8]. These corrections, whiare proportional tom tanb, are sufficiently important topermit bottom-tau unification at large tanb for m . 0.(The casem , 0 is completely excluded at large tanb.)These finite corrections were omitted in the analysisRef. [14], which came to a different conclusion.

For n5 ­ 1, the value ofeg is not significantly affectedby changes in tanb or MyL. At the smallest values oftanb, eg increases by about 0.5%, while forMyL ­ 104,eg increases by about 0.2%. The parametereb is moresensitive to changes inMyL. For MyL ­ 104, the eb

curve is 2.5% to 3% higher at intermediate tanb, andrises to120% at tanb . 40.

In Fig. 2 we plot eg and eb for the case ofn5 ­n10 ­ 1, versusL and tanb, respectively. The otherparameters are fixed as in Fig. 1, except that in Fig. 2(L ­ 50 TeV (to keep the scalar masses unchanged)MyL ­ 100. (Two MyL ­ 2 curves are shown in dottedlines.) Figure 2(a) shows that everything shifts because

VOLUME 78, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 10 FEBRUARY 1997

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FIG. 2. The same as Fig. 1, except thatn5 ­ n10 ­ 1, andin (b) L ­ 50 TeV and MyL ­ 100. The dot-dashed linesindicateeb and its lower limit in the MMD model forMyL ­2. The line in the upper right-hand corner of (b) describestop-quark threshold,et , for MyL ­ 100.

the largeraGUT , but the overlap between the band frothe MMD model and the allowed region forassMZd isstill almost complete. In this case, increasingMyL to104 significantly changes Fig. 2(a). The central valueeg runs from24% for L ­ 20 TeV to 21.5% for L ­100 TeV. The band for the MMD model is such that threquired value ofeg lies entirely within the band. (Notehowever, thatn5 ­ n10 ­ 1 gives rise to nonperturbativecouplings aboveMGUT in the MMD case.)

The change in Fig. 2(b) as compared to Fig. 1(b)more dramatic. Because the gauge couplings are elarger than in the previous case, bottom-tau unificatturns out to be barely possible forMyL ­ 2 (dot-dashedlines). Note, however, that there is still a significaregion for unification in the missing-doublet model witMyL ­ 100. In Fig. 2(b) we also show the necessathreshold,et , for top-tau Yukawa unification. We sefrom Fig. 2(b) that the top, bottom, and tau couplinunify at the largest values of tanb (in the region whereB . 0). Such a unification is expected in SO(10) modeHowever, the thresholds in any particular SO(10) momust be calculated to be sure the model is consiswith data.

This work was supported by Department of EnerContract No. DE-AC03-76SF00515 and by the U.National Science Foundation, Grant No. NSF-PH9404057.

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