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Precision corrections in the minimal supersymmetric standard model
ELSEVIER
Nuclear Physics B 491 (1997) 3-67
Precision corrections in the minimal supersymmetric standard model* Damien M. Pierce a, Jonathan A. Bagger b, Konstantin T. Matchev b, Ren-Jie Zhangb a Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA b Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA
Received 4 June 1996; revised 13 November 1996; accepted 26 November1996
Abstract In this paper we compute one-loop corrections to masses and couplings in the minimal super° symmetric standard model. We present explicit formulae for the complete corrections and a set of compact approximations which hold over the unified parameter space associated with radiative electroweak symmetry breaking. We illustrate the importance of the corrections and the accuracy of our approximations by scanning over the parameter space. We calculate the supersymmetric one-loop corrections to the W-boson mass, the effective weak mixing angle, and the quark and lepton masses, and discuss implications for gauge and Yukawa coupling unification. We also compute the one-loop corrections to the entire superpartner and Higgs-boson mass spectrum. We find significant corrections over much of the parameter space, and illustrate that our approximations are good to 69(1%) for many of the superparticle masses. (~) 1997 Elsevier Science B.V. PACS: 11.30.Pb; 12.10.Dm; 12.10.Kt; 12.15.Lk; 14.80.Ly Keywords: Supersymmetry;Radiative corrections;Precision measurements;Grand unification
1. Introduction Most precision measurements of electroweak parameters agree quite well with the predictions of the standard model [ 1 ]. These experiments rule out many possibilities for physics beyond the standard model, but they have not touched supersymmetry, which evades precision constraints because it decouples from standard-model physics if the * Work supported by Departmentof Energy contract DE-AC03-76SF00515and by the U.S. NationalScience Foundation,grant NSF-PHY-9404057. 0550-3213/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. Pll S0550-3213 (96)00683-9
4
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67
scale of supersymmetry breaking is more than a few hundred GeV. Definitive tests of supersymmetry will probably have to wait for direct searches at future colliders. Once supersymmetry is discovered, a host of new questions arise. For example, one would like to test the low-energy supersymmetric relations between the particle masses and couplings by making precision measurements of the supersymmetric parameters. One would like to measure the supersymmetric masses as accurately as possible to shed light on the origin of supersymmetry breaking. Furthermore, one would also like to know whether weak-scale supersymmetry sheds any light on physics at even higher energies. Indeed, the successful unification of gauge couplings encourages hope that other supersymmetric parameters might unify as well. It is important to measure these parameters precisely at low energies so that they can be extrapolated with confidence to higher energies. It is in this spirit that we present our calculation of one-loop corrections to the minimal supersymmetric standard model (MSSM). We define the MSSM to be the minimally supersymmetrized standard model, with no right-handed neutrinos, and all possible softbreaking terms. We believe that this minimal model provides an appropriate framework for analyzing the phenomenology of supersymmetry and supersymmetric unification. We approach our calculation in the standard fashion associated with precision electroweak measurements. We take as inputs the electromagnetic coupling at zero momentum, O~em= 1/137.036, the Fermi constant, G~ = 1.16639 × 10 -5 GeV -2, the Z-boson pole mass, M z = 91.188 GeV, the strong coupling in the MS scheme at the scale Mz, a . ~ ( M z ) = 0.118, as well as the quark and lepton masses, mt = 175 GeV, mb = 4.9 GeV, and mr = 1.777 GeV [2]. From these inputs, for any tree-level supersymmetric spectrum, we compute the oneloop W-boson pole mass, M w , as well as the one-loop values of the effective weak lep t mixing angle . . tin2 . . . t ~eft, and the DR [3] weak mixing angle, ~2. We also compute the one-loop corrections to the quark and lepton Yukawa couplings, as well as the masses of all the supersymmetric and Higgs particles. We work in the DR scheme, and take the tree-level masses to be given in terms of the running DR parameters. For each (bosonic) particle, we determine the one-loop pole mass, M 2 = hT/2(Q) - R e / / ( M 2) ,
(1)
where h?/(Q) is the tree-level DR mass, evaluated at the DR scale Q, and H ( p 2) is the one-loop self-energy. (As usual, H ( p 2) depends on Q and on the masses and couplings of the particles in the loop. There is a similar expression for the fermion pole mass.) In all our computations we include the full self-energies, which contain both logarithmic and finite contributions. The logarithmic corrections can be absorbed by changes in the scale Q. Therefore we checked our logarithmic results against the one-loop supersymmetric renormalization group equations. Since we write our results using PassarinoVeltman functions [4], some of our finite terms are automatically correct. As a further check, we verified that our corrections decouple from electroweak observables.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
5
We present our complete calculations in a series of appendices. These appendices include the full one-loop corrections to the gauge and Yukawa couplings, as well as the complete one-loop corrections to the entire MSSM mass spectrum. While some of these results are not new (the gauge-boson [5,6], Higgs-boson [6,7] and gluino [8-11 ] self-energies and the gauge-coupling corrections [ 5,12 ] already appear in the literature), we include the full set of corrections in order to provide a complete, self-contained and more useful reference. In Appendix A we write the tree-level masses in terms of the parameters of the MSSM, and in Appendix B we define the Passarino-Veltman functions that we use to present our one-loop results. In Appendix C we compute the one-loop radiative corrections to the gauge couplings of the MSSM, and in Appendix D we write the one-loop corrections to the masses. Where appropriate, we evaluate the corrections to the mass matrices to account for full one-loop superparticle mixing. This allows for an accurate determination of the masses and mixing through the entire parameter space. Finally, in Appendix E we discuss the radiative corrections to the Higgs-boson masses. The results in the appendices hold for the MSSM with the most general pattern of (flavor diagonal) soft supersymmetry breaking, t The parameter space is huge because of the large number of operators that softly break supersymmetry. Therefore in the body of the paper we illustrate our results in a reduced parameter space, obtained by assuming that the soft breaking parameters unify at some high scale. The unification assumption is useful because it reduces the size of the parameter space. Moreover, it implies certain mass relations that can be tested once supersymmetry is discovered. In addition, for any set of parameters, it allows us to determine the unification scale thresholds that are necessary to achieve unification. As we will see, the present set of precision measurements is sufficient to begin to constrain the physics at the unification scale. We implement the unification assumption by solving the two-loop supersymmetric renormalization group equations subject to two-sided boundary conditions. At the weak scale, we assume a supersymmetric spectrum, and for a given value of the ratio of vacuum expectation values, tan fl, we use our one-loop corrections to extract the DR couplings g~, g2, g3, At, Ab, and ,~ at the scale M z . We then use the two-loop DR renormalization group equations [13] to run these six parameters to the scale MGtJT, which we define to be the scale where gl and g2 meet. We require that the soft breaking parameters unify at the scale MGUT. Therefore at MGu3- we fix a universal scalar mass, M0, a universal gaugino mass, M1/2, and a universal trilinear scalar coupling, A0. We then run all the soft parameters back down to the scale M~2 = M 2 + 4M~/2, where we calculate the supersymmetric spectrum using the full one-loop threshold corrections that we present in this paper. In Section 4 we show that this scale is essentially the scale of the squark masses, and that the other scalar masses and the Higgsino mass are correlated with it as well. I Our results can be readily extended to include inter-generational mixing at the cost of additional mixing matrices.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
We require radiative electroweak symmetry breaking [14], so the CP-odd Higgs I/~1, are determined in a full one-loop calculation at the scale M#. The sign o f / z is left undetermined. We then iterate the entire procedure to determine a self-consistent solution. Typically, convergence to an accuracy of better than 10 -4 is achieved after four iterations. Once we have a consistent solution, we use the results of the appendices to illustrate the one-loop corrections in the reduced parameter space associated with unification. We display results for a randomly chosen sample of 4000 points. Our sample is chosen with a logarithmic measure in the range 1 < tan/3 < 60, 50 < Mj/2 < 500 GeV, 10 < M0 < 1000 GeV, and with a linear measure in the range - 3 M # < A0 < 3M~. (The upper limits on M0 and Ml/2 are chosen so that the squark masses are less than about 1 TeV. While larger squark masses are certainly possible, they reintroduce the fine tunings that supersymmetry is designed to avoid.) Each of these points corresponds to a (local) minimum of the one-loop scalar potential with the correct electroweak symmetry breaking, and each passes a series of phenomenological constraints: We require the first- and second-generation squark masses to be larger than 220 GeV [15], the gluino mass to be greater than 170 GeV [15], the light Higgs mass 2 to be greater than 60 GeV [2], the slepton masses to be greater than 45 GeV [2], and the chargino masses to be greater than 65 GeV [16]. We also require all the Yukawa couplings to remain perturbative (/l < 3.5) up to the unification scale, and since we assume that R-parity is unbroken, we enforce the cosmological requirement that the lightest supersymmetric particle be neutral. We derive approximations to the radiative corrections that hold with reasonable accuracy over the unified parameter space. Where appropriate, we use scatter plots to illustrate the effectiveness of our approximations. The approximations consist of two parts. First we identify the most important contributions to the one-loop corrections. In most cases these are the loops that involve the strong and/or third generation Yukawa couplings. Then we derive approximations to the loop expressions that hold over the unified parameter space. In the next section, we discuss the radiative corrections to the effective weak mix~qlept ing angle, sin 2 %ff, and the W-boson pole mass, Mw. We illustrate the magnitudes of the different supersymmetric contributions to these observables. We also discuss the renormalization of the DR weak mixing angle, ~2, and comment on the way that it affects the gauge thresholds at the unification scale. In Section 3 we examine radiative corrections to the third generation quark and lepton masses. We illustrate the different contributions and present approximations which hold to a few percent. We also examine Yukawa unification and demonstrate the size of the unification-scale Yukawa thresholds. In Section 4 we present our results for the radiative corrections to the supersymmetric and Higgs-boson particle masses. We find large corrections to the masses of the light superparticles. We compare our results with those of the leading logarithmic approximamass, m a , and the supersymmetric Higgs mass parameter,
2The light Higgs boson is similar to that of the standard model in almost all of our parameterspace, so we apply the standard-modelbound.
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67
7
tion and find significant improvements over much of the unified parameter space. These corrections are important for unraveling the underlying supersymmetric structure from the supersymmetric mass spectrum.
2. The weak mixing angle and the W-boson mass The calculation of supersymmetric contributions to electroweak observables began in 1984 [5]. Since then, the precise confrontation of electroweak data with theoretical predictions of the MSSM has been an active area of study [17,12]. Global fits to precision data in the MSSM have been performed by several groups [18]. In this section we display our results for two electroweak observables over the parameter space associated with radiative electroweak symmetry breaking and universal unification-scale boundary conditions. We extract the contributions from the various superpartners, and illustrate the manner in which the different contributions decouple from the low-energy observables. 2.1. Effective weak mixing angle
'olept• The full oneWe start by considering the effective weak mixing angle, s~ = sin 2 ~eff loop calculation is presented 3 for completeness in Appendix C. The complete result is rather involved; for now we simply say that the calculation follows the outline presented above: We take aem, G~, M z , a,. ( M z ) , and the fermion masses as inputs, and compute s~ as a function of the supersymmetric masses. Because we compute the experimental observable s~ in terms of other low-energy observables, its one-loop supersymmetric corrections decouple as the supersymmetric masses become larger than Mz. From Fig. 1 we see that s 2 is especially sensitive to light sleptons, and that the sum of the supersymmetric corrections is always negative. We did not plot the Higgs boson and first two generation squark contributions. They are negligible, less than 1 × 10 - 4 and 4 × 10 -5 in magnitude, respectively. The corrections to #-decay and the corrections to the Z-g+-e - vertex comprise the non-universal corrections to s~. The former contributes between - 3 and 1.5 x 10 -4, the latter between ± 1.5 × 10 -4, and their sum is in the range - 4 to 1 x 10 -4. With m, = 175 GeV, we find the standard-model value of s~ varies between 0.2311 to 0.2315 for Higgs masses in the range 60 < mh < 130 GeV. This is subject to an error of 2.5 x 10 -4 from the experimental uncertainty in the electromagnetic coupling evaluated at Mz, and to corrections of this same order from higher loop effects [19]. Furthermore, increasing mt by 10 GeV decreases s 2 by 3.3 x 10 -4. These predictions for se2 should be compared with the LEP and SLD average 4 [1] of 0.23165 :t: 0.00024. Clearly, the standard-model calculation agrees quite well with 3 We do not include the supersymmetricnon-universal Z-vertex contribution in the appendix. It is a negligible correction in the parameter space we consider. 4 The number quoted here assumes lepton universality.
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experiment. The additional contribution from supersymmetry can lower the value of s~2 slightly below 0.2300, or about 60- below the experimental central value. However, we note that higher-order standard-model corrections, changes in O~emand mt, and other precision observables should all be systematically taken into account to delineate which regions of parameter space are ruled out by these measurements. We do not attempt such a study here. The corrections to s~ diminish rapidly as the superpartner masses become heavy. For example, if we require m ) , , m b , mh ~> 90 GeV, we find that s~ is shifted by at most - 8 x 10 -4 relative to the standard-model value. 2.2. W-boson mass
We now turn to our second precision electroweak observable, and compute the oneloop correction to the W-boson pole mass, Mw. In Fig. 2 we illustrate some of the finite corrections. The full finite correction increases the prediction for Mw by up to 250 MeV. As with s~, the contributions from Higgs bosons and the first two generations of squarks are small, less than 12 and 8 MeV, respectively. The non-universal correction is also small, between - 6 and 15 MeV. For large supersymmetric masses, the prediction reduces to that of the standard model because of decoupling.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
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For mt = 175 GeV we find the standard-model value of Mw in the range 80.39 to 80.43 GeV, with an error of +13 MeV from the experimental uncertainty of the electromagnetic coupling. This is subject to additional corrections of the same order from higher-loop effects [ 19]. If we increase mt by 10 GeV, we find a 65 MeV increase in Mw. From our calculations we find that the MSSM value for Mw ranges from 80.39 to 80.64 GeV. These numbers can be compared to the current world average, 80.33 ± 0.15 GeV [ 2 ]. With the current experimental error, all of the supersymmetric parameter space lies within 20" of the central value. By the end of the decade, the error on Mw is expected to be about 50 MeV. If supersymmetry is not discovered by that time, one might think that a much more exacting test could be performed. However, the limits on the superpartner spectrum will also have increased to the point where the effects on weakscale observables from virtual supersymmetry will be greatly diminished. For example, imposing the limits m~,, m~÷, mh > 90 GeV, we find that typically 8Mw < 50 MeV, and at most 8Mw = 100 MeV.
2.3. Gauge coupling unification We are now ready to study gauge coupling unification [20]. We start by computing the DR electromagnetic coupling constant, &, and the D----Rweak mixing angle, ~2, as described in Appendix C. The DR weak mixing angle is closely related to the effective weak mixing angle, s~. The main difference is that ~2 is not an experimental observable,
10
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67
so its radiative corrections involve non-decoupling logarithms of supersymmetric masses. However, once we subtract these logarithms we find finite corrections to ~2 which are quantitatively similar to the corrections to s~ shown in Fig. 1. In the context of gauge coupling unification, the finite corrections to g2 are very important [21-23]. They play a significant role in determining the required unificationscale thresholds, which are formally of the same order in perturbation theory. As we will see, precision measurements already limit the size of these thresholds and place constraints on unified models. We determine the unification thresholds as follows: we calculate the full one-loop corrections to t~ and ~2, and use them to determine the DR couplings gl and g2. We take a s ( M z ) = 0.118 from experiment [2], and apply the supersymmetric threshold corrections to fix the DR coupling g3 at the scale M z , (2)
g~(Mz) _ as(Mz) 4or 1 - Aas '
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(4)
In Fig. 3a we plot the threshold, eg, versus the squark mass scale, M 0, which we define to be M 02 = M~ + 4M~/e. From the figure we see that for a s ( M z ) = 0.118, unification requires a negative unification-scale threshold correction of between - 1 % and - 3 % , depending on the weak-scale supersymmetric spectrum. For small M 0, the finite corrections to the gauge couplings are comparable in size to the logarithmic corrections; they both decrease eg at small M o. In order for a unified model to be consistent with gauge coupling unification, it must be able to accommodate values of eg ~- - 2 % . Different unified models give rise to different unification-scale threshold corrections. In the minimal [24] and missing doublet SU(5) [25] models, eg depends only on the triplet Higgs mass [23], the same mass that enters the nucleon decay rate formulae. The maximum and minimum values of eg in these two models are shown in Fig. 3b. The two thin regions correspond to the maximum values of eg, obtained by setting the triplet Higgs mass to 1019 GeV. The two larger regions show the minimum values of eg in each model. These values are found
II
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by setting the triplet Higgs mass as small as possible, consistent with the bounds from nucleon decay [26]. From the figure we see that it is difficult to achieve the necessary thresholds in minimal SU(5), but that missing doublet SU(5) has unification-scale thresholds in the right range. The unification-scale threshold eg necessary for unification is directly correlated with a s ( M z ) . Increasing a s ( M z ) by 5% increases eg by about 1%, as expected from the one-loop relation 6as(Mz) as(Mz) = 2 - a.~ ( M z ) aGUT
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3. Q u a r k and lepton masses
The full set of radiative corrections to the quark and lepton masses is presented in Appendix D. In this section we derive approximations to these formulae, valid for the third generation.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
12
3.1. Top quark mass The top quark mass provides an important input for radiative electroweak symmetry breaking. It receives strong and electroweak radiative corrections [27,11 ]. Our approximation begins by eliminating the small electroweak corrections, setting g = g~ = 0 and At = ab = 0. We then simplify the resulting expressions by setting p2 = 0 because mt is much smaller than a typical squark or gluino mass. In this limit, the physical top quark mass is given by m t = lht(Q)
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2 2 with M = max(ml,m2), m = min(ml,m2), and x = m2/m 1. The full B functions are written in Appendix B; the formulae presented here are simplifications that hold when the first argument is zero. In Fig. 4 we show the complete correction to the top quark mass as well as the contributions from the squark/gluino and electroweak loops. The tree-level mass is defined to be lht(mt). We see that the squark/gluino loop contribution can be as large as the gluon contribution for TeV-scale gluino and squark masses. The electroweak 5 Here and in the following, we implicitly perform DR renormalization, so the 1/~ poles are subtracted. 6 We have included the two-loop M"'S contribution Amt = 1.l lot2mt [28], which we assume is close to the DR value.
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I .... I,,,, I,, ,,I,.,,I.,,,I,...I
300 500 M~ (GeV)
i000
Fig. 4. Corrections to the top quark mass, versus M0. Figure (a) shows the full one-loop correction. Figure (b) illustrates the correction from the squark/gluino loop; the solid line shows the gluon contribution for comparison. Figure (c) shows the electroweak corrections. In Fig. (d) we plot the difference between the full one-loop result and the approximation given in the text. corrections are small because of cancellations. In the figure we also plot the difference between the full correction and our approximation. We see that our approximation is good to typically ±1%.
3.2. Bottom quark mass Corrections to the bottom quark mass in the MSSM have received much attention because they can contain significant enhanced supersymmetric contributions [29,30]. These large contributions play an important role in Yukawa coupling unification. Previous studies have included only the enhanced contributions. In this paper we present our results for the full one-loop correction. Moreover, we systematically develop approximations to the supersymmetric corrections. In this way we can see the importance of the enhanced contributions relative to the full result. The corrections to the bottom quark mass are found as follows. Because the bottom quark is light, o~,(mb) is large and we must resum the gluon contribution. We start with the bottom-quark pole mass, mb. We find the standard-model DR bottom quark mass at the scale mb using the two-loop QCD correction,
[ (Amb~bg] thb(mb) TM =mb 1 -- \ m b / J ,
(ll)
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
14 where 7 [28]
( Amb ~ bg _ 5as(mb) 3 7r mb /
+12.4 (as(rob))2 '
(12)
and c~s(mb) is the five-flavor three-loop running DR coupling. We then evolve this mass to the scale Mz using a numerical solution to the two-loop (plus three-loop O(ce~)) standard-model renormalization group equations [31]. Taking the bottom quark pole mass mb = 4.9 GeV, and ees(Mz) = 0.118, we find the standard-model DR value r~b(Mz) TM = 2.92 GeV. The final step is to add the one-loop corrections from massive particles,
lhb(Mz)=thb(Mz)SM I1-- (Amb~massive mb / ]
(13)
We approximate these corrections as follows. We ignore the small W, Z, Higgs, and neutralino contributions. This leaves the squark/gluino and squark/chargino loops, (Am--~b']massive
\ mb .I
= ( Amb x) b~ + ( Am-----~b "] '-'U . \ mb / \ mb .I
(14)
We then set p2 = 0. The squark/gluino contribution is again given by (8), with the obvious substitution t ---, b. To approximate the squark/chargino contribution, we set g = g' = ab = at = 0, except for terms that are enhanced by the Higgsino mass parameter IX or by tan ft. We simplify our expressions by setting the chargino masses to M2 and Ix, respectively. In this case the squark/chargino loops give rise to the following terms: --4
\ mb /
- 16~r "" 2 # -.5----.-5 m?l - rnf2
Bo(O, IX, m~, ) - Bo(O, ix, m~)
IX2 _ M~ +(IX +-~ M 2 ) } ,
(15)
where Bo(O, ml,m2) is defined in (9), and ct (st) is cos0, (sin0,). In Fig. 5 we show the corrections to the DR bottom quark mass, ~nb(Mz), plotted against tan/3. Fig. 5a shows the full one-loop correction, while 5b and 5c illustrate the predominantly finite corrections from squark/gluino and squark/chargino loops. At large tan/3, the top branches in Figs. 5a and 5b correspond to Ix < 0, while for Fig. 5c the bottom branch corresponds to Ix < 0. The contributions from Figs. 5b and 5c tend 7 We do not know the two- and three-loop correctionsto mb in the ~ scheme. Similarly, we do not know the DR three-loop QCD contribution to the running of the strong coupling. In both cases we use the MS values. Alternatively. we could have used MS equations to run up to Mz, then convertto DR. The difference between the two approaches, Athb(Mz ) < 0.05 GeV, is nearly an order of magnitude smaller than the experimental uncertainty in the bottom quark mass.
D.M. Pierceet al./Nuclear Physics B 491 (1997) 3-67 I
'
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',
,
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........ .-~.~,~.~i,~: : . . . .
<~ -2o -40
(o) I
2
(d) a f t e r a p p r o x ,
I
5
....
[
i0
tanfl
I
20
i
i
50
L
I
1
2
,
I
....
5
i.
50
tanfl
Fig. 5. Corrections to the DR bottom quark mass ~hb(Mz), plotted versus tanfl. Figure (a) shows the full one-loop correction; (b) illustrates the full correction from the bottom squark/gluino loops; (c) shows the correction from the top squark/chargino loops. Figure (d) plots the differencebetween the full one-loop result and the approximation given in the text. to cancel. Because of the cancellations and large corrections, previous approximations to the bottom quark mass appearing in the literature can be substantially different from the full one-loop result. In Fig. 5d we see that the approximation (14) typically agrees with the full one-loop result to within a few percent.
3.3. Tau lepton mass The corrections to the tau lepton mass are of course much smaller than those of the quarks. After resumming the two-loop QED corrections which relate the tau pole mass to the DR running mass at Mz [31], we obtain the DR mass ~hr(Mz) = 1.7463 GeV. We approximate the remaining corrections by setting p2 = 0 and keeping only those terms proportional to g2 and enhanced by /z or tan ft. The only such terms arise from the chargino loops. They give
mr /
16rr 2 /z---g_~2~ Bo(O, M 2 , m ~ , ) - B o ( O , # , m r , , )
]
,
(16)
where Bo(O, mt,m2) is given in (9). We illustrate the tau corrections in Fig. 6. The full correction ranges from - 1 0 % to ÷6%, while the approximation is good to within a few percent. For large tan/3, the top branch corresponds t o / x > 0.
D.M. Pierceet al./Nuclear Physics B 491 (1997)3-67
16 i
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,,,,I
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,
I
(b) a f t e r a p p r o x
,
I
50
2
,
,
I
5
tan~
....
50
tan{{ m
Fig. 6. Supersymmetric corrections to the DR tau lepton mass fnr(Mz ), plotted against tan B. Figure (a) gives the full one-loop correction; (b) illustrates the difference between the full one-loop result and the approximation given in the text.
3.4. Yukawa coupling unification In many supersymmetric unified theories, the DR bottom and tan Yukawa couplings are predicted to unify at the scale M6UT [20]. To test this hypothesis, one must first extract the running DR Yukawa couplings Ab and Ar from the DR bottom and tan masses. Our procedure is as follows. We first use the formulae in Appendix D to find DR masses for the bottom and tau. We then use the following relations to find the DR Yukawa couplings at the scale Mz: 1
tnb( M z ) = --~ ,~b(g z ) v( M z ) cos fl( Mz) , 1
Pnr( M z ) = ~ Ar( M z ) v ( M z ) c o s f l ( M z ) . x/2
(17)
We determine the full one-loop DR vev v ( M z ) from the relation
l(
M2z + ReHrz(M2z) = -~ gZ(Mz) + g'2(Mz)
)
v2(Mz) ,
(18)
where g and g' are the D--Rcouplings, and H~z is the transverse Z-boson DR self-energy. Alternatively, the DR vev v ( M z ) can be taken from the following empirical fit:
v(Mz)=
[
248.6+0.91n
Mzz
GeV.
(19)
This expression gives the correct one-loop vev to an accuracy of better than 1%. Once we have the DR Yukawa couplings at Mz, we run them to the unification scale MGUT. At that scale we define the unification threshold eb to be the discrepancy between the couplings, Ab(MGuT) ----Ar(MGuT) (1 + co) •
(20)
17
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 I
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I
,
50
tanfl
I
2
,
I
5
. . . .
I
I
10
20
~
,
I,
50
tanfl
Fig. 7. The unification-scale correction, e9, that is necessary to obtain bottom-tau unification with the given values of as (Mz) and mb, plotted versus tan ft. The bottom quark pole mass is labeled in GeV. Of course, the relation between the bottom quark mass and the DR Yukawa coupling Ab depends strongly on the QCD coupling as ( M z ) . Therefore we compute eb assuming that eg has already been chosen so that a s ( M z ) is some fixed value. Such an analysis is illustrated in Fig. 7a, where we plot eb versus tanfl for a s ( M z ) = 0.118. From the figure we see the well-known feature that Yukawa unification is possible with eb ~-- 0 for small tanfl (1.2 < tanfl < 1.7) and large tanfl (15 < tanfl < 40). In the large tan fl region we distinguish the two cases, /x < 0 a n d / z > 0. F o r / z < 0 we see that eb is always far from zero, in the range - 2 4 to - 6 0 % . For /z > 0 we find points which permit bottom-tan unification with eb ~-- O. However, they are not generic; eb depends sensitively on the parameter space, and varies between - 2 0 to +30%. The discrepancy eb is sensitive to the value of a s ( M z ) , as well as the input value for the bottom quark pole mass. We illustrate this in Figs. 7b-d, for the ( a s , m9) values (0.118, 5.2), (0.112, 4.9), and (0.124, 4.9), with mb in GeV. We note that setting a s ( M z ) = 0.112 and mb = 5.2 GeV, we find solutions with [eb[ < 0.05 over the whole range 1.3 < tan fl < 30.
4. S u p e r s y m m e t r i c
and Higgs-boson
masses
We will begin our analysis of the supersymmetric spectrum by discussing some of its general features. We will use some of these features when we derive our approximations for the radiative corrections. We will be careful to note when we do, so that one can
18
D.M. Pierce et al./Nuclear Physics B 491 (1997)3-67
lOOO
2# ~ 500
2
0
0
0
~
,ooo k
+1ooo ~
~
50
200
500 1000
2oOloo 50200
~ 500 1000
M~
/...-,~2;'~,
5oo f
"
!
zoo 200
500 1000
M~
Fig. 8. The masses (a) reaL, (b) m~z, (c) I~1, (d) ma, (e) m~+, and (f) m)?~,versusM#. The lines indicate (a) M,;, (b) M#/3 and M,7, (c) Mq/2 and 2M#, (d) M#/2 and 2M#, (e) Mq/2, and (f) MJ2 and 2Mq. The units for both axes are in GeV. assess the validity of our approximations in other scenarios. Perhaps the most striking consequence of the universal boundary conditions is the fact that they produce a low-energy spectrum which is tightly correlated with the magnitude of the squark mass scale M#2 = M 2 4- 4M~/2. In Fig. 8 we show the masses m~L, m~t, I~1, mm, m~, and m ~ versus M,;. From the figure we see that the squark masses are nearly equal to M,;, while the other masses are generally within a factor of two or three. The exceptions to this degeneracy include the Higgs boson, h, which is always light, and the additional possibilities of a light top squark, a light Higgs sector, and/or light gauginos. Of course, the gaugino masses are nearly proportional to M1/2. We find that typically mr0 -~ 0.4M1/2, m~o -~ mk~ --~ 0.8M1/2, and m~ " 2.4M1/2, but there are substantial variations from weak-scale threshold corrections (and from mixing for the charginos/neutralinos). We show the ratios m~/Ml/2 and m~/Ml/2 versus M,; in Fig. 9. In the rest of this section we discuss the one-loop corrections to the masses of the gluino, the charginos and neutralinos, the squarks, the sleptons, and the Higgs bosons. In the following expressions for the mass corrections we implicitly take the real part of the Passarino-Veltman functions. 4.1. Gluino mass The gluino mass corrections are perhaps the simplest of all the mass renormalizations. They have previously been studied in Refs. [ 8-11 ]; for completeness we list the corrections in Appendix D. The gluino mass corrections arise from gluon/gluino and
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
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M~ (b)
m£o/Ml/2,
(c)
m£~/Ml/2,
and
(d)
mg/Mu2,
versus
M 0.
quark/squark loops. The corrections can be rather large, so we include them in a way which automatically incorporates the one-loop renormalization group resummation, [
m~=M3(Q)
(AM3~g~'
1- \--MT]
~ A M 3
x~qq
- \-~-3,]
]
-I
J
(21)
The gluon/gluino loop gives --~3 J
=~
[ 2 B o ( M 3 , M 3 , 0 ) - BI(M3, M3,0) ]
= 167r2 3In
+5
.
(22)
The quark/squark loop can be simplified by assuming that all quarks have zero mass, and that all squarks have a common mass, which we take to be MQ~,the soft mass of the first generation of left-handed squarks. We find { A M 3 ~ q?l = - 3g~3B I ( M 3 , 0 , MQt) \ M3 ] 47r2 Here
(23)
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
20
30
. . . .
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B,(p,O,m)=--~ln +~O(x-
+l-~x
1+
x
lnlx- 11
l) lnx,
(24)
pZ/m2.
where M 2 = max(p2,m 2) and x = As usual, the full mass renormalization contains logarithmic and finite contributions. The gluino mass corrections are shown in Fig. 10. In the figure we define the tree-level gluino mass to be M3(M3), and we evaluate the one-loop mass at the scale Q = Because we resum the correction, varying the scale from to 2M 0 changes the one-loop mass by at most + 1%. From the figure we see that the leading logarithmic correction can be as large as 20%, while the finite correction ranges from 3 to 10%. The finite contribution is largest in the region where the logarithm is largest, so the leading logarithm approximation is nowhere good. On the other hand, the approximation we provide typically holds to a few percent. It is off by as much as 6% in the region where the full correction is 30%. In this region we expect the two-loop correction to be of order 6%.
MO/2
MO.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
21
4.2. Neutralino and chargino masses The complete set of corrections to the neutralino and chargino masses [ 32,9] is given in Appendix D. In this section we present a set of approximations to these corrections. These approximations are more involved than those discussed above because there are no color corrections that would dominate the results. Our approximation is as follows. We start by assuming that IIXl > Mj,M2, Mz. (We find that Mz/IX 2 2 and M2/IX2 2/ are less than 0.53; see Fig. 8). We work with the undiagonalized tree-level (chargino or neutralino) mass matrix, and correct the diagonal entries only, that is, the parameters MI, M2, and /z. This approximation neglects the corrections to the off-diagonal entries of the mass matrices, which leads to an error of order (a/47r) m2z/IX 2 in the masses. We simplify our expressions by setting all loop masses and external momenta to their diagonal values, i.e. we set myd = M1, etc. We neglect all Yukawa couplings except At and ah. We also ignore the mixings of the charginos and neutralinos in the radiative corrections. This also leads to an error of order (a/47r)M2z/IX 2 in our final result. We further simplify our expressions by setting all quark masses to zero, and by assuming that all squarks are degenerate with mass MQ~, and that all sleptons are degenerate as well with mass ML~. We also take the Higgs masses to be mh = Mz and ,,,,, = mH~ = ma. This means that we also neglect terms of order (ce/4zr)M2z/m2A. In this limit the dominant correction to MI comes from quark/squark, chargino/ charged-Higgs and neutralino/neutral-Higgs loops. We find 1~--~2 llBI(Mj,O, MQ,)+9BI(M1,0, ML,)
--~-l J -
+-~1 sin(2fl)(Bo(Ml,ix, mm) -Bo(MI,IX, M z ) ) +BI (M j, IX, ma) + B1 (Ml, IX, Mz ) 1 "
(25)
Since Mz, Mi << IX, we can simplify this expression by setting Mz -- M1 = 0 inside the B functions. This gives AMI "]
--~-j j = ~
g,2 {
110M~MO.~+ 90M1M~,~+ OM~UMz-- 2B1 (0, IX, mA)
2 # sin(2/3) (B0(0, IX, 0 ) B- o ( 0 , +M-7
IX, mA ) ) -- -2-3 } ,
(26)
where Bo(0, ml, m2) and Bl (0, ml, m2) were defined in (9) and (10), and 0m~...m2 ~In(M2/Q 2) with M 2 = max(ml2. . . . . m2). The form of the finite corrections depends on the assumed hierarchy in the low-energy spectrum, but the leading logarithms are always correctly given by the 0 terms. We set the first subscript of a 0 term, ml, equal to the external momentum. Note that when the renormalization scale equals the
D.M. Pierceet al./NuclearPhysics B 491 (1997)3-67
22
external momentum, Q = ml, the theta function reduces to the familiar form Ore,m2 =
ln( m~/ Q2) O( m 2 _ Q2). The leading logarithmic corrections are easy to read from Eq. (26). 8 Note that the terms proportional to sin(2fl) are enhanced by the ratio tz/Ml. These finite terms are completely missed in the run-and-match approach because they do not contribute to the beta function. In a similar way, we approximate the corrections to M2 from quark/squark and Higgs loops. They are 1 ~ 2 9BI(M2,0, MQ1) + 3BI(M2,0, MLI)
-~2 J
/ + /x sin(2/3) I Bo( M2, tz, mA ) - Bo( M2, #, M z ) ) M2 +Bl(M2,~,ma) + Bl(M2, tx, Mz) I
(27)
o
Setting Mz = M2 = 0 inside the B functions, we find ---M-~-2/
~
90M2MQ~+30M2m,~, +OM2gMz -2Bl(O, tx, mA)
2/z sin(2/3) (B0(0,/z,0) - Bo(O, Iz, ma ) ) - ~ - ~ } +~-~2
.
(28)
There are additional corrections to M2 from gauge boson loops. Because M2 enters both the chargino and the neutralino mass matrices, the corrections differ slightly for the two cases. However, to the order of interest, it suffices to use the neutralino result, M2 /
= 47r2 2B0(M2, M2, M w )
-
B1 (M2, M2, Mw)
.
(29)
Because M2 is of order Mw, one must use the full B functions in this expression. Alternatively, one can use the following empirical fit which works to better than 1%: 2,,w + 0 ( M w - M2)
-O(M2-Mw)
.57~--~
[0.541n (MM2w--0.8)+ 1.15] } .
(30)
The corrections to /x are obtained in a similar manner. In the limit g~2 << g2, we find -
32~-2 (a~ + a~)B~(tz,0,Ma~) + a';Bl(u,O, Mt,~) + a~,B~(~,O, Mo~)
The logarithmicpart of --2Bl(0, be,mA) in Eq. (26) is givenby OM~u,nA.
D.M.Pierceet al./NuclearPhysicsB 491 (1997)3-67 I-'
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200
(CeV)
Fig. I 1. Corrections to the lightest neutralino mass, as in Fig. 10. 3g2 64~ 2
[B](Ix, M2, mm) + B1(#,M2, Mz)
+2B1 (/z,/z, M z ) -- 4B0(/z,/z, M z ) ] . As above, we set
-
Mz
32
(31)
= M2 = 0 inside the B function, in which case (31 ) reduces to
3[ -2
+AZB~(tz, O, MD3)] + ~ 3 g 2 [~1 0 ~M2Mz_
30,o.Mz--Bl(Iz, O,mA)+4]
(32)
The expression for B1 (p, 0, m) is given in Eq. (24). In Fig. 11 we show the corrections to the lightest neutralino mass. In Fig. 1 la we show the full correction in percent, with the tree-level mass defined as the eigenvalue of the mass matrix, where the running parameters Ms, M2, a n d / z are evaluated at their own scale. (The tree-level mass matrices also contain tan fl at Mz and the W- and Z-boson pole masses.) The one-loop masses have negligible scale dependence. As usual, the full corrections are made up of logarithmic and finite pieces. The logarithmic corrections are shown in Fig. 1 lb and the finite corrections are shown in Fig. l lc. Note that the finite corrections can be more than half as large as the logarithmic
24
D.M. Pierce et a l . / N u c l e a r Physics B 491 (1997) 3 - 6 7 ":'I"'T'"['"'I
.
.
.
.
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15 ÷.10
. . . .
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~
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,, I..,,I,..,I,,,,I
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.
.
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approx
. . - ~ : . : . ' . :!~ , .~ :.:~"~, . z a . - ~ , ~ ~ . , ~ , ~
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.
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loo
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.
zoo
mg~ (CeV)
....
I
, ..I,...I....I....I
.
loo
.
I
. . . .
I
....
I
zoo
mg+ (GeV)
Fig. 12. Corrections to the lightest chargino mass, as in Fig. 10. corrections. Indeed, the finite corrections can be larger than 5% in the small Ml region, primarily because of the Higgsino-loop term proportional to ~. In Fig. 1 ld we show the difference between the full one-loop result and our approximation. Here, and in the following two figures, the logarithmic corrections (Fig. 1 lb) include an explicit sum over the soft squark and slepton masses, while the approximations (Fig. l l d ) use a single soft squark or slepton mass. Fig. 12 shows, similarly, the corrections to the lightest chargino mass. Again there is a term proportional to /x which dominates the finite corrections when M2 is small. In this region, the finite corrections can be as large as 10%. In Fig. 12d we show that the difference between our approximate correction and the full one-loop mass is less than 2%. These corrections are quantitatively similar to the corrections to the second-lightest neutralino mass. The corrections to the heavy chargino mass are shown in Fig. 13. These corrections are less than a few percent, as are the corrections for the two heaviest neutralino masses. The logarithmic corrections are in the range 0 to 2.5%, and the finite corrections are in the range 0 to - 3 % . Fig. 13d shows that our approximation for the heavy chargino mass generally holds to better than 0.5%. Our approximation also works to typically better than 1% for the two heaviest neutralino masses, but it can be off by nearly 2%.
D . M . Pierce et a l . / N u c l e a r
3
'
'
I
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....
I " " I ' " T " T ' - I " ' T " ' I
. . . .
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2 +,~
. . . .
P h y s i c s B 491 (1997) 3 - 6 7 I
•
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[
(a) full
1
<".
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25
I''"1""1'"'1'"'1'"'1''1
. . . .
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,, .
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•
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.
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,
2000 200
I....I....I....I....I....I....I
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500 1000 m2~ (GeV)
,
,
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2000
Fig. 13. C o r r e c t i o n s to the h e a v i e s t c h a r g i n o m a s s , as in Fig. 10.
4.3. Squark masses The first two generations of squarks receive QCD [33] and electroweak corrections. However, it is a very good approximation to ignore the electroweak graphs, since the dominant corrections come from gluon/squark and gluino/quark loops. Neglecting the quark masses, these corrections are as follows:
mo:Fn2(Q)
1+
t
m~
)J
(33)
'
where
)
L
+
(1
3 l+3x+(x_l)21nlx_ll_x21nx+2xln 6~2
,
(34)
2 2 and x = mp,/m O. For the case of universal boundary conditions, the gluino mass is less than or roughly equal to the squark mass, so the correction (34) is essentially finite at Q = m O. From Fig. 14 we see that it varies from around 1% for x << 1 to between 4 and 5% for x -~ 1. We also see that the electroweak corrections are small, less than 0.5%.
D.M. Pierce et al./Nuclear Physics B 491 (1997)3-67
26
~-~
6
•
'
I
. . . .
I
(a) f
~
E
~
. . . .
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u
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l
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•
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. . . .
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. . . .
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I " ' T " T ' " I
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I ....
I,.,
(b) EW
4 2
z#
o •
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....
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0.3
I ....
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I,,,A,.,I,,,,I
0.5
,
,
1.0
,
[
. . . .
....
0.3
0.5
m~/m~
.l....I....I
1.0
m~/m~
L
Fig. 14. (a) Full one-loop corrections to the first generation squark mass, m~L,versus the ratio mffmaL. (b) The difference between the full corrections and the approximation in the text, versus mffmaL. (These are essentially the electroweak corrections.)
'
i
. . . .
I
....
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I ' ' " I ' " T " T " T ' " I
'
(a) f u l l
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. . . .
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approx
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,
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....
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....
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500 m~
(GeV)
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<1
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,
,
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,
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,
.:
.
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/'..
,I,.~',11,1",1,,.,I,.,,I,..,I.,..I,...I
200
500 m~
I000
(GeV)
Fig. 15. (a) The corrections to the heavy top squark mass versus its mass. (b) The difference between the full one-loop heavy top squark mass and the approximation,Eq. (35). (c) Same as (a), for hi. (d) Same as (a), for b2. The third generation squark masses receive Yukawa corrections on the order of, and opposite in sign to, the QCD corrections. In Fig. 15 we show the full corrections to the third generation heavy squark masses. As usual, the tree-level masses are defined in terms of the gauge-boson and quark pole masses, as well as the soft masses MQ(MQ), M y ( M y ) , and M D ( M o ) . The tree-level mass matrices also contain t a n f l ( M z ) , / z ( / z ) , and Ai(max( IAil, M z ) ) , where A i denotes the top or bottom A-term. (Our convention for the third generation squarks is to associate the subscript 1 with the mostly left-handed
D.M. Pierceet al./Nuclear Physics B 491 (1997) 3-67
27
squark. Since the light top squark is predominantly right-handed, its mass is denoted mh.) From Fig. 15 we see that the heavy top squark mass receives corrections in the range - 5 to 2%, while the bottom squark masses receive corrections mostly in the 0 to 3% range. We note that in none of these cases does the leading logarithm approximation work well: as is the case for all the squarks and sleptons, these corrections are essentially non-logarithmic. (The light top squark mass does receive some substantial logarithmic corrections, but they are generally not larger than the finite corrections.) We will now present our approximation for the top squark mass matrix. We will derive our approximation for the case of the light top squark, but it also works quite well for the heavy top squark (see Fig. 15b). The mass of the light top squark receives potentially large additive corrections proportional to the strong coupling and the top and bottom Yukawa couplings. We approximate the corrections to m h by neglecting g, g' and the Yukawa couplings of the first two generations. We also neglect all quark masses except rnt, which eliminates all sfermion mixing except for that of the top squarks. We neglect the mixing of charginos and neutralinos, so the two heavy neutralinos and the heavy chargino all have mass I/zl. We also make the approximations mh = Mz and rnH = rna,, = mA. Finally, we set p = 0 in the B-functions if any of the other arguments is much bigger than rnh. This gives the following expressions for the one-loop corrections to the top squark mass matrix:
2 ~ AM2L AM2R ) jk4~ = 3)I~(Q) + \ A M 2 R AM2R_ •
(35)
The A M 2 entries are as follows:
A M%L = 6s--3--r--52m22 [c2B I (m h , m~,, O) + s 2B1 (m h , m h , 0)]
2 __ m~)Oo(O, m~, mt) } +ao(mg,) 4- Ao(mt) - (m~2 - rag, 1 [22 167.r2 AtstAo(mh ) + ,~2Ao(mb)
-2(At2 4- A~)Ao(/z) 4- (A~c2~ 4- h6s~)Ao(mA )
1
A2 [a(ot,~J)Bo(O, mh,ma) + a ( o t - 2,~)Bo(O,O, mA)
32rr 2
+ A ( O t , f l - -)B°(0'mh'0)qr 4- A ( O , - ~2 , 8 - -2
1
1 [ 222 16rr 2 (atmtc~ + a2(/*c~ -- Abs/~)2) B0(0, rnb, mA) 2 2 2 + (Atmts~ 4- Az(lzs# 4- abc#) 2) Bo(O, mb, O)]] ,
(36)
28
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 g2 2 AM2R = ---~-~2CtSt [(m~ + m~2)Bo(mT2,m~,O ) + 2 m~2Bo(mh,m~z , 0)] 3,~2 ctstAo( m~l ) -- 3g~2 mt m~Bo (0, mt, m~,) - 1--~2 ,~2t 32.n-2
[g2(Ot, fl)Bo(O,m~,,ma) + .(2(--tgt,fl)Bo(O,O, ma)
+n(O,,
+
,o) + gl(-o,, ~ + 2
1
l~7.r2 [-- (,~mtc/3(IzsB-- atcB) + ,~2bmtsp(#cB-- abs~))Bo(O, mb, mm) +,~mts~(Izc~ + Ats~)Bo(O, m~,,0) 1 ,
(37)
AMZR = 6~2 {2m~z [s~Bl(mr2,m~,,O) +c2tBl(mT2,m~2,0)]
+Ao(m~) + Ao(mt) - (m}2 - m~2 -- mZ)Bo(O, m~, mt) } 16~r2 [c~Ao(m~,) + ao(mb) - 4Ao(/z) + 2c~ao(mA)] 3 ~ 2 A(
- Ot,fl)Bo(O,m~,,ma) + A(-Ot, fl)Bo(O,O, ma)
"17" 77" 7"1" ] + a ( g - 0,, ~ - -~ )Bo(O, m~,, O) + a ( - o , , ¢~ - ~ )Bo(m h, m s, mz)
1
1
16~r2
(abm,s~ + 2
2 2
a~(
~s~
-
a,c~) 2) Bo(O, m~, mA)
+,~(l~c~ + Ats~)2Bo(O, m b, O) 1 .
(38)
We have defined the two functions
A ( Or, fl) = ( 2mr cos fl cos Ot - (tz sin fl - At cos fl) sin Ot) 2 +(/,t sin fl
-
At
COS/3) 2 sin2 Or,
$2(0t, fl) = 2m 2 cos 2 fl sin 20t
--
2mr cos fl(/~ sin fl - At cos fl) .
(39)
(40)
Note that the running mass matrix A~I2(Q) in Eq. (35) contains the soft m a s s e s M Q and My (as well as /z, A t , etc.) at some common scale, Q. In the limit ho ~ 0, these expressions are equivalent to the results of Ref. [ 11 ], with certain external momenta set to zero. These approximations depend on the mass of the light top squark. Normally, one would take it to be the tree-level mass. For the case at hand, however, the choice is more subtle because for very light top squarks, the radiative corrections can be quite large. In fact, the radiative correction can change the top squark mass squared from
D.M. Pierceet al./Nuclear Physics B 491 (1997)3-67 200
....
i
....
~'
i0
'::.~
I
I
'
I
. . . .
150
I
(b)
>
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5o
200
10o z o o
m~2(tree) • •
---:'....... ~:
3oo
. . . .
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:a.' "-~.: ;';- ,~,t-._'-."
....
1 O0
,,.i
;'"
"-I
"~'
.:.
' ~ ....
; o 5 0 ::
I ....
v) r'"l""r'-r'"r"r"'l
\,
\
-,-' - 5 0 "~:~~::~ . . . .
-2
-i
0
1
2
3
"x
(d)
-100
0 -3
I
500 l o 0 o
(c
2oo
~100
,
. . . .
100
Ao/M~
" -I ....
I ....
300
I,..,
I.,, ,I,.,,I.,J..,,I.J
500
1000
Q (GeV)
Fig. 16. (a) The full one-loop light top squark mass, versus the tree-levelmass (in GeV). On the x-axis we plot sign(m-2t2)lm2-t2fl/2' so a negativetree-levelmass correspondsto m~2< 0. (b) The differencebetween the full correction and the approximationin the text, versus the one-loop mass. (c) The light top squark mass at one loop, versus Ao/M#. The solid line correspondsto point (I) in the text, the dashed to point (II). (d) The running and one-loop light top squark mass versus the renormalizationscale Q, for the choice of parameters (I) (solid) and (II) (dashed) in the text, with Ao/M# = -1.83 and -2.2, respectively.The running mass curves each have points where m~2 becomes negative.In these cases we plot the signed square-root of the mass-squared, as in (a). negative to positive. Therefore we shall take m 5 to be the one-loop pole mass, which we find by iteration. (We find the one-loop top squark mixing angle by iteration as well.) We show the light top squark one-loop pole mass versus the tree-level mass in Fig. 16a, where the tree-level mass is the eigenvalue of the mass matrix which contains the running parameters M 2 and M~ evaluated at their own scale (or Mz, whichever is larger; the tree-level mass matrix also contains the top quark and Z-boson pole masses,
tanfl(Mz), # ( / z ) , and At(max([Atl, M z ) ) . In Fig. 16b we see that our approximation for the light top squark mass holds to within 10 GeV. With the present unification assumptions, a top squark with mass less than Mz requires that the RR term in the mass matrix be small and that the LR element, proportional to the A-term, be large. The light top squark mass results from a cancellation between the diagonal and off-diagonal terms, which requires a fine tuning. We illustrate this in Fig. 16c, where we plot the light top squark one-loop mass versus Ao/M#. On the same plot we show the curves corresponding to two choices of parameters, (I) tan fl = 20, M0 = 500 GeV, M]/2 = 100 GeV, and /x < 0, and (II) t a n f l = 5, M0 = 100 GeV, M~/2 = 200 GeV, and tz > 0. Whether at tree level or one loop, the parameter A0 must
30
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 .... I'"T"T"I
.-.
. . . .
I
. . . .
I ....
I ' '" '1"
' T"T"T"T"I
•
(
. . . .
I
I
•
(a) ~L
2
I
. . . .
I
(b) ~e
i
..
..~.. . . . .
o
,,
<1 -2 ,I,.,,I,,.,1,.,I
4
~
"I'"T"T'"I
. . . .
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. . . .
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I,,,,
. . . .
I
. . . .
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l ' ' ' T
I,,,,I,..I,,,,I,..I...
d
,
' ' T "'I'"'I'"T'"I
•
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,
-1
,
,,I I
. . . . •
.
.
- -
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,
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. . . .
,
,, I
. . . .
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-
(e) ~,
v
z~
::,~~;%,.. "¢~ ~ ¢".'I.;-:'.'.L .
2
?" f~;.
o .
-2
, L,..I..I,,,.I
100
. . . .
I
. . . .
I ....
200 m 7 (GeV)
I.,,,
I,, ,.I.,,,I,...I*.,L.,.I
500
1000
,
I
ao
.
.
, , , , 1 "
•
.
.
-
1
ioo
200
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500 iooc
m 7 (GeV)
Fig. 17. (a) The full set of corrections to the left-handed selectron mass versus its mass. (b) The complete corrections to the electron sneutrino mass versus its mass. The complete corrections to the predominantly (c) left-handed and (d) right-handed tan slepton masses. be tuned to one part in 75 to obtain a light top squark mass below 50 GeV. We see from Fig. 16a that the light top squark mass-squared can be raised from - ( 100 GeV) 2 at tree level to over (100 GeV) 2 at one loop. For such large corrections, it is important to keep in mind that two-loop effects might be important. The size o f these effects can be estimated by the scale dependence o f the one-loop mass. In Fig. 16d we show the scale dependence at the points (I) and ( I I ) , with A0 = - 9 8 5 GeV and A0 = - 9 0 7 GeV, respectively. We see that as the renormalization scale increases from 100 to 1000 GeV, the running masses vary over a wide range. In contrast, the scale dependence of the one-loop masses is quite mild. 4.4. S l e p t o n m a s s e s
Corrections to the S U ( 2 ) mass sum rules were studied in Ref. [ 34]. They suggest that the corrections to the slepton masses are small. We find that the corrections to the lefthanded electron and muon slepton masses are typically in the range 4-1%, as illustrated in Fig. 17a. The right-handed electron and muon slepton mass corrections are larger, but still less than 1.7%. The sneutrino mass corrections are essentially identical for all three generations. The full correction is typically in the range + 1%, and reaches at most - 2 . 5 % at my -~ 50 GeV, as shown in Fig. 17b. The (predominantly) left- and righthanded tau slepton corrections are similar to the corrections of the first two generation charged slepton masses. However, from Figs. 17c,d we see that the scatter plots show
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
31
less uniformity because of the additional Yukawa coupling corrections. We emphasize that in the corrections to the slepton masses, the leading logarithmic approximation [ 35 ] typically gives zero correction. (The mass m h receives some logarithmic corrections. However, the finite corrections are typically of the same order as or larger than the logarithmic corrections.)
4.5. Higgs-boson masses We first discuss the corrections to the heavy Higgs-boson masses (mA, mr4, mu~), and then we consider the one-loop light Higgs boson mass. The full one-loop corrections to the Higgs-boson masses appear in Ref. [6]. As usual, we parametrize all the Higgs-boson masses at tree level in terms of the CP-odd Higgs-boson mass, ma, and tan/3. To compute the one-loop mass ma, we first take the soft masses m ~ (Q) and m22 (Q) as outputs from the renormalization group equations, and apply corrections from the electroweak symmetry breaking conditions to obtain the D----Rrunning parameters th2(Q) and/z2(Q) (see Appendix E). We then apply further corrections to obtain the CP-odd Higgs boson pole mass, ma, from the running mass, mA(Q),
m2=rh2A(Q )_RelIAA(m2A ) + c2t2 + 2tl B U2
SB 0--7 '
(41)
where tj and t2 are the tadpole contributions listed in Appendix E. Note that we treat the Higgs mass analogously to the superpartner masses, in that we compare the pole mass with the running mass. However, the Higgs mass is different because the tadpole, or effective potential, corrections must be added to the "tree-level" running mass to obtain the DR running mass, m a ( Q ) . The difference between the running mass and the pole mass can be quite substantial; as for the light top squark, the radiative corrections can change a negative mass-squared running mass into a positive mass-squared pole mass. In Fig. 18a we show the one-loop pole mass, mA, evaluated at Q = M 0' versus the running mass, &A(ma). From Fig. 18a we see that there are points in parameter space where the running mass-squared is - 1 TeV 2, while the one-loop mass is over 300 GeV. (If one considers the running mass at the scale Mz, the largest corrections are even more extreme.) These large corrections arise from terms enhanced by tan/3. In fact, all of the points in Fig. 18a with frl2a(ma) < 0 Occur for tan/3 > 25. The tan/3 enhanced contributions come only from the last term in (41) since l/vl scales like tan/3 for large tan/3. Therefore the tan/3 enhanced corrections are simply
m2a -- rh2A( Q ) -
3 s~/.z tan /3 ( 167r2 ~ a'2a'
B0(0, m~,, m~.) + a~ab B0(0, m~,, m~2)
Jz2 J
(42)
32
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
500
. . . .
I
. . . .
I
6
. . . .
(a)
4OO
2
200 100
,"
0 -1500
,
,
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I ' ' ' ' I ' " T " T " T " T ' I
. . . .
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4
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. . . . . . . . .
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.
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. . . .
-500
&A(mA)
o
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.
.
500
<]
-2 i00
. .j~.
. • -
?
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>..
,.:...., ..,.z
:
.
- ~ i ~ - ~ . L : . . ~-~,~-.-;-.-...... .... ~-,f, 800
] ....
500
1000
In.+
Fig. 18. (a) The one-loop pole mass, mA, versus the running mass at the scale mA (the ordinate is the signed square-root of the mass-squared, sign(fn2a)]£n2A[1/2); (b) the corrections to the charged Higgs mass, m u~, v e r s u s
m H ~ .
where B0(0, m l , m2) is given in (9). These terms account for the large corrections seen in Fig. 18a. We parametrize the other heavy Higgs-boson masses in terms of the pole mass, m A. The corrections to the heavy Higgs-boson masses, m m and mm turn out to be quite small. For example, the corrections to mH~ are typically less than 1%, as shown in Fig. 18b. Corrections to the charged Higgs mass are the subject of Ref. [36]. The corrections to the light Higgs-boson mass, m], have been studied extensively in the literature [ 37,6,7 ]. Here we show our results for the full one-loop pole mass over the parameter space associated with radiative electroweak symmetry breaking and universal unification-scale boundary conditions. We also show the size of a set of corrections which are often neglected, and the accuracy of Dabelstein's approximation [7]. In Fig. 19a we show the one-loop light Higgs-boson mass versus tan fl, as well as the upper limit at tree level, mh = Mz[ cos 2fl I. The upper limit on the one-loop Higgs mass depends sensitively on the top quark mass, and somewhat less sensitively on the squark masses. In the parameter space we consider the squark masses are less than about 1 TeV. With m t = 175 GeV, we find mh < 130 GeV. In Fig. 19b we show the gauge/Higgs/gaugino/Higgsino contribution to the Higgs mass, which is typically - 2 GeV. In Fig. 19c we show the difference between the oneloop result and Dabelstein's approximation, Eq. (4.9) of Ref. [7], which only includes the top sector. We see that this approximation is typically 2 to 6 GeV larger than the full one-loop mass. In any pole mass, the scale dependence formally cancels. However, at any given order, there are usually higher-order corrections which do not cancel. For example, when we vary the scale in the Higgs mass calculation, we change the tree-level DR mass. To one-loop order, this variation is canceled by the change in the self-energy. However, as the scale varies, the couplings and masses in the self-energies also change. For the case of the Higgs mass, the change in the top quark Yukawa coupling gives rise to a two-loop O(A 4) scale dependence in our one-loop results. In Fig. 19d we show the difference between the full one-loop result evaluated at the scale MO and M # / 2 . We see that the scale dependence is usually small, in the range 0 to - 1 GeV. The fact that it
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
. : :,;-.:....,,:.. . ~ ! ~ .
100
....
~.~.:1:.-,-~
"-'"
33
0
"
"~'~q
"
":-:-.~;"°"'::"......... . . . . . .
<1
L/
t '
'
I
'
'I
'
'
I
I
, I
,,,I
I
I
I
'
I
'''1
I
I
,
I
, ,,I
I
0 :./"-~.4:7"-.'-'..~.--..
. .
Q) v
~-r'~~Z':~:.
~-5
~-.:
Q-I
I::I?-:".V :::"!::T .~- :: . - ? i .... .... . .:-..- " . . . - ( e )
:: "1
-i0 1
2
,.
, i~
,,';1"
5 tan
: "
t
10 20 fl
,
~-....-. .-
(d) ,,
I
50
I
-2 1
2
•
5
10 20
,
I
,
50
tan
Fig. 19. Figure (a) shows the full one-loop light Higgs mass v e n u s tan/3. The line indicates the tree-level bound mh < Mzl cos 2/31. Figure (b) shows the contribution from the g a u g e / H i g g s / g a u g i n o / H i g g s i n o loops; (c) shows the difference between the full one-loop result and Dabelstein's approximation; and (d) shows the difference between the full one-loop mass evaluated at the scales M 0 and M#/2.
is negative follows from the dependence of tan fl on Q, and from the dependence of At on tan ft.
5. Conclusions In this paper we computed one-loop radiative corrections in the minimal supersymmetric standard model. We took as inputs the electromagnetic coupling at zero momentum, Ofem, the Fermi constant, Gu, the Z-boson pole mass, Mz, the strong coupling in the MS scheme at the scale Mz, ces( M z ) , and the quark and lepton masses. From these we computed the W-boson mass, Mw, as well as the one-loop value of the effective weak mixing angle, sin 2 Dlept veff , as a function of the supersymmetric parameters. We studied the size of the corrections in a reduced parameter space associated with the unification of the soft breaking parameters and radiative electroweak symmetry breaking. We found that supersymmetric radiative corrections can reduce sin 2 0 ~ t by as much as - 1 . 6 × 10 - 3 with respect to the standard-model value. Similarly, we found that they can increase Mw by as much as 250 MeV. Because of decoupling, the points with the largest deviations are also the points with the lightest superpartners. As direct searches increase the limits on the superparticle masses, the size of the supersymmetric radiative corrections will decrease. Indeed, if superparticles are not discovered at LEP 2, we found
34
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
that the maximum size of the supersymmetric radiative corrections will be reduced by a factor of two. The apparent unification of the SU(3), SU(2), and U(1) coupling constants is a major piece of evidence in favor of supersymmetry. At next-to-leading order, the weakand unification-scale threshold corrections come into play. The weak-scale thresholds decrease the one-loop weak mixing angle. This leads to an increase in the predicted value of the strong coupling, a s ( M z ) . As we have seen, for squark masses less than one TeV, a unification-scale threshold of - 1 to - 3 % is necessary to bring a s ( M z ) into accord with experiment. The size of the unification-scale thresholds places an important constraint on unified model building. In any unified model, the unification-scale thresholds can be calculated as a function of the grand unification parameters. One can see whether the model is consistent with a unification-scale threshold of about - 2 % . In this paper we studied the minimal SU(5) model and the missing doublet SU(5) model. We found that the former was not compatible with gauge coupling unification, while the latter was. Grand unified theories also predict the unification of certain Yukawa couplings, and in a similar fashion, the mismatch of the Yukawa couplings at the unification scale can be used to constrain unification-scale physics. To this end it is necessary to extract as precisely as possible the DR Yukawa couplings from the fermion pole masses. In this paper we presented full one-loop relations between the two, as well as approximations that work at the (.9(1%) level. We studied the substantial (up to 50%) tan/3-enhanced corrections to the bottom quark mass, as well as the corrections to the top and tau masses, which are of order 5%. Supersymmetry also predicts relations between the masses and couplings of the supersymmetric particles. Indeed, if new particles are discovered at future colliders, it will be necessary to check these relations to see whether the new particles are in fact supersymmetric partners [38]. The radiative corrections to the supersymmetric mass spectrum presented in this paper will be an essential element in these determinations. The corrections to the supersymmetric mass spectrum will be used in (at least) two ways. First, they will be used to correct the tree-level mass sum rules [ 34,39] which test supersymmetry at the weak scale. Second, they will be needed to extract the underlying soft parameters from the physical observables. The soft parameters can then be run to higher scales, to test for unification and possibly to shed light on the origin of the supersymmetry breaking. The corrections to the supersymmetric masses in the spin-½ sector include potentially 30% corrections to the gluino mass, as well as (_9(5%) corrections to the neutralino and chargino masses. In the spin-0 sector, the famous quadratic divergences give rise to large corrections to the scalar masses. These corrections can lift the running mass-squared of the light top squark from - ( 100 GeV) 2 to (100 GeV) 2. Even more dramatically, large tan/3 corrections can lift the mass-squared of the CP-odd Higgs boson from, e.g., - ( 1 TeV) 2 to (300 GeV) 2. Radiative corrections also have an important effect on the mass of the lightest Higgs boson, h. In the parameter space we consider, they effectively change the sign of the
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
35
tree-level bound from m h < Mzl cos 2ill to m h > Mz[ cos 2/31. We found the light Higgs mass was raised to at most 130 GeV. The corrections to the rest of the scalar masses are smaller. For example, we found I to 5% corrections to the first two generation squark masses, and (.9(1%) corrections to the slepton masses. In the paper we presented approximations to many of the formulae for the supersymmetric mass corrections. These approximations, often good to better than a couple of percent, provide useful substitutes for the full corrections.
Acknowledgements D.M.E thanks M. Peskin, T. Rizzo and J. Wells for useful discussions.
Appendix A. Tree-level masses In this appendix we define the tree-level masses. These tree-level relations also hold for the running DR parameters at a common scale, Q. For the most part we follow the conventions of Ref. [40]. The up- and down-type quark and charged-lepton masses are related to the Yukawa couplings and the vev's vl and v2 by
1 mu = --~)tuV2 ,
md =
1 -'~/[dUl .
(A.1)
The ratio of vev's vz/vl is denoted tan ft. The tree-level gauge boson masses are M ~ v = ~1g 2 (v~+v22),
M2z= 1 (g,Z +g2)(v2 + v2),
(A.2)
where g and g' are the SU(2) and U(1) gauge couplings. The Lagrangian contains the neutralino mass matrix as -~°TA4ffo~ ° + h.c., where
~o = (-i[~, -ivY3, hi, h2) T and .A4~o =
MI 0 -Mzc/3sw Mzs/3sw I 0 M2 MzCl3CW - M z s p c w -Mzcl3sw Mzc/3cw 0 Iz " Mzs/3sw -Mzs/zcw tz 0
(A.3)
We use s and c for sine and cosine, so that s# - sinfl, c2~ - cos2fl, etc. MI and M2 are the soft supersymmetry-breaking bino and wino gaugino masses, /z is the supersymmetric Higgsino mass, and sw (Cw) is the sine (cosine) of the weak mixing angle. The neutralino masses are found by acting on the matrix .Ad¢7o with a unitary matrix N, so that N*.Adc;oNt is a diagonal matrix which contains the physical neutralino masses, m~o. In the usual case that one of the eigenvalues of (A.3) is negative, the matrix N is complex even if the elements of .Ad~0 are real.
36
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
The Lagrangian contains the chargino mass matrix as _ ~ - r . M e T ~ + + h.c., where ~+ = ( _ i # + , ,~-)r, ~ - = ( - i ~ : - , h~-)r and
M2 v/2 MwSB) .MET' =
x/~ Mwc#
-At
(A.4)
"
The chargino masses are found by acting on the matrix .A4eT+ with a biunitary transformation, so that U*A/te7+Vt is a diagonal matrix containing the two chargino mass eigenvalues, m ff. The matrices U and V are easily found, as they diagonalize, respectively, the matrices A4eT~*.Me7 ,T and .M~+.MeT,. At tree level the gluino mass, m~, is given by the soft mass, M3. The tree-level squark masses are found by diagonalizing the following mass matrices: M2Q + m u2 + guLM2c2B mu ( A , + Atcot fl)
( Here
m, (A, + Atcot fl)
(A.5)
M2v + m2u + guRM2zc2l~ ,] '
M2Q + m d + gdL Mzc2B
rnd ( Ad + At tan fl)
md ( Aa + Attanfl)
M~ + m2d + gdRM2 c2B
'~
)
(A.6)
m
MQ,
Mu, and MD are the soft supersymmetry-breaking squark masses, and the the soft supersymmetry-breaking A-terms. The slepton mass matrices are analogous. The soft slepton masses are denoted ML and ME. We have defined the weak neutral-current couplings
Af's
are
gf = I f - efs 2 .
(A.V)
The electric charge, hypercharge, and third component of isospin of the sfermions are 9
~L
~R
dL
~
~L
~
ef
g
2
_2
3
--3
3
1
0
-1
1
Ys 1f
31 ~l
- 34
~1 - 31
2
--1
--1
2
0
~l
- 31
0
0
1
dR
(A.8)
A symbol without an L or R subscript refers to the L-field (e.g. eu = 2/3). The matrix which diagonalizes a sfermion mass matrix is denoted by Q C f S f ) ,Cf __Sf
(A.9)
where cy is the cosine of the sfermion mixing angle, cos Of, and sf the sine. These angles are given by 9 Our convention for the right-handed sfermion fields is the charge-conjugate of that of Ref. [40].
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67
2mu (Au + I z c o t f l ) 2 2 ' _ _ 2e,sw) Mzc2~ 2md ( Ad +/x tan fl) Q _ ME + (_½ _ 2eas2)M2zc2 .
t a n ( 2 0 " ) = M2°
tan(20a)=M2
37
M2v+(½
(A.10)
(A.11)
Since there is no right-handed sneutrino, the slepton m i x i n g angle for ~ satisfies c~ = 1, and the sneutrino mass is mo2 = a4~ + a 4 ~ / 2 . Given values for tan fl and the C P - o d d Higgs-boson mass, are given, at tree level, by
m2H,h
1 = -~
mA, the
other Higgs masses
(m~A+ M~ + j( m~A+ a4~) 2-4mAMZc2~2)
(A.12)
and m~+ = m~ + M2w .
(A.13)
The C P - e v e n gauge eigenstates ( s l , s2) are rotated by the angle a into the mass eigenstates ( H , h) as follows:
(H)
= ( c~ ca s,)
s2
.
(A.14)
At tree level, the angle o~ is given by tan 2 a - m2 +-- M 2 tan 2ft. m2 M2
(A.|5)
Appendix B. O n e - l o o p scalar functions The f o l l o w i n g integrals appear at one loop in a self-energy calculation [4] : l0
Ao(m)
= 167r2Q 4-n
Bo(p, mj,m2)
= 16~'2Q 4-n
/
dnq 1 i (2~-) n q2 _ m 2 + dnq /(27r) n
dnq puB,(p, ml,m2)
= 167-r2Q4-n
i (2~.)n
ie '
(B.I)
1 [q2_m21+ie][(q_p)Z_m~+ie], (B.2) qs, [qZ--m~+ie][(q---p)Z--mZ+ie] (B.3)
m Our A and B functions differ from those of Ref. [4] since we use the Minkowski metric. Also, A0, BI and B22 differ by a sign. Eqs. (B.2)-(B.4) contain an abuse of notation. The first argument of B-functions is the square root of the scalar p2, whereas elsewhere the p represents the external momentum four-vector.
38
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3 - 6 7
PupvB21 (P, ml, m2) ÷ gjz~,B22(p, ml, m2) = 16,rr2 Q4-n f
d"q
q~qu
i(27r), [q2_ m~ + ie] [ ( q - p ) 2
_ m2 + i e ] ' (B.4)
where Q is the renormalization scale and we regularize by integrating in n = 4 - 2e dimensions. The expression for A0 can be integrated to give
Ao(m) = m 2
(B.5)
+l-In
where 1/~ = 1/e - YE + In 4~-. The function Bo can be written in the form
,/
1
Bo(p, ml,m2) = -~ -
dx In
(l-x)
m2 + x mZ - x( l - x) p2 - ie
(B.6)
OR
o
It has the analytic expression
Bo(p, ml,m2) = ~ - In
- fB(x+) - f s ( x - )
(S.7)
,
where S z~z ~ / S 2 -x± =
fs(x)=ln(l--x)--
4p2(m~
- ie)
2p2 (B.8)
xln(1-x -1)-1,
and s = p 2 _ m~ + m 2. All the other functions can be written in terms of Ao and Bo. For example,
BI (p, ml,m2) = ~p2 Ao(m2) - Ao(ml) + (p2 + m~ - mZ)Bo(p, ml,m2)
, (B.9)
and
B22(p, m l , m 2 ) = 6 { ~l ( A o ( m l ) + A o ( m 2 ) ) + ( m 2 + m ~ - ½ P Z ) B o ( p , m2_- ml
We also define
ml,m2)
Ao(m2) - Ao(ml) - (m~ - m~)Bo(p, ml,m2)
D.M. Pierceet al./NuclearPhysicsB 491 (1997)3-67
39
F(p, ml,m2) =Ao(ml) - 2Ao(m2) - (2p 2 4- 2m 2 - m~)Bo(p, ml,m2) ,
(B.I 1)
G(p, ml,m2) = (p2 _ m 2 _ mZ)Bo(p, ml,m2) - ao(ml) - ao(m2) ,
(B.12)
H(p, ml ,m2) =4Bz2(p, ml ,m2) + G(p, ml,m2) ,
(B.13)
1
/~22(P, ml, m2) = B22(p, ml, m2) -
Ao(ml) - ~Ao(m2) .
(B.14)
The functions F and G arise in scalar self-energies, with either a vector boson and a scalar or fermions in the loop, while H and /~22 occur in vector-boson self-energies, with either fermions or scalars in the loop.
Appendix C. The gauge couplings In the remaining appendices we denote ~2 =- sin 2 t~w, where ~w is the DR weak mixing angle, and S 2 ~ sin 20w = 1 -Mw/M 2 2z, where Ow is the "on-shell" weak mixing angle and Mw, Mz are the gauge-boson pole masses. The DR electromagnetic coupling is given by O/ern & -- 1 -- Age '
aem
1 137.036 '
(C.1)
where 11 Age=0.0682±0.0007-
_ ~ { -71n ( M w ) 1 6+9 ~
2 4 u~Zln
1 (mH~
+ 3 in \ Mz ] + -9
1 2
i-I
(md~ ~
+9ZZln\M---zzJ d i=1
+3
In
( mMzz r )
(m~,)
MZZ
1 ~e ~-~ln(m~ ) 4 2 (rn2+~} Mzz + 3 Z l n \ M z J ' i=1 i=1
(C.2)
and ~ , indicates a sum over u, c, t, and similarly for )-'~d, )--]~e"In this expression, the number 0.0682 includes the two-loop QED and QCD corrections given in Ref. [41 ], as well as the five-flavor contribution " O~ha (5)(M2z) = 0.0280 4- 0.0007 of Ref. [42] d The DR weak mixing angle is given by [43] ~2~2 =
,/.gge
v/2M2zG~(1 - A~) '
T , ( O) ~ Ar=pH ~w
T 2 Re Hzz ( Mz ) M2z +8VB,
(C.3)
where /5 is defined to be C2/~"2, and tSvu denotes the non-universal vertex and box diagram corrections given below. The W and Z gauge-boson self-energies are given in Eqs. (D.4) and (D.9), We compute/5 via [43] 1 / 5 - 1 - A/5'
A/5 = Re [ llrz(M2z) [ /5M2z
II~vw( r Mw 2 )] M2
].
11The coefficientsof In(Mw/Mz) in the expressionsfor A& in Refs. 122,231 are both incorrect.
(C.4)
40
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67
We deduce the leading two-loop standard-model corrections to A? and A/) from [41],
as[2.145m2+o.5751n(mt) M2z Mzz
A? 2_loop - 4~s--2c2 ~ ~ [
- 0.224- 0.144 M~] m
12 (2) (mr) --SxtP ~ (1 -- A~)~,
(C.5)
&2 as-2.145 A/) 2-loop - 4~-~ 7r [ m~ +1.2621n ( mMzz ,) +3 xt
- 2 . 2 4 - 0 . 8 5 _~t~]
-~t '
(C.6)
where xt = 3Ggm2t/8~r2v~and m r is the standard-model Higgs-boson mass. For r ~< 1.9, p(2)(r) is well approximated by [44] p(Z)(r)=19-~r+]-~
43r2
+ ( 3 3 + 1 0 r3-Trx/~ 4 - ~ r +
13)
r 2+2--~r
--Tr2(2--2r+~r2)--lnr(gr--~r2) ,
(C.7)
while, for r/> 1.9, we use
p(2)(r)=ln2r(3-9r -l-15r-2-48r-3-168r-4-612r -5) -
In r ( _ ~ + 4r-1
125 r-2 4
558 ,.-3 5
8307 r.-4 20
109321 -5"~
--~ r )
+Tr2 (1 - 4r -1 - 5r -2 - 16r -3 - 56r -4 - 204r -5) 49 2 -1 1613r-2 8757 -3 341959r_ 4 9737663 -5 +-~- + ~r + 48 + l i - ~ r + 1200 + 980--------~r
(c.8) For the case of the MSSM, we replace the function p(2)(m~o/mt) with //COSCe'~ 2 p , 2f. ~[~mh~ _ _ __
~,sinflJ
~mtJ"
(C.9)
We have not computed the corresponding G2um4 higher-order contributions from the heavy Higgs bosons, but we know that they must decouple. Using the ansatz A/9 heavy Higgs
=~x~{ fsina~2 mn'~J ~,s~nflJ p(2)f~--~t
(ta-~ ) 2p(2)(m~A' ]}' \ mt /
(C.10)
we find these contributions are negligible. We do not include them in our results. The non-universal contribution to A~ is made up of two parts, one from the standard model and the other from supersymmetry, 6VB = 6"Svr~+ b'svUB sv .
(C.11)
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
41
The standard-model part is given by the well-known formula [43]
a S v M =^p ~ 2&{
--(5
6 + lnc2
3C2"~1
The supersymmetric part appears in Ref. [5], and more recently in Ref. [ 12]. We include it here for completeness. It includes box diagram contributions, vertex corrections, and external wave-function renormalizations. We neglect the mixing between different generations of sleptons, and we ignore the left-right slepton mixing in the first two generations, in which case the right-handed sleptons eR, tXR do not contribute. We find
6Z~+6Z~ + S Z . + S Z ~ .
xsusY=___M2zReal+6v~+Sv~+ ~vB 27r&
.
(C.13)
The wave-function and vertex corrections are 2
4
b*t~,~L 2B'(O'm*/'moL) - Z
167r26Z~=-Z
b,o,,,r,, 2Bl(O,m,o,m~,) ,
,j=l
i=1
(C.14) 2
2
16rr28Ze=- Z
4
ayri~er,~ Bl(O,m~i,,mr,~) - Z
i=1
2
b2oe& Bl(O, myd,m~L),
j=l
(C.15) 2
4
16~'23ve = Z
.
v~
y ~ b~,+~ ,~ , ~tb~oe~ xj ~ [,[---a~o~+wm~+m~,o g AjAi Ai .'tj Co(mo,,mydi ,m2oj ) "
i=l
+-~gb2O~j~ w Bo(O, m~;., m~o) + m~,.. Co(mo,,.. m~,+,.., m~,o).,j
-
4
-- ~
a~+er ,e b~% ~ Ai A j e ~1
~ ~
i=l
j=l
%
- - - be ~ o " U a~~wm~,, t , m~o Co(m~,, m2;~, m2o) o
+----~--a~ox~+ w Bo(O, m~/, m~o) + m~, x/2g ~j ' +'21 4Z
b**OeoLb2°d',~', ,
, m~o) -
[Bo(O,m~t,mr'~) +m2oCo(myro,m~,,mr,,) + ~1 .
j=l
(C.16) The corrections SZ~, SZ~,, and 6v u are obtained from these expressions by replacing e --~/x. The ,~-fermion-sfermion couplings ay(iffj and by(,ffj are listed in Eqs. (D.20)(D.22), while the chargino-neutralino-W couplings a2o,x,~w and b~9~w are defined in Eqs. (D.12), (D.13). In these expressions, the B0, B1, and Co functions are evaluated at zero momentum,
D.M. Pierceet al./NuclearPhysicsB 491 (1997)3-67
42
,
Bo(O, ml,m2) = ~ + 1 + In
1[1
+
(Q2)
B,(O,m,,m2)=-~
+l+ln
,
m2
in (m2"~
m~-m 2
(
m ~ ~2 ln(m~'~
~222 + \ m 2 _ m ] j
1____~[ 2m~ 21n(m2) m ] t m , _ m 2 \mZJ
Co(m, m2,m3)-m2
(C.17)
\mf,]'
1 (ml2+m~'~]
\m~/ + 2 \m2--mZJ] ' (c.18)
m~ ln(m~'~],
(C.19)
m,Z---m32 \ m 2 J
where Q is the renormalization scale. The box diagram contributions are 2 1 16rr2 aj = -~ Z
4
~ a~,~,,~,,~b*.-+.x,e*,. , ~ b~.ojv, , ~ b~oee.~j,. m2i~rn~?oD0 (m~L, rn~, m~?+,m)?o)
i=1 .j=l
1
2
4
+-2 Z ~ " a*-+ _ b-~p - b*-o - b*-o - rn~+m-o Xi ePe Xi ,~]zL XjPel]e Xj]£]£1. Xj Do(mr, L, m~, m U, m~o) i=1 j=l 2
+ Z
4
~
b-+p - b*+ - b*o - b-oe~ Xi ;~IZL Xi PeeL Xj~IZL Xj L D27(m#,, ms,., mL.+,m~,o)
i=1 j=l 2
4
+ Z ~ - - ' ~ a * "~i ~ .I&Uga ~ a~.+er, ~ D27(mr,,,mr, e,m2+,m~o) Ai e b~.ov * j tz r,I*b*.0. l[jVel*e
(C.20)
i=1 j=l
where the functions Do and D27 are
Do(ml,m2,m3,m4)
m2
'[
m2 Co(ml,m3,m4)
]
Co(m2,m3,m4) ,
O27(m! ,m2,m3,m4) - 4(m2 -l [ m m22 )C o ( m l , m 3 , m n ) - m 2 C o ( m 2 ,
(C.21)
m3,m4)]. (C.22)
We checked our box diagram calculation against Refs. [5,12], and we checked our formulas for 8Z and c$v with those of Ref. [ 12]. Here (there) the formulas are written in terms of the couplings corresponding to vertices with incoming (outgoing) charginos and neutralinos. To compare we must make the transformation a ~ f f ~ bygff, except for couplings involving the chargino and down-type fermions, which remain unchanged• Also, their ,~'+.~°W coupling differs from ours by a sign. The effective weak mixing angle is given in terms of the DR weak mixing angle, #2, via • 2 ~ l e p t = ~2
sm eelf
Re/ce
(C.23)
43
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
where [45] ~:~ = 1 + ~
Hz~(M2z) - IIz~(O) &~2 In c 2 & 2 M ~Z'T + 7r~-5--- 47r~2 Ve ( M z ) ,
(C.24)
with l (~2) (~2) ~(M2z) = ~ f +4~2g Ref(x)
g(x)=
x
2
3+
1 - 6se + 8s4 4c 2 f(1),
lnx
(1 1)(tan~ly ) 9 +
(C.25)
-
l (1)
1 +8+2xx-
l+~x
4(tan__1 y)2
.
x
(C.26) Here y =_ V / X / ( 4 - x), Li2 is the Spence function, and l l z r is listed in Eq. (D.15). We do not include here the non-universal Z-vertex supersymmetric contribution to sin 2 veff . The largest contributions can be obtained from Ref. [46].
jqlept
Appendix D. One-loop self-energies In this appendix we list all the relevant self-energy functions which allow us to determine the one-loop fermion, gauge-boson, and superpartner masses. We explicitly include all of the necessary couplings. We perform our calculations in the 't HooftFeynman gauge, in which the Goldstone bosons and the ghosts have the same masses as the corresponding gauge bosons. The gauge couplings g', g and g3, and the Yukawa couplings /~f are all DR couplings. The neutralino mixing matrix N, the chargino mixing matrices U and V, the Higgs mixing angles te and /3, and the sfermion mixing angles Of are described in Appendix A, as are the normalizations of the Yukawa couplings/If. The self-energies are given in terms of the Passarino-Veltman functions A0, B0, Bl, F, G, H, and Bee listed in Appendix B, Eqs. (B.5)-(B.14). To streamline notation we do not write explicitly the external momentum dependence of these functions, e.g., we write Bo(p, m l , m 2 ) as B o ( m l , m 2 ) . Throughout this appendix we write s for sin, c for cos and t for tan, so that s¢~ - sin/3, c2o, = 2Or, etc., and for the sfermion mixing angles, c, = cos 0,, etc. Sub- or superscripts f denote a quark or lepton, and q denotes a quark. Inside a summation ~f,,, the subscript or superscript u denotes all up-type (s)fermions, u, c, t, ~'e, v~,, ~'~, and similarly inside a summation y'~f,,, the script d denotes all down-type (s)fermions, d, s , b , e , tz, and z. The sum ~-]~L,/fd denotes a summation over (s)quark and (s)lepton doublets, and the sum ~'q denotes a sum over (s)quarks. Some terms are zero, for example a~ = 0, and terms involving the right-handed sneutrino are absent.
cos
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
44
In the self-energies listed below the 1/# poles are canceled by counterterms which relate the bare mass to the running mass. So, in the following DR self-energies we implicitly subtract the 1/# poles.
D.1. Z and Wbosons The full one-loop MSSM gauge-boson self-energies appear in Ref. [5] and subsequently in Ref. [6]. The supersymmetric contributions are listed in Refs. [47,22]. The self-energies of the gauge bosons can be separated into transverse and longitudinal pieces, e.g., ,a~, T 2 [gUV p/~pV] L 2 pUpV Hzz(p2) =Hzz(P )_ p2 J + Hzz(p ) p2
(D.1)
The physical gauge boson masses are the poles of the corresponding propagators, which involve only the transverse part of the gauge-boson self-energy,
M2z =Mz(Q)^ 2
- Re Hrzz(M2z),
(D.2)
M2w = ]Q2w(Q) - Re Hww(Mw). T
2
(D.3)
Here Mz(Q) and ~iw(Q) denote the DR running masses which are related to the D--R gauge couplings and vev's, as in Eq. (A.2). The gauge-boson self-energies are evaluated at the renormalization scale Q. The transverse part of the Z-boson self-energy is 16~ 2 ~
T 2 ~ Hzz( p 2 ) = -s,~fl B22(mA,mta) + B22(Mz,mh) - M2zBo(Mz,mh)
2 --CAB
I~22(Mz,m~l)+~22(mA,mh)_M2zBo(Mz,mH) 1
--2c4(2p2-~M2-M2~)no(Mw,
-(8~ 4 +
Mw,
2 ~ 2 ~ c20w)B22( Mw, Mw) - c20wB22(mH~ , mn~ )
2 4N~s 2g fijB22 ( m ]?
m.?j)
f i,j=l
+~f Nf{(g2fL+g2fR)H(mf, mf)-4gfLgf~m~Bo(mf, mf)} ~2 4 } +~g2 ~-~{ f~ijzH(m~(°' m~o) + 2 g°jz m2~m~,?Bo(m~o,m~:~) i,j=l " C2~If+z +g~ -i,.j=l ~"
H(m2+,m-+ ) + 2 giiz + m~mr,~:Bo(m~,:~,mr,. +) } , X] -~-- ,,, ,,j ,,t "'1 (D.4)
D.M. Pierce et a l . / N u c l e a r Physics B 491 (1997) 3 - 6 7
45
where the summation ~-~f is over all quarks and leptons, and the color factor N[ is 3 for (s)quarks and 1 for (s)leptons. The notation s,~rl denotes s i n ( a - f l ) , and carl refers to cos(a - / 3 ) . The sfermion-sfermion-Z couplings can be written in terms of the weak neutralcurrent couplings defined in Eq. (A.7): V f l l = g f ~ c f2
_ gf, sZf ,
2 vf22 = gf~cZf - gf, sf
V f l 2 = Uf21 = ( g f L -~ g f R ) C f S f
(D.5)
•
The neutralino-neutralino-Z-boson couplings are defined by
~ z• = [a~?~°zl2 +
Ib~?~°zl 2,
. = 2Re ( b*.-o~oTa~o~Oz) giOz xixJ l~ AiAJ / ,
(D.6)
and analogous definitions hold for fij+z and gijz. + We write the Feynman rule for the )~)~Zu vertex, where ,~ is a chargino or neutralino, as --iyu(a79L + bT~R), where T'L,R are the usual chiral projectors, 7~L,R= (1 :F Y5)/2. The couplings involving the unrotated ~o and ~+ fields satisfy b~;o$oz = -a~o~o z and b~?~j~z = a(,/~jz. The non-zero a-type couplings are
g
a ~//3~//3 -0 -07 -0 -07 ~ = - a ~//4~4 ~ = ~-~ ,
a(q~; z -
a~(,~z = g~,
gc2ow 2~
(D.7)
For an incoming f(o and incoming ~(+ we have
a Zi-o-o Xj Z
= Ni*k N i l a(,o~,oz ,
aL,2~z = Vi*k Vjt a~;~? z ,
b2°yc°z = Nik Nj*Ib(,o(,oz b~] z
,
= Uik U~.l b~;~+ z .
(D.8)
(Here and in the following formulae which specify rotations, we adopt the summation convention for repeated indices.) For the transverse part of the W-boson self-energy, we find 167"r2g2 H~w(p2) = -s,~rl2 I~22(mn,mrt~ ) + B22(mh, Mw) _ MwB02 (mh, Mw) 1
2 [B22(mh,mH+)+ B22(mH, M w ) - M2Bo(mH, Mw)] --Carl --O22(mA, m n ~ ) - ( 1 + 8~'2)/~22(Mz, Mw) -~2 [8/~22(Mw, O)+4p2Bo(Mw, O) 1 - I ( 4 p 2 + M 2 + M 2 ) ~ "2 - M 2 s 4] Bo(Mz,Mw) +
I-I(m.,m,~) - ~ ,,
i,j= 1
2N cI w ~fqB2z( mr,. rod2)
46
D.M. Pierce et a l . / N u c l e a r Physics B 491 (1997) 3 - 6 7
--
+g2
fijw
i=1 j=l "
H (re,o, m,f ) + 2 gijw m~o m ~j~.+Bo (m "4i~o, m,~ ) --i
}
'
(D.9) where the s u m m a t i o n ~--~f,,/fd is over quark and lepton doublets, and W f l I "= CuC d ,
Wfl2
= CuS d ,
Wf21 = Sued ,
W f 2 2 _m SuSd .
(D.IO) The neutralino-chargino-W-boson couplings are f i j•w
= la,o,fwl
2 -'b
gijw = 2 Re
Ib,y,;wl 2 ,
(b*o-+ x, xj w a-o-+ z, xj w )
•
(D.11)
We write the Feynman rule for the neutralino-chargino-W u vertex as - i y u (aT'L + bT~R), and the non-zero couplings are
= b4'°4"~w= - g '
W
a~'°~w = -b~°&+-w= ~22"
(D.12)
For an incoming ~o we have the couplings to mass eigenstates,
a~o~ w = N~k Vjla(,o(,/w,
byc°2~w= Nik Uflb(,°~?w'
(D.13)
while for an incoming f(+ we have the couplings
a~o~ w = Ni~ Vj*l a4,o(,i,w ,
bycosc~w= N*k Ujtb(,o~j~w"
(D. 14)
Finally, we write the mixed Z - T self-energy as ^
167r2C-~-Hzr(p2) = ( 12~ 2 - 10)/~22(Mw, Mw) - 2(MZw + 2d2p2)Bo(Mw, Mw) eg -b~Nfef(gf,.-gfR)[4B22(mf,
mf)+p2Bo(mf, my)]
f
- 2c2bwB22(mn+ , mn+ ) +~ ~-~( IVa 12+ lUll I2 + 2c2&) 4/~22(m~, ms(~ ) + p2Bo(m2?, m2~ ) i=1 s ef [ ( g f L c f ~ - gs.s}~ NSc
-4 ~ f
J~22(m/l, m~ )
I. I
+(gy, s2f - gf~c})B22(mk, mA) 1 • /
(D.15)
D.2. Quarks and leptons The fermion masses are defined as the poles of the corresponding fermion propagators. They are related to the D----Rmasses, ~hf, by the self-energies, Zf(p2), as follows:
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
47
mf = r~f(O ) - Re 2f(m2f ) .
(D.16)
The DR fermion mass /'~/f is related to the DR Yukawa coupling and vev as shown in Eq. (A.I). Care must be taken in evaluating the DR vev. After evaluating the DR gauge couplings g' and g as outlined in Appendix C, we determine the DR vev via v2(O) = 4 M2z + ReHrz(M2) g,2(Q) _+_g2(Q)
(D.17)
,
where Q is the renormalization scale (the argument of the Z self-energy is the external momentum; it implicitly depends on the scale Q as well). For the top quark, ~Vt(p2) is
{
16~'2 Xt(PZ)m, - 4g~3 Bi(m~,m~,) + Bl(m~,m~2) -
(
°2)
5 +31n~7
-szo,~(Bo(m~,m~,)-Bo(m~,m~2))}
+c213[Bl(mt,mA) --Bo(mt, mA)] +s2~[BI(mr, Mz) -Bo(mt, M z ) ] } +-~ (.,~2s2~q- At cB)Bl(mb, mH~ ) + (g2 + .,~2C213q_ .~t SB) B l(mb, M w) +/I2c2~[Bo(mh, mH,)--Bo(mb,mw) 1 - (ee,)2(5 + 31n m~ ) +~_~
gt2L+&2 Bl(m,,Mz) +4g,Lg,~Bo(mt, Mz)
+~
f~aj B1 (re,o, r% ) + g~t~j i=1 j=l
fitbjBl(m~+,,mbj)
+2
+ git~j mt
mt
Bo (rn,o, m~j ) mL* , mbj) ]
(D.18)
i,j=l
The neutral current couplings gf are defined in Eq. (A.7). We write the Feynman rules for the )(iffj couplings as -i(aPL + bPR) (for vertices involving the chargino and down-type fermions the Feynman rule is iC -1 (aPE -4-bPR), where C is the charge-conjugation matrix). We define L f L = la*,fLI 2 +
Ib~,,fL 12'
gif}j = 2Re (b,,f?j air,T?j) .
In the unrotated ~0, ~+ basis, we have g!
I
b&of },. = -~2 Yf " '
(D.19)
48
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 b ~ f f L = v / 2 g l ft- '
a(~doL = b(,~udL = g,
a(joddt. = b~odd R = - - b ~ d~L = - b ( q u d R = hal, a(~ou~L = b~ou~ ~ = - a ~ ud~ = - - a ~ d~R = au ,
(D.20)
where the quantum numbers Yf and I f are listed in the table of Eq. (A.8). These couplings correspond to vertices with incoming neutralinos and incoming charginos. To obtain the couplings to the mass eigenstates j,0 and )~+, we specify the rotations a 2 o f f = N~ a ~ o f f ,
a ~ f f f , = Vi~ a ~ + f f , ,
b2off = Nij b ~ o f f ,
(D.21)
bscff f, = Uij b~; f f ' "
(D.22)
The couplings to the sfermion mass eigenstates are found by rotating these couplings (both a- and b-type) by the sfermion mixing matrix,
a2ff~
--s f, c f,
a 2 f f ,~
The self-energies £ y ( p 2 ) for the other up-type quarks and leptons can be obtained from the previous formulae by obvious substitutions. For the bottom quark (and similarly for all down-type fermions), one interchanges t ~ b, c~ ~ s~, and c# ~ sty. D.3. Charginos and neutralinos
The complete one-loop self-energies for charginos and neutralinos are given in [ 32,9] ; we present them here in a matrix formulation. For the Higgs-boson contributions, refers to H, h, G°, and A, while H + represents G + and H +. The GO and G + are the Goldstone bosons; in the 't Hooft-Feynman gauge their masses are equal to M z and M w , respectively. We now describe the full one-loop neutralino and chargino mass matrices, from which we determine the one-loop masses. The one-loop neutralino mass matrix has the form .h.4~0 + ~1 (6.Me;0 (p2) +SA/tr_o(p2) ) g ,
,
(D.24)
where ~.A/[ffo ( p 2) = --~R(pZ).A,'[~0 -- .A/[ffO~0L(p 2) -- ~ s ( p 2) .
(D.25)
+0 2 Here A.4~o is the tree-level neutralino mass matrix of Eq. (A.3), and the ~r,'R,s(P ) are matrix corrections. They allow us to determine the one-loop masses and mixing angles for arbitrary tree-level parameters. The one-loop chargino mass matrix is as follows: .h4~ - 2 + ( p 2) .h4~+ - .MgT+ ~:~-(p2) _ 2:s+ ( p 2) ,
(D.26)
where .h4~ is the tree-level chargino mass matrix of Eq. (A.4). The elements of .M~o and .A.44;, contain DR parameters at the scale Q. In particular, they include corrections
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3 - 6 7
49
corresponding to replacing Mz with /l~/z, obtained from Eq. (D.2). Similarly, tan/3 in the tree-level matrices is tan/3(Q). The self-energies 2L,R,S are also evaluated at the scale Q. To obtain the mass for a given neutralino or chargino, for example ,~0, we first evaluate the matrix of Eq. (D.24) with the momenta p2 = m~0. We then solve for the eigenvalues of that matrix. So, in determining four neutralino and two chargino masses, we construct a total of six different matrices. We compute the mass matrix corrections by evaluating two-point diagrams with unrotated neutralinos or charginos on external legs, and mass eigenstates inside the loop. We obtain the couplings associated with these diagrams by the following method. The neutralino mass corrections involve the couplings ago.., which we obtain from the various couplings aL0.. (for incoming X °) by leaving off one factor of Ni*k. The neutralino mass corrections also involve the couplings bg~)., which we obtain from the couplings b)?l).. (for incoming ~(o) by leaving off one rotation Nik. We obtain the couplings agk`... which appear in the chargino mass corrections from the couplings a)?/... (for incoming ~',.+) by leaving off one factor of V/~, and we determine the couplings bg~ ... from the couplings b t/... (for incoming ~(i+) by leaving off one factor of Ua,. For the neutralinos, we have the one-loop correction 2
16"rr2~Li.j( p 2) =Z~-~'~ NJc f agoff * k ag~f}kRe B1 ( m f, mfk ) f
k=l 2
a~°2;w ag°,~<~iwReBj (m~<2 , Mw)
+2 Z k=l 4
+ Z a~°2°z a~°~°z ~u)at ReB1 (m2o, Mz) k=l 2
+ Z a~?22n,~,ag°,~:H+Re Bl (m2;, MH,+' ) k,n=l 4
1
+2 Z a;°j<°n°,,ag%?°H,°,Renl (myc°t' MH°,,) ;
(D.27)
k,n=l
is obtained from .sol by replacing the couplings ago.., with bg0.... The ~ss(P 2) correction is given by 2
16~"2 .~ss~;(p 2) = 2 Z
Z Nf b~°,IA ag°ffk mfRe Bo(mf, mfk )
f
k=l 2
-+w a},o ~-jx, - my<2 Re Bo (m22, Mw) -8 Z b*0 ,/,;xk -+w k=l
50
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67 4
- 4 Z b*o -o~ a.r.o:oz m~.o Re Bo(mko, Mz ) qti X,~L ~j a',~ ak k=l 2 b ~ ° ~ H ~ a,~,o -, t4~ m , ~ R e B 0 ( m , ~ , MH,+' )
+2 E
wi Xk n
k,n=l
wj Xk .
4 b,~;o ~ouo a.r,o r,Ono,, m :,~k R e B o ( m : , ,'k MH..o) •"iak",, "j~k
+ Z
Q
(D.28)
k,n=l
The chargino mass corrections are given by similar formulae, 2 2 + 2 1 167r X L i j ( P ) = 2 E
, E NTa(~+f?~ a~fff~ f
Re
B, (mf, mf~)
k=l
4
+ Z ax°CW a*°(,;wReB' (m2o, Mw) k=l 2
+ Z a~/~;z a¢2~zRe B1 (m~(~,Mz ) k=l 2
+ Z a;+2~, a-+ -+r Re Bl (m~'1' 0) O~xk k=l 4
1
2
+2 Z E a~°k¢:H,+,as(°¢';H,,+Re B, (m2o, MH+' ) k=l n=l 2
1 +~ ~XT"~
4
a*+,,X,~'+H0,a.z+~j , *k~+H°Re,, B1 (m~2, MHO,), ;
(D.29)
k=l n=l
X~(p 2) is obtained from X+(p 2) by substituting a(,:.., with bff,~.... X~-(p 2) is given by the following formula: 2
16~rZX+ii(PZ)=ZZ * ff~ asf ff~ mfRe Bo( mf, my; ) • N¢f b~,: f
k=l 4
-4 E b*k°d,/wa~o(,]w m~o Re Bo (m2o, Mw) k=l 2
-4 ~
b*,-. . . . a.~+~+ z m~,~ R e B o ( m 2 2 , M z ) gti Xk L ~'j ,~k ak
k=l 2
-4 E b~.)~+r *,r, a.r,+~+,,m~+ReBo(m~,O) ~.j,,~, ,,k ,,k k=l
D.M. Pierce et al. /Nuclear Physics B 491 (1997) 3-67
51
(D.30)
In these expressions, $ff
couplings
the color factor NJ is 3 for (s)quarks,
are listed in Eqs. (D.20)-(D.23),
are given in Eqs. (D.7), from the following
(DJ),
equations,
(D.12)-(D.14).
and 1 for (s)leptons.
and the &jZ We determine
The
and $iW
couplings
the $+j+y
couplings
which apply for incoming 1::
where we write the chargino-chargino-photon We next list the if-Higgs-boson couplings.
Feynman rule as -iy,( aPL + bPR). We write these couplings in the unrotated
Higgs basis (st , sz), (~1 ,p2), and (ht, hz). These fields are rotated to obtain the mass eigenstate fields. The (H, h) rotation is given in Eq. (A.14)) while for (Cc’, A) and (G+, H+) we have
We write the Feynman
rules for the j”josk
as (aPL + bPR) . These couplings
couplings
as -i(aPL+bPR)
and for j”topk
are symmetric under i H j and satisfy bG;lc,oS,= aJlpqyPst
and b&oq,,,k = -a$p+oyk. The non-vanishing 1I 1I g’ - agp$;,, = qjpps2 = 7j- ,
a-couplings
a$:6:.s, = -agp:,,
are g = z ,
(D.33) (D.34)
The couplings ajp,+,
to incoming neutralino
mass eigenstates
,$ are
= Ni;, Nj; a+;$+,, 7
b-0 -0Sr= Nik Njl b~;~~s,, , x, x, and likewise for pn couplings. by rotating these couplings,
(D.35) The couplings
to Higgs-boson
and likewise for the b-couplings. We write the Feynman rules for the f+j+--neutral-Higgs for couplings with CP-even s-fields, and (up, + bP,)
mass eigenstates
are found
couplings as -i( aPL+bPR) for couplings with CP-odd p-
D.M. Pierce et a l . / N u c l e a r Physics B 491 (1997) 3 - 6 7
52
fields. These couplings satisfy b~? ~.,, = a6j~6, s,, and bd;, 4~ p,, = -ad) (,;>,. The non-zero a-couplings are g
a('i' 0~2s~ = a67~4'i' .32 = a~b~(,~ t,~ = - a ~ 4+ m = ~ "
(D.37)
The couplings to incoming X(+ are obtained from these as follows:
a2/2) ,,,, = Vi~ U)*ta6~4;s,,'
b2) 2~.,.,, = Uik Vjt b(,~? ~,, ,
(D.38)
and the same rotations apply for the p,-couplings. To find the couplings to Higgs-boson mass eigenstates, we rotate these couplings by the angle c~ or fl, just as for the ,~'°~'°s and ,~0~0p couplings in Eq. (D.36). The )~'°)~,+-charged-Higgs-boson vertex Feynman rules are written as --i(aPL + bT~n), where, for incoming ~o, we have
g' a&°(,2' h~ = bg,o&j h~ = - ~ '
g a&o(,j h~ = b&°6]h~ = - - ~ ,
a,r.o,z tq = -b6°4~ h~ = - g "
(D.39)
To obtain the couplings to chargino and neutralino mass eigenstates with an incoming neutralino .~0, we rotate these couplings as
a~',~2~h,', = Ni*k U/*l a(,o(,/h;l '
b.-,~-~,~j ~, h',, = Nik Vii. b(,°~l h;', ,
(D.40)
while for an incoming chargino ..~.+, we rotate them as
a-o-~ XiXj h',, = Ni*k V~" .1l b-o O~l-~ h ,~, '
b~o~,h+ . x i x ) ,, = Nik Ujla,~,o,;,,h~ .r~.r~
(D.41)
To find the couplings to charged-Higgs mass eigenstates, we rotate both a- and bcouplings by the angle/3,
(a2°2'~) a2,,2,14-,
:(
c# s/3) ( a ~ ° ' + / q ) - sfl c# a2"2~lq
.
(D.42)
D.4. Gluino The gluino self-energy appears in Refs. [ 8 - l l ]. The physical gluino mass satisfies m~, = M 3 ( Q ) - Re 2;g,(m~) ,
(D.43)
where X~(p2)=l--~ 2 -m~
15+91n
+ZZm~Bl(mq, q
q
where (2 is the renormalization scale.
i=1
mo,)
D.M.Pierceet al./NuclearPhysicsB 491 (1997)3-67
53
D.5. Squarks and sleptons We find the sfermion masses by taking the real part of the poles of the propagator matrix Det [p/2- 3,t~(p/2)] = 0 ,
m~i = Re(p/2) ,
(D.45)
where M}(p2)
=
( M2I I,. -Hf'f'~(p2) M2. _ Hf~f,.(p2) flefl,
M2"M 2- . fRfR
)
- u},t.(P2
lIf, fR
(p2)
.
(D.46)
The matrix formalism allows us to determine the one-loop masses and mixing angles for arbitrary tree-level parameters. In this expression, the Mf, f, (i,j = L, R) are the DR tree-level mass matrix entries given in Eqs. (A.5), (A.6): all the entries contain running DR parameters at a common scale Q. In particular, the DR tree-level matrix contains corrections from the replacements M2z --+ M2z = M2z + Re Hrz (M2z) and mf --~ fnf = mf + ReXf(m2f). (The arguments of these self-energy functions are external momenta, not the scale Q.) The lI], L, (i,j = L, R) are the sfermion selfenergy functions evaluated at the scale Q. Of course, for the first two generations of sfermions, both the tree-level and one-loop contributions to the off-diagonal elements of the mass matrices are negligible. Note F~,},. ¢ Hf,,}~ because of the absorptive part, which contributes to the mass-squared at O(~2). For a t'c squark we have 167r2HhjL (p 2 )
= 4g23[2G(m~,m,)+c~F(mh,O ) +s:F(m~,,O). +c:Ao(mh)+s:Ao(m~)l. +Af(s2tAo(mh) +c~Ao(mh) ) +A~(s2Ao(mb~) +c~Ao(mb2))
,4(
2)
+~ ~-~ a~ D.. - g202 g'---LG Ao(m.o,,) n=l
4
n=3 4
\ 2~2 2
.... 2
q- Z ~~( An°h-?,)aB{'(mH°'m?,) + Z ( AH,+,~,.b,)2Bo(mb,'mHd) n=l
i=1
i,n=l
q-4g2~--(gtt.)2AO(Mz)+2g2AO(Mw)+(ete)2(c~F(mh,O)+s~F(mh,O))
+g=j(gtt)2Ic~F(mh , Mz) -t-s~F(mh,Mz) ]
D.M.Pierceetal./NuclearPhysicsB491(1997)3-67
54
+-~
+ s2,ao(mf~)
ftf (cfAo(mf,)+s~Ao(mf~)) 2 +g2~f NJcI313 +gl-~-4(YtL)2(c2tAo(m~,)+s~Ao(m~2)) +gZ~--~YtL~fNf[YfL(c2fAo(mf,)+s}Ao(m,2)) +YfR(s2fAo(mfl) +c~Ao(m?2))l 4 +i~__l[fit~LLG(my~°,mt)--2git?L,.m2°mtBo(m~°,mt) 1 + Z fib~L,G(m~i'' m~) - 2 gib?LLmL~mt,Bo(m~i',mb) ,
(D.47)
i=1 and similarly for a tR squark, 16~-2 HTR~R(p2)
4g~
2G(m~,m,) + sZ,F(m~,,O) + c~F(mT~,O) + s, Ao(m~) + c2,Ao(m~)
3 +AZt(c2tAo(m~,)+s2Ao(m~2)+c2bAo(mb,)+s2Ao(m~,2)) 4 g2gtR +.~-~(A2Dnu_' g2gt~cn)Ao(mHo,)WZ(A2Dnd_l_--~TCn)Ao(mH ,,-2) n=l 4
n=3 " 2
2
+ Z Z (AH%?i)2B°(mH°'m?,) + Z(AH,+?,b,)2Bo(mb,,mH+) n=l i=1 i.n=l 4g2 +--~(gt~)ZAo(Mz) + (e,e) 2 ( s~F(mT~,O)+ cZF(m~2,0)) g2 (gt,~) 2 [s~F(mTl q--~-~ ,Mz)
+c~F(m~z'Mz)]
gt2 +gt--~4Ytk~fNf[Yf,.(c2fAo(m?,)+s2fAo(m,z))
D.M. Pierceet al./NuclearPhysicsB 491 (1997)3-67
55
+Yf~(s2fAo(mfl) +c2fAo(mf2))] 4
-+-~=l[fit?RRG(m2o,mt)--2giffkRm2°mtBo(m2°,mt)] + Z2 [[fib~RRG ( m~)~,mb ) - 2 gib?Rnm2; mbBo ( mL', mb ) ] .
(D.48)
i=1
The off-diagonal self-energy is 16rr2Hh~ (p2)
= 4g~I4mgmB'°(mgm' )t3 4
+stct(F(m?,,O)-
2
F ( m T 2 , 0 ) - A o ( m ? , ) + Ao(m~,))]
2
+ Z Z "ln%7,ata°,,~R~,B°(mH°,,'mTi) + Z at4,tbS,hH,~?,bflO(mb,'mH,+,) n=l
i=1
i,n=l
gt2
/"
.
'+-@~f NfAuS2o,,(Ao(m~l)-Ao(mr4,.))-f---~-Yt, Ytkstct~Ao(m?,)-Ao(m?2) ) +(ete)2stct(F(mil,0) -F(m?2,0) ) g2 gt~gtt¢stct (F(mh,Mz)_F(m~2,Mz)) 4
+~'~[fit[rRG(m2o,mt)--2giffm, m2omtBo(m2o,mt)] i=1
[ + Z2 [fib~.,G(m~i~' mb) -- 2 gib?LRm2{mbBo(m2/, mb) ] •
(D.49)
i=1
Inside the sum ~ f , the sub- or superscript f refers to (s)quarks and (s)leptons, and in the sum y~'
f it?t., = ax~,,,?,,a2°,tTR--k b~& b,oaR , giff~.g= b*2o& a2otTk + a,off,, * b2offR ,
(D.50)
fn, b~ - ,, hrz.a~/b~ + bk~' h~,.bydbr~ , .
--
a*
-- b*
*
*
(D.51)
56
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
with analogous definitions for the LL and RR couplings. The 2 f f couplings are listed in Eqs. (D.20)-(D.22). The Higgs bosons ~ refer to H, h, G°, and A, and H + refer to H ÷, G ÷. The H , H,,-sfermion-sfermion couplings involve C, and Dnf , and are given in the following table:
n
Cn
l
--C2a
2 C2a 3 -c2~ 4 c2~
Dnu
Dnd
2 Sa 2 Ca
2 Ca 2 Sa
s~ c2~
c~ s~
(D.52)
We write the Feynman rules associated with the CP-even-Higgs-sfermion-sfermion vertices as -iA, and list the couplings Asff in the following table: SI
$2
~L~L
gMz ---~---gu,C~
- ~gMz c guLS/3+ v/2aumu
fiRfiR
gMz e 'guRC#
- ~gMz c guRs#+ v/-2Aumu
hu
UL~R
A~ A.
"~ ~
V~
(D.53)
dtdc
gMz
g&c/3 + V~Admd
gMz ----'~--C gdLS#
dRdR
gMz gd~CB -F V ~ Admd
gMz gdRSfl --~-ff'--
-
Ad
-
dLdR
ad
v~Ad
"-'~#
We find the couplings in the fl,2 sfermion basis via (D.54) /~s,,./r2f, /~s,,f2f2
--Sf
Cf
\AS,,fRfL
/~s,,ykfg
Sf
Cf
'
we obtain the couplings in the mixed fL,Rfl,2 basis by omitting the left-most matrix on the right-hand side of the above equation. The couplings to the CP-even Higgs-boson eigenstates (H, h) are obtained from the couplings to (sj, s2) using the rotation
"~hf,ij
-s,~ c,~
A~2i,ij
'
(D.55)
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 The couplings ,t6.0i,fj and
/~Aflfj
=
57
"~Afifj vanish for i = j, while for i 4: j they satisfy
--/~Afyfi" We write the Feynman rules for these couplings for incoming fL as
A. They are 6~
A
+ a.s.)
dLdR
V5
A,I (IZS~ + Aac~)
(D.56)
-
ha (/zc# - Ads~)
v5
We obtain these couplings in the fc,Rfl,2 basis by a rotation as described after (D.54). We also write the Feynman rules for the charged-Higgs-sfermion-sfermion vertices in the form -i,~. The couplings Ac, Y,d~ and an~ ~,~; are
G+ F~LdL
gMw v/~ce# -- Aum, SB + AdmdC#
~RdR
H+ gMw ~ $213 A,m,,c# - ,tdmas~
0
- -
-- A,,mdcB -- Admus B
fiLdR
Aa (~S~ + Zac~)
Aj (lzc~ - A~Is~)
tTRdL
-- Au (t.zc~ + Aus~)
h. (#s~ - auc#) (D.57)
These couplings are obtained in the fl,2 basis via
( Att~'d' '~n~'d2 ) = ( cu Su ) ( AH+r~"dL/~t4+~LdR) ( cd -sd ) AH,g~2dI l~H+Fted2 --Su Cu /~Ht~t~dtI~H+~RdR Sd Cd "
(D.58)
We obtain the mixed sfermion basis couplings to ~L,Rdl,: (fil,2dL.R) by leaving off the left-most (right-most) matrix on the right-hand side of the above equation. The expressions for//b,b: are obtained from//~,~j by interchanging the indices t ~ b, replacing u ---, d, and substituting c~ +--* s¢~. The self-energy of a charged slepton (sneutrino) is given by a formula similar to that for a b-squark (t-squark), with the SU(3) correction set to zero and with the appropriate SU(2) x U ( I ) quantum-number substitutions.
D.6. Higgs bosons The full one-loop MSSM Higgs-boson self-energies appear in Refs. [6,7]. Corrections to the Higgs-boson masses are the subject of Refs. [36,37]. We discuss the relations between the self-energies and the pole masses of the Higgs bosons in Appendix E. Here we list the self-energies. The Higgs-boson contributions to the Higgs-boson self-energies involve the trilinear and quartic couplings, which we denote .~tf,o,:4o~k,Au,,' • ..... , and AH,,H,,H.,N°,,AH,,U,,U~U,, ,
58
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
where the ~ refer to the H, h, G°, and A Higgs bosons, and the H + refer to the G-and H + Higgses. The GO and G + are the neutral and charged Goldstone bosons, which in the 't Hooft-Feynman gauge have masses Mz and Mw, respectively. For the two CP-even Higgs bosons, we have 16¢r2/Ts, s, (p2)
4m2d)Bo(md, md) -- 2Ao(md)
+
Ao(md~) + Ao(m&)]
Nmr (c>+m.~,~+s>o,,m::,) f 2
+gf,(s2fAo(mfi)+c2fAo(mfz))l+ZZN{h:if~?:Bo(m?i.m?:) f
+
+c~[2F(Mw, Mw) +
i,j=l
"<.ma,,"'.'l,...,2
F(Mz>Mz)] -~ j}
7 2 2[
-~g c13 2M2Bo(Mw, Mw) +
MZBo(Mz'Mz)] ~2
+g2[2Zo(Mw) + Ao(Mz) ~-----~] ' mHo" ) + ,~HOHOslslAO(mH~) 1 + 1 4 [ D .~2HOHO s' Bo ( mHo. "2 = _ ".,
-'~
[Z
,'/=1 Lm=l
+~ ~
A 2H+H£ sl B O ( m H + , m H + ) +
t~H,+H,.s,,,Ao(mn+)
f~Js,,G(m~?,m~7) - 2g°is. mem~TBo(mx.7,ms(~)
i,j=l
+ ~_.[f~.,G(m~.+,,m.~ - 2 gijsN + m~, m~f Bo (m2)~,m2j ) ,
(D.59)
i j=l
16~'2IIs2s2(p2)
: Z'~:c"~[<; -4m>'o
2C 2
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
59
2
+gf~(s~Ao(mji)+c~ao(mj2))]+ZZN{~2f~fjBo(mj~,m]j) f i,j=l
F(Mz, Mz) +Zg2s~4 [2M~Bo(Mw, Mw) + M2zB°(Mz'ao Mz)] +g2[ 2A°(MW) + Ao(Mz) ~-------5~] ~ 4 q_~1
2 [~-~ a,o..o BO(mHo,mHo) + /~HOH%~s.Ao( .... ..... mHo ) ]J *~n r a m $2
n
m
n=l Lm=l 2
+~
I...d
2
2
[~--" A. . . . . ~
£1 n 1"1m S 2
Bo(mn4, mH,;) -kn
n=l Lm=l
]
AH;~,H,Ss2szAO(mH,I) ]
,4[
1
i,j=l 2
+Z[f-i]:s2zG(mu,m,+)-2g+s22m,+m,]Bo(m~),m,j~) ]
(D.60)
i,j=l
2
16"trZ/ls,s2(p2) = Z Z
Nf'~s,f, fjAszi,YjB°(m], ' mfj)
f i,.j=l
÷~g2s/3c~{2F(Mw, Mw) - 2F(mH,~,Mw) q
F(Mz,
Mz) - F(ma, Mz) ~2
+7[ 2M2B°(MW'MW) + M2zBo(Mz, &2 Mz)] }
1
q'-~
AHOHO~,.,", , '~HOHOs-BO(mH ., ~ ,, mH°.,) + AHOHOs~szAo(mH~,,) 2
+~
2
i,j=l L
o 2 mx,omgBo(m~o, s,2G(ms°,mg) - 2gijs, , • , m~)
]
60
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67 2
+Z[ft+s~,G(m2i,,myG,)-
(D.61)
2 &is,~ + m2,~ m~f Bo ( m ~ , my~f ) ] ,
i,.j=1
where N f is the number of colors, which is 3 if f is a (s)quark and 1 if f is a (s)lepton. The neutral current couplings g f are defined in Eq. (A.7). The j'i~'.j-Higgs couplings fij+,k~, gim,,+ fq.~, and gijs~,° are defined by f i.#'k, = a*2,2jsk a2~2Js, + b x~2jsk bk,2:s, ,
(D.62)
g~#~l = bx~2j.~ka2~2jsl + a2~2jsk b2,2js~ ,
and the a2~2j,k and b2,2js k couplings are defined in Eqs. (D.33), (D.35)-(D.38). The Higgs-squark-squark couplings .tt4?~f~ and Ah?jfk are given in Eqs. (D.53), (D.54). We write the Feynman rules for the relevant quartic Higgs couplings as -iA, and define AI4,,H,,H,,H,,, = g2/(4O2)-XFt,,H,,H,,I4,,. We list the necessary ~n,,H,,H,,/t,, couplings in the following two tables: SISI
HH hh G°G ° AA G+G H+H -
$2S2
3s~ -3c] -
3c~ - s a2 3s] - c a2
S1S2
C2a S a2
C2fl
--C2~
-c2~
c2~
--S2a S2a 0
0
~2-.}-~2C2 ~
~2--j2C2B
--~2S2fl
e2-~2c2~
G+g2c2B
~2s2# (D.63)
G°G° AA G+G H+H
AA
H+H -
3s2~ - 1 3c2~ e2(l + s~#) - g2c~#
~2(1 + s~#) - ~2c2# c2# 2s~# - 1
-
(D.64) For the couplings involving (sl, s2), we obtain the corresponding couplings in the (H, h) eigenstate basis by the following rotations:
AH,,H,,Hh An,,n,,hh J
--Sa Ca
\ AH,,H,,sb~2 An,,n,,s2s2
Sa
Ca
We write the Feynman rules for the trilinear Higgs-boson couplings as -iA, and define ,tH,,N.,,, = gMz/(2O)-XH,,H.,s,. We list the AH,,Ho,s, in the following two tables:
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3 - 6 7
HH s,
c ~ ( 3 c ~ - s ] ) - s~s2~
s~
s~(3s]-c])
G°G°
- c~s2~
AA
G° A
61
hh
Hh
s~(3c~ - s~) + c~s2.
-2cfls2a -- sflc2~ 2s fls2a -- c flc2a
G+G -
H+ H -
(D.66)
G+ H -
sl c2~c~ -c2~c~ -s2#c# c2~c~ -c2~c~ + 2,~2ce --S2flCfl "3c-6"2Sfl $2 --C2flSfl C2flSfl S2flSfl --C2flS,8 C2flSfl÷ 2~2Sfl S2flSfl-- C2Cfl
(D.67)
To obtain the couplings involving (sl, s2) in the (H, h) eigenstate basis, we rotate the (sl, s2) couplings by the angle a, as in Eq. (D.55). The CP-odd Higgs boson A and charged Higgs H + self-energies are
N c A. p2Bo(m.,m.) - 2Ao(m.) f,,
(
f,,
AucB - -~I3gLc2B
)[
l
c2uAo(m~,)+ s2uAo(m~2)
2 2 -~ g2iu3gRc2B] u "~ [s2ua°(m~l) + c2uAo(m~:)] + Z Nf ( AucBf. +Z
~
f,,
N[A~.Bo(m~,m~j) +
i.j=l
g2 s~B cab .~. +--~ 2 F ( m m , M w ) + -~-F(mH, Mz) + - ~ - r t m h , M z ) +~
A2AHOfl~,Bo( m~o. , mHo,) + .~AAH~No,Ao( mHo,) n=l
-
g2 M 2 0 , +-----~-ootMw,
+~ mH, )
) ,AAAH,~H,sAo(mH,+
n=l
Ao( Mz) +g2[2Ao(Mw) + d------U---] +-2
AG(m2°'m~°) -- 2gO/A X~ X:
,ri ' m2°)
i,j=l t. 2
÷ Z f f i + A G(m~i~,m~+) + J - 2 giiam~':m~'~B°(m~':~'m~ )1' i , j = l t.
•"
"LI
''3
"'1
"'J
(D.68)
D.M. Pierceet al./Nuclear PhysicsB 491 (1997)3-67
62
16rr2//H+H- (p2)
2 +AdS# 22 )G(mu'md) -- 2AuAdmumdS2~Bo(mu,md) Njcf [ 2(~uCp
=Z
]
f,,/f,! 2
f 2 ZNc'~,+~,djB°(mu"mdi )
+ Z
.f. / f ,t i,.]=l _k_{~f. g[ (l~2dS2a ~13gLC2a+-2C2[t) g2 u u g2 [c2uaO(m~l)+ s2uAo(mft2)l +ZN
[
f
(
22a -auC
)[
](
gURC2a s2Ao(m~l) + c2Ao(m~2) +
sau~-+d +--+ca
)}
f,, g2 C2A +---~ s2aF(mH, Mw) + c2aF(mh, Mw) + F(mA, Mw) + -20w ~z F(mH+, Mz)]j g2c2^ +e2F(mH, ,0) + 2g2Ao(Mw) + ~ A o ( M z )
2
C _2M2
q- Z ~H.~ 2 HOH~,Bo (mHo,,,mH[' ) + ~L~.W__Bo(Mw,mA) /?,n/=l 4 2 I q--2 Z ~-n~H-H~.H° ao(mHo) + Z ~'H+H-H+H,SaO(mH+) n=l n=] 2 4 IfijH+a(mx'i' 'm;d) - 2 gijn, m•+, m•oBo(mx,+,m•o) ] , i=1 j=l
+Z Z
(D.69)
where c,,a (s~a) denotes cos(a-/?) (sin(a-fl)). The gft., gfR are defined in Eq. (A.7), and the 1f are listed in the table of Eq. (A.8). N[ denotes the number of colors, which is 3 for a (s)quark. The )(i)(jA couplings f,~+A, gija, + fijA' and gOijA are defined by
fZjA=la,,,,AI 2 +[ ,i.y(jAI ,
gqA=2Re byci2jAa2~2jA ,
(D.70)
and similarly for the fijH', gijn+ couplings. The a2,~ja, bT~2ja couplings are given in Eqs. (D.34)-(D.38), and those of the charged Higgs are listed in Eqs. (D.39)-(D.42). The Higgs-sfermion-sfermion couplings AA]jA and an+fjA are given in Eqs. (D.56)(D.58). The fL,R basis couplings aA]/ of Eq. (D.56) also apply in the fl,2 basis.
Appendix E. One-loop Higgs-boson masses In this appendix we will present the formalism necessary to obtain accurate Higgsboson masses at the one-loop level. The tadpole diagrams play an important role in determining the masses. The one-loop tadpole contributions are listed in Refs. [6,48].
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
63
At any given order in perturbation theory, minimizing the scalar potential is equivalent to requiring that the tadpoles vanish. At tree level, we have T1 = T2 = 0, with the tadpoles given by the DR relations T1 = ~1 a~/2 z c 2/~+m 2 , + /z2 +B/xtan/3,
(E.I)
T2
(E.2)
~ l~,12zC2B+ m2H2_k_tz2 + Btzcot /3 '
U2
where the Higgs-sector soft supersymmetry-breaking potential is Vsoft = m2n, ]Hll2 + m22 IH212 +
(Btz,ijH{H ~ + h.c.) .
(E.3)
At one-loop level, the total (tree-level plus one-loop) tadpole must vanish, so Ti - t~ = 0, 7"2 - t2 = 0, with 2
167r2v~-~= - ~
i , f eYltSlfifi A ~" 2Nf A2A°(ma) + Z 2-a'Vc ~ o,,,,],)
( g2ca~8o f~t
2
f
i=1
Ao(ma) +2Ao(mM+)
-
-
)g2 + TAo(mH+)
tanB/Ao(m~)
4
-- i~=l g MwCl ~ e N/3(Ni2 -
- )_..£ x/2g ~ R e
[V/1Ui2
Nil tan0w)
]
Ao(m~)
Ao(msc+)
i=1
+ 3T g z ( 2 A o ( M w ) + Ao(Mz) - - )e+2 - - - ~ - ~ g2c2~ /2 Ao(Mw)+Ao(Mz )) (E.4) and 167r2 t2 U2
2 _q._y 2 Z2 ~'y g~s2H_____~,~_ = - Z 2Ncf AuA°(mu) , _, f,, f i=l ~v~ 2MwsB~aotmfjl
+g2c2~(Ao(ma, ~ + 2A0(mH+,) q--~Ao(mH~ ) + - ~ 3c2 -- s2 -4-s2acot/3 mo(mh) - c, -
cot/3) A0(mH)
64
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
4
m'°
[
+~--~.g2~Re lvl w s 3
-
m
v~g i=1
]
Ni4(Ni2 - Nil tant~w)
Ao(m~o)
El
Re Vi2Uil A o ( m ~ ? )
T
~2
8~2
'
(E.5) where N f is the number of colors, 3 if f is a (s)quark and 1 otherwise. The A0 function is given in Eq. (B.5); ~ denotes cos 0w and c~ = cos/3, etc. The matrices N, U, and V are described in Appendix A and the couplings A,,f,~? As2Y,fj are given by Eqs. (D.53), (D.54). The DR (tree-level) CP-odd Higgs mass is given by rh2a = -B/x(tan/3 + cot/3), and Eqs. (E.1), (E.2) allow us to solve for the DR parameter, /x2, and the pole mass 12, mA
2:1 [tan2 ( m 2A. c_~. (--2 m H2 .
cot )
N . 2)
M2z
Re H zrz ( M 2z ) - R e n a a ( m 2 ) +
ba,
(E.6)
where ~ 2 = m~, - t , / v l , -m~2 = m22 - t2/v2. The self-energies H r z and I1AA are given in Eqs. (D.4) and (D.68), respectively, and b A = $213t l / U1 "~- C2.13t2 / U2. Having determined the physical CP-odd Higgs-boson mass mA, we are in a position to compute the remaining Higgs masses. The physical mass for the charged Higgs boson H + is
E
T 21 ,
m2~ = m 2 + M~v + Re HAA(m~) -- HH+H- (m2+) + H w w ( M w )
(E.7)
where the W-boson self-energy is given in Eq. (D.9), and the charged-Higgs self-energy in Eq. (D.69). The CP-even Higgs-boson masses are obtained from the real part of the poles of the propagator matrix, Det [p,21 - . A d s 2 ( Pi)] 2
=
0,
m 2 = Re(p2),
(E.8)
where the matrix .Ad2(p 2) is
12In case mA is very close to M z , there is an additional O(ot) correction to ma from the off-diagonal element of the CP-odd mass matrix.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
65
.A42( ,~ P 2) ( IVI2 c2B + rn2As2/3-- Hsls, (p2) + tl/V, \
+
- ( ~42z + en2 )s~c~ -/zs,,2(p
- a,2 ,
+
-- IIs2
p2
2) '~
+
(E.9) In this e x p r e s s i o n , ~ / 2 a n d &2z are the Z - and A - b o s o n D R m a s s e s (/(/2z = M ~ +
ReH~z(M2z),
&2 = m2A + R e H a a ( m 2) _ bz). T h e self-energies Hs, sj are g i v e n in
Eqs. ( D . 5 9 ) - ( D . 6 1 ) .
A t o n e loop, t h e a n g l e a d i a g o n a l i z e s the m a t r i x A,42(p 2) for
s o m e c h o i c e o f m o m e n t u m p2; w e c h o o s e p2 = m 2h .
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Nuclear Physics B 491 (1997) 3-67
Precision corrections in the minimal supersymmetric standard model* Damien M. Pierce a, Jonathan A. Bagger b, Konstantin T. Matchev b, Ren-Jie Zhangb a Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA b Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA
Received 4 June 1996; revised 13 November 1996; accepted 26 November1996
Abstract In this paper we compute one-loop corrections to masses and couplings in the minimal super° symmetric standard model. We present explicit formulae for the complete corrections and a set of compact approximations which hold over the unified parameter space associated with radiative electroweak symmetry breaking. We illustrate the importance of the corrections and the accuracy of our approximations by scanning over the parameter space. We calculate the supersymmetric one-loop corrections to the W-boson mass, the effective weak mixing angle, and the quark and lepton masses, and discuss implications for gauge and Yukawa coupling unification. We also compute the one-loop corrections to the entire superpartner and Higgs-boson mass spectrum. We find significant corrections over much of the parameter space, and illustrate that our approximations are good to 69(1%) for many of the superparticle masses. (~) 1997 Elsevier Science B.V. PACS: 11.30.Pb; 12.10.Dm; 12.10.Kt; 12.15.Lk; 14.80.Ly Keywords: Supersymmetry;Radiative corrections;Precision measurements;Grand unification
1. Introduction Most precision measurements of electroweak parameters agree quite well with the predictions of the standard model [ 1 ]. These experiments rule out many possibilities for physics beyond the standard model, but they have not touched supersymmetry, which evades precision constraints because it decouples from standard-model physics if the * Work supported by Departmentof Energy contract DE-AC03-76SF00515and by the U.S. NationalScience Foundation,grant NSF-PHY-9404057. 0550-3213/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. Pll S0550-3213 (96)00683-9
4
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67
scale of supersymmetry breaking is more than a few hundred GeV. Definitive tests of supersymmetry will probably have to wait for direct searches at future colliders. Once supersymmetry is discovered, a host of new questions arise. For example, one would like to test the low-energy supersymmetric relations between the particle masses and couplings by making precision measurements of the supersymmetric parameters. One would like to measure the supersymmetric masses as accurately as possible to shed light on the origin of supersymmetry breaking. Furthermore, one would also like to know whether weak-scale supersymmetry sheds any light on physics at even higher energies. Indeed, the successful unification of gauge couplings encourages hope that other supersymmetric parameters might unify as well. It is important to measure these parameters precisely at low energies so that they can be extrapolated with confidence to higher energies. It is in this spirit that we present our calculation of one-loop corrections to the minimal supersymmetric standard model (MSSM). We define the MSSM to be the minimally supersymmetrized standard model, with no right-handed neutrinos, and all possible softbreaking terms. We believe that this minimal model provides an appropriate framework for analyzing the phenomenology of supersymmetry and supersymmetric unification. We approach our calculation in the standard fashion associated with precision electroweak measurements. We take as inputs the electromagnetic coupling at zero momentum, O~em= 1/137.036, the Fermi constant, G~ = 1.16639 × 10 -5 GeV -2, the Z-boson pole mass, M z = 91.188 GeV, the strong coupling in the MS scheme at the scale Mz, a . ~ ( M z ) = 0.118, as well as the quark and lepton masses, mt = 175 GeV, mb = 4.9 GeV, and mr = 1.777 GeV [2]. From these inputs, for any tree-level supersymmetric spectrum, we compute the oneloop W-boson pole mass, M w , as well as the one-loop values of the effective weak lep t mixing angle . . tin2 . . . t ~eft, and the DR [3] weak mixing angle, ~2. We also compute the one-loop corrections to the quark and lepton Yukawa couplings, as well as the masses of all the supersymmetric and Higgs particles. We work in the DR scheme, and take the tree-level masses to be given in terms of the running DR parameters. For each (bosonic) particle, we determine the one-loop pole mass, M 2 = hT/2(Q) - R e / / ( M 2) ,
(1)
where h?/(Q) is the tree-level DR mass, evaluated at the DR scale Q, and H ( p 2) is the one-loop self-energy. (As usual, H ( p 2) depends on Q and on the masses and couplings of the particles in the loop. There is a similar expression for the fermion pole mass.) In all our computations we include the full self-energies, which contain both logarithmic and finite contributions. The logarithmic corrections can be absorbed by changes in the scale Q. Therefore we checked our logarithmic results against the one-loop supersymmetric renormalization group equations. Since we write our results using PassarinoVeltman functions [4], some of our finite terms are automatically correct. As a further check, we verified that our corrections decouple from electroweak observables.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
5
We present our complete calculations in a series of appendices. These appendices include the full one-loop corrections to the gauge and Yukawa couplings, as well as the complete one-loop corrections to the entire MSSM mass spectrum. While some of these results are not new (the gauge-boson [5,6], Higgs-boson [6,7] and gluino [8-11 ] self-energies and the gauge-coupling corrections [ 5,12 ] already appear in the literature), we include the full set of corrections in order to provide a complete, self-contained and more useful reference. In Appendix A we write the tree-level masses in terms of the parameters of the MSSM, and in Appendix B we define the Passarino-Veltman functions that we use to present our one-loop results. In Appendix C we compute the one-loop radiative corrections to the gauge couplings of the MSSM, and in Appendix D we write the one-loop corrections to the masses. Where appropriate, we evaluate the corrections to the mass matrices to account for full one-loop superparticle mixing. This allows for an accurate determination of the masses and mixing through the entire parameter space. Finally, in Appendix E we discuss the radiative corrections to the Higgs-boson masses. The results in the appendices hold for the MSSM with the most general pattern of (flavor diagonal) soft supersymmetry breaking, t The parameter space is huge because of the large number of operators that softly break supersymmetry. Therefore in the body of the paper we illustrate our results in a reduced parameter space, obtained by assuming that the soft breaking parameters unify at some high scale. The unification assumption is useful because it reduces the size of the parameter space. Moreover, it implies certain mass relations that can be tested once supersymmetry is discovered. In addition, for any set of parameters, it allows us to determine the unification scale thresholds that are necessary to achieve unification. As we will see, the present set of precision measurements is sufficient to begin to constrain the physics at the unification scale. We implement the unification assumption by solving the two-loop supersymmetric renormalization group equations subject to two-sided boundary conditions. At the weak scale, we assume a supersymmetric spectrum, and for a given value of the ratio of vacuum expectation values, tan fl, we use our one-loop corrections to extract the DR couplings g~, g2, g3, At, Ab, and ,~ at the scale M z . We then use the two-loop DR renormalization group equations [13] to run these six parameters to the scale MGtJT, which we define to be the scale where gl and g2 meet. We require that the soft breaking parameters unify at the scale MGUT. Therefore at MGu3- we fix a universal scalar mass, M0, a universal gaugino mass, M1/2, and a universal trilinear scalar coupling, A0. We then run all the soft parameters back down to the scale M~2 = M 2 + 4M~/2, where we calculate the supersymmetric spectrum using the full one-loop threshold corrections that we present in this paper. In Section 4 we show that this scale is essentially the scale of the squark masses, and that the other scalar masses and the Higgsino mass are correlated with it as well. I Our results can be readily extended to include inter-generational mixing at the cost of additional mixing matrices.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
We require radiative electroweak symmetry breaking [14], so the CP-odd Higgs I/~1, are determined in a full one-loop calculation at the scale M#. The sign o f / z is left undetermined. We then iterate the entire procedure to determine a self-consistent solution. Typically, convergence to an accuracy of better than 10 -4 is achieved after four iterations. Once we have a consistent solution, we use the results of the appendices to illustrate the one-loop corrections in the reduced parameter space associated with unification. We display results for a randomly chosen sample of 4000 points. Our sample is chosen with a logarithmic measure in the range 1 < tan/3 < 60, 50 < Mj/2 < 500 GeV, 10 < M0 < 1000 GeV, and with a linear measure in the range - 3 M # < A0 < 3M~. (The upper limits on M0 and Ml/2 are chosen so that the squark masses are less than about 1 TeV. While larger squark masses are certainly possible, they reintroduce the fine tunings that supersymmetry is designed to avoid.) Each of these points corresponds to a (local) minimum of the one-loop scalar potential with the correct electroweak symmetry breaking, and each passes a series of phenomenological constraints: We require the first- and second-generation squark masses to be larger than 220 GeV [15], the gluino mass to be greater than 170 GeV [15], the light Higgs mass 2 to be greater than 60 GeV [2], the slepton masses to be greater than 45 GeV [2], and the chargino masses to be greater than 65 GeV [16]. We also require all the Yukawa couplings to remain perturbative (/l < 3.5) up to the unification scale, and since we assume that R-parity is unbroken, we enforce the cosmological requirement that the lightest supersymmetric particle be neutral. We derive approximations to the radiative corrections that hold with reasonable accuracy over the unified parameter space. Where appropriate, we use scatter plots to illustrate the effectiveness of our approximations. The approximations consist of two parts. First we identify the most important contributions to the one-loop corrections. In most cases these are the loops that involve the strong and/or third generation Yukawa couplings. Then we derive approximations to the loop expressions that hold over the unified parameter space. In the next section, we discuss the radiative corrections to the effective weak mix~qlept ing angle, sin 2 %ff, and the W-boson pole mass, Mw. We illustrate the magnitudes of the different supersymmetric contributions to these observables. We also discuss the renormalization of the DR weak mixing angle, ~2, and comment on the way that it affects the gauge thresholds at the unification scale. In Section 3 we examine radiative corrections to the third generation quark and lepton masses. We illustrate the different contributions and present approximations which hold to a few percent. We also examine Yukawa unification and demonstrate the size of the unification-scale Yukawa thresholds. In Section 4 we present our results for the radiative corrections to the supersymmetric and Higgs-boson particle masses. We find large corrections to the masses of the light superparticles. We compare our results with those of the leading logarithmic approximamass, m a , and the supersymmetric Higgs mass parameter,
2The light Higgs boson is similar to that of the standard model in almost all of our parameterspace, so we apply the standard-modelbound.
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67
7
tion and find significant improvements over much of the unified parameter space. These corrections are important for unraveling the underlying supersymmetric structure from the supersymmetric mass spectrum.
2. The weak mixing angle and the W-boson mass The calculation of supersymmetric contributions to electroweak observables began in 1984 [5]. Since then, the precise confrontation of electroweak data with theoretical predictions of the MSSM has been an active area of study [17,12]. Global fits to precision data in the MSSM have been performed by several groups [18]. In this section we display our results for two electroweak observables over the parameter space associated with radiative electroweak symmetry breaking and universal unification-scale boundary conditions. We extract the contributions from the various superpartners, and illustrate the manner in which the different contributions decouple from the low-energy observables. 2.1. Effective weak mixing angle
'olept• The full oneWe start by considering the effective weak mixing angle, s~ = sin 2 ~eff loop calculation is presented 3 for completeness in Appendix C. The complete result is rather involved; for now we simply say that the calculation follows the outline presented above: We take aem, G~, M z , a,. ( M z ) , and the fermion masses as inputs, and compute s~ as a function of the supersymmetric masses. Because we compute the experimental observable s~ in terms of other low-energy observables, its one-loop supersymmetric corrections decouple as the supersymmetric masses become larger than Mz. From Fig. 1 we see that s 2 is especially sensitive to light sleptons, and that the sum of the supersymmetric corrections is always negative. We did not plot the Higgs boson and first two generation squark contributions. They are negligible, less than 1 × 10 - 4 and 4 × 10 -5 in magnitude, respectively. The corrections to #-decay and the corrections to the Z-g+-e - vertex comprise the non-universal corrections to s~. The former contributes between - 3 and 1.5 x 10 -4, the latter between ± 1.5 × 10 -4, and their sum is in the range - 4 to 1 x 10 -4. With m, = 175 GeV, we find the standard-model value of s~ varies between 0.2311 to 0.2315 for Higgs masses in the range 60 < mh < 130 GeV. This is subject to an error of 2.5 x 10 -4 from the experimental uncertainty in the electromagnetic coupling evaluated at Mz, and to corrections of this same order from higher loop effects [19]. Furthermore, increasing mt by 10 GeV decreases s 2 by 3.3 x 10 -4. These predictions for se2 should be compared with the LEP and SLD average 4 [1] of 0.23165 :t: 0.00024. Clearly, the standard-model calculation agrees quite well with 3 We do not include the supersymmetricnon-universal Z-vertex contribution in the appendix. It is a negligible correction in the parameter space we consider. 4 The number quoted here assumes lepton universality.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 •
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experiment. The additional contribution from supersymmetry can lower the value of s~2 slightly below 0.2300, or about 60- below the experimental central value. However, we note that higher-order standard-model corrections, changes in O~emand mt, and other precision observables should all be systematically taken into account to delineate which regions of parameter space are ruled out by these measurements. We do not attempt such a study here. The corrections to s~ diminish rapidly as the superpartner masses become heavy. For example, if we require m ) , , m b , mh ~> 90 GeV, we find that s~ is shifted by at most - 8 x 10 -4 relative to the standard-model value. 2.2. W-boson mass
We now turn to our second precision electroweak observable, and compute the oneloop correction to the W-boson pole mass, Mw. In Fig. 2 we illustrate some of the finite corrections. The full finite correction increases the prediction for Mw by up to 250 MeV. As with s~, the contributions from Higgs bosons and the first two generations of squarks are small, less than 12 and 8 MeV, respectively. The non-universal correction is also small, between - 6 and 15 MeV. For large supersymmetric masses, the prediction reduces to that of the standard model because of decoupling.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
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.
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.
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.
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' ' V"T"T'"]'"']
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. . . .
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";4
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,
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h,,
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7O
. . . .
100
I
. . . .
200
I ....
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rnTI (GeV)
Fig. 2. Finite corrections to Mw, in MeV. ( a ) - ( d ) are as in Fig. 1.
For mt = 175 GeV we find the standard-model value of Mw in the range 80.39 to 80.43 GeV, with an error of +13 MeV from the experimental uncertainty of the electromagnetic coupling. This is subject to additional corrections of the same order from higher-loop effects [ 19]. If we increase mt by 10 GeV, we find a 65 MeV increase in Mw. From our calculations we find that the MSSM value for Mw ranges from 80.39 to 80.64 GeV. These numbers can be compared to the current world average, 80.33 ± 0.15 GeV [ 2 ]. With the current experimental error, all of the supersymmetric parameter space lies within 20" of the central value. By the end of the decade, the error on Mw is expected to be about 50 MeV. If supersymmetry is not discovered by that time, one might think that a much more exacting test could be performed. However, the limits on the superpartner spectrum will also have increased to the point where the effects on weakscale observables from virtual supersymmetry will be greatly diminished. For example, imposing the limits m~,, m~÷, mh > 90 GeV, we find that typically 8Mw < 50 MeV, and at most 8Mw = 100 MeV.
2.3. Gauge coupling unification We are now ready to study gauge coupling unification [20]. We start by computing the DR electromagnetic coupling constant, &, and the D----Rweak mixing angle, ~2, as described in Appendix C. The DR weak mixing angle is closely related to the effective weak mixing angle, s~. The main difference is that ~2 is not an experimental observable,
10
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67
so its radiative corrections involve non-decoupling logarithms of supersymmetric masses. However, once we subtract these logarithms we find finite corrections to ~2 which are quantitatively similar to the corrections to s~ shown in Fig. 1. In the context of gauge coupling unification, the finite corrections to g2 are very important [21-23]. They play a significant role in determining the required unificationscale thresholds, which are formally of the same order in perturbation theory. As we will see, precision measurements already limit the size of these thresholds and place constraints on unified models. We determine the unification thresholds as follows: we calculate the full one-loop corrections to t~ and ~2, and use them to determine the DR couplings gl and g2. We take a s ( M z ) = 0.118 from experiment [2], and apply the supersymmetric threshold corrections to fix the DR coupling g3 at the scale M z , (2)
g~(Mz) _ as(Mz) 4or 1 - Aas '
where Ac~.
s
-
Ces(Mz) _ _
2or
[~
2
-~ln
( mt ) ( m~ ) 1 Z~-~ln Mzz - 2 1 n M z - - 6 O i=1
(m 0')] " Mzz
(3)
The sum indexed by ~ runs over the six squark flavors. The constant in (3) transforms the MS coupling as ( M z ) into the DR coupling g3. We also calculate the one-loop DR Yukawa couplings At(Mz), Ab(Mz), and a¢(Mz), as described in the next section. We then run the six coupled two-loop renormalization group equations [ 13] up to the unification scale, MGUT, which we define to be the point where gl and g2 meet. At that scale we define the unification threshold, eg, to be the discrepancy between g3 and the electroweak couplings gl and g2, g3(MGuT) = gl (MGuT) (1 + 8g) .
(4)
In Fig. 3a we plot the threshold, eg, versus the squark mass scale, M 0, which we define to be M 02 = M~ + 4M~/e. From the figure we see that for a s ( M z ) = 0.118, unification requires a negative unification-scale threshold correction of between - 1 % and - 3 % , depending on the weak-scale supersymmetric spectrum. For small M 0, the finite corrections to the gauge couplings are comparable in size to the logarithmic corrections; they both decrease eg at small M o. In order for a unified model to be consistent with gauge coupling unification, it must be able to accommodate values of eg ~- - 2 % . Different unified models give rise to different unification-scale threshold corrections. In the minimal [24] and missing doublet SU(5) [25] models, eg depends only on the triplet Higgs mass [23], the same mass that enters the nucleon decay rate formulae. The maximum and minimum values of eg in these two models are shown in Fig. 3b. The two thin regions correspond to the maximum values of eg, obtained by setting the triplet Higgs mass to 1019 GeV. The two larger regions show the minimum values of eg in each model. These values are found
II
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 I
. . . .
(a) a s
I ....
r ' " T " T " T " T ' " I
'
I
'
t
~ ' '"1"
' T " T " T ' " I
(b)
.118
:
i ....
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~J
0 •
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,
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. . . .
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. . . .
[
4
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...
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. . , I ....
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[ ....
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(d) ct s = .124
112
=
"
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"
. . . . - . ~ , : . , - - ~ , ~ , - . . ~ : ~ ¢ ~ . ~ . >__.~.~..-:, ...
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0
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_
~
~
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"
'
:
-2 -
"" I
-1 . . . .
200
I
. . . .
I ....
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500 M~, (GeV)
I
1000
.
200
,
,
, I . , , , I
....
I,.,,I,,,.I,,.,k,.,L.,I
500
1000
M S (GeV)
Fig. 3. (a) The unification-scale correction, eg, necessary to obtain a s ( M z ) = 0.118, plotted versus M O. (b) The maximum and minimum eg allowed in minimal SU(5) (top two regions) and missing doublet SU(5) (bottom two regions), against M O. (c) Same as (a) with a s ( M z ) = 0.112. (d) Same as (a) with a s ( M z ) = 0.124.
by setting the triplet Higgs mass as small as possible, consistent with the bounds from nucleon decay [26]. From the figure we see that it is difficult to achieve the necessary thresholds in minimal SU(5), but that missing doublet SU(5) has unification-scale thresholds in the right range. The unification-scale threshold eg necessary for unification is directly correlated with a s ( M z ) . Increasing a s ( M z ) by 5% increases eg by about 1%, as expected from the one-loop relation 6as(Mz) as(Mz) = 2 - a.~ ( M z ) aGUT
t~:g .
(5)
We illustrate this in Figs. 3c,d, where we plot eg for a s ( M z ) = 0.112 and 0.124.
3. Q u a r k and lepton masses
The full set of radiative corrections to the quark and lepton masses is presented in Appendix D. In this section we derive approximations to these formulae, valid for the third generation.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
12
3.1. Top quark mass The top quark mass provides an important input for radiative electroweak symmetry breaking. It receives strong and electroweak radiative corrections [27,11 ]. Our approximation begins by eliminating the small electroweak corrections, setting g = g~ = 0 and At = ab = 0. We then simplify the resulting expressions by setting p2 = 0 because mt is much smaller than a typical squark or gluino mass. In this limit, the physical top quark mass is given by m t = lht(Q)
1 + - -
(6)
,
mt J
where the one-loop correction receives two important contributions. The first is the well-known gluon correction, 5 tg g2 (Amt~ =_~2[2Bo(mt, mt,O)_Bl(mt, mt,O)] \mtJ
= ~127r2 [ 3 1 n ( m ~ ) + 5 ]
.
(7)
Note that this contains a logarithmic and a finite piece; the latter gives a 6.6% contribution. 6 The second correction comes from the top squark/gluino loops,
(Amt~ mt /
g2{
=-1232
Bl(O,me, mT,) + Bl(O,m~,mr2)
-sin(2Ot)(~tt)[Bo(O,m,,m,,)-Bo(O,m,,m~2)]},
(8)
where 0t is the top-squark mixing angle, and
Bo(O,ml,m2) = - I n
(M_~)
Bl(O, ml,m2)=~1 [ --In
m2 ( M 2) + 1 + m2_ M~ In ~ ,
(M_~)
1 +~+
1 log 1_~--~+ (l _ x)--~---
(9)
O(l-x) lnx]
(10)
2 2 with M = max(ml,m2), m = min(ml,m2), and x = m2/m 1. The full B functions are written in Appendix B; the formulae presented here are simplifications that hold when the first argument is zero. In Fig. 4 we show the complete correction to the top quark mass as well as the contributions from the squark/gluino and electroweak loops. The tree-level mass is defined to be lht(mt). We see that the squark/gluino loop contribution can be as large as the gluon contribution for TeV-scale gluino and squark masses. The electroweak 5 Here and in the following, we implicitly perform DR renormalization, so the 1/~ poles are subtracted. 6 We have included the two-loop M"'S contribution Amt = 1.l lot2mt [28], which we assume is close to the DR value.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 '
'
I
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. . . .
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- ,/.:L:= .. :.. .2,-:~:.:"::~:?~,.'~o'.~,"-
...., .:..:..: ........ ~ . ~ ~ . : ~ , - , = : .
. .:.~'.!i,. ~-
<:1
--4
, I ....
I
200 300
,,.I
.... I.,,,I,.,,I.,-I,,.,I.,.I
500 Mg (GeV)
i000
• I ....
200
i ....
I .... I,,,, I,, ,,I,.,,I.,,,I,...I
300 500 M~ (GeV)
i000
Fig. 4. Corrections to the top quark mass, versus M0. Figure (a) shows the full one-loop correction. Figure (b) illustrates the correction from the squark/gluino loop; the solid line shows the gluon contribution for comparison. Figure (c) shows the electroweak corrections. In Fig. (d) we plot the difference between the full one-loop result and the approximation given in the text. corrections are small because of cancellations. In the figure we also plot the difference between the full correction and our approximation. We see that our approximation is good to typically ±1%.
3.2. Bottom quark mass Corrections to the bottom quark mass in the MSSM have received much attention because they can contain significant enhanced supersymmetric contributions [29,30]. These large contributions play an important role in Yukawa coupling unification. Previous studies have included only the enhanced contributions. In this paper we present our results for the full one-loop correction. Moreover, we systematically develop approximations to the supersymmetric corrections. In this way we can see the importance of the enhanced contributions relative to the full result. The corrections to the bottom quark mass are found as follows. Because the bottom quark is light, o~,(mb) is large and we must resum the gluon contribution. We start with the bottom-quark pole mass, mb. We find the standard-model DR bottom quark mass at the scale mb using the two-loop QCD correction,
[ (Amb~bg] thb(mb) TM =mb 1 -- \ m b / J ,
(ll)
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
14 where 7 [28]
( Amb ~ bg _ 5as(mb) 3 7r mb /
+12.4 (as(rob))2 '
(12)
and c~s(mb) is the five-flavor three-loop running DR coupling. We then evolve this mass to the scale Mz using a numerical solution to the two-loop (plus three-loop O(ce~)) standard-model renormalization group equations [31]. Taking the bottom quark pole mass mb = 4.9 GeV, and ees(Mz) = 0.118, we find the standard-model DR value r~b(Mz) TM = 2.92 GeV. The final step is to add the one-loop corrections from massive particles,
lhb(Mz)=thb(Mz)SM I1-- (Amb~massive mb / ]
(13)
We approximate these corrections as follows. We ignore the small W, Z, Higgs, and neutralino contributions. This leaves the squark/gluino and squark/chargino loops, (Am--~b']massive
\ mb .I
= ( Amb x) b~ + ( Am-----~b "] '-'U . \ mb / \ mb .I
(14)
We then set p2 = 0. The squark/gluino contribution is again given by (8), with the obvious substitution t ---, b. To approximate the squark/chargino contribution, we set g = g' = ab = at = 0, except for terms that are enhanced by the Higgsino mass parameter IX or by tan ft. We simplify our expressions by setting the chargino masses to M2 and Ix, respectively. In this case the squark/chargino loops give rise to the following terms: --4
\ mb /
- 16~r "" 2 # -.5----.-5 m?l - rnf2
Bo(O, IX, m~, ) - Bo(O, ix, m~)
IX2 _ M~ +(IX +-~ M 2 ) } ,
(15)
where Bo(O, ml,m2) is defined in (9), and ct (st) is cos0, (sin0,). In Fig. 5 we show the corrections to the DR bottom quark mass, ~nb(Mz), plotted against tan/3. Fig. 5a shows the full one-loop correction, while 5b and 5c illustrate the predominantly finite corrections from squark/gluino and squark/chargino loops. At large tan/3, the top branches in Figs. 5a and 5b correspond to Ix < 0, while for Fig. 5c the bottom branch corresponds to Ix < 0. The contributions from Figs. 5b and 5c tend 7 We do not know the two- and three-loop correctionsto mb in the ~ scheme. Similarly, we do not know the DR three-loop QCD contribution to the running of the strong coupling. In both cases we use the MS values. Alternatively. we could have used MS equations to run up to Mz, then convertto DR. The difference between the two approaches, Athb(Mz ) < 0.05 GeV, is nearly an order of magnitude smaller than the experimental uncertainty in the bottom quark mass.
D.M. Pierceet al./Nuclear Physics B 491 (1997) 3-67 I
'
I
....
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'
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'
"
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15
....
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.:~-.:%7: : ::.7i;.~:':~....
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20
<~ o <~ -2o -40
: :::";,:s
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~.
4O
v
20
(b) ~ g
full .
I
....
I
,
,
I
,
'
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....
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.
ii
L .'1
";
:':
I
,
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....
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i
, .
I
'
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....
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'
"
t.
I
i
,
,
10
20
I
'
- ~.... .....
',
,
-.-.
........ .-~.~,~.~i,~: : . . . .
<~ -2o -40
(o) I
2
(d) a f t e r a p p r o x ,
I
5
....
[
i0
tanfl
I
20
i
i
50
L
I
1
2
,
I
....
5
i.
50
tanfl
Fig. 5. Corrections to the DR bottom quark mass ~hb(Mz), plotted versus tanfl. Figure (a) shows the full one-loop correction; (b) illustrates the full correction from the bottom squark/gluino loops; (c) shows the correction from the top squark/chargino loops. Figure (d) plots the differencebetween the full one-loop result and the approximation given in the text. to cancel. Because of the cancellations and large corrections, previous approximations to the bottom quark mass appearing in the literature can be substantially different from the full one-loop result. In Fig. 5d we see that the approximation (14) typically agrees with the full one-loop result to within a few percent.
3.3. Tau lepton mass The corrections to the tau lepton mass are of course much smaller than those of the quarks. After resumming the two-loop QED corrections which relate the tau pole mass to the DR running mass at Mz [31], we obtain the DR mass ~hr(Mz) = 1.7463 GeV. We approximate the remaining corrections by setting p2 = 0 and keeping only those terms proportional to g2 and enhanced by /z or tan ft. The only such terms arise from the chargino loops. They give
mr /
16rr 2 /z---g_~2~ Bo(O, M 2 , m ~ , ) - B o ( O , # , m r , , )
]
,
(16)
where Bo(O, mt,m2) is given in (9). We illustrate the tau corrections in Fig. 6. The full correction ranges from - 1 0 % to ÷6%, while the approximation is good to within a few percent. For large tan/3, the top branch corresponds t o / x > 0.
D.M. Pierceet al./Nuclear Physics B 491 (1997)3-67
16 i
r
'
I
.
.
.
I
.
I
5
'
I
'
L
'
'
1
....
I
I
'
r
I
,
10
20
I
'
I
,
~;$<.. -:
..,.,~,~..~,~:~:,:~
0 •
" -:';~".'~k~,:.
"
.
~ -5 <1 -10
''ta) full I
2
,
..... I
,,,,I
5
I
10
20
,
I
(b) a f t e r a p p r o x
,
I
50
2
,
,
I
5
tan~
....
50
tan{{ m
Fig. 6. Supersymmetric corrections to the DR tau lepton mass fnr(Mz ), plotted against tan B. Figure (a) gives the full one-loop correction; (b) illustrates the difference between the full one-loop result and the approximation given in the text.
3.4. Yukawa coupling unification In many supersymmetric unified theories, the DR bottom and tan Yukawa couplings are predicted to unify at the scale M6UT [20]. To test this hypothesis, one must first extract the running DR Yukawa couplings Ab and Ar from the DR bottom and tan masses. Our procedure is as follows. We first use the formulae in Appendix D to find DR masses for the bottom and tau. We then use the following relations to find the DR Yukawa couplings at the scale Mz: 1
tnb( M z ) = --~ ,~b(g z ) v( M z ) cos fl( Mz) , 1
Pnr( M z ) = ~ Ar( M z ) v ( M z ) c o s f l ( M z ) . x/2
(17)
We determine the full one-loop DR vev v ( M z ) from the relation
l(
M2z + ReHrz(M2z) = -~ gZ(Mz) + g'2(Mz)
)
v2(Mz) ,
(18)
where g and g' are the D--Rcouplings, and H~z is the transverse Z-boson DR self-energy. Alternatively, the DR vev v ( M z ) can be taken from the following empirical fit:
v(Mz)=
[
248.6+0.91n
Mzz
GeV.
(19)
This expression gives the correct one-loop vev to an accuracy of better than 1%. Once we have the DR Yukawa couplings at Mz, we run them to the unification scale MGUT. At that scale we define the unification threshold eb to be the discrepancy between the couplings, Ab(MGuT) ----Ar(MGuT) (1 + co) •
(20)
17
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 I
I
. . . .
I
I
'
I
'
25
w -25
• .~ ..~2;(i.v:
-50
50I'
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. . . .
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. . . .
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, .:
(c) a~=.l12 mb=4.9
~-~
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. . . .
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. . . .
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0
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(d) a,=.124 mb=4.9
" •
~k
,
I
(b) c%=.118 rob=5.2 • .-
:~.
0
~.
I
I
(a) a , = . l 1 8 mb=4.9
5O ~-~ #t
'
-'. ?...':~
.'"
~.i~,
~-" •
.... ~i~:
w -25 -..: ?-'~.~:
-50 I
I
1
2
,
I
5
. . . .
I
I0
I
20
,
,
I
,
50
tanfl
I
2
,
I
5
. . . .
I
I
10
20
~
,
I,
50
tanfl
Fig. 7. The unification-scale correction, e9, that is necessary to obtain bottom-tau unification with the given values of as (Mz) and mb, plotted versus tan ft. The bottom quark pole mass is labeled in GeV. Of course, the relation between the bottom quark mass and the DR Yukawa coupling Ab depends strongly on the QCD coupling as ( M z ) . Therefore we compute eb assuming that eg has already been chosen so that a s ( M z ) is some fixed value. Such an analysis is illustrated in Fig. 7a, where we plot eb versus tanfl for a s ( M z ) = 0.118. From the figure we see the well-known feature that Yukawa unification is possible with eb ~-- 0 for small tanfl (1.2 < tanfl < 1.7) and large tanfl (15 < tanfl < 40). In the large tan fl region we distinguish the two cases, /x < 0 a n d / z > 0. F o r / z < 0 we see that eb is always far from zero, in the range - 2 4 to - 6 0 % . For /z > 0 we find points which permit bottom-tan unification with eb ~-- O. However, they are not generic; eb depends sensitively on the parameter space, and varies between - 2 0 to +30%. The discrepancy eb is sensitive to the value of a s ( M z ) , as well as the input value for the bottom quark pole mass. We illustrate this in Figs. 7b-d, for the ( a s , m9) values (0.118, 5.2), (0.112, 4.9), and (0.124, 4.9), with mb in GeV. We note that setting a s ( M z ) = 0.112 and mb = 5.2 GeV, we find solutions with [eb[ < 0.05 over the whole range 1.3 < tan fl < 30.
4. S u p e r s y m m e t r i c
and Higgs-boson
masses
We will begin our analysis of the supersymmetric spectrum by discussing some of its general features. We will use some of these features when we derive our approximations for the radiative corrections. We will be careful to note when we do, so that one can
18
D.M. Pierce et al./Nuclear Physics B 491 (1997)3-67
lOOO
2# ~ 500
2
0
0
0
~
,ooo k
+1ooo ~
~
50
200
500 1000
2oOloo 50200
~ 500 1000
M~
/...-,~2;'~,
5oo f
"
!
zoo 200
500 1000
M~
Fig. 8. The masses (a) reaL, (b) m~z, (c) I~1, (d) ma, (e) m~+, and (f) m)?~,versusM#. The lines indicate (a) M,;, (b) M#/3 and M,7, (c) Mq/2 and 2M#, (d) M#/2 and 2M#, (e) Mq/2, and (f) MJ2 and 2Mq. The units for both axes are in GeV. assess the validity of our approximations in other scenarios. Perhaps the most striking consequence of the universal boundary conditions is the fact that they produce a low-energy spectrum which is tightly correlated with the magnitude of the squark mass scale M#2 = M 2 4- 4M~/2. In Fig. 8 we show the masses m~L, m~t, I~1, mm, m~, and m ~ versus M,;. From the figure we see that the squark masses are nearly equal to M,;, while the other masses are generally within a factor of two or three. The exceptions to this degeneracy include the Higgs boson, h, which is always light, and the additional possibilities of a light top squark, a light Higgs sector, and/or light gauginos. Of course, the gaugino masses are nearly proportional to M1/2. We find that typically mr0 -~ 0.4M1/2, m~o -~ mk~ --~ 0.8M1/2, and m~ " 2.4M1/2, but there are substantial variations from weak-scale threshold corrections (and from mixing for the charginos/neutralinos). We show the ratios m~/Ml/2 and m~/Ml/2 versus M,; in Fig. 9. In the rest of this section we discuss the one-loop corrections to the masses of the gluino, the charginos and neutralinos, the squarks, the sleptons, and the Higgs bosons. In the following expressions for the mass corrections we implicitly take the real part of the Passarino-Veltman functions. 4.1. Gluino mass The gluino mass corrections are perhaps the simplest of all the mass renormalizations. They have previously been studied in Refs. [ 8-11 ]; for completeness we list the corrections in Appendix D. The gluino mass corrections arise from gluon/gluino and
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
0.5
I ....
I ....
I''"I'"'I"T"I"T"'I
• .
:
'
-...:/....~
.....
.
'
'
I
1.0
.
.~.!5,~...i.:.~.:.~: $'...*~.:'.'.::, ~:e"~ ",:...:. 4".¢a~. . _•
¢q
~0.4
"':s~'-.:~ : .2...
~.:~-~:
I ....
•
I ' ' ' ' I ""
..
T'"I"'I'"T'"I
'
•
....,. ~..-.:.-:.. _. ~.....-: -
::,.,z...-.::.:..-..-.:.----:..-.,:--..:.-..
t'4
....
. . . .
19
:~
0.8
~
0.6
.:-...--,.
"-::.~-~" -
.:...~" o.a %.
(b) I ....
I ....
I..,.I,...I,,..I..,,I,.,.[....I
,
,
I ....
I ....
I ' ' ' ' I ' ' ' T '"P"T"T'I
'
'
0.4 3.5
1.0
I ....
I ....
I *, •, I, *, ,I,,,,I,,,,I,,,,I,,.I
,
,
I ....
I ....
I ' ' ' T ' ' '1 " " I ' " T " T ' " I
'
'
,
,
¢,/
-: -~.~: .'~#,~.~-
0.8 0.6
. ~ : ~
-
:. ..~,
~
.
-=
9.
I ....
200
The
I,...I,.,,I..,.I.,..1....I....1
500
ratios
- '.." i . ~ . . ? (
2.5
.
.
2.0
1000
.
I ....
I ....
(a)
my(o/M1/2,
I.,..I.,,,I.,..I...,I,,..i,,.,I
500
200
M~ Fig.
"
(o)
I ....
0.4
• -
3.0
i000
M~ (b)
m£o/Ml/2,
(c)
m£~/Ml/2,
and
(d)
mg/Mu2,
versus
M 0.
quark/squark loops. The corrections can be rather large, so we include them in a way which automatically incorporates the one-loop renormalization group resummation, [
m~=M3(Q)
(AM3~g~'
1- \--MT]
~ A M 3
x~qq
- \-~-3,]
]
-I
J
(21)
The gluon/gluino loop gives --~3 J
=~
[ 2 B o ( M 3 , M 3 , 0 ) - BI(M3, M3,0) ]
= 167r2 3In
+5
.
(22)
The quark/squark loop can be simplified by assuming that all quarks have zero mass, and that all squarks have a common mass, which we take to be MQ~,the soft mass of the first generation of left-handed squarks. We find { A M 3 ~ q?l = - 3g~3B I ( M 3 , 0 , MQt) \ M3 ] 47r2 Here
(23)
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
20
30
. . . .
I
. . . .
I ....
I ....
I ' ' ' '1 '"
'1'"'1'"'1
(a)
-
. . . .
I
. . . .
I ....
I ....
r
' " r " T ' " r ' " l
(b) log
full
2O lo .~
0 .
30
. . . .
. I ....
I ....
f....l....L...I....I
....
I
....
I ....
I ....
l,,..h.,.h.,,l.,.,l
I ....
I ....
I '"'I'"'P"'I'"'I
....
I
....
I ....
I ....
I' '"I'"'I'"T'"I
(¢) f i n i t e
(d) a f t e r a p p r o x
2O
z~ <3
0 L
,
,
,
I
. . . .
[
0.2
....
I ....
,
I,.,,I,,,.I,.,.I.,.I
0.5 m~/M~
,
0.2
1.0
. "1
. . . .
I ....
I...,IH..I..-I....i....I
0.5 m[/M~
1.0
Fig. 10. Corrections to the gluino mass versus m~/Mq. Figure (a) shows the complete one-loop corrections; (b) shows the leading logarithmic corrections; (c) shows the finite corrections; and (d) shows the difference between the full one-loop result and the approximation given in the text.
B,(p,O,m)=--~ln +~O(x-
+l-~x
1+
x
lnlx- 11
l) lnx,
(24)
pZ/m2.
where M 2 = max(p2,m 2) and x = As usual, the full mass renormalization contains logarithmic and finite contributions. The gluino mass corrections are shown in Fig. 10. In the figure we define the tree-level gluino mass to be M3(M3), and we evaluate the one-loop mass at the scale Q = Because we resum the correction, varying the scale from to 2M 0 changes the one-loop mass by at most + 1%. From the figure we see that the leading logarithmic correction can be as large as 20%, while the finite correction ranges from 3 to 10%. The finite contribution is largest in the region where the logarithm is largest, so the leading logarithm approximation is nowhere good. On the other hand, the approximation we provide typically holds to a few percent. It is off by as much as 6% in the region where the full correction is 30%. In this region we expect the two-loop correction to be of order 6%.
MO/2
MO.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
21
4.2. Neutralino and chargino masses The complete set of corrections to the neutralino and chargino masses [ 32,9] is given in Appendix D. In this section we present a set of approximations to these corrections. These approximations are more involved than those discussed above because there are no color corrections that would dominate the results. Our approximation is as follows. We start by assuming that IIXl > Mj,M2, Mz. (We find that Mz/IX 2 2 and M2/IX2 2/ are less than 0.53; see Fig. 8). We work with the undiagonalized tree-level (chargino or neutralino) mass matrix, and correct the diagonal entries only, that is, the parameters MI, M2, and /z. This approximation neglects the corrections to the off-diagonal entries of the mass matrices, which leads to an error of order (a/47r) m2z/IX 2 in the masses. We simplify our expressions by setting all loop masses and external momenta to their diagonal values, i.e. we set myd = M1, etc. We neglect all Yukawa couplings except At and ah. We also ignore the mixings of the charginos and neutralinos in the radiative corrections. This also leads to an error of order (a/47r)M2z/IX 2 in our final result. We further simplify our expressions by setting all quark masses to zero, and by assuming that all squarks are degenerate with mass MQ~, and that all sleptons are degenerate as well with mass ML~. We also take the Higgs masses to be mh = Mz and ,,,,, = mH~ = ma. This means that we also neglect terms of order (ce/4zr)M2z/m2A. In this limit the dominant correction to MI comes from quark/squark, chargino/ charged-Higgs and neutralino/neutral-Higgs loops. We find 1~--~2 llBI(Mj,O, MQ,)+9BI(M1,0, ML,)
--~-l J -
+-~1 sin(2fl)(Bo(Ml,ix, mm) -Bo(MI,IX, M z ) ) +BI (M j, IX, ma) + B1 (Ml, IX, Mz ) 1 "
(25)
Since Mz, Mi << IX, we can simplify this expression by setting Mz -- M1 = 0 inside the B functions. This gives AMI "]
--~-j j = ~
g,2 {
110M~MO.~+ 90M1M~,~+ OM~UMz-- 2B1 (0, IX, mA)
2 # sin(2/3) (B0(0, IX, 0 ) B- o ( 0 , +M-7
IX, mA ) ) -- -2-3 } ,
(26)
where Bo(0, ml, m2) and Bl (0, ml, m2) were defined in (9) and (10), and 0m~...m2 ~In(M2/Q 2) with M 2 = max(ml2. . . . . m2). The form of the finite corrections depends on the assumed hierarchy in the low-energy spectrum, but the leading logarithms are always correctly given by the 0 terms. We set the first subscript of a 0 term, ml, equal to the external momentum. Note that when the renormalization scale equals the
D.M. Pierceet al./NuclearPhysics B 491 (1997)3-67
22
external momentum, Q = ml, the theta function reduces to the familiar form Ore,m2 =
ln( m~/ Q2) O( m 2 _ Q2). The leading logarithmic corrections are easy to read from Eq. (26). 8 Note that the terms proportional to sin(2fl) are enhanced by the ratio tz/Ml. These finite terms are completely missed in the run-and-match approach because they do not contribute to the beta function. In a similar way, we approximate the corrections to M2 from quark/squark and Higgs loops. They are 1 ~ 2 9BI(M2,0, MQ1) + 3BI(M2,0, MLI)
-~2 J
/ + /x sin(2/3) I Bo( M2, tz, mA ) - Bo( M2, #, M z ) ) M2 +Bl(M2,~,ma) + Bl(M2, tx, Mz) I
(27)
o
Setting Mz = M2 = 0 inside the B functions, we find ---M-~-2/
~
90M2MQ~+30M2m,~, +OM2gMz -2Bl(O, tx, mA)
2/z sin(2/3) (B0(0,/z,0) - Bo(O, Iz, ma ) ) - ~ - ~ } +~-~2
.
(28)
There are additional corrections to M2 from gauge boson loops. Because M2 enters both the chargino and the neutralino mass matrices, the corrections differ slightly for the two cases. However, to the order of interest, it suffices to use the neutralino result, M2 /
= 47r2 2B0(M2, M2, M w )
-
B1 (M2, M2, Mw)
.
(29)
Because M2 is of order Mw, one must use the full B functions in this expression. Alternatively, one can use the following empirical fit which works to better than 1%: 2,,w + 0 ( M w - M2)
-O(M2-Mw)
.57~--~
[0.541n (MM2w--0.8)+ 1.15] } .
(30)
The corrections to /x are obtained in a similar manner. In the limit g~2 << g2, we find -
32~-2 (a~ + a~)B~(tz,0,Ma~) + a';Bl(u,O, Mt,~) + a~,B~(~,O, Mo~)
The logarithmicpart of --2Bl(0, be,mA) in Eq. (26) is givenby OM~u,nA.
D.M.Pierceet al./NuclearPhysicsB 491 (1997)3-67 I-'
10
"•1
....
I"'T"T"T"T"'I
•
..
'
'1
....
I ....
r'"
F"T"T"T'"I
. . . .
I
(b) log
".,...
• ...-~..'
5
I
( a ) full
-i.:~..'::f:":%2
t~
. . . .
,.
23
~,',:.',~_:-.:~,.-...:.,
..
.
,
. . . . . .
.- ~-;... - ..:;~
~
o -,
.,
Q .
~
E <3
O --5
I,,, I'
''
[ i , ~ . [....I
....I....I....I....I
. . . .
I
.
I
....
I ....
I....I....I....I....I....I
,
I ....
"'T"T"T'"I
. . . .
I
'
I ....
I ....
I ' ' ' '1 '"'I'"T"T'"I
. . . .
1 ' ' "1
10
,
,
.
I I
(d) a f t e r a p p r o x
(e) f i n i t e
5 ..~
o~
~
...
~
0
~
~i
~
• ',.~ . . ~ : > , . ~
- ._
.....
~
III
. I
II.
<3 --5
I ....
I ....
I,,,
,I ,, ,,I,.,I,..I,,.I
50
m~
. . . .
100
I
,
I ....
200
(GeV)
I ....
I...,
I. ,. J,,.I,,,,I,.,.I
50
m~?
. . . .
100
I
,
200
(CeV)
Fig. I 1. Corrections to the lightest neutralino mass, as in Fig. 10. 3g2 64~ 2
[B](Ix, M2, mm) + B1(#,M2, Mz)
+2B1 (/z,/z, M z ) -- 4B0(/z,/z, M z ) ] . As above, we set
-
Mz
32
(31)
= M2 = 0 inside the B function, in which case (31 ) reduces to
3[ -2
+AZB~(tz, O, MD3)] + ~ 3 g 2 [~1 0 ~M2Mz_
30,o.Mz--Bl(Iz, O,mA)+4]
(32)
The expression for B1 (p, 0, m) is given in Eq. (24). In Fig. 11 we show the corrections to the lightest neutralino mass. In Fig. 1 la we show the full correction in percent, with the tree-level mass defined as the eigenvalue of the mass matrix, where the running parameters Ms, M2, a n d / z are evaluated at their own scale. (The tree-level mass matrices also contain tan fl at Mz and the W- and Z-boson pole masses.) The one-loop masses have negligible scale dependence. As usual, the full corrections are made up of logarithmic and finite pieces. The logarithmic corrections are shown in Fig. 1 lb and the finite corrections are shown in Fig. l lc. Note that the finite corrections can be more than half as large as the logarithmic
24
D.M. Pierce et a l . / N u c l e a r Physics B 491 (1997) 3 - 6 7 ":'I"'T'"['"'I
.
.
.
.
I
15 ÷.10
. . . .
I
....
I ' !
....
.
.
.
[
. . . .
-i~,~..~L.[.
4.:~-,
~ ...
-_
I
....
I'
(b) log i~.~ s.-:....
..
.,..~'.~e#~.-o:r,:~<-k::~".:;...,.--:--:
~
~
.
I'"'I'"T'"I
(a) m n
~.
~
~,
~•
,
~
" i
'
" ~""
J~ . . . i.
" • -- :
.
0
<~ - 5 l..,,I,,,,I.,..I,,.,I ' ' ' I " " I ' " T ' " I
"-" 15 +.10
5 + ~
I
. . . .
i
....
I ....
I
.
I
. . . .
I
I
.
.
.
. . . .
,, I..,,I,..,I,,,,I
(e) f i n i t e
~ : ' . . ' ~ .: . ? . ~ - : , . . . . . : . _ , . a , ... - ~-.,~g~it~-~..-~
"r
-.:'."."
.
.
I
. . . .
I ....
I
approx
. . - ~ : . : . ' . :!~ , .~ :.:~"~, . z a . - ~ , ~ ~ . , ~ , ~
"" " - "
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
J
I.[ . . . .
L
"
I
loo
.
(d) a f t e r
.
F.~"
.
zoo
mg~ (CeV)
....
I
, ..I,...I....I....I
.
loo
.
I
. . . .
I
....
I
zoo
mg+ (GeV)
Fig. 12. Corrections to the lightest chargino mass, as in Fig. 10. corrections. Indeed, the finite corrections can be larger than 5% in the small Ml region, primarily because of the Higgsino-loop term proportional to ~. In Fig. 1 ld we show the difference between the full one-loop result and our approximation. Here, and in the following two figures, the logarithmic corrections (Fig. 1 lb) include an explicit sum over the soft squark and slepton masses, while the approximations (Fig. l l d ) use a single soft squark or slepton mass. Fig. 12 shows, similarly, the corrections to the lightest chargino mass. Again there is a term proportional to /x which dominates the finite corrections when M2 is small. In this region, the finite corrections can be as large as 10%. In Fig. 12d we show that the difference between our approximate correction and the full one-loop mass is less than 2%. These corrections are quantitatively similar to the corrections to the second-lightest neutralino mass. The corrections to the heavy chargino mass are shown in Fig. 13. These corrections are less than a few percent, as are the corrections for the two heaviest neutralino masses. The logarithmic corrections are in the range 0 to 2.5%, and the finite corrections are in the range 0 to - 3 % . Fig. 13d shows that our approximation for the heavy chargino mass generally holds to better than 0.5%. Our approximation also works to typically better than 1% for the two heaviest neutralino masses, but it can be off by nearly 2%.
D . M . Pierce et a l . / N u c l e a r
3
'
'
I
I
....
I " " I ' " T " T ' - I " ' T " ' I
. . . .
'.
2 +,~
. . . .
P h y s i c s B 491 (1997) 3 - 6 7 I
•
'
[
(a) full
1
<".
. . . .
I ....
25
I''"1""1'"'1'"'1'"'1''1
. . . .
..:;~.,:..ci,'. .~ "...."
: . ~ ' "-~W .; #~~~-;-~-.
,, .
......
o +~ ~ --1
--2
"
-3 ~.
.. '-.;;~:;..':- ---2_':f.,.;,-""~. " "
"
-
log
(b)
':'.7~.FL.'~." q ..'~ i- " i ;:
•
,
I
....
I ....
I,,,'./,,,.I,,,,I.,,I,,.I,.,1:
. . . .
•
I
. . . .
I ....
l..l,,,.l...l,.,l..,d,..,l
,
'
'
I
. . . .
I
I ' ' ' '1 ' ' " I " ' T " T " T " ' I
. . . .
'
I
. . . .
]
I ' ' ' ' I'
. . . .
.
I
. . . .
I ....
....
....
"'1'"'1'"'1'"'1''1
.
,
,
(c) finite
2
v
+~
1
o Z;~
<1
--1
(d) after approx
-2 •
-3
.
I
. . . .
2O0
I ....
I....f....I....I....I...I....I
.
500 1000 m2~ (CeV)
.
.
~
I
,
2000 200
I....I....I....I....I....I....I
~
*
500 1000 m2~ (GeV)
,
,
I
2000
Fig. 13. C o r r e c t i o n s to the h e a v i e s t c h a r g i n o m a s s , as in Fig. 10.
4.3. Squark masses The first two generations of squarks receive QCD [33] and electroweak corrections. However, it is a very good approximation to ignore the electroweak graphs, since the dominant corrections come from gluon/squark and gluino/quark loops. Neglecting the quark masses, these corrections are as follows:
mo:Fn2(Q)
1+
t
m~
)J
(33)
'
where
)
L
+
(1
3 l+3x+(x_l)21nlx_ll_x21nx+2xln 6~2
,
(34)
2 2 and x = mp,/m O. For the case of universal boundary conditions, the gluino mass is less than or roughly equal to the squark mass, so the correction (34) is essentially finite at Q = m O. From Fig. 14 we see that it varies from around 1% for x << 1 to between 4 and 5% for x -~ 1. We also see that the electroweak corrections are small, less than 0.5%.
D.M. Pierce et al./Nuclear Physics B 491 (1997)3-67
26
~-~
6
•
'
I
. . . .
I
(a) f
~
E
~
. . . .
I ....
u
I ....
l
I'"'I'"T'"I
•
~
'
I
. . . .
I
. . . .
I ....
I ....
I " ' T " T ' " I
I ....
I ....
I,.,
(b) EW
4 2
z#
o •
I
....
I ....
0.3
I ....
I ....
I,,,A,.,I,,,,I
0.5
,
,
1.0
,
[
. . . .
....
0.3
0.5
m~/m~
.l....I....I
1.0
m~/m~
L
Fig. 14. (a) Full one-loop corrections to the first generation squark mass, m~L,versus the ratio mffmaL. (b) The difference between the full corrections and the approximation in the text, versus mffmaL. (These are essentially the electroweak corrections.)
'
i
. . . .
I
....
I ....
I ' ' " I ' " T " T " T ' " I
'
(a) f u l l
'
I
. . . .
I ....
(b) a f t e r
I ....
I'"'I'"'I"'T"T'"I
approx
v
<1 - 5 6
,
I
,
'
I
. . . .
,
, , I , , . . I .... I
....
I ....
l.H.l....l...,l,,,,l..l I ' ' ' '1 ' ' "1'"'1'"'1'"'1
?i.....: ."
4
i -"..
:......:..
" "
'
'
I
'
'
'
'.1
....
I'"'I'"T"T"T'"I
(d) b'2
(e) ~
- ..
.....:..:~c.. -,:-~: .:;, ~ . o :::io::L~:;:,i& _-...~..
~,,,a
~j.
2
"
~':";r~.~.*.~
~ . t. ~-~"~r-.'~Z'.~:(;::;.;:
~~;~r
o -2
:
,
I
, , ,.
i.....i
200
....
i..,,i,..h....i...,I.,,,i
500 m~
(GeV)
.~_. ...
,t'?...~ -r:':"
- -
] :i::!. ~ : ~ ~ : - :
l,.o
<1
I ....
i000
,
,
I
,
i
,
.:
.
.
/'..
,I,.~',11,1",1,,.,I,.,,I,..,I.,..I,...I
200
500 m~
I000
(GeV)
Fig. 15. (a) The corrections to the heavy top squark mass versus its mass. (b) The difference between the full one-loop heavy top squark mass and the approximation,Eq. (35). (c) Same as (a), for hi. (d) Same as (a), for b2. The third generation squark masses receive Yukawa corrections on the order of, and opposite in sign to, the QCD corrections. In Fig. 15 we show the full corrections to the third generation heavy squark masses. As usual, the tree-level masses are defined in terms of the gauge-boson and quark pole masses, as well as the soft masses MQ(MQ), M y ( M y ) , and M D ( M o ) . The tree-level mass matrices also contain t a n f l ( M z ) , / z ( / z ) , and Ai(max( IAil, M z ) ) , where A i denotes the top or bottom A-term. (Our convention for the third generation squarks is to associate the subscript 1 with the mostly left-handed
D.M. Pierceet al./Nuclear Physics B 491 (1997) 3-67
27
squark. Since the light top squark is predominantly right-handed, its mass is denoted mh.) From Fig. 15 we see that the heavy top squark mass receives corrections in the range - 5 to 2%, while the bottom squark masses receive corrections mostly in the 0 to 3% range. We note that in none of these cases does the leading logarithm approximation work well: as is the case for all the squarks and sleptons, these corrections are essentially non-logarithmic. (The light top squark mass does receive some substantial logarithmic corrections, but they are generally not larger than the finite corrections.) We will now present our approximation for the top squark mass matrix. We will derive our approximation for the case of the light top squark, but it also works quite well for the heavy top squark (see Fig. 15b). The mass of the light top squark receives potentially large additive corrections proportional to the strong coupling and the top and bottom Yukawa couplings. We approximate the corrections to m h by neglecting g, g' and the Yukawa couplings of the first two generations. We also neglect all quark masses except rnt, which eliminates all sfermion mixing except for that of the top squarks. We neglect the mixing of charginos and neutralinos, so the two heavy neutralinos and the heavy chargino all have mass I/zl. We also make the approximations mh = Mz and rnH = rna,, = mA. Finally, we set p = 0 in the B-functions if any of the other arguments is much bigger than rnh. This gives the following expressions for the one-loop corrections to the top squark mass matrix:
2 ~ AM2L AM2R ) jk4~ = 3)I~(Q) + \ A M 2 R AM2R_ •
(35)
The A M 2 entries are as follows:
A M%L = 6s--3--r--52m22 [c2B I (m h , m~,, O) + s 2B1 (m h , m h , 0)]
2 __ m~)Oo(O, m~, mt) } +ao(mg,) 4- Ao(mt) - (m~2 - rag, 1 [22 167.r2 AtstAo(mh ) + ,~2Ao(mb)
-2(At2 4- A~)Ao(/z) 4- (A~c2~ 4- h6s~)Ao(mA )
1
A2 [a(ot,~J)Bo(O, mh,ma) + a ( o t - 2,~)Bo(O,O, mA)
32rr 2
+ A ( O t , f l - -)B°(0'mh'0)qr 4- A ( O , - ~2 , 8 - -2
1
1 [ 222 16rr 2 (atmtc~ + a2(/*c~ -- Abs/~)2) B0(0, rnb, mA) 2 2 2 + (Atmts~ 4- Az(lzs# 4- abc#) 2) Bo(O, mb, O)]] ,
(36)
28
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 g2 2 AM2R = ---~-~2CtSt [(m~ + m~2)Bo(mT2,m~,O ) + 2 m~2Bo(mh,m~z , 0)] 3,~2 ctstAo( m~l ) -- 3g~2 mt m~Bo (0, mt, m~,) - 1--~2 ,~2t 32.n-2
[g2(Ot, fl)Bo(O,m~,,ma) + .(2(--tgt,fl)Bo(O,O, ma)
+n(O,,
+
,o) + gl(-o,, ~ + 2
1
l~7.r2 [-- (,~mtc/3(IzsB-- atcB) + ,~2bmtsp(#cB-- abs~))Bo(O, mb, mm) +,~mts~(Izc~ + Ats~)Bo(O, m~,,0) 1 ,
(37)
AMZR = 6~2 {2m~z [s~Bl(mr2,m~,,O) +c2tBl(mT2,m~2,0)]
+Ao(m~) + Ao(mt) - (m}2 - m~2 -- mZ)Bo(O, m~, mt) } 16~r2 [c~Ao(m~,) + ao(mb) - 4Ao(/z) + 2c~ao(mA)] 3 ~ 2 A(
- Ot,fl)Bo(O,m~,,ma) + A(-Ot, fl)Bo(O,O, ma)
"17" 77" 7"1" ] + a ( g - 0,, ~ - -~ )Bo(O, m~,, O) + a ( - o , , ¢~ - ~ )Bo(m h, m s, mz)
1
1
16~r2
(abm,s~ + 2
2 2
a~(
~s~
-
a,c~) 2) Bo(O, m~, mA)
+,~(l~c~ + Ats~)2Bo(O, m b, O) 1 .
(38)
We have defined the two functions
A ( Or, fl) = ( 2mr cos fl cos Ot - (tz sin fl - At cos fl) sin Ot) 2 +(/,t sin fl
-
At
COS/3) 2 sin2 Or,
$2(0t, fl) = 2m 2 cos 2 fl sin 20t
--
2mr cos fl(/~ sin fl - At cos fl) .
(39)
(40)
Note that the running mass matrix A~I2(Q) in Eq. (35) contains the soft m a s s e s M Q and My (as well as /z, A t , etc.) at some common scale, Q. In the limit ho ~ 0, these expressions are equivalent to the results of Ref. [ 11 ], with certain external momenta set to zero. These approximations depend on the mass of the light top squark. Normally, one would take it to be the tree-level mass. For the case at hand, however, the choice is more subtle because for very light top squarks, the radiative corrections can be quite large. In fact, the radiative correction can change the top squark mass squared from
D.M. Pierceet al./Nuclear Physics B 491 (1997)3-67 200
....
i
....
~'
i0
'::.~
I
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I
. . . .
150
I
(b)
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m~2(tree) • •
---:'....... ~:
3oo
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I
:a.' "-~.: ;';- ,~,t-._'-."
....
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,,.i
;'"
"-I
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.:.
' ~ ....
; o 5 0 ::
I ....
v) r'"l""r'-r'"r"r"'l
\,
\
-,-' - 5 0 "~:~~::~ . . . .
-2
-i
0
1
2
3
"x
(d)
-100
0 -3
I
500 l o 0 o
(c
2oo
~100
,
. . . .
100
Ao/M~
" -I ....
I ....
300
I,..,
I.,, ,I,.,,I.,J..,,I.J
500
1000
Q (GeV)
Fig. 16. (a) The full one-loop light top squark mass, versus the tree-levelmass (in GeV). On the x-axis we plot sign(m-2t2)lm2-t2fl/2' so a negativetree-levelmass correspondsto m~2< 0. (b) The differencebetween the full correction and the approximationin the text, versus the one-loop mass. (c) The light top squark mass at one loop, versus Ao/M#. The solid line correspondsto point (I) in the text, the dashed to point (II). (d) The running and one-loop light top squark mass versus the renormalizationscale Q, for the choice of parameters (I) (solid) and (II) (dashed) in the text, with Ao/M# = -1.83 and -2.2, respectively.The running mass curves each have points where m~2 becomes negative.In these cases we plot the signed square-root of the mass-squared, as in (a). negative to positive. Therefore we shall take m 5 to be the one-loop pole mass, which we find by iteration. (We find the one-loop top squark mixing angle by iteration as well.) We show the light top squark one-loop pole mass versus the tree-level mass in Fig. 16a, where the tree-level mass is the eigenvalue of the mass matrix which contains the running parameters M 2 and M~ evaluated at their own scale (or Mz, whichever is larger; the tree-level mass matrix also contains the top quark and Z-boson pole masses,
tanfl(Mz), # ( / z ) , and At(max([Atl, M z ) ) . In Fig. 16b we see that our approximation for the light top squark mass holds to within 10 GeV. With the present unification assumptions, a top squark with mass less than Mz requires that the RR term in the mass matrix be small and that the LR element, proportional to the A-term, be large. The light top squark mass results from a cancellation between the diagonal and off-diagonal terms, which requires a fine tuning. We illustrate this in Fig. 16c, where we plot the light top squark one-loop mass versus Ao/M#. On the same plot we show the curves corresponding to two choices of parameters, (I) tan fl = 20, M0 = 500 GeV, M]/2 = 100 GeV, and /x < 0, and (II) t a n f l = 5, M0 = 100 GeV, M~/2 = 200 GeV, and tz > 0. Whether at tree level or one loop, the parameter A0 must
30
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 .... I'"T"T"I
.-.
. . . .
I
. . . .
I ....
I ' '" '1"
' T"T"T"T"I
•
(
. . . .
I
I
•
(a) ~L
2
I
. . . .
I
(b) ~e
i
..
..~.. . . . .
o
,,
<1 -2 ,I,.,,I,,.,1,.,I
4
~
"I'"T"T'"I
. . . .
I
. . . .
I ....
I,,,,
. . . .
I
. . . .
I ....
l ' ' ' T
I,,,,I,..I,,,,I,..I...
d
,
' ' T "'I'"'I'"T'"I
•
I
,
-1
,
,,I I
. . . . •
.
.
- -
I
,
I
,
I
'
I
. . . .
,
,, I
. . . .
I
-
(e) ~,
v
z~
::,~~;%,.. "¢~ ~ ¢".'I.;-:'.'.L .
2
?" f~;.
o .
-2
, L,..I..I,,,.I
100
. . . .
I
. . . .
I ....
200 m 7 (GeV)
I.,,,
I,, ,.I.,,,I,...I*.,L.,.I
500
1000
,
I
ao
.
.
, , , , 1 "
•
.
.
-
1
ioo
200
I
500 iooc
m 7 (GeV)
Fig. 17. (a) The full set of corrections to the left-handed selectron mass versus its mass. (b) The complete corrections to the electron sneutrino mass versus its mass. The complete corrections to the predominantly (c) left-handed and (d) right-handed tan slepton masses. be tuned to one part in 75 to obtain a light top squark mass below 50 GeV. We see from Fig. 16a that the light top squark mass-squared can be raised from - ( 100 GeV) 2 at tree level to over (100 GeV) 2 at one loop. For such large corrections, it is important to keep in mind that two-loop effects might be important. The size o f these effects can be estimated by the scale dependence o f the one-loop mass. In Fig. 16d we show the scale dependence at the points (I) and ( I I ) , with A0 = - 9 8 5 GeV and A0 = - 9 0 7 GeV, respectively. We see that as the renormalization scale increases from 100 to 1000 GeV, the running masses vary over a wide range. In contrast, the scale dependence of the one-loop masses is quite mild. 4.4. S l e p t o n m a s s e s
Corrections to the S U ( 2 ) mass sum rules were studied in Ref. [ 34]. They suggest that the corrections to the slepton masses are small. We find that the corrections to the lefthanded electron and muon slepton masses are typically in the range 4-1%, as illustrated in Fig. 17a. The right-handed electron and muon slepton mass corrections are larger, but still less than 1.7%. The sneutrino mass corrections are essentially identical for all three generations. The full correction is typically in the range + 1%, and reaches at most - 2 . 5 % at my -~ 50 GeV, as shown in Fig. 17b. The (predominantly) left- and righthanded tau slepton corrections are similar to the corrections of the first two generation charged slepton masses. However, from Figs. 17c,d we see that the scatter plots show
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
31
less uniformity because of the additional Yukawa coupling corrections. We emphasize that in the corrections to the slepton masses, the leading logarithmic approximation [ 35 ] typically gives zero correction. (The mass m h receives some logarithmic corrections. However, the finite corrections are typically of the same order as or larger than the logarithmic corrections.)
4.5. Higgs-boson masses We first discuss the corrections to the heavy Higgs-boson masses (mA, mr4, mu~), and then we consider the one-loop light Higgs boson mass. The full one-loop corrections to the Higgs-boson masses appear in Ref. [6]. As usual, we parametrize all the Higgs-boson masses at tree level in terms of the CP-odd Higgs-boson mass, ma, and tan/3. To compute the one-loop mass ma, we first take the soft masses m ~ (Q) and m22 (Q) as outputs from the renormalization group equations, and apply corrections from the electroweak symmetry breaking conditions to obtain the D----Rrunning parameters th2(Q) and/z2(Q) (see Appendix E). We then apply further corrections to obtain the CP-odd Higgs boson pole mass, ma, from the running mass, mA(Q),
m2=rh2A(Q )_RelIAA(m2A ) + c2t2 + 2tl B U2
SB 0--7 '
(41)
where tj and t2 are the tadpole contributions listed in Appendix E. Note that we treat the Higgs mass analogously to the superpartner masses, in that we compare the pole mass with the running mass. However, the Higgs mass is different because the tadpole, or effective potential, corrections must be added to the "tree-level" running mass to obtain the DR running mass, m a ( Q ) . The difference between the running mass and the pole mass can be quite substantial; as for the light top squark, the radiative corrections can change a negative mass-squared running mass into a positive mass-squared pole mass. In Fig. 18a we show the one-loop pole mass, mA, evaluated at Q = M 0' versus the running mass, &A(ma). From Fig. 18a we see that there are points in parameter space where the running mass-squared is - 1 TeV 2, while the one-loop mass is over 300 GeV. (If one considers the running mass at the scale Mz, the largest corrections are even more extreme.) These large corrections arise from terms enhanced by tan/3. In fact, all of the points in Fig. 18a with frl2a(ma) < 0 Occur for tan/3 > 25. The tan/3 enhanced contributions come only from the last term in (41) since l/vl scales like tan/3 for large tan/3. Therefore the tan/3 enhanced corrections are simply
m2a -- rh2A( Q ) -
3 s~/.z tan /3 ( 167r2 ~ a'2a'
B0(0, m~,, m~.) + a~ab B0(0, m~,, m~2)
Jz2 J
(42)
32
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
500
. . . .
I
. . . .
I
6
. . . .
(a)
4OO
2
200 100
,"
0 -1500
,
,
,
I ' ' "
I ' ' ' ' I ' " T " T " T " T ' I
. . . .
(b)
4
30O
,
. . . . . . . . .
I
,
.
-1000
.
,
g
I
:>.
. . . .
-500
&A(mA)
o
,
I
0
.
.
.
.
500
<]
-2 i00
. .j~.
. • -
?
.
>..
,.:...., ..,.z
:
.
- ~ i ~ - ~ . L : . . ~-~,~-.-;-.-...... .... ~-,f, 800
] ....
500
1000
In.+
Fig. 18. (a) The one-loop pole mass, mA, versus the running mass at the scale mA (the ordinate is the signed square-root of the mass-squared, sign(fn2a)]£n2A[1/2); (b) the corrections to the charged Higgs mass, m u~, v e r s u s
m H ~ .
where B0(0, m l , m2) is given in (9). These terms account for the large corrections seen in Fig. 18a. We parametrize the other heavy Higgs-boson masses in terms of the pole mass, m A. The corrections to the heavy Higgs-boson masses, m m and mm turn out to be quite small. For example, the corrections to mH~ are typically less than 1%, as shown in Fig. 18b. Corrections to the charged Higgs mass are the subject of Ref. [36]. The corrections to the light Higgs-boson mass, m], have been studied extensively in the literature [ 37,6,7 ]. Here we show our results for the full one-loop pole mass over the parameter space associated with radiative electroweak symmetry breaking and universal unification-scale boundary conditions. We also show the size of a set of corrections which are often neglected, and the accuracy of Dabelstein's approximation [7]. In Fig. 19a we show the one-loop light Higgs-boson mass versus tan fl, as well as the upper limit at tree level, mh = Mz[ cos 2fl I. The upper limit on the one-loop Higgs mass depends sensitively on the top quark mass, and somewhat less sensitively on the squark masses. In the parameter space we consider the squark masses are less than about 1 TeV. With m t = 175 GeV, we find mh < 130 GeV. In Fig. 19b we show the gauge/Higgs/gaugino/Higgsino contribution to the Higgs mass, which is typically - 2 GeV. In Fig. 19c we show the difference between the oneloop result and Dabelstein's approximation, Eq. (4.9) of Ref. [7], which only includes the top sector. We see that this approximation is typically 2 to 6 GeV larger than the full one-loop mass. In any pole mass, the scale dependence formally cancels. However, at any given order, there are usually higher-order corrections which do not cancel. For example, when we vary the scale in the Higgs mass calculation, we change the tree-level DR mass. To one-loop order, this variation is canceled by the change in the self-energy. However, as the scale varies, the couplings and masses in the self-energies also change. For the case of the Higgs mass, the change in the top quark Yukawa coupling gives rise to a two-loop O(A 4) scale dependence in our one-loop results. In Fig. 19d we show the difference between the full one-loop result evaluated at the scale MO and M # / 2 . We see that the scale dependence is usually small, in the range 0 to - 1 GeV. The fact that it
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
. : :,;-.:....,,:.. . ~ ! ~ .
100
....
~.~.:1:.-,-~
"-'"
33
0
"
"~'~q
"
":-:-.~;"°"'::"......... . . . . . .
<1
L/
t '
'
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,,,I
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'''1
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0 :./"-~.4:7"-.'-'..~.--..
. .
Q) v
~-r'~~Z':~:.
~-5
~-.:
Q-I
I::I?-:".V :::"!::T .~- :: . - ? i .... .... . .:-..- " . . . - ( e )
:: "1
-i0 1
2
,.
, i~
,,';1"
5 tan
: "
t
10 20 fl
,
~-....-. .-
(d) ,,
I
50
I
-2 1
2
•
5
10 20
,
I
,
50
tan
Fig. 19. Figure (a) shows the full one-loop light Higgs mass v e n u s tan/3. The line indicates the tree-level bound mh < Mzl cos 2/31. Figure (b) shows the contribution from the g a u g e / H i g g s / g a u g i n o / H i g g s i n o loops; (c) shows the difference between the full one-loop result and Dabelstein's approximation; and (d) shows the difference between the full one-loop mass evaluated at the scales M 0 and M#/2.
is negative follows from the dependence of tan fl on Q, and from the dependence of At on tan ft.
5. Conclusions In this paper we computed one-loop radiative corrections in the minimal supersymmetric standard model. We took as inputs the electromagnetic coupling at zero momentum, Ofem, the Fermi constant, Gu, the Z-boson pole mass, Mz, the strong coupling in the MS scheme at the scale Mz, ces( M z ) , and the quark and lepton masses. From these we computed the W-boson mass, Mw, as well as the one-loop value of the effective weak mixing angle, sin 2 Dlept veff , as a function of the supersymmetric parameters. We studied the size of the corrections in a reduced parameter space associated with the unification of the soft breaking parameters and radiative electroweak symmetry breaking. We found that supersymmetric radiative corrections can reduce sin 2 0 ~ t by as much as - 1 . 6 × 10 - 3 with respect to the standard-model value. Similarly, we found that they can increase Mw by as much as 250 MeV. Because of decoupling, the points with the largest deviations are also the points with the lightest superpartners. As direct searches increase the limits on the superparticle masses, the size of the supersymmetric radiative corrections will decrease. Indeed, if superparticles are not discovered at LEP 2, we found
34
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
that the maximum size of the supersymmetric radiative corrections will be reduced by a factor of two. The apparent unification of the SU(3), SU(2), and U(1) coupling constants is a major piece of evidence in favor of supersymmetry. At next-to-leading order, the weakand unification-scale threshold corrections come into play. The weak-scale thresholds decrease the one-loop weak mixing angle. This leads to an increase in the predicted value of the strong coupling, a s ( M z ) . As we have seen, for squark masses less than one TeV, a unification-scale threshold of - 1 to - 3 % is necessary to bring a s ( M z ) into accord with experiment. The size of the unification-scale thresholds places an important constraint on unified model building. In any unified model, the unification-scale thresholds can be calculated as a function of the grand unification parameters. One can see whether the model is consistent with a unification-scale threshold of about - 2 % . In this paper we studied the minimal SU(5) model and the missing doublet SU(5) model. We found that the former was not compatible with gauge coupling unification, while the latter was. Grand unified theories also predict the unification of certain Yukawa couplings, and in a similar fashion, the mismatch of the Yukawa couplings at the unification scale can be used to constrain unification-scale physics. To this end it is necessary to extract as precisely as possible the DR Yukawa couplings from the fermion pole masses. In this paper we presented full one-loop relations between the two, as well as approximations that work at the (.9(1%) level. We studied the substantial (up to 50%) tan/3-enhanced corrections to the bottom quark mass, as well as the corrections to the top and tau masses, which are of order 5%. Supersymmetry also predicts relations between the masses and couplings of the supersymmetric particles. Indeed, if new particles are discovered at future colliders, it will be necessary to check these relations to see whether the new particles are in fact supersymmetric partners [38]. The radiative corrections to the supersymmetric mass spectrum presented in this paper will be an essential element in these determinations. The corrections to the supersymmetric mass spectrum will be used in (at least) two ways. First, they will be used to correct the tree-level mass sum rules [ 34,39] which test supersymmetry at the weak scale. Second, they will be needed to extract the underlying soft parameters from the physical observables. The soft parameters can then be run to higher scales, to test for unification and possibly to shed light on the origin of the supersymmetry breaking. The corrections to the supersymmetric masses in the spin-½ sector include potentially 30% corrections to the gluino mass, as well as (_9(5%) corrections to the neutralino and chargino masses. In the spin-0 sector, the famous quadratic divergences give rise to large corrections to the scalar masses. These corrections can lift the running mass-squared of the light top squark from - ( 100 GeV) 2 to (100 GeV) 2. Even more dramatically, large tan/3 corrections can lift the mass-squared of the CP-odd Higgs boson from, e.g., - ( 1 TeV) 2 to (300 GeV) 2. Radiative corrections also have an important effect on the mass of the lightest Higgs boson, h. In the parameter space we consider, they effectively change the sign of the
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
35
tree-level bound from m h < Mzl cos 2ill to m h > Mz[ cos 2/31. We found the light Higgs mass was raised to at most 130 GeV. The corrections to the rest of the scalar masses are smaller. For example, we found I to 5% corrections to the first two generation squark masses, and (.9(1%) corrections to the slepton masses. In the paper we presented approximations to many of the formulae for the supersymmetric mass corrections. These approximations, often good to better than a couple of percent, provide useful substitutes for the full corrections.
Acknowledgements D.M.E thanks M. Peskin, T. Rizzo and J. Wells for useful discussions.
Appendix A. Tree-level masses In this appendix we define the tree-level masses. These tree-level relations also hold for the running DR parameters at a common scale, Q. For the most part we follow the conventions of Ref. [40]. The up- and down-type quark and charged-lepton masses are related to the Yukawa couplings and the vev's vl and v2 by
1 mu = --~)tuV2 ,
md =
1 -'~/[dUl .
(A.1)
The ratio of vev's vz/vl is denoted tan ft. The tree-level gauge boson masses are M ~ v = ~1g 2 (v~+v22),
M2z= 1 (g,Z +g2)(v2 + v2),
(A.2)
where g and g' are the SU(2) and U(1) gauge couplings. The Lagrangian contains the neutralino mass matrix as -~°TA4ffo~ ° + h.c., where
~o = (-i[~, -ivY3, hi, h2) T and .A4~o =
MI 0 -Mzc/3sw Mzs/3sw I 0 M2 MzCl3CW - M z s p c w -Mzcl3sw Mzc/3cw 0 Iz " Mzs/3sw -Mzs/zcw tz 0
(A.3)
We use s and c for sine and cosine, so that s# - sinfl, c2~ - cos2fl, etc. MI and M2 are the soft supersymmetry-breaking bino and wino gaugino masses, /z is the supersymmetric Higgsino mass, and sw (Cw) is the sine (cosine) of the weak mixing angle. The neutralino masses are found by acting on the matrix .Ad¢7o with a unitary matrix N, so that N*.Adc;oNt is a diagonal matrix which contains the physical neutralino masses, m~o. In the usual case that one of the eigenvalues of (A.3) is negative, the matrix N is complex even if the elements of .Ad~0 are real.
36
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
The Lagrangian contains the chargino mass matrix as _ ~ - r . M e T ~ + + h.c., where ~+ = ( _ i # + , ,~-)r, ~ - = ( - i ~ : - , h~-)r and
M2 v/2 MwSB) .MET' =
x/~ Mwc#
-At
(A.4)
"
The chargino masses are found by acting on the matrix .A4eT+ with a biunitary transformation, so that U*A/te7+Vt is a diagonal matrix containing the two chargino mass eigenvalues, m ff. The matrices U and V are easily found, as they diagonalize, respectively, the matrices A4eT~*.Me7 ,T and .M~+.MeT,. At tree level the gluino mass, m~, is given by the soft mass, M3. The tree-level squark masses are found by diagonalizing the following mass matrices: M2Q + m u2 + guLM2c2B mu ( A , + Atcot fl)
( Here
m, (A, + Atcot fl)
(A.5)
M2v + m2u + guRM2zc2l~ ,] '
M2Q + m d + gdL Mzc2B
rnd ( Ad + At tan fl)
md ( Aa + Attanfl)
M~ + m2d + gdRM2 c2B
'~
)
(A.6)
m
MQ,
Mu, and MD are the soft supersymmetry-breaking squark masses, and the the soft supersymmetry-breaking A-terms. The slepton mass matrices are analogous. The soft slepton masses are denoted ML and ME. We have defined the weak neutral-current couplings
Af's
are
gf = I f - efs 2 .
(A.V)
The electric charge, hypercharge, and third component of isospin of the sfermions are 9
~L
~R
dL
~
~L
~
ef
g
2
_2
3
--3
3
1
0
-1
1
Ys 1f
31 ~l
- 34
~1 - 31
2
--1
--1
2
0
~l
- 31
0
0
1
dR
(A.8)
A symbol without an L or R subscript refers to the L-field (e.g. eu = 2/3). The matrix which diagonalizes a sfermion mass matrix is denoted by Q C f S f ) ,Cf __Sf
(A.9)
where cy is the cosine of the sfermion mixing angle, cos Of, and sf the sine. These angles are given by 9 Our convention for the right-handed sfermion fields is the charge-conjugate of that of Ref. [40].
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67
2mu (Au + I z c o t f l ) 2 2 ' _ _ 2e,sw) Mzc2~ 2md ( Ad +/x tan fl) Q _ ME + (_½ _ 2eas2)M2zc2 .
t a n ( 2 0 " ) = M2°
tan(20a)=M2
37
M2v+(½
(A.10)
(A.11)
Since there is no right-handed sneutrino, the slepton m i x i n g angle for ~ satisfies c~ = 1, and the sneutrino mass is mo2 = a4~ + a 4 ~ / 2 . Given values for tan fl and the C P - o d d Higgs-boson mass, are given, at tree level, by
m2H,h
1 = -~
mA, the
other Higgs masses
(m~A+ M~ + j( m~A+ a4~) 2-4mAMZc2~2)
(A.12)
and m~+ = m~ + M2w .
(A.13)
The C P - e v e n gauge eigenstates ( s l , s2) are rotated by the angle a into the mass eigenstates ( H , h) as follows:
(H)
= ( c~ ca s,)
s2
.
(A.14)
At tree level, the angle o~ is given by tan 2 a - m2 +-- M 2 tan 2ft. m2 M2
(A.|5)
Appendix B. O n e - l o o p scalar functions The f o l l o w i n g integrals appear at one loop in a self-energy calculation [4] : l0
Ao(m)
= 167r2Q 4-n
Bo(p, mj,m2)
= 16~'2Q 4-n
/
dnq 1 i (2~-) n q2 _ m 2 + dnq /(27r) n
dnq puB,(p, ml,m2)
= 167-r2Q4-n
i (2~.)n
ie '
(B.I)
1 [q2_m21+ie][(q_p)Z_m~+ie], (B.2) qs, [qZ--m~+ie][(q---p)Z--mZ+ie] (B.3)
m Our A and B functions differ from those of Ref. [4] since we use the Minkowski metric. Also, A0, BI and B22 differ by a sign. Eqs. (B.2)-(B.4) contain an abuse of notation. The first argument of B-functions is the square root of the scalar p2, whereas elsewhere the p represents the external momentum four-vector.
38
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3 - 6 7
PupvB21 (P, ml, m2) ÷ gjz~,B22(p, ml, m2) = 16,rr2 Q4-n f
d"q
q~qu
i(27r), [q2_ m~ + ie] [ ( q - p ) 2
_ m2 + i e ] ' (B.4)
where Q is the renormalization scale and we regularize by integrating in n = 4 - 2e dimensions. The expression for A0 can be integrated to give
Ao(m) = m 2
(B.5)
+l-In
where 1/~ = 1/e - YE + In 4~-. The function Bo can be written in the form
,/
1
Bo(p, ml,m2) = -~ -
dx In
(l-x)
m2 + x mZ - x( l - x) p2 - ie
(B.6)
OR
o
It has the analytic expression
Bo(p, ml,m2) = ~ - In
- fB(x+) - f s ( x - )
(S.7)
,
where S z~z ~ / S 2 -x± =
fs(x)=ln(l--x)--
4p2(m~
- ie)
2p2 (B.8)
xln(1-x -1)-1,
and s = p 2 _ m~ + m 2. All the other functions can be written in terms of Ao and Bo. For example,
BI (p, ml,m2) = ~p2 Ao(m2) - Ao(ml) + (p2 + m~ - mZ)Bo(p, ml,m2)
, (B.9)
and
B22(p, m l , m 2 ) = 6 { ~l ( A o ( m l ) + A o ( m 2 ) ) + ( m 2 + m ~ - ½ P Z ) B o ( p , m2_- ml
We also define
ml,m2)
Ao(m2) - Ao(ml) - (m~ - m~)Bo(p, ml,m2)
D.M. Pierceet al./NuclearPhysicsB 491 (1997)3-67
39
F(p, ml,m2) =Ao(ml) - 2Ao(m2) - (2p 2 4- 2m 2 - m~)Bo(p, ml,m2) ,
(B.I 1)
G(p, ml,m2) = (p2 _ m 2 _ mZ)Bo(p, ml,m2) - ao(ml) - ao(m2) ,
(B.12)
H(p, ml ,m2) =4Bz2(p, ml ,m2) + G(p, ml,m2) ,
(B.13)
1
/~22(P, ml, m2) = B22(p, ml, m2) -
Ao(ml) - ~Ao(m2) .
(B.14)
The functions F and G arise in scalar self-energies, with either a vector boson and a scalar or fermions in the loop, while H and /~22 occur in vector-boson self-energies, with either fermions or scalars in the loop.
Appendix C. The gauge couplings In the remaining appendices we denote ~2 =- sin 2 t~w, where ~w is the DR weak mixing angle, and S 2 ~ sin 20w = 1 -Mw/M 2 2z, where Ow is the "on-shell" weak mixing angle and Mw, Mz are the gauge-boson pole masses. The DR electromagnetic coupling is given by O/ern & -- 1 -- Age '
aem
1 137.036 '
(C.1)
where 11 Age=0.0682±0.0007-
_ ~ { -71n ( M w ) 1 6+9 ~
2 4 u~Zln
1 (mH~
+ 3 in \ Mz ] + -9
1 2
i-I
(md~ ~
+9ZZln\M---zzJ d i=1
+3
In
( mMzz r )
(m~,)
MZZ
1 ~e ~-~ln(m~ ) 4 2 (rn2+~} Mzz + 3 Z l n \ M z J ' i=1 i=1
(C.2)
and ~ , indicates a sum over u, c, t, and similarly for )-'~d, )--]~e"In this expression, the number 0.0682 includes the two-loop QED and QCD corrections given in Ref. [41 ], as well as the five-flavor contribution " O~ha (5)(M2z) = 0.0280 4- 0.0007 of Ref. [42] d The DR weak mixing angle is given by [43] ~2~2 =
,/.gge
v/2M2zG~(1 - A~) '
T , ( O) ~ Ar=pH ~w
T 2 Re Hzz ( Mz ) M2z +8VB,
(C.3)
where /5 is defined to be C2/~"2, and tSvu denotes the non-universal vertex and box diagram corrections given below. The W and Z gauge-boson self-energies are given in Eqs. (D.4) and (D.9), We compute/5 via [43] 1 / 5 - 1 - A/5'
A/5 = Re [ llrz(M2z) [ /5M2z
II~vw( r Mw 2 )] M2
].
11The coefficientsof In(Mw/Mz) in the expressionsfor A& in Refs. 122,231 are both incorrect.
(C.4)
40
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67
We deduce the leading two-loop standard-model corrections to A? and A/) from [41],
as[2.145m2+o.5751n(mt) M2z Mzz
A? 2_loop - 4~s--2c2 ~ ~ [
- 0.224- 0.144 M~] m
12 (2) (mr) --SxtP ~ (1 -- A~)~,
(C.5)
&2 as-2.145 A/) 2-loop - 4~-~ 7r [ m~ +1.2621n ( mMzz ,) +3 xt
- 2 . 2 4 - 0 . 8 5 _~t~]
-~t '
(C.6)
where xt = 3Ggm2t/8~r2v~and m r is the standard-model Higgs-boson mass. For r ~< 1.9, p(2)(r) is well approximated by [44] p(Z)(r)=19-~r+]-~
43r2
+ ( 3 3 + 1 0 r3-Trx/~ 4 - ~ r +
13)
r 2+2--~r
--Tr2(2--2r+~r2)--lnr(gr--~r2) ,
(C.7)
while, for r/> 1.9, we use
p(2)(r)=ln2r(3-9r -l-15r-2-48r-3-168r-4-612r -5) -
In r ( _ ~ + 4r-1
125 r-2 4
558 ,.-3 5
8307 r.-4 20
109321 -5"~
--~ r )
+Tr2 (1 - 4r -1 - 5r -2 - 16r -3 - 56r -4 - 204r -5) 49 2 -1 1613r-2 8757 -3 341959r_ 4 9737663 -5 +-~- + ~r + 48 + l i - ~ r + 1200 + 980--------~r
(c.8) For the case of the MSSM, we replace the function p(2)(m~o/mt) with //COSCe'~ 2 p , 2f. ~[~mh~ _ _ __
~,sinflJ
~mtJ"
(C.9)
We have not computed the corresponding G2um4 higher-order contributions from the heavy Higgs bosons, but we know that they must decouple. Using the ansatz A/9 heavy Higgs
=~x~{ fsina~2 mn'~J ~,s~nflJ p(2)f~--~t
(ta-~ ) 2p(2)(m~A' ]}' \ mt /
(C.10)
we find these contributions are negligible. We do not include them in our results. The non-universal contribution to A~ is made up of two parts, one from the standard model and the other from supersymmetry, 6VB = 6"Svr~+ b'svUB sv .
(C.11)
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
41
The standard-model part is given by the well-known formula [43]
a S v M =^p ~ 2&{
--(5
6 + lnc2
3C2"~1
The supersymmetric part appears in Ref. [5], and more recently in Ref. [ 12]. We include it here for completeness. It includes box diagram contributions, vertex corrections, and external wave-function renormalizations. We neglect the mixing between different generations of sleptons, and we ignore the left-right slepton mixing in the first two generations, in which case the right-handed sleptons eR, tXR do not contribute. We find
6Z~+6Z~ + S Z . + S Z ~ .
xsusY=___M2zReal+6v~+Sv~+ ~vB 27r&
.
(C.13)
The wave-function and vertex corrections are 2
4
b*t~,~L 2B'(O'm*/'moL) - Z
167r26Z~=-Z
b,o,,,r,, 2Bl(O,m,o,m~,) ,
,j=l
i=1
(C.14) 2
2
16rr28Ze=- Z
4
ayri~er,~ Bl(O,m~i,,mr,~) - Z
i=1
2
b2oe& Bl(O, myd,m~L),
j=l
(C.15) 2
4
16~'23ve = Z
.
v~
y ~ b~,+~ ,~ , ~tb~oe~ xj ~ [,[---a~o~+wm~+m~,o g AjAi Ai .'tj Co(mo,,mydi ,m2oj ) "
i=l
+-~gb2O~j~ w Bo(O, m~;., m~o) + m~,.. Co(mo,,.. m~,+,.., m~,o).,j
-
4
-- ~
a~+er ,e b~% ~ Ai A j e ~1
~ ~
i=l
j=l
%
- - - be ~ o " U a~~wm~,, t , m~o Co(m~,, m2;~, m2o) o
+----~--a~ox~+ w Bo(O, m~/, m~o) + m~, x/2g ~j ' +'21 4Z
b**OeoLb2°d',~', ,
, m~o) -
[Bo(O,m~t,mr'~) +m2oCo(myro,m~,,mr,,) + ~1 .
j=l
(C.16) The corrections SZ~, SZ~,, and 6v u are obtained from these expressions by replacing e --~/x. The ,~-fermion-sfermion couplings ay(iffj and by(,ffj are listed in Eqs. (D.20)(D.22), while the chargino-neutralino-W couplings a2o,x,~w and b~9~w are defined in Eqs. (D.12), (D.13). In these expressions, the B0, B1, and Co functions are evaluated at zero momentum,
D.M. Pierceet al./NuclearPhysicsB 491 (1997)3-67
42
,
Bo(O, ml,m2) = ~ + 1 + In
1[1
+
(Q2)
B,(O,m,,m2)=-~
+l+ln
,
m2
in (m2"~
m~-m 2
(
m ~ ~2 ln(m~'~
~222 + \ m 2 _ m ] j
1____~[ 2m~ 21n(m2) m ] t m , _ m 2 \mZJ
Co(m, m2,m3)-m2
(C.17)
\mf,]'
1 (ml2+m~'~]
\m~/ + 2 \m2--mZJ] ' (c.18)
m~ ln(m~'~],
(C.19)
m,Z---m32 \ m 2 J
where Q is the renormalization scale. The box diagram contributions are 2 1 16rr2 aj = -~ Z
4
~ a~,~,,~,,~b*.-+.x,e*,. , ~ b~.ojv, , ~ b~oee.~j,. m2i~rn~?oD0 (m~L, rn~, m~?+,m)?o)
i=1 .j=l
1
2
4
+-2 Z ~ " a*-+ _ b-~p - b*-o - b*-o - rn~+m-o Xi ePe Xi ,~]zL XjPel]e Xj]£]£1. Xj Do(mr, L, m~, m U, m~o) i=1 j=l 2
+ Z
4
~
b-+p - b*+ - b*o - b-oe~ Xi ;~IZL Xi PeeL Xj~IZL Xj L D27(m#,, ms,., mL.+,m~,o)
i=1 j=l 2
4
+ Z ~ - - ' ~ a * "~i ~ .I&Uga ~ a~.+er, ~ D27(mr,,,mr, e,m2+,m~o) Ai e b~.ov * j tz r,I*b*.0. l[jVel*e
(C.20)
i=1 j=l
where the functions Do and D27 are
Do(ml,m2,m3,m4)
m2
'[
m2 Co(ml,m3,m4)
]
Co(m2,m3,m4) ,
O27(m! ,m2,m3,m4) - 4(m2 -l [ m m22 )C o ( m l , m 3 , m n ) - m 2 C o ( m 2 ,
(C.21)
m3,m4)]. (C.22)
We checked our box diagram calculation against Refs. [5,12], and we checked our formulas for 8Z and c$v with those of Ref. [ 12]. Here (there) the formulas are written in terms of the couplings corresponding to vertices with incoming (outgoing) charginos and neutralinos. To compare we must make the transformation a ~ f f ~ bygff, except for couplings involving the chargino and down-type fermions, which remain unchanged• Also, their ,~'+.~°W coupling differs from ours by a sign. The effective weak mixing angle is given in terms of the DR weak mixing angle, #2, via • 2 ~ l e p t = ~2
sm eelf
Re/ce
(C.23)
43
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
where [45] ~:~ = 1 + ~
Hz~(M2z) - IIz~(O) &~2 In c 2 & 2 M ~Z'T + 7r~-5--- 47r~2 Ve ( M z ) ,
(C.24)
with l (~2) (~2) ~(M2z) = ~ f +4~2g Ref(x)
g(x)=
x
2
3+
1 - 6se + 8s4 4c 2 f(1),
lnx
(1 1)(tan~ly ) 9 +
(C.25)
-
l (1)
1 +8+2xx-
l+~x
4(tan__1 y)2
.
x
(C.26) Here y =_ V / X / ( 4 - x), Li2 is the Spence function, and l l z r is listed in Eq. (D.15). We do not include here the non-universal Z-vertex supersymmetric contribution to sin 2 veff . The largest contributions can be obtained from Ref. [46].
jqlept
Appendix D. One-loop self-energies In this appendix we list all the relevant self-energy functions which allow us to determine the one-loop fermion, gauge-boson, and superpartner masses. We explicitly include all of the necessary couplings. We perform our calculations in the 't HooftFeynman gauge, in which the Goldstone bosons and the ghosts have the same masses as the corresponding gauge bosons. The gauge couplings g', g and g3, and the Yukawa couplings /~f are all DR couplings. The neutralino mixing matrix N, the chargino mixing matrices U and V, the Higgs mixing angles te and /3, and the sfermion mixing angles Of are described in Appendix A, as are the normalizations of the Yukawa couplings/If. The self-energies are given in terms of the Passarino-Veltman functions A0, B0, Bl, F, G, H, and Bee listed in Appendix B, Eqs. (B.5)-(B.14). To streamline notation we do not write explicitly the external momentum dependence of these functions, e.g., we write Bo(p, m l , m 2 ) as B o ( m l , m 2 ) . Throughout this appendix we write s for sin, c for cos and t for tan, so that s¢~ - sin/3, c2o, = 2Or, etc., and for the sfermion mixing angles, c, = cos 0,, etc. Sub- or superscripts f denote a quark or lepton, and q denotes a quark. Inside a summation ~f,,, the subscript or superscript u denotes all up-type (s)fermions, u, c, t, ~'e, v~,, ~'~, and similarly inside a summation y'~f,,, the script d denotes all down-type (s)fermions, d, s , b , e , tz, and z. The sum ~-]~L,/fd denotes a summation over (s)quark and (s)lepton doublets, and the sum ~'q denotes a sum over (s)quarks. Some terms are zero, for example a~ = 0, and terms involving the right-handed sneutrino are absent.
cos
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
44
In the self-energies listed below the 1/# poles are canceled by counterterms which relate the bare mass to the running mass. So, in the following DR self-energies we implicitly subtract the 1/# poles.
D.1. Z and Wbosons The full one-loop MSSM gauge-boson self-energies appear in Ref. [5] and subsequently in Ref. [6]. The supersymmetric contributions are listed in Refs. [47,22]. The self-energies of the gauge bosons can be separated into transverse and longitudinal pieces, e.g., ,a~, T 2 [gUV p/~pV] L 2 pUpV Hzz(p2) =Hzz(P )_ p2 J + Hzz(p ) p2
(D.1)
The physical gauge boson masses are the poles of the corresponding propagators, which involve only the transverse part of the gauge-boson self-energy,
M2z =Mz(Q)^ 2
- Re Hrzz(M2z),
(D.2)
M2w = ]Q2w(Q) - Re Hww(Mw). T
2
(D.3)
Here Mz(Q) and ~iw(Q) denote the DR running masses which are related to the D--R gauge couplings and vev's, as in Eq. (A.2). The gauge-boson self-energies are evaluated at the renormalization scale Q. The transverse part of the Z-boson self-energy is 16~ 2 ~
T 2 ~ Hzz( p 2 ) = -s,~fl B22(mA,mta) + B22(Mz,mh) - M2zBo(Mz,mh)
2 --CAB
I~22(Mz,m~l)+~22(mA,mh)_M2zBo(Mz,mH) 1
--2c4(2p2-~M2-M2~)no(Mw,
-(8~ 4 +
Mw,
2 ~ 2 ~ c20w)B22( Mw, Mw) - c20wB22(mH~ , mn~ )
2 4N~s 2g fijB22 ( m ]?
m.?j)
f i,j=l
+~f Nf{(g2fL+g2fR)H(mf, mf)-4gfLgf~m~Bo(mf, mf)} ~2 4 } +~g2 ~-~{ f~ijzH(m~(°' m~o) + 2 g°jz m2~m~,?Bo(m~o,m~:~) i,j=l " C2~If+z +g~ -i,.j=l ~"
H(m2+,m-+ ) + 2 giiz + m~mr,~:Bo(m~,:~,mr,. +) } , X] -~-- ,,, ,,j ,,t "'1 (D.4)
D.M. Pierce et a l . / N u c l e a r Physics B 491 (1997) 3 - 6 7
45
where the summation ~-~f is over all quarks and leptons, and the color factor N[ is 3 for (s)quarks and 1 for (s)leptons. The notation s,~rl denotes s i n ( a - f l ) , and carl refers to cos(a - / 3 ) . The sfermion-sfermion-Z couplings can be written in terms of the weak neutralcurrent couplings defined in Eq. (A.7): V f l l = g f ~ c f2
_ gf, sZf ,
2 vf22 = gf~cZf - gf, sf
V f l 2 = Uf21 = ( g f L -~ g f R ) C f S f
(D.5)
•
The neutralino-neutralino-Z-boson couplings are defined by
~ z• = [a~?~°zl2 +
Ib~?~°zl 2,
. = 2Re ( b*.-o~oTa~o~Oz) giOz xixJ l~ AiAJ / ,
(D.6)
and analogous definitions hold for fij+z and gijz. + We write the Feynman rule for the )~)~Zu vertex, where ,~ is a chargino or neutralino, as --iyu(a79L + bT~R), where T'L,R are the usual chiral projectors, 7~L,R= (1 :F Y5)/2. The couplings involving the unrotated ~o and ~+ fields satisfy b~;o$oz = -a~o~o z and b~?~j~z = a(,/~jz. The non-zero a-type couplings are
g
a ~//3~//3 -0 -07 -0 -07 ~ = - a ~//4~4 ~ = ~-~ ,
a(q~; z -
a~(,~z = g~,
gc2ow 2~
(D.7)
For an incoming f(o and incoming ~(+ we have
a Zi-o-o Xj Z
= Ni*k N i l a(,o~,oz ,
aL,2~z = Vi*k Vjt a~;~? z ,
b2°yc°z = Nik Nj*Ib(,o(,oz b~] z
,
= Uik U~.l b~;~+ z .
(D.8)
(Here and in the following formulae which specify rotations, we adopt the summation convention for repeated indices.) For the transverse part of the W-boson self-energy, we find 167"r2g2 H~w(p2) = -s,~rl2 I~22(mn,mrt~ ) + B22(mh, Mw) _ MwB02 (mh, Mw) 1
2 [B22(mh,mH+)+ B22(mH, M w ) - M2Bo(mH, Mw)] --Carl --O22(mA, m n ~ ) - ( 1 + 8~'2)/~22(Mz, Mw) -~2 [8/~22(Mw, O)+4p2Bo(Mw, O) 1 - I ( 4 p 2 + M 2 + M 2 ) ~ "2 - M 2 s 4] Bo(Mz,Mw) +
I-I(m.,m,~) - ~ ,,
i,j= 1
2N cI w ~fqB2z( mr,. rod2)
46
D.M. Pierce et a l . / N u c l e a r Physics B 491 (1997) 3 - 6 7
--
+g2
fijw
i=1 j=l "
H (re,o, m,f ) + 2 gijw m~o m ~j~.+Bo (m "4i~o, m,~ ) --i
}
'
(D.9) where the s u m m a t i o n ~--~f,,/fd is over quark and lepton doublets, and W f l I "= CuC d ,
Wfl2
= CuS d ,
Wf21 = Sued ,
W f 2 2 _m SuSd .
(D.IO) The neutralino-chargino-W-boson couplings are f i j•w
= la,o,fwl
2 -'b
gijw = 2 Re
Ib,y,;wl 2 ,
(b*o-+ x, xj w a-o-+ z, xj w )
•
(D.11)
We write the Feynman rule for the neutralino-chargino-W u vertex as - i y u (aT'L + bT~R), and the non-zero couplings are
= b4'°4"~w= - g '
W
a~'°~w = -b~°&+-w= ~22"
(D.12)
For an incoming ~o we have the couplings to mass eigenstates,
a~o~ w = N~k Vjla(,o(,/w,
byc°2~w= Nik Uflb(,°~?w'
(D.13)
while for an incoming f(+ we have the couplings
a~o~ w = Ni~ Vj*l a4,o(,i,w ,
bycosc~w= N*k Ujtb(,o~j~w"
(D. 14)
Finally, we write the mixed Z - T self-energy as ^
167r2C-~-Hzr(p2) = ( 12~ 2 - 10)/~22(Mw, Mw) - 2(MZw + 2d2p2)Bo(Mw, Mw) eg -b~Nfef(gf,.-gfR)[4B22(mf,
mf)+p2Bo(mf, my)]
f
- 2c2bwB22(mn+ , mn+ ) +~ ~-~( IVa 12+ lUll I2 + 2c2&) 4/~22(m~, ms(~ ) + p2Bo(m2?, m2~ ) i=1 s ef [ ( g f L c f ~ - gs.s}~ NSc
-4 ~ f
J~22(m/l, m~ )
I. I
+(gy, s2f - gf~c})B22(mk, mA) 1 • /
(D.15)
D.2. Quarks and leptons The fermion masses are defined as the poles of the corresponding fermion propagators. They are related to the D----Rmasses, ~hf, by the self-energies, Zf(p2), as follows:
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
47
mf = r~f(O ) - Re 2f(m2f ) .
(D.16)
The DR fermion mass /'~/f is related to the DR Yukawa coupling and vev as shown in Eq. (A.I). Care must be taken in evaluating the DR vev. After evaluating the DR gauge couplings g' and g as outlined in Appendix C, we determine the DR vev via v2(O) = 4 M2z + ReHrz(M2) g,2(Q) _+_g2(Q)
(D.17)
,
where Q is the renormalization scale (the argument of the Z self-energy is the external momentum; it implicitly depends on the scale Q as well). For the top quark, ~Vt(p2) is
{
16~'2 Xt(PZ)m, - 4g~3 Bi(m~,m~,) + Bl(m~,m~2) -
(
°2)
5 +31n~7
-szo,~(Bo(m~,m~,)-Bo(m~,m~2))}
+c213[Bl(mt,mA) --Bo(mt, mA)] +s2~[BI(mr, Mz) -Bo(mt, M z ) ] } +-~ (.,~2s2~q- At cB)Bl(mb, mH~ ) + (g2 + .,~2C213q_ .~t SB) B l(mb, M w) +/I2c2~[Bo(mh, mH,)--Bo(mb,mw) 1 - (ee,)2(5 + 31n m~ ) +~_~
gt2L+&2 Bl(m,,Mz) +4g,Lg,~Bo(mt, Mz)
+~
f~aj B1 (re,o, r% ) + g~t~j i=1 j=l
fitbjBl(m~+,,mbj)
+2
+ git~j mt
mt
Bo (rn,o, m~j ) mL* , mbj) ]
(D.18)
i,j=l
The neutral current couplings gf are defined in Eq. (A.7). We write the Feynman rules for the )(iffj couplings as -i(aPL + bPR) (for vertices involving the chargino and down-type fermions the Feynman rule is iC -1 (aPE -4-bPR), where C is the charge-conjugation matrix). We define L f L = la*,fLI 2 +
Ib~,,fL 12'
gif}j = 2Re (b,,f?j air,T?j) .
In the unrotated ~0, ~+ basis, we have g!
I
b&of },. = -~2 Yf " '
(D.19)
48
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 b ~ f f L = v / 2 g l ft- '
a(~doL = b(,~udL = g,
a(joddt. = b~odd R = - - b ~ d~L = - b ( q u d R = hal, a(~ou~L = b~ou~ ~ = - a ~ ud~ = - - a ~ d~R = au ,
(D.20)
where the quantum numbers Yf and I f are listed in the table of Eq. (A.8). These couplings correspond to vertices with incoming neutralinos and incoming charginos. To obtain the couplings to the mass eigenstates j,0 and )~+, we specify the rotations a 2 o f f = N~ a ~ o f f ,
a ~ f f f , = Vi~ a ~ + f f , ,
b2off = Nij b ~ o f f ,
(D.21)
bscff f, = Uij b~; f f ' "
(D.22)
The couplings to the sfermion mass eigenstates are found by rotating these couplings (both a- and b-type) by the sfermion mixing matrix,
a2ff~
--s f, c f,
a 2 f f ,~
The self-energies £ y ( p 2 ) for the other up-type quarks and leptons can be obtained from the previous formulae by obvious substitutions. For the bottom quark (and similarly for all down-type fermions), one interchanges t ~ b, c~ ~ s~, and c# ~ sty. D.3. Charginos and neutralinos
The complete one-loop self-energies for charginos and neutralinos are given in [ 32,9] ; we present them here in a matrix formulation. For the Higgs-boson contributions, refers to H, h, G°, and A, while H + represents G + and H +. The GO and G + are the Goldstone bosons; in the 't Hooft-Feynman gauge their masses are equal to M z and M w , respectively. We now describe the full one-loop neutralino and chargino mass matrices, from which we determine the one-loop masses. The one-loop neutralino mass matrix has the form .h.4~0 + ~1 (6.Me;0 (p2) +SA/tr_o(p2) ) g ,
,
(D.24)
where ~.A/[ffo ( p 2) = --~R(pZ).A,'[~0 -- .A/[ffO~0L(p 2) -- ~ s ( p 2) .
(D.25)
+0 2 Here A.4~o is the tree-level neutralino mass matrix of Eq. (A.3), and the ~r,'R,s(P ) are matrix corrections. They allow us to determine the one-loop masses and mixing angles for arbitrary tree-level parameters. The one-loop chargino mass matrix is as follows: .h4~ - 2 + ( p 2) .h4~+ - .MgT+ ~:~-(p2) _ 2:s+ ( p 2) ,
(D.26)
where .h4~ is the tree-level chargino mass matrix of Eq. (A.4). The elements of .M~o and .A.44;, contain DR parameters at the scale Q. In particular, they include corrections
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3 - 6 7
49
corresponding to replacing Mz with /l~/z, obtained from Eq. (D.2). Similarly, tan/3 in the tree-level matrices is tan/3(Q). The self-energies 2L,R,S are also evaluated at the scale Q. To obtain the mass for a given neutralino or chargino, for example ,~0, we first evaluate the matrix of Eq. (D.24) with the momenta p2 = m~0. We then solve for the eigenvalues of that matrix. So, in determining four neutralino and two chargino masses, we construct a total of six different matrices. We compute the mass matrix corrections by evaluating two-point diagrams with unrotated neutralinos or charginos on external legs, and mass eigenstates inside the loop. We obtain the couplings associated with these diagrams by the following method. The neutralino mass corrections involve the couplings ago.., which we obtain from the various couplings aL0.. (for incoming X °) by leaving off one factor of Ni*k. The neutralino mass corrections also involve the couplings bg~)., which we obtain from the couplings b)?l).. (for incoming ~(o) by leaving off one rotation Nik. We obtain the couplings agk`... which appear in the chargino mass corrections from the couplings a)?/... (for incoming ~',.+) by leaving off one factor of V/~, and we determine the couplings bg~ ... from the couplings b t/... (for incoming ~(i+) by leaving off one factor of Ua,. For the neutralinos, we have the one-loop correction 2
16"rr2~Li.j( p 2) =Z~-~'~ NJc f agoff * k ag~f}kRe B1 ( m f, mfk ) f
k=l 2
a~°2;w ag°,~<~iwReBj (m~<2 , Mw)
+2 Z k=l 4
+ Z a~°2°z a~°~°z ~u)at ReB1 (m2o, Mz) k=l 2
+ Z a~?22n,~,ag°,~:H+Re Bl (m2;, MH,+' ) k,n=l 4
1
+2 Z a;°j<°n°,,ag%?°H,°,Renl (myc°t' MH°,,) ;
(D.27)
k,n=l
is obtained from .sol by replacing the couplings ago.., with bg0.... The ~ss(P 2) correction is given by 2
16~"2 .~ss~;(p 2) = 2 Z
Z Nf b~°,IA ag°ffk mfRe Bo(mf, mfk )
f
k=l 2
-+w a},o ~-jx, - my<2 Re Bo (m22, Mw) -8 Z b*0 ,/,;xk -+w k=l
50
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67 4
- 4 Z b*o -o~ a.r.o:oz m~.o Re Bo(mko, Mz ) qti X,~L ~j a',~ ak k=l 2 b ~ ° ~ H ~ a,~,o -, t4~ m , ~ R e B 0 ( m , ~ , MH,+' )
+2 E
wi Xk n
k,n=l
wj Xk .
4 b,~;o ~ouo a.r,o r,Ono,, m :,~k R e B o ( m : , ,'k MH..o) •"iak",, "j~k
+ Z
Q
(D.28)
k,n=l
The chargino mass corrections are given by similar formulae, 2 2 + 2 1 167r X L i j ( P ) = 2 E
, E NTa(~+f?~ a~fff~ f
Re
B, (mf, mf~)
k=l
4
+ Z ax°CW a*°(,;wReB' (m2o, Mw) k=l 2
+ Z a~/~;z a¢2~zRe B1 (m~(~,Mz ) k=l 2
+ Z a;+2~, a-+ -+r Re Bl (m~'1' 0) O~xk k=l 4
1
2
+2 Z E a~°k¢:H,+,as(°¢';H,,+Re B, (m2o, MH+' ) k=l n=l 2
1 +~ ~XT"~
4
a*+,,X,~'+H0,a.z+~j , *k~+H°Re,, B1 (m~2, MHO,), ;
(D.29)
k=l n=l
X~(p 2) is obtained from X+(p 2) by substituting a(,:.., with bff,~.... X~-(p 2) is given by the following formula: 2
16~rZX+ii(PZ)=ZZ * ff~ asf ff~ mfRe Bo( mf, my; ) • N¢f b~,: f
k=l 4
-4 E b*k°d,/wa~o(,]w m~o Re Bo (m2o, Mw) k=l 2
-4 ~
b*,-. . . . a.~+~+ z m~,~ R e B o ( m 2 2 , M z ) gti Xk L ~'j ,~k ak
k=l 2
-4 E b~.)~+r *,r, a.r,+~+,,m~+ReBo(m~,O) ~.j,,~, ,,k ,,k k=l
D.M. Pierce et al. /Nuclear Physics B 491 (1997) 3-67
51
(D.30)
In these expressions, $ff
couplings
the color factor NJ is 3 for (s)quarks,
are listed in Eqs. (D.20)-(D.23),
are given in Eqs. (D.7), from the following
(DJ),
equations,
(D.12)-(D.14).
and 1 for (s)leptons.
and the &jZ We determine
The
and $iW
couplings
the $+j+y
couplings
which apply for incoming 1::
where we write the chargino-chargino-photon We next list the if-Higgs-boson couplings.
Feynman rule as -iy,( aPL + bPR). We write these couplings in the unrotated
Higgs basis (st , sz), (~1 ,p2), and (ht, hz). These fields are rotated to obtain the mass eigenstate fields. The (H, h) rotation is given in Eq. (A.14)) while for (Cc’, A) and (G+, H+) we have
We write the Feynman
rules for the j”josk
as (aPL + bPR) . These couplings
couplings
as -i(aPL+bPR)
and for j”topk
are symmetric under i H j and satisfy bG;lc,oS,= aJlpqyPst
and b&oq,,,k = -a$p+oyk. The non-vanishing 1I 1I g’ - agp$;,, = qjpps2 = 7j- ,
a-couplings
a$:6:.s, = -agp:,,
are g = z ,
(D.33) (D.34)
The couplings ajp,+,
to incoming neutralino
mass eigenstates
,$ are
= Ni;, Nj; a+;$+,, 7
b-0 -0Sr= Nik Njl b~;~~s,, , x, x, and likewise for pn couplings. by rotating these couplings,
(D.35) The couplings
to Higgs-boson
and likewise for the b-couplings. We write the Feynman rules for the f+j+--neutral-Higgs for couplings with CP-even s-fields, and (up, + bP,)
mass eigenstates
are found
couplings as -i( aPL+bPR) for couplings with CP-odd p-
D.M. Pierce et a l . / N u c l e a r Physics B 491 (1997) 3 - 6 7
52
fields. These couplings satisfy b~? ~.,, = a6j~6, s,, and bd;, 4~ p,, = -ad) (,;>,. The non-zero a-couplings are g
a('i' 0~2s~ = a67~4'i' .32 = a~b~(,~ t,~ = - a ~ 4+ m = ~ "
(D.37)
The couplings to incoming X(+ are obtained from these as follows:
a2/2) ,,,, = Vi~ U)*ta6~4;s,,'
b2) 2~.,.,, = Uik Vjt b(,~? ~,, ,
(D.38)
and the same rotations apply for the p,-couplings. To find the couplings to Higgs-boson mass eigenstates, we rotate these couplings by the angle c~ or fl, just as for the ,~'°~'°s and ,~0~0p couplings in Eq. (D.36). The )~'°)~,+-charged-Higgs-boson vertex Feynman rules are written as --i(aPL + bT~n), where, for incoming ~o, we have
g' a&°(,2' h~ = bg,o&j h~ = - ~ '
g a&o(,j h~ = b&°6]h~ = - - ~ ,
a,r.o,z tq = -b6°4~ h~ = - g "
(D.39)
To obtain the couplings to chargino and neutralino mass eigenstates with an incoming neutralino .~0, we rotate these couplings as
a~',~2~h,', = Ni*k U/*l a(,o(,/h;l '
b.-,~-~,~j ~, h',, = Nik Vii. b(,°~l h;', ,
(D.40)
while for an incoming chargino ..~.+, we rotate them as
a-o-~ XiXj h',, = Ni*k V~" .1l b-o O~l-~ h ,~, '
b~o~,h+ . x i x ) ,, = Nik Ujla,~,o,;,,h~ .r~.r~
(D.41)
To find the couplings to charged-Higgs mass eigenstates, we rotate both a- and bcouplings by the angle/3,
(a2°2'~) a2,,2,14-,
:(
c# s/3) ( a ~ ° ' + / q ) - sfl c# a2"2~lq
.
(D.42)
D.4. Gluino The gluino self-energy appears in Refs. [ 8 - l l ]. The physical gluino mass satisfies m~, = M 3 ( Q ) - Re 2;g,(m~) ,
(D.43)
where X~(p2)=l--~ 2 -m~
15+91n
+ZZm~Bl(mq, q
q
where (2 is the renormalization scale.
i=1
mo,)
D.M.Pierceet al./NuclearPhysicsB 491 (1997)3-67
53
D.5. Squarks and sleptons We find the sfermion masses by taking the real part of the poles of the propagator matrix Det [p/2- 3,t~(p/2)] = 0 ,
m~i = Re(p/2) ,
(D.45)
where M}(p2)
=
( M2I I,. -Hf'f'~(p2) M2. _ Hf~f,.(p2) flefl,
M2"M 2- . fRfR
)
- u},t.(P2
lIf, fR
(p2)
.
(D.46)
The matrix formalism allows us to determine the one-loop masses and mixing angles for arbitrary tree-level parameters. In this expression, the Mf, f, (i,j = L, R) are the DR tree-level mass matrix entries given in Eqs. (A.5), (A.6): all the entries contain running DR parameters at a common scale Q. In particular, the DR tree-level matrix contains corrections from the replacements M2z --+ M2z = M2z + Re Hrz (M2z) and mf --~ fnf = mf + ReXf(m2f). (The arguments of these self-energy functions are external momenta, not the scale Q.) The lI], L, (i,j = L, R) are the sfermion selfenergy functions evaluated at the scale Q. Of course, for the first two generations of sfermions, both the tree-level and one-loop contributions to the off-diagonal elements of the mass matrices are negligible. Note F~,},. ¢ Hf,,}~ because of the absorptive part, which contributes to the mass-squared at O(~2). For a t'c squark we have 167r2HhjL (p 2 )
= 4g23[2G(m~,m,)+c~F(mh,O ) +s:F(m~,,O). +c:Ao(mh)+s:Ao(m~)l. +Af(s2tAo(mh) +c~Ao(mh) ) +A~(s2Ao(mb~) +c~Ao(mb2))
,4(
2)
+~ ~-~ a~ D.. - g202 g'---LG Ao(m.o,,) n=l
4
n=3 4
\ 2~2 2
.... 2
q- Z ~~( An°h-?,)aB{'(mH°'m?,) + Z ( AH,+,~,.b,)2Bo(mb,'mHd) n=l
i=1
i,n=l
q-4g2~--(gtt.)2AO(Mz)+2g2AO(Mw)+(ete)2(c~F(mh,O)+s~F(mh,O))
+g=j(gtt)2Ic~F(mh , Mz) -t-s~F(mh,Mz) ]
D.M.Pierceetal./NuclearPhysicsB491(1997)3-67
54
+-~
+ s2,ao(mf~)
ftf (cfAo(mf,)+s~Ao(mf~)) 2 +g2~f NJcI313 +gl-~-4(YtL)2(c2tAo(m~,)+s~Ao(m~2)) +gZ~--~YtL~fNf[YfL(c2fAo(mf,)+s}Ao(m,2)) +YfR(s2fAo(mfl) +c~Ao(m?2))l 4 +i~__l[fit~LLG(my~°,mt)--2git?L,.m2°mtBo(m~°,mt) 1 + Z fib~L,G(m~i'' m~) - 2 gib?LLmL~mt,Bo(m~i',mb) ,
(D.47)
i=1 and similarly for a tR squark, 16~-2 HTR~R(p2)
4g~
2G(m~,m,) + sZ,F(m~,,O) + c~F(mT~,O) + s, Ao(m~) + c2,Ao(m~)
3 +AZt(c2tAo(m~,)+s2Ao(m~2)+c2bAo(mb,)+s2Ao(m~,2)) 4 g2gtR +.~-~(A2Dnu_' g2gt~cn)Ao(mHo,)WZ(A2Dnd_l_--~TCn)Ao(mH ,,-2) n=l 4
n=3 " 2
2
+ Z Z (AH%?i)2B°(mH°'m?,) + Z(AH,+?,b,)2Bo(mb,,mH+) n=l i=1 i.n=l 4g2 +--~(gt~)ZAo(Mz) + (e,e) 2 ( s~F(mT~,O)+ cZF(m~2,0)) g2 (gt,~) 2 [s~F(mTl q--~-~ ,Mz)
+c~F(m~z'Mz)]
gt2 +gt--~4Ytk~fNf[Yf,.(c2fAo(m?,)+s2fAo(m,z))
D.M. Pierceet al./NuclearPhysicsB 491 (1997)3-67
55
+Yf~(s2fAo(mfl) +c2fAo(mf2))] 4
-+-~=l[fit?RRG(m2o,mt)--2giffkRm2°mtBo(m2°,mt)] + Z2 [[fib~RRG ( m~)~,mb ) - 2 gib?Rnm2; mbBo ( mL', mb ) ] .
(D.48)
i=1
The off-diagonal self-energy is 16rr2Hh~ (p2)
= 4g~I4mgmB'°(mgm' )t3 4
+stct(F(m?,,O)-
2
F ( m T 2 , 0 ) - A o ( m ? , ) + Ao(m~,))]
2
+ Z Z "ln%7,ata°,,~R~,B°(mH°,,'mTi) + Z at4,tbS,hH,~?,bflO(mb,'mH,+,) n=l
i=1
i,n=l
gt2
/"
.
'+-@~f NfAuS2o,,(Ao(m~l)-Ao(mr4,.))-f---~-Yt, Ytkstct~Ao(m?,)-Ao(m?2) ) +(ete)2stct(F(mil,0) -F(m?2,0) ) g2 gt~gtt¢stct (F(mh,Mz)_F(m~2,Mz)) 4
+~'~[fit[rRG(m2o,mt)--2giffm, m2omtBo(m2o,mt)] i=1
[ + Z2 [fib~.,G(m~i~' mb) -- 2 gib?LRm2{mbBo(m2/, mb) ] •
(D.49)
i=1
Inside the sum ~ f , the sub- or superscript f refers to (s)quarks and (s)leptons, and in the sum y~'
f it?t., = ax~,,,?,,a2°,tTR--k b~& b,oaR , giff~.g= b*2o& a2otTk + a,off,, * b2offR ,
(D.50)
fn, b~ - ,, hrz.a~/b~ + bk~' h~,.bydbr~ , .
--
a*
-- b*
*
*
(D.51)
56
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
with analogous definitions for the LL and RR couplings. The 2 f f couplings are listed in Eqs. (D.20)-(D.22). The Higgs bosons ~ refer to H, h, G°, and A, and H + refer to H ÷, G ÷. The H , H,,-sfermion-sfermion couplings involve C, and Dnf , and are given in the following table:
n
Cn
l
--C2a
2 C2a 3 -c2~ 4 c2~
Dnu
Dnd
2 Sa 2 Ca
2 Ca 2 Sa
s~ c2~
c~ s~
(D.52)
We write the Feynman rules associated with the CP-even-Higgs-sfermion-sfermion vertices as -iA, and list the couplings Asff in the following table: SI
$2
~L~L
gMz ---~---gu,C~
- ~gMz c guLS/3+ v/2aumu
fiRfiR
gMz e 'guRC#
- ~gMz c guRs#+ v/-2Aumu
hu
UL~R
A~ A.
"~ ~
V~
(D.53)
dtdc
gMz
g&c/3 + V~Admd
gMz ----'~--C gdLS#
dRdR
gMz gd~CB -F V ~ Admd
gMz gdRSfl --~-ff'--
-
Ad
-
dLdR
ad
v~Ad
"-'~#
We find the couplings in the fl,2 sfermion basis via (D.54) /~s,,./r2f, /~s,,f2f2
--Sf
Cf
\AS,,fRfL
/~s,,ykfg
Sf
Cf
'
we obtain the couplings in the mixed fL,Rfl,2 basis by omitting the left-most matrix on the right-hand side of the above equation. The couplings to the CP-even Higgs-boson eigenstates (H, h) are obtained from the couplings to (sj, s2) using the rotation
"~hf,ij
-s,~ c,~
A~2i,ij
'
(D.55)
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67 The couplings ,t6.0i,fj and
/~Aflfj
=
57
"~Afifj vanish for i = j, while for i 4: j they satisfy
--/~Afyfi" We write the Feynman rules for these couplings for incoming fL as
A. They are 6~
A
+ a.s.)
dLdR
V5
A,I (IZS~ + Aac~)
(D.56)
-
ha (/zc# - Ads~)
v5
We obtain these couplings in the fc,Rfl,2 basis by a rotation as described after (D.54). We also write the Feynman rules for the charged-Higgs-sfermion-sfermion vertices in the form -i,~. The couplings Ac, Y,d~ and an~ ~,~; are
G+ F~LdL
gMw v/~ce# -- Aum, SB + AdmdC#
~RdR
H+ gMw ~ $213 A,m,,c# - ,tdmas~
0
- -
-- A,,mdcB -- Admus B
fiLdR
Aa (~S~ + Zac~)
Aj (lzc~ - A~Is~)
tTRdL
-- Au (t.zc~ + Aus~)
h. (#s~ - auc#) (D.57)
These couplings are obtained in the fl,2 basis via
( Att~'d' '~n~'d2 ) = ( cu Su ) ( AH+r~"dL/~t4+~LdR) ( cd -sd ) AH,g~2dI l~H+Fted2 --Su Cu /~Ht~t~dtI~H+~RdR Sd Cd "
(D.58)
We obtain the mixed sfermion basis couplings to ~L,Rdl,: (fil,2dL.R) by leaving off the left-most (right-most) matrix on the right-hand side of the above equation. The expressions for//b,b: are obtained from//~,~j by interchanging the indices t ~ b, replacing u ---, d, and substituting c~ +--* s¢~. The self-energy of a charged slepton (sneutrino) is given by a formula similar to that for a b-squark (t-squark), with the SU(3) correction set to zero and with the appropriate SU(2) x U ( I ) quantum-number substitutions.
D.6. Higgs bosons The full one-loop MSSM Higgs-boson self-energies appear in Refs. [6,7]. Corrections to the Higgs-boson masses are the subject of Refs. [36,37]. We discuss the relations between the self-energies and the pole masses of the Higgs bosons in Appendix E. Here we list the self-energies. The Higgs-boson contributions to the Higgs-boson self-energies involve the trilinear and quartic couplings, which we denote .~tf,o,:4o~k,Au,,' • ..... , and AH,,H,,H.,N°,,AH,,U,,U~U,, ,
58
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
where the ~ refer to the H, h, G°, and A Higgs bosons, and the H + refer to the G-and H + Higgses. The GO and G + are the neutral and charged Goldstone bosons, which in the 't Hooft-Feynman gauge have masses Mz and Mw, respectively. For the two CP-even Higgs bosons, we have 16¢r2/Ts, s, (p2)
4m2d)Bo(md, md) -- 2Ao(md)
+
Ao(md~) + Ao(m&)]
Nmr (c>+m.~,~+s>o,,m::,) f 2
+gf,(s2fAo(mfi)+c2fAo(mfz))l+ZZN{h:if~?:Bo(m?i.m?:) f
+
+c~[2F(Mw, Mw) +
i,j=l
"<.ma,,"'.'l,...,2
F(Mz>Mz)] -~ j}
7 2 2[
-~g c13 2M2Bo(Mw, Mw) +
MZBo(Mz'Mz)] ~2
+g2[2Zo(Mw) + Ao(Mz) ~-----~] ' mHo" ) + ,~HOHOslslAO(mH~) 1 + 1 4 [ D .~2HOHO s' Bo ( mHo. "2 = _ ".,
-'~
[Z
,'/=1 Lm=l
+~ ~
A 2H+H£ sl B O ( m H + , m H + ) +
t~H,+H,.s,,,Ao(mn+)
f~Js,,G(m~?,m~7) - 2g°is. mem~TBo(mx.7,ms(~)
i,j=l
+ ~_.[f~.,G(m~.+,,m.~ - 2 gijsN + m~, m~f Bo (m2)~,m2j ) ,
(D.59)
i j=l
16~'2IIs2s2(p2)
: Z'~:c"~[<; -4m>'o
2C 2
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
59
2
+gf~(s~Ao(mji)+c~ao(mj2))]+ZZN{~2f~fjBo(mj~,m]j) f i,j=l
F(Mz, Mz) +Zg2s~4 [2M~Bo(Mw, Mw) + M2zB°(Mz'ao Mz)] +g2[ 2A°(MW) + Ao(Mz) ~-------5~] ~ 4 q_~1
2 [~-~ a,o..o BO(mHo,mHo) + /~HOH%~s.Ao( .... ..... mHo ) ]J *~n r a m $2
n
m
n=l Lm=l 2
+~
I...d
2
2
[~--" A. . . . . ~
£1 n 1"1m S 2
Bo(mn4, mH,;) -kn
n=l Lm=l
]
AH;~,H,Ss2szAO(mH,I) ]
,4[
1
i,j=l 2
+Z[f-i]:s2zG(mu,m,+)-2g+s22m,+m,]Bo(m~),m,j~) ]
(D.60)
i,j=l
2
16"trZ/ls,s2(p2) = Z Z
Nf'~s,f, fjAszi,YjB°(m], ' mfj)
f i,.j=l
÷~g2s/3c~{2F(Mw, Mw) - 2F(mH,~,Mw) q
F(Mz,
Mz) - F(ma, Mz) ~2
+7[ 2M2B°(MW'MW) + M2zBo(Mz, &2 Mz)] }
1
q'-~
AHOHO~,.,", , '~HOHOs-BO(mH ., ~ ,, mH°.,) + AHOHOs~szAo(mH~,,) 2
+~
2
i,j=l L
o 2 mx,omgBo(m~o, s,2G(ms°,mg) - 2gijs, , • , m~)
]
60
D.M. Pierce et aL/Nuclear Physics B 491 (1997) 3-67 2
+Z[ft+s~,G(m2i,,myG,)-
(D.61)
2 &is,~ + m2,~ m~f Bo ( m ~ , my~f ) ] ,
i,.j=1
where N f is the number of colors, which is 3 if f is a (s)quark and 1 if f is a (s)lepton. The neutral current couplings g f are defined in Eq. (A.7). The j'i~'.j-Higgs couplings fij+,k~, gim,,+ fq.~, and gijs~,° are defined by f i.#'k, = a*2,2jsk a2~2Js, + b x~2jsk bk,2:s, ,
(D.62)
g~#~l = bx~2j.~ka2~2jsl + a2~2jsk b2,2js~ ,
and the a2~2j,k and b2,2js k couplings are defined in Eqs. (D.33), (D.35)-(D.38). The Higgs-squark-squark couplings .tt4?~f~ and Ah?jfk are given in Eqs. (D.53), (D.54). We write the Feynman rules for the relevant quartic Higgs couplings as -iA, and define AI4,,H,,H,,H,,, = g2/(4O2)-XFt,,H,,H,,I4,,. We list the necessary ~n,,H,,H,,/t,, couplings in the following two tables: SISI
HH hh G°G ° AA G+G H+H -
$2S2
3s~ -3c] -
3c~ - s a2 3s] - c a2
S1S2
C2a S a2
C2fl
--C2~
-c2~
c2~
--S2a S2a 0
0
~2-.}-~2C2 ~
~2--j2C2B
--~2S2fl
e2-~2c2~
G+g2c2B
~2s2# (D.63)
G°G° AA G+G H+H
AA
H+H -
3s2~ - 1 3c2~ e2(l + s~#) - g2c~#
~2(1 + s~#) - ~2c2# c2# 2s~# - 1
-
(D.64) For the couplings involving (sl, s2), we obtain the corresponding couplings in the (H, h) eigenstate basis by the following rotations:
AH,,H,,Hh An,,n,,hh J
--Sa Ca
\ AH,,H,,sb~2 An,,n,,s2s2
Sa
Ca
We write the Feynman rules for the trilinear Higgs-boson couplings as -iA, and define ,tH,,N.,,, = gMz/(2O)-XH,,H.,s,. We list the AH,,Ho,s, in the following two tables:
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3 - 6 7
HH s,
c ~ ( 3 c ~ - s ] ) - s~s2~
s~
s~(3s]-c])
G°G°
- c~s2~
AA
G° A
61
hh
Hh
s~(3c~ - s~) + c~s2.
-2cfls2a -- sflc2~ 2s fls2a -- c flc2a
G+G -
H+ H -
(D.66)
G+ H -
sl c2~c~ -c2~c~ -s2#c# c2~c~ -c2~c~ + 2,~2ce --S2flCfl "3c-6"2Sfl $2 --C2flSfl C2flSfl S2flSfl --C2flS,8 C2flSfl÷ 2~2Sfl S2flSfl-- C2Cfl
(D.67)
To obtain the couplings involving (sl, s2) in the (H, h) eigenstate basis, we rotate the (sl, s2) couplings by the angle a, as in Eq. (D.55). The CP-odd Higgs boson A and charged Higgs H + self-energies are
N c A. p2Bo(m.,m.) - 2Ao(m.) f,,
(
f,,
AucB - -~I3gLc2B
)[
l
c2uAo(m~,)+ s2uAo(m~2)
2 2 -~ g2iu3gRc2B] u "~ [s2ua°(m~l) + c2uAo(m~:)] + Z Nf ( AucBf. +Z
~
f,,
N[A~.Bo(m~,m~j) +
i.j=l
g2 s~B cab .~. +--~ 2 F ( m m , M w ) + -~-F(mH, Mz) + - ~ - r t m h , M z ) +~
A2AHOfl~,Bo( m~o. , mHo,) + .~AAH~No,Ao( mHo,) n=l
-
g2 M 2 0 , +-----~-ootMw,
+~ mH, )
) ,AAAH,~H,sAo(mH,+
n=l
Ao( Mz) +g2[2Ao(Mw) + d------U---] +-2
AG(m2°'m~°) -- 2gO/A X~ X:
,ri ' m2°)
i,j=l t. 2
÷ Z f f i + A G(m~i~,m~+) + J - 2 giiam~':m~'~B°(m~':~'m~ )1' i , j = l t.
•"
"LI
''3
"'1
"'J
(D.68)
D.M. Pierceet al./Nuclear PhysicsB 491 (1997)3-67
62
16rr2//H+H- (p2)
2 +AdS# 22 )G(mu'md) -- 2AuAdmumdS2~Bo(mu,md) Njcf [ 2(~uCp
=Z
]
f,,/f,! 2
f 2 ZNc'~,+~,djB°(mu"mdi )
+ Z
.f. / f ,t i,.]=l _k_{~f. g[ (l~2dS2a ~13gLC2a+-2C2[t) g2 u u g2 [c2uaO(m~l)+ s2uAo(mft2)l +ZN
[
f
(
22a -auC
)[
](
gURC2a s2Ao(m~l) + c2Ao(m~2) +
sau~-+d +--+ca
)}
f,, g2 C2A +---~ s2aF(mH, Mw) + c2aF(mh, Mw) + F(mA, Mw) + -20w ~z F(mH+, Mz)]j g2c2^ +e2F(mH, ,0) + 2g2Ao(Mw) + ~ A o ( M z )
2
C _2M2
q- Z ~H.~ 2 HOH~,Bo (mHo,,,mH[' ) + ~L~.W__Bo(Mw,mA) /?,n/=l 4 2 I q--2 Z ~-n~H-H~.H° ao(mHo) + Z ~'H+H-H+H,SaO(mH+) n=l n=] 2 4 IfijH+a(mx'i' 'm;d) - 2 gijn, m•+, m•oBo(mx,+,m•o) ] , i=1 j=l
+Z Z
(D.69)
where c,,a (s~a) denotes cos(a-/?) (sin(a-fl)). The gft., gfR are defined in Eq. (A.7), and the 1f are listed in the table of Eq. (A.8). N[ denotes the number of colors, which is 3 for a (s)quark. The )(i)(jA couplings f,~+A, gija, + fijA' and gOijA are defined by
fZjA=la,,,,AI 2 +[ ,i.y(jAI ,
gqA=2Re byci2jAa2~2jA ,
(D.70)
and similarly for the fijH', gijn+ couplings. The a2,~ja, bT~2ja couplings are given in Eqs. (D.34)-(D.38), and those of the charged Higgs are listed in Eqs. (D.39)-(D.42). The Higgs-sfermion-sfermion couplings AA]jA and an+fjA are given in Eqs. (D.56)(D.58). The fL,R basis couplings aA]/ of Eq. (D.56) also apply in the fl,2 basis.
Appendix E. One-loop Higgs-boson masses In this appendix we will present the formalism necessary to obtain accurate Higgsboson masses at the one-loop level. The tadpole diagrams play an important role in determining the masses. The one-loop tadpole contributions are listed in Refs. [6,48].
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
63
At any given order in perturbation theory, minimizing the scalar potential is equivalent to requiring that the tadpoles vanish. At tree level, we have T1 = T2 = 0, with the tadpoles given by the DR relations T1 = ~1 a~/2 z c 2/~+m 2 , + /z2 +B/xtan/3,
(E.I)
T2
(E.2)
~ l~,12zC2B+ m2H2_k_tz2 + Btzcot /3 '
U2
where the Higgs-sector soft supersymmetry-breaking potential is Vsoft = m2n, ]Hll2 + m22 IH212 +
(Btz,ijH{H ~ + h.c.) .
(E.3)
At one-loop level, the total (tree-level plus one-loop) tadpole must vanish, so Ti - t~ = 0, 7"2 - t2 = 0, with 2
167r2v~-~= - ~
i , f eYltSlfifi A ~" 2Nf A2A°(ma) + Z 2-a'Vc ~ o,,,,],)
( g2ca~8o f~t
2
f
i=1
Ao(ma) +2Ao(mM+)
-
-
)g2 + TAo(mH+)
tanB/Ao(m~)
4
-- i~=l g MwCl ~ e N/3(Ni2 -
- )_..£ x/2g ~ R e
[V/1Ui2
Nil tan0w)
]
Ao(m~)
Ao(msc+)
i=1
+ 3T g z ( 2 A o ( M w ) + Ao(Mz) - - )e+2 - - - ~ - ~ g2c2~ /2 Ao(Mw)+Ao(Mz )) (E.4) and 167r2 t2 U2
2 _q._y 2 Z2 ~'y g~s2H_____~,~_ = - Z 2Ncf AuA°(mu) , _, f,, f i=l ~v~ 2MwsB~aotmfjl
+g2c2~(Ao(ma, ~ + 2A0(mH+,) q--~Ao(mH~ ) + - ~ 3c2 -- s2 -4-s2acot/3 mo(mh) - c, -
cot/3) A0(mH)
64
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
4
m'°
[
+~--~.g2~Re lvl w s 3
-
m
v~g i=1
]
Ni4(Ni2 - Nil tant~w)
Ao(m~o)
El
Re Vi2Uil A o ( m ~ ? )
T
~2
8~2
'
(E.5) where N f is the number of colors, 3 if f is a (s)quark and 1 otherwise. The A0 function is given in Eq. (B.5); ~ denotes cos 0w and c~ = cos/3, etc. The matrices N, U, and V are described in Appendix A and the couplings A,,f,~? As2Y,fj are given by Eqs. (D.53), (D.54). The DR (tree-level) CP-odd Higgs mass is given by rh2a = -B/x(tan/3 + cot/3), and Eqs. (E.1), (E.2) allow us to solve for the DR parameter, /x2, and the pole mass 12, mA
2:1 [tan2 ( m 2A. c_~. (--2 m H2 .
cot )
N . 2)
M2z
Re H zrz ( M 2z ) - R e n a a ( m 2 ) +
ba,
(E.6)
where ~ 2 = m~, - t , / v l , -m~2 = m22 - t2/v2. The self-energies H r z and I1AA are given in Eqs. (D.4) and (D.68), respectively, and b A = $213t l / U1 "~- C2.13t2 / U2. Having determined the physical CP-odd Higgs-boson mass mA, we are in a position to compute the remaining Higgs masses. The physical mass for the charged Higgs boson H + is
E
T 21 ,
m2~ = m 2 + M~v + Re HAA(m~) -- HH+H- (m2+) + H w w ( M w )
(E.7)
where the W-boson self-energy is given in Eq. (D.9), and the charged-Higgs self-energy in Eq. (D.69). The CP-even Higgs-boson masses are obtained from the real part of the poles of the propagator matrix, Det [p,21 - . A d s 2 ( Pi)] 2
=
0,
m 2 = Re(p2),
(E.8)
where the matrix .Ad2(p 2) is
12In case mA is very close to M z , there is an additional O(ot) correction to ma from the off-diagonal element of the CP-odd mass matrix.
D.M. Pierce et al./Nuclear Physics B 491 (1997) 3-67
65
.A42( ,~ P 2) ( IVI2 c2B + rn2As2/3-- Hsls, (p2) + tl/V, \
+
- ( ~42z + en2 )s~c~ -/zs,,2(p
- a,2 ,
+
-- IIs2
p2
2) '~
+
(E.9) In this e x p r e s s i o n , ~ / 2 a n d &2z are the Z - and A - b o s o n D R m a s s e s (/(/2z = M ~ +
ReH~z(M2z),
&2 = m2A + R e H a a ( m 2) _ bz). T h e self-energies Hs, sj are g i v e n in
Eqs. ( D . 5 9 ) - ( D . 6 1 ) .
A t o n e loop, t h e a n g l e a d i a g o n a l i z e s the m a t r i x A,42(p 2) for
s o m e c h o i c e o f m o m e n t u m p2; w e c h o o s e p2 = m 2h .
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