Light MSSM Higgs boson mass to three-loop accuracy

Transcription

1 SFB/CPP-0-34 TTP/0-3 HU-EP-0/5 WUB/0-3 Ligt MSSM Higgs boson mass to tree-loop accuracy P. Kant a, R.V. Harlander b, L. Miaila c, M. Steinauser c a Institut für Pysik, Humboldt-Universität zu Berlin 489 Berlin, Germany b Facbereic C, Teoretisce Pysik, Universität Wuppertal 4097 Wuppertal, Germany c Institut für Teoretisce Teilcenpysik, Karlsrue Institute of Tecnology KIT 768 Karlsrue, Germany Abstract Te ligt CP even Higgs boson mass, M, is calculated to tree-loop accuracy witin te Minimal Supersymmetric Standard Model MSSM. Te result is expressed in terms of DR parameters and implemented in te computer program H3m. Te calculation is based on te proper approximations and teir combination in various regions of te parameter space. Te tree-loop effects to M are typically of te order of a few undred MeV and opposite in sign to te two-loop corrections. Te remaining teory uncertainty due to iger order perturbative corrections is estimated to be less tan GeV. PACS numbers:.60.jv, 4.80.Da,.38.Bx

2 Introduction Among te main expectations in view of te CERN Large Hadron Collider LHC is to provide clear penomena beyond te Standard Model SM of particle pysics. A very promising candidate for an extension of te SM is te so-called Minimal Supersymmetric Standard Model MSSM wic relies on an extended symmetry between fermions and bosons [, ]. It is constructed in suc a way tat in te low-energy limit te SM is recovered tus leading to te same penomena in te energy range around te electroweak scale. In particular, te MSSM is in accordance wit te electroweak precision data [3]. At te same time it provides a dark matter candidate, solves te ierarcy problem and provides a platform were also gravitational interactions can be included. An appealing feature of te MSSM is te quite restrictive Higgs sector wic is described at leading order by two independent parameters. In particular, te mass of te ligtest CP-even Higgs boson, M, is not a free parameter, like in te SM, but a prediction wic can be used in order to test tis minimal supersymmetric extension of te SM. M is very sensitive to radiative corrections. In lowest order it is bound from above by te Z boson mass wic is already excluded by experiment. Already quite some time ago it as been observed tat large one-loop corrections, in particular from te top quark and top squark sector can raise M to about 40 GeV [4 6]. In te meantime a number of iger order corrections ave been computed including even CP-violating couplings and improvements from renormalization group considerations see Refs. [7 9] for a review. In tis paper we consider neiter CP violation nor te resummation of iger order logaritms. Let us neverteless mention tat in particular CP violating pases can lead to a sift of a few GeV in M, see, e.g., Refs. [?,?]. In Ref. [0] a large class of two-loop corrections to te ligtest Higgs boson mass ave been considered and in Ref. [] leading logaritmic corrections at tree-loop order ave been computed. Te first complete tree-loop calculation of te leading quartic top quark mass terms witin supersymmetric QCD SQCD; more precisely, tis means supersymmetric six-flavor QCD coupled to te MSSM Higgs sector as been performed in Ref. [] for a degenerate supersymmetric mass spectrum. It is te aim of tis paper to provide details and extend tis calculation. At te moment tere are two computer programs publicly available wic include most of te iger order corrections. FeynHiggs as been available already since 998 [3, 4, 9] and as been continuously improved since ten [5, 6]. In particular, it contains all numerically important two-loop corrections and accepts bot real and complex MSSM input parameters. Te second program, CPSuperH [7,8], is based on a renormalization group improved diagrammatic calculation and allows for explicit CP violation. Bot programs compute te mass spectrum as well as te decay widt of te neutral and carged Higgs bosons. In tis paper we discuss te tree-loop corrections originating from te strong sector of te MSSM wic are proportional to te quartic top quark mass. At tree-loop order

3 several mass scales enter te Feynman diagrams making teir evaluation quite involved. In addition to te top quark mass tere are te top squark masses, te gluino mass and te masses of te remaining squarks. An exact evaluation of te tree-loop integrals is currently out of range. However, it is possible to apply expansion tecniques for various limits wic allow to cover a large part of te supersymmetric SUSY parameter space. In particular, we can construct precise approximations for te Snowmass Points and Slopes SPS [9,0]. Our set-up is easily extendable to oter regions of parameter space wic may become interesting in future. Togeter wit tis paper we provide a Matematica program, H3m [], wic contains all our tree-loop results. Furtermore, H3m constitutes an interface to FeynHiggs [] and various SUSY spectrum generators wic allows for precise predictions of M on te basis of realistic SUSY scenarios. Te remainder of te paper is organized as follows: In te next Section we revisit te two-loop corrections. In particular, we construct approximations wic are also available at tree-loop order and compare wit te exact result. In Section 3 we provide details on our tree-loop calculations for te various ierarcies. In particular we discuss te renormalization and te asymptotic expansion. Section 4 describes te implementation of our results in te computer program H3m and te penomenological implementations are discussed in Section 5. We present a summary and te conclusions in Section 6. In te Appendices additional material is provided, in particular all te one- and two-loop counterterms tat ave entered in our calculation. M in te MSSM. Higgs boson sector of te MSSM Te mass of te Higgs boson is obtained from te quadratic terms in te corresponding potential wic for te MSSM as te following form: V H = µ SUSY + m H + µ SUSY + m H m ǫ ab HH a b + ǫ ab H a H b + [ g 8 + g H H ] + g H H. wit ǫ = ǫ = and ǫ = ǫ = 0. µ SUSY is te Higgs-Higgsino bilinear coupling from te super potential and m, m and m are soft breaking parameters. Note tat te quartic terms are fixed by te SU and U gauge couplings g and g. Te parameters 3

4 in are related to te masses of te gauge bosons and te pseudoscalar Higgs via M W = g v + v, M Z = g + g v + v, MA = m tanβ + cot β, were tanβ = v /v. After spontaneous symmetry breaking te neutral components of te doublets H and H acquire te vacuum expectation values v and v, and we write v + H = φ + iχ, φ φ + H = v +, 3 φ + iχ wic leads to te following representation of te Higgs boson mass matrix at tree level: M H,tree = V sin β M = Z cot β + MA tanβ M Z M A φ φ MZ M A MZ tan β + M A cotβ, 4 were we restrict ourselves to te two CP-even Higgs bosons. In tis paper we restrict ourselves to te numerically dominant m 4 t corrections. As a consequence we can set te electroweak gauge couplings to zero and furtermore nullify te external momentum in te occurring two-point functions. Te formalism presented in te following is adapted to tis framework. It is convenient to evaluate te quantum corrections in te {φ, φ } basis wic requires te evaluation of self energy corrections Σ φ and Σ φ involving φ and φ. Denoting renormalized quantities wit a at, Eq. 4 gets modified to M H = M H,tree ˆΣφ ˆΣφ φ, 5 ˆΣ φ φ ˆΣφ 4

5 wit [5] ˆΣ φ = Σ φ Σ A sin β e + t φ cosβ + sin β M W sin ϑ W e t φ cos β sin β, M W sin ϑ W ˆΣ φ = Σ φ Σ A cos β e t φ sin β cosβ M W sin ϑ W e + t φ sin β + cos β, M W sin ϑ W ˆΣ φ φ = Σ φ φ + Σ A sin β cosβ e + t φ sin 3 β M W sin ϑ W e + t φ cos 3 β. M W sin ϑ W 6 In tis equation, ϑ W is te weak mixing angle, Σ A denotes te self energy of te pseudoscalar Higgs boson and t φi te tadpole contributions of te field φ i. Typical diagrams to te individual contributions can be found in Fig.. Since we are only interested in te leading corrections proportional to m 4 t we evaluate te quantum corrections in te limit of vanising external momentum [3 5]. For te evaluation of te ligtest Higgs boson mass we consider in a first step te matrix element of M H to a given order in perturbation teory. Subsequently, we determine te eigenvalues and assign te smaller one to M. We perform tis procedure at tree level and at one-, two-, and tree-loop order wic leads to te corresponding approximations of te Higgs boson mass. It is convenient to introduce te quantity M i = M i loop M tree, 7 representing te difference between te Higgs boson mass evaluated wit i-loop accuracy and te tree-level result.. Top squark sector of te MSSM In tis paper we only consider strong corrections wic means tat apart from te quarks and gluons also te corresponding superpartners, te squarks and gluinos, are present. Te leading contribution proportional to G F m 4 t is generated by te top quark Yukawa coupling wic distinguises te top squark sector from te oter squark parts of te MSSM Lagrange density. In order to fix te notation let us discuss in more detail te mass matrix of te left- and rigt-anded component of te top squark, L and R, wic 5

6 t ε φ / A g g φ / A φ / A t t φ / A t a ε b t g φ / t c g t φ / d g g t Figure : Sample diagrams contributing to Σ φ, Σ φ, Σ φ φ, Σ A, t φ and t φ. Internal solid, dased, dotted and curly lines correspond to top quarks, top squarks, ε-scalar and gluons, respectively. Gluinos are depicted wit as curly lines wit an additional solid line in te middle. Te external dased line corresponds to te Higgs boson. is given by m M = t + MZ 3 sin ϑ W cos β + M m Q t A t µ SUSY cot β m t A t µ SUSY cot β m t + 3 M Z sin ϑ W cos β + M Ũ m L m t X t, 8 m t X t m R wit X t = A t µ SUSY cot β. M Q and MŨ are soft SUSY breaking masses, and A t is te soft SUSY breaking tri-linear coupling between te Higgs boson and te top squark fields. Diagonalization of Eq. 8 leads to te mass eigenstates and wit masses m, = m L + m R m L m R + 4m t X t. 9 Te mixing angle is defined troug te unitary transformation m 0 0 m = R cos θt sin θ MR, wit R = t sin θ t cos θ t, 0 6

7 and sin θ t = m ta t µ SUSY cotβ m m..3 Leading m 4 t corrections in te on-sell and DR sceme As already mentioned above, te numerically dominant contribution arises from te self energy diagrams evaluated for vanising external momentum. Tus, it is convenient to introduce te following notation for te i-loop corrections to te Higgs boson mass M i = m4 t M i + rem M i, were m4 i t M comprises te complete SQCD contribution of order α t αs i α t is te top Yukawa coupling originating from te top quark/squark sector for vanising external momentum wic is proportional to m 4 t. At one-loop order only top quarks and top squarks are present in te loops. Te two- and tree-loop corrections, m4 t M and m4 3 t M, are obtained by adding gluon, gluino, quark and squark contributions. rem M i represents te remaining part wic is only available at one- and two-loop order. In our approac tese corrections are taken from FeynHiggs wic includes te complete one-loop corrections and all available two-loop terms. At tree-loop order only te contribution m4 3 t M is considered. In te following we discuss te relative contribution to te Higgs boson mass comparing M i and m4 i t M at one- and two-loop order i =, were bot te on-sell and DR sceme for te mass parameters and te mixing angle are considered. For illustration we adopt te scenarios SPSa and SPS and sow te Higgs boson mass as a function of m /. Te solid lines in Fig. sow te results for M i using on-sell parameters for te masses and θ t as it is provided by FeynHiggs. Te dased lines correspond to te m 4 t approximation m4 i t M wic can be found in Refs. [5,6] and as been confirmed by us by an independent calculation. Te panels a and b correspond to te SPSa and SPS bencmark scenarios, respectively, were te input parameters ave been generated wit te elp of SOFTSUSY [7]. Te small differences of te solid and dased lines 3 demonstrate tat te leading top quark mass term approximates te full result to a ig accuracy. Tis statement is also true in te DR sceme. Te corresponding two-loop results are sown in Fig. as das-dotted and dotted curves were te former corresponds to te full and te latter to te approximate result. It is well known tat te perturbative series can exibit a bad convergence beaviour in For a detailed description we refer to te FeynHiggs ome page []. For te on-sell renormalization of te mixing angle we adopt te convention of Ref. [6]. 3 In te case of SPSa te two-loop dased line is almost on top of te solid one. 7

8 M GeV 40 0 SPSa M GeV 40 0 SPS m / GeV a m / GeV b Figure : Comparison of complete and approximate one- and two-loop corrections to te Higgs boson mass for SPSa a and SPS b. Te solid full result and dased lines m 4 t approximation represent te results in te on-sell sceme were te upper and lower curves correspond to te one- and two-loop results, respectively. Te two-loop DR results are sown as das-dotted full result and dotted m 4 t approximation curves. case it is parametrized in terms of te on-sell quark masses 4 wic is due to intrinsically large contributions related to te infra-red beaviour of te teory. Tus, it is tempting to re-parametrize te results for te Higgs boson mass in terms of DR parameters for te top quark mass, te masses of te SUSY particles and te top squark mixing angle. it is sufficient to consider only tis term and take over rem M i from te output of FeynHiggs. In te remainder of te paper we will refer to tis renormalization sceme as DR sceme altoug it contains a mixture of on-sell and DR parameters. Let us mention already at tis point tat from outside tis mixture does not pose any complication in te practical use since te spectrum generator produces bot on-sell and DR parameters wic ten serve as input for te evaluation of M. Since te corrections are dominated by m4 t M i Furter below we will discuss te sceme dependence of M and indeed sow tat te loop corrections are in general smaller in te DR sceme cf. Fig. 8. Similar studies can also be found in te literature [4,8]. Te considerations of tis subsection motivates te following procedure at tree loops: It is certainly sufficient to consider only te approximation m4 3 t M since te size of te remaining term is expected to be below 00 MeV. Furtermore, we adopt te DR sceme since we expect tat te perturbative series sows a better convergence beaviour. In addition te evaluation of te counterterms temselves is significantly simpler. Actually, most of tem are already available in te literature and te computation of te remaining ones is quite straigtforward as we discuss in Appendix A. We provide te analytical 4 For a typical example we refer to te electroweak ρ parameter. Using te on-sell top quark mass te four-loop corrections [8 30] are larger by a factor 50 as compared to te MS sceme. 8

9 results for te two-loop renormalization constants of te top squark masses and mixing angle, tat can be easily expanded for te mass ierarcies considered in tis paper. Te multiplicative DR renormalization constants of te top quark and gluino mass are mass independent and terefore valid for all ierarcies..4 Construction of approximations Considering te many different mass parameters entering te formula for te Higgs boson mass an exact calculation of te tree-loop corrections is currently not feasible. However, due to te various ierarcies among te particle masses it is promising to consider expansions in properly cosen small parameters. As a guideline for te latter we follow te SPS scenarios as defined in Refs. [9,0]. In order to construct approximations covering all SPS cases it is sufficient to consider te following ierarcies among te SUSY masses 5 3 m q m m m g, 4 m q m m m g, 5 m q m m m g, 6 m q m m g m, 6b m q m m g m, 9 m q m m m g, 3 were in te case of an asymptotic expansion in te corresponding ierarcy is performed. In te case of a naive Taylor expansion in te difference of te particle masses is sufficient. Trougout tis paper, q denotes any squark oter tan, and we assume a common mass value m q m q = m q for all of tese eavy squarks. In all ierarcies we assume tat te SUSY masses are larger tan te top quark mass and perform an asymptotic expansion in te corresponding ratio. In te numerical results discussed below we include for te various ierarcies te expansion terms as given in 5 We decided to keep te nomenclature for te ierarcies as tey are in our internal computations and documents. Te non-continuous numeration results from te fact tat for testing purposes we ave computed furter ieracies wic, owever, are not included in te program H3m cf. Section 4. 9

10 ierarcy expansion dept 3 x 3, x g 3, x q 3 4 x 8 q 5 x g, x 4, x 4 q 6 x 3, x g, x 4 q 6b x 3, x g, x q 9 x 3, /x 4, x q 3 Table : Expansion terms available for te individual ierarcies as defined in Eq. 3 at tree-loop order. Tab. were te following notation as been introduced x = m m, x g = m m g, x g = x x g = m m g, x q = m m q, x q = m. 4 m q Note tat at two-loop order te contributions involving te squarks q wit q {u, d, s, c, b} cancel in te sum of all diagrams. At tree-loop level, owever, te results depend on m q. In tose cases were m q is muc larger tan te oter masses at least tree expansion terms are computed and a good convergence even up to m q m is observed. In order to demonstrate tis point we consider te ierarcies 3 and 4 and sow in Fig. 3 te tree-loop prediction for M. Te dased and solid lines correspond to 4 including successively iger orders in /m q were for illustration te following input parameters ave been cosen: 6 m SUSY m g = m = m = 800 GeV, A t = µ SUSY = θ t = 0, M A = 500 GeV. 5 Te orizontal m q -independent dotted line corresponds to te scenario 3 wit m q = m SUSY fixed at 800 GeV. One observes a crossing of te latter and te 4-curve including /m 8 q corrections for m q m SUSY wic nicely demonstrates te rapid convergence in te 6 If not stated oterwise we set te renormalization scale equal to te on-sell top quark mass and evaluate all DR parameters at tat scale. Note, owever, tat H3m is not restricted to tis coice. 0

11 3.9 M GeV m q~ GeV Figure 3: Dependence of te tree-loop corrections on te eavy squark mass m q. Te dased and solid lines correspond to te ierarcy 4 were successively iger order terms in m SUSY /m q ave been included Te solid curve contains terms of order m SUSY /m q 4.. Te dotted and das-dotted curves correspond to te ierarcy 3; for te dotted line m q as been kept fixed at 800 GeV, te das-dotted curve includes terms of order m SUSY /m q 3. large-m q expansion. Fig. 3 also sows tat te expansion around m q = m SUSY leads to good approximations even if m q is two to tree times as big as m SUSY, see das-dotted curve. Note tat for tis plot we computed te DR top quark mass for m q = 800 GeV and kept it fixed. Let us in a next step compare te approximate SQCD corrections according to our ierarcies wit te full prediction for M from Ref. [6]. For illustration we adopt in te remainder of tis Section a minimal supergravity msugra scenario wit tan β = 0, A 0 = 0, µ SUSY > 0, 6

12 m m m0 a m0 c m m m0 b m0 d Figure 4: Comparison of approximate and full two-loop result for ierarcy 3 a and te combination of 3, 5, 6, 6b and 9 b. Te contour lines indicate te deviations in MeV. In c and d te results of a and b are normalized to te genuine two-loop contributions were te contour lines indicate te deviations in per cent. Te bencmark points and slopes are sown as wite dots and lines. and vary m 0 and m / as follows 60 GeV < m 0 < 600 GeV, 00 GeV < m / < 800 GeV. 7 We ave cecked tat very similar results are obtained for oter coices of tanβ, A 0, and signµ SUSY. Tus, our conclusions are at least valid for all msugra SPS scenarios cf. Appendix C. In Fig. 4a and b te absolute value of te difference between te full and te approximate two-loop prediction for te Higgs boson mass, M M,3lcut, is sown in te m 0 -m / plane were M,3lcut includes te same number of expansion terms wic are available at tree loops. In Fig. 4a we only include te results from ierarcy 3 wereas in b also 5, 6, 6b and 9 enter. For eac ierarcy we compute te difference to te exact result and plot in Fig. 4b te minimum. We define te relative uncertainty troug δ = M M,3lcut M M, 8

13 optimal GeV GeV ierarcy SPSa b SPSa b SPSb SPS SPS SPS SPS SPS SPS M M,appr Table : Comparison of full and approximate two-loop prediction for M for te different bencmark points. wic is sown in Fig. 4c and d. In Eq. 8 M i corresponds to te exact i-loop prediction. For reference we sow in Fig. 4 te msugra SPS bencmark points and slopes as wite dots and lines, aving in mind tat for some of tem te values of tan β and A 0 are different from te ones cosen in Eq. 6. Te assignment of te individual scenarios to te corresponding dot is easily done wit te elp of te table in Appendix C. Already for 3 alone one observes a good coverage in te wole m 0 -m / plane wit deviations smaller tan 50 MeV. Tis gets furter improved after including te oter ierarcies. For lower values of m / one as relative deviations also above 0%, owever, te absolute difference between te full result and te approximation is below 00 MeV. In Tab. we directly compare te two-loop predictions for M for te bencmark points listed in te table in Appendix C and SPS7 and SPS8 gauge-mediated supersymmetry breaking [9]. As before, te full results are based on FeynHiggs and Ref. [6], and for M,appr we use te approximation incorporated in H3m. An impressive agreement is found, often even below 00 MeV. Te results discussed in tis Subsection are very promising in view of te tree-loop approximation. At two-loop order te expansion terms specified in Tab. provide an excellent approximation to te full result. Tus, it can be assumed tat te corresponding terms at tree loops approximate te unknown result wit ig precision. 3 Tecnical details to te tree-loop calculation Te tree-loop calculation of te individual Green s functions contributing to M is organized as follows: All Feynman diagrams are generated wit QGRAF [3]. In order to properly take into account te Majorana caracter of te gluino, te output is subse- 3

14 quently manipulated by a PERL script [3] wic applies te rules given in Ref. [33]. Te various diagram topologies are identified and transformed to FORM [34] wit te elp of qe and exp [35, 36]. Te program exp is also used in order to apply te asymptotic expansion see, e.g., Ref. [37] in te various mass ierarcies. Te actual evaluation of te integrals is performed wit te package MATAD [38], resulting in an expansion in d 4 for eac diagram, were d is te space-time dimension. Te total number of tree-loop diagrams amounts to 6706 and 7670 for te φ and φ self energies, respectively, and 845 and 98 for te corresponding tadpole contributions. Te computation of te off-diagonal matrix element Σ φ involves 636 diagrams, and te propagator of te pseudoscalar Higgs anoter Te application of te asymptotic expansion significantly enlarges tese numbers leading to about for 3 or even for 6 subdiagrams. Note tat tere are diagrams were, depending on te ierarcy, up to 5 subdiagrams ave to be considered. A typical example is sown in Fig. a. A subtlety arises from diagrams as te one sown in Fig. b. If bot te external momentum and te ε-scalar mass are set to zero from te beginning, an infra-red divergence occurs and cancels te ultra-violet divergence of te integral. In effect, te diagram will be of order d 4 due to te ε-scalar algebra. In order to avoid tis, we keep te external momentum q non-zero, toug muc smaller tan all oter scales. Te ultra-violet pole multiplied by te algebraic factor of d 4 ten produces a finite contribution, wile te infra-red divergence leads to d 4 lnq and vanises as d 4. Instead of te requirement q 0 one could also introduce a nonzero mass for te ε-scalars in order to regulate te infra-red divergences. In te final result we again observe tat te regulator is multiplied by an additional factor d 4 leading to a finite result for M ε 0. We ave cecked tat te latter prescription leads to identical results as te one wit q 0. We refrain from presenting all available analytical results for te expansion in te various regions. Tey are implemented in te program H3m and tus easily accessible if necessary. However, for te convenience of te reader we provide in tis Section te result for 4, see Eq. 3, wic could be useful for oter applications. We will present te results expressed in terms of te DR parameters α s, m t, m, m, m g and θ t. Te corresponding counterterms can be found in Appendix A. Before providing explicit expressions a comment concerning te DR renormalization constants for te top squarks is in order. Due to diagrams involving eavy squarks q, for example Fig. 5a, te squared Higgs boson mass receives contributions wic are proportional to m q and tus can lead to unnatural large corrections. For tis reason we adopt te on-sell sceme for tese contributions to δz m and δz m cf. Eq. 37 and Fig. 5b for a sample diagram. Tis avoids te potentially large terms m q from te tree-loop diagrams. We follow tis procedure also in te case were te top squarks and te eavy squarks are degenerate in mass. Te renormalization of te mixing angle is free of suc enanced contributions and we can stick to te pure DR sceme in tat case. We asten 4

15 q q φ / A φ / A q a q b Figure 5: a Feynman diagram involving a eavy virtual squark contributing to te Higgs boson self energy. b Counterterm diagram related to te diagram in a. Te same notation as in Fig. as been adopted. to add tat tis discussion only concerns te internal structure of H3m and as no direct consequences for te user. Te input parameters of H3m are te DR ones as tey appear, e.g., in te output of SOFTSUSY. As already noticed in Refs. [5, 6], a similar beaviour is observed wen te gluino is muc eavier tan te top squarks. In tis case, te two- and tree-loop corrections to te Higgs masses computed in te DR sceme contain terms proportional to m g and m g. Tese contributions are canceled in te on-sell sceme by te finite parts of te relevant counterterms. Tus, in order to avoid unnatural large radiative corrections to te Higgs masses, we adopt for scenarios wit eavy gluino masses a modified renormalization sceme for te top squark masses. We call tis sceme modified DR MDR and it is caracterized by te non minimal renormalization of te top squark masses. Te additional finite sifts of top squark masses are cosen suc tat tey cancel te powerlike beaviour of te gluino contributions. Again, te renormalization of te mixing angle will not be modified as compared to te genuine DR sceme. Te relevant finite sifts for te scenarios considered in tis paper are explicitly given in Appendix B. As can be noticed from Eq. 3 te scenarios 4, 5 and 6 display eavy squark mass contributions wereas eavy gluino terms are specific only for te scenarios 6 and 6b. In te practical calculation we use te DR top squark mass parameters as provided by te spectrum generators and transform tem wit te elp of te formulae of Appendix B to te corresponding parameters in te MDR sceme wic constitute te input for our analytic expressions. In te following we present results for te renormalized two-point functions ˆΣ φ, ˆΣ φ and 5

16 ˆΣ φ φ for te ierarcy 4 wic for equal top squark and gluino masses take te form 7 ˆΣ φ = G [ Fm 4 t 3 π sin β l ts + α s 4π αs { π l µt 6 7 l µt 36 3 l µt + m q l ts + 4ltS + A t 4 + 8l µt + 4l ts m SUSY lts l µt + 4ltS l t q 00 3 l t q 0 3 l3 t q 0ζ 80l t qζ ζ l µt 38 3 l µt 40l t q 0lt q 40ζ + 6ζ3 l ts + m q l m µt + 80l t q + 80ζ SUSY + m SUSY 4356 m q 5 + 8l ts 8 45 l t q 76 3 l t q l t q l ts ζ + A tm SUSY 80 + l ts l t q l t q l t q ζ + A t m SUSY l µt l µt l ts + l ts l µt 60 3 l t q l t q + 40lt q + 80ζ 3 ζ3 + A t 349 m SUSY l µt l µt l µt l ts + 94 } ] m 4 3 ζ3 + O SUSY, m 4 q ˆΣ φ = G Fm 4 t αs [ A t 349 π cos β 4π m SUSY l µt l µt l µt 7 We only include terms up to order /m q. l ts l ts l ts ζ3 + O m 4 SUSY m 4 q ], 6

17 ˆΣ φ = G Fm 4 t π cosβ sin β αs { A + t 4π m SUSY [ αs A t 4l µt l ts 4π m SUSY l µt 3 9 l µt l µt l t q 40 3 l t q + l ts + A tm SUSY m q + A t m SUSY + l ts l t q l ts 3 9 l ts ζ l µt 00 3 l µt l ts l t q 0lt q } l µt l t q 40ζ ζ3 3 ζ3 m 4 + O SUSY m 4 q ], wit m t = m t µ r, m SUSY = m SUSY µ r = m µ r = m µ r = m g µ r, l µt = lnµ r/m t, l ts = lnm t /m SUSY and l t q = lnm t /m q were µ r is te renormalization scale. Te on-sell result corresponding to ˆΣ φ as been presented in Ref. [] for A t = 0. We refrain from providing more analytic results since all of tem come along wit te program H3m wic is discussed in te next section. 4 Description of H3m In tis Section we describe te implementation of our tree-loop results in a user-friendly computer program wic allows te evaluation of te ligt CP even Higgs boson mass M to tree-loop accuracy. Te program is implemented in te form of a Matematica package. To set te input parameters for te calculation, i.e. te SUSY spectrum and SM parameters, te SUSY Les Houces Accord sla [39] is used. For ease of use, we provide functions tat call a spectrum generator from Matematica to produce an sla spectrum file. To produce te plots in tis publication, we ave cosen SOFTSUSY [7], but it is possible to use any spectrum generator tat provides te DR parameters in addition to te on-sell mass spectrum like SuSpect [40] or SPeno [4]. Te advantage of SOFTSUSY is tat te renormalization scale of te DR parameters can be cosen independently of te electroweak symmetry breaking. Te corrections m4 t M to M, being proportional to te fourt power of te mass of te top quark m t, are very sensitive to bot te definition and te uncertainty of m t. Tus, it is important to use te most precise value of m t available. For tis reason we take into account te full two-loop SQCD corrections between te on-sell and DR top quark mass given in Ref. [4]. 8 In tis paper, te relation of te on-sell top mass M t and DR top mass m t was derived as a function of te DR masses. Solving tis equation iteratively, 8 We tank Steven Martin for providing us wit te relevant formulae from Ref. [4] in electronic form, and for allowing us to include is code in our program. 7

18 we get m t as a function of M t. Te integrals appearing in [4] are evaluated using te C library TSIL [43]. Tis relation is available for general renormalization scale µ r wic enables us to obtain m t µ r in te DR sceme using te on-sell mass M t as measured at te Tevatron [44] as input. Anoter critical parameter for te evaluation of M is te strong coupling α s. We use α s M Z as input and follow Ref. [45 47] in order to evaluate α s in te DR sceme wit all SUSY particles contributing to te running. First, α s 5,MS M Z = 0.84 [48] is run up to te decoupling scale, wic we set to te average value of te SUSY particles, using te four-loop β function [49, 50]. Tere, we perform te transition to te DR sceme and te full teory. Te two-loop matcing coefficients from [47] are used in tis step. To obtain α s full,dr µ r for arbitrary values of te renormalization scale µ r, we use te tree-loop SQCD β function given in [5,3]. Te remaining input parameters comprise te ones for te SUSY breaking scenario, wic we summarise in te table of Appendix C and te SM parameters M Z, G F and α wic also serve as input for te spectrum. Te default values set in H3m read M Z = GeV, M t = 73. GeV, G F = GeV, /αm Z = 7.934, α s M Z = α 5,MS s M Z = Of course, it is possible to modify tese default values. Note tat te top squark masses and mixing angle are obtained from te soft breaking parameters according to Eqs. 9 and. In order to include all te known corrections to M at te one- and two-loop level, te spectrum file is passed to FeynHiggs [9, 3 5]. FeynHiggs uses on-sell parameters tat are given in te spectrum file and provides te neutral Higgs mass matrix up to te two-loop level. Since we prefer to use te DR sceme, we need to perform a conversion before adding our tree-loop results. Tis we do by subtracting te on-sell expression for m4 t M up to two loops, witout any expansions in te masses 9 and adding it back in te DR sceme. Tus, we use te DR sceme to evaluate m4 t M, wic are dominant and sensitive to te top quark mass, and te on-sell sceme for rem M. Te next step is to coose a suitable mass ierarcy for te expansion of te treeloop corrections. Tis is done by comparing, at te two-loop level, te full result from Ref. [6] wit te expansions in all te mass ierarcies and coosing te one minimizing te error. Finally, te tree-loop corrections are added, te neutral Higgs mass matrix is diagonalized, and te mass of te ligt Higgs is returned to te user. Te interface of te program is outlined in Fig. 6. Te parameters are set up by a call of 9 We tank Pietro Slavic for sending us te compact formulae from Ref. [6] in electronic form. 8

19 Figure 6: Flowcart of H3m. First, te user calls H3SetSLHA or one of its descendants to set te parameters. A subsequent call to H3mcomputes M. te function H3SetSLHA, wic passes its arguments to te spectrum generator and parses its output to get te relevant input parameters for te calculation. Te top mass and strong coupling are calculated as described above. Alternatively, te function H3GetSLHA uses an existing spectrum file instead of running a generator. For te user interested in te Snowmass Points and Slopes, we provide convenient wrapper functions H3SetSPS<x> wic call H3SetSLHA wit parameters according to a specific bencmark scenario. Te main calculation is organized by te function H3m, wic calls FeynHiggs, does te conversion to te DR sceme described above, cooses an appropriate mass ierarcy, adds te tree-loop corrections, and returns M. In Fig. 7, a typical Matematica session wit H3m is sown. A more detailed description of H3m comes along wit te program wic can be found at te web page []. 5 Te Higgs boson mass to tree-loop accuracy In tis Section we use, if not stated oterwise, te input parameters as listed in Eq. 9 and furtermore adopt for te renormalization scale µ r = M t as our default value. In Fig. 8 we sow te renormalization sceme dependence of M as a function of m / for te SPS scenario. Tis is convenient since we ave te same abscissa bot for te on-sell 9

20 Matematica 7.0 for Linux x86 64-bit Copyrigt Wolfram Researc, Inc. In[]:= Needs["H3 "]; RunDec: a Matematica package for running and decoupling of te strong coupling and quark masses by K.G. Cetyrkin, J.H. Kun and M. Steinauser January 000 In[]:= H3SetSPSa[ 300.]; H3GetSLHA::TSIL: Using TSIL by S.P. Martin FeynHiggs.6.5 built on Dec 0, 008 T. Han, S. Heinemeyer, W. Hollik, H. Rzeak, G. Weiglein ttp:// FHHiggsCorr contains code by: P. Slavic et al. -loop rmssm Higgs self-energies Loading Results for ierarcy 3 Loading Results for ierarcy 3 Loading Results for ierarcy 6bqg Loading Results for ierarcy 6bqg In[3]:= H3m[] Loading Results for ierarcy 6bqg Out[3]= {m -> 4.76} Figure 7: A typical Matematica session wit H3m. and DR result. Note, owever, tat te tree-loop on-sell result is only available for a degenerate mass spectrum of te SUSY particles and vanising parameter A t []. Tus, we restrict ourselves to tis limit also for te DR result. For tis reason te following discussion sould be considered in a less quantitative but more qualitative sense and sould not be used, e.g., for estimating a teoretical uncertainty. In te left panel of Fig. 8 te upper dotted, dased and solid curve correspond to te one-, two- and tree-loop prediction of M in te on-sell sceme wereas te corresponding lower tree curves are obtained in te DR sceme. In te on-sell sceme one observes large positive one-loop corrections wic get reduced by 0 to 0 GeV after including te 0

21 40 0 M GeV 30 0 M GeV m / GeV m / GeV Figure 8: Renormalization sceme dependence of M as a function of m / adopting SPS. Dotted, dased and solid curves correspond to one-, two- and tree-loop results. Te DR on-sell results correspond to te lower upper tree curves. In te rigt panel te interesting part of te left one is magnified. two-loop terms. Te tree-loop corrections amount to several undred MeV. Tey are positive or negative depending on te value of m /. Te situation is completely different for DR mass parameters: te one-loop corrections are significantly smaller and lead to values of M wic are already of te order of te twoand tree-loop on-sell prediction. Te two-loop term leads to a small sift of te order of GeV and te tree-loop term to a positive sift of about te same order of magnitude. Te final prediction for M is very close to te one obtained after incorporating tree-loop on-sell results. 0 Comparing te DR and on-sell results in Fig. 8 one observes a nice reduction of te sceme dependence wen incorporating iger order corrections. Wereas tere is a uge gap between te two one-loop curves dotted te difference in te two-loop prediction of M is below GeV wic gets furter reduced by about a factor ten after incorporating te tree-loop corrections to rougly 00 MeV. Fig. 9 extends Fig. 4 to tree loops. In a and b we again discuss te ierarcy 3 and te combination of te ierarcies 3, 5, 6, 6b and 9, respectively, and sow te difference between our best prediction and te one were te expansion parameters are cut by one unit. In te wole parameter plane we observe small absolute corrections reacing at most about 00 MeV. Tis leads to te conclusion tat as a conservative estimate of te uncertainty of our approximation procedure one can assign about 00 MeV. 0 Te relatively large tree-loop corrections as compared to te two-loop ones do not pose any problem since we use simplified formulae as mentioned before. Furtermore, tere are regions in te parameter space were te two-loop corrections are accidentally small in te DR sceme leading to large relative tree-loop terms. Neverteless te overall size of te two- and tree-loop corrections is small. Up to two-loop order te sceme dependence as already been discussed in Fig.. We only cut in parameters originating from asymptotic expansion: wen counting powers of mass ratios, we leave sin θ t untouced and do not replace it by Eq..

22 m m0 a 50 m m0 b m m0 c m m0 d Figure 9: Comparison of te tree-loop predictions for M using te maximal available expansion terms and reduced input as described in te text. In a and b te absolute deviations are sown for te ierarcy 3 and te combination of 3, 5, 6, 6b and 9, respectively. Te contour lines indicate te deviations in MeV. In c and d te results of a and b are normalized to te genuine tree-loop contributions were te contour lines indicate te deviations in per cent. Te bencmark points and slopes are sown as wite dots and lines. In Fig. 9c and d we sow relative deviations defined troug δ 3 = M3 M 3,cut M 3 M, 0 were M 3 is te prediction were at treeloop order te expansion dept of eac parameter cf. Tab. is reduced by one unit, is our best tree-loop prediction, M 3,cut and M corresponds to te full two-loop term. Similar to te two-loop case, larger corrections are only observed for small values of m 0 and m / wic is a consequence of a small denominator in Eq. 0. Te tree-loop correction terms, owever, are stable as can be seen from te panels a and b. Tus, as in te two-loop case, we are able to cover te wole m 0 -m / plane and are in particular able to produce precise values for M for all SPS scenarios. Te slopes for tree SPS scenarios are sown in Fig. 0a, b and c were te dotted, dased and solid lines correspond to te one-, two- and tree-loop predictions, respectively. For all tree cases one observes negative corrections between and 4 GeV at two loops

23 M GeV.5 0 SPSa m / GeV a M GeV.5 0 SPS m / GeV b M GeV.5 0 SPS m / GeV c M GeV SPSa µ r GeV d Figure 0: M for te different slopes of te bencmark scenarios SPSa a, SPS b and SPS3 c. Dotted, dased and solid lines correspond to te one-, two- and tree-loop predictions. Te dased line wit longer dases at two loops correspond to te full results, te one wit te sorter dases to te approximation implemented in H3m. In d te dependence of M on te renormalization scale is sown were te dotted, dased and solid line corresponds to te one-, two- and tree-loop prediction. and positive contributions from te tree-loop term wic amount up to about GeV. In Fig. 0d we sow te dependence of te prediction for M on te renormalization scale µ r. As an example we adopt te SPSa scenario wit m / = 50 GeV and exploit tat SOFTSUSY allows te evaluation of all DR parameters at te scale µ r. One observes a strong dependence at one-loop order wic gets significantly reduced at two loops. Te treeloop curve is even more flat resulting in a stable prediction for M. Around µ r = 50 GeV te two-loop correction sows a local maximum and is furtermore very small wereas te tree-loop term still amounts to about 500 MeV. Around µ r = M t, wic is often used as a default coice, one as negative two-loop corrections of about GeV and a sligtly larger tree-loop contribution tan for µ r = 50 GeV. Te corresponding plots for SPS and SPS3 look very similar. Tus, we refrain from presenting tem ere; tey can easily be generated wit te elp of H3m. 3

24 35 30 M GeV A t GeV Figure : Dependence of M on A t. Te dotted, dased and solid line corresponds to te one-, two- and tree-loop prediction. It is interesting to investigate te dependence of M on te soft breaking parameter A t. In Fig. we sow te result for M were te following values for te parameters ave been cosen m = 500 GeV, m = 000 GeV, m g = 500 GeV, m q = 000 GeV, µ SUSY = 800 GeV, tanβ = 0, M A = 500 GeV. Furtermore, we employ only te m 4 t corrections since it is not possible to transmit te parameters of Eq. directly to FeynHiggs and evaluate te corresponding Higgs boson mass. It is interesting to note tat te tree-loop corrections are quite sizeable, amounting 4

25 m m m m0 Figure : Genuine two- upper panel and tree-loop lower panel corrections to M in te m 0 -m / plane. up to about 3 GeV. In contrast to te two-loop terms tey are positive and lead to a compensation. For A t = TeV and A t = 0 te tree-loop prediction is even above te one-loop value for M. In order to get an impression on te size of te tree-loop corrections we sow in Fig. 5

26 m m0 Figure 3: Prediction of M to tree-loop accuracy using H3m. Te same conventions as in Fig. ave been adopted. lower panel te difference between our best tree-loop prediction and te full two-loop result as a function of m 0 and m /. We observe tat te corrections are always positive and vary for our parameters between a few undred MeV and about GeV. Tey sow only a mild dependence on m 0, but vary strongly wit m /. In particular te corrections become larger for increasing values of m /. For comparison, we sow in Fig. upper panel te corresponding quantity at two-loop order, i.e., te difference between te twoloop and te one-loop result. In contrast to te tree-loop contributions tey are negative and amount to about twice te tree-loop terms in a large region of te parameter space. However, tere are also regions were te tree-loop corrections are larger tan te twoloop ones. Note, owever, tat in te wole m 0 -m / plane te one-loop corrections are more tan ten times bigger. Furtermore, te occurrence of tree-loop corrections, wic are large compared to te two-loop ones, can also be seen from Fig. 0d: wereas te two-loop corrections vary between 4 GeV and +4 GeV in te considered range of µ r te tree-loop term is almost constant and amounts to about GeV relative to te two-loop result. In Fig. 3 we finally sow te tree-loop prediction of M including te tree-loop effects discussed in tis paper. Again we restrict ourselves to te parameter space defined above. Values for oter input parameters are easily obtained wit te elp of H3m. One can see tat for increasing m / also te Higgs boson mass gets larger and values well above 0 GeV can be reaced. Tis is already observed at one-loop order and is due to te fact tat for larger values of m / te wole supersymmetric spectrum becomes eavier. 6

27 optimal GeV GeV ierarcy SPSa b SPSa b SPSb SPS SPS SPS SPS SPS SPS M 3 M 3,cut Table 3: Comparison of te best tree-loop prediction to te one were some expansion parameters are cut see text. Te last column sows te cosen ierarcy. scenario loop loops 3 loops m max no-mixing gluopobic small α eff Table 4: Results for M in GeV for te bencmark scenarios defined in Ref. [5] were tan β = 0 and M A = 500 GeV as been cosen. For definiteness we sow in Tab. 3 te tree-loop prediction of M for te SPS points. We sow te best prediction M 3 and, for comparison, te result obtained by cutting parameters as for Fig. 9 M 3,cut. Furtermore, we indicate te optimal ierarcy as cosen by H3m. In all cases te uncertainty due to our approximation can be estimated to be below 00 MeV. As a last penomenological application we consider te bencmark points identified in Ref. [5] in order to perform te MSSM Higgs boson searc at adron colliders. In Tab. 4 te one-, two- and tree-loop predictions for M are sown for te four scenarios m max, no-mixing, gluopobic and small α eff. Wereas for no-mixing significant tree-loop effects are observed more moderate, owever, still important contributions are obtained for te remaining tree scenarios. In Tab. 4 tanβ = 0 and M A = 500 GeV as been cosen, owever, a similar conclusion can be drawn for oter values. Let us at te end of tis Section estimate te teoretical uncertainty on te prediction of M after including te tree-loop corrections. We divide te uncertainty into two parts: i te teoretical error due to missing iger order corrections and oter as of yet uncalculated corrections, and ii te parametric uncertainty, mainly due to te top quark mass, α s and te supersymmetric masses. At two-loop level, a toroug investigation 7

28 of te teoretical uncertainties i as been performed in Ref. [8] for earlier work, see Ref. [4]. It was found tat missing two-loop corrections most importantly electro-weak and finite-momentum effects can be assumed to be well below GeV, wile te as of ten unknown tree-loop effects could amount to -3 GeV, as indicated by te variation of te two-loop results wit te renormalization scale. Wit te α t αs corrections of our calculation, we sould terefore be able to reduce te teory uncertainty. Instead of renormalization scale variations, owever, we want to adopt a more conservative attitude and assume a geometric progression of te perturbative series. As can be confirmed at two-loop level, a conservative estimate of te uncertainty is 50% of te difference to te next lower order. Tus we assign 50% of te tree-loop contribution to M as a teoretical error. For te msugra scenarios tis leads to an uncertainty of about MeV for m / = 00 GeV and to about GeV for m / = TeV. Tese estimates cover also corrections from renormalization group improvements wic at two-loop order lead to sifts in M of te order of a few GeV [?]. Earlier in tis section, we ave identified two more contributions to te uncertainties of type i: Te corrections beyond te quartic top quark mass contribution, rem M 3, as been estimated to about 00 MeV and te uncertainty due to our approximation procedure also amounts to at most 00 MeV. Bot contributions are smaller tan te one due to missing iger order corrections discussed above. Te parametric uncertainties can be easily estimated wit te elp of H3m. For definiteness we adopt in te following SPS and vary M t and α s M Z as follows M t = 73. ±.3 GeV, α s M Z = 0.84 ± In te case of te top quark mass we observe for m / = 00 GeV a variation of M by about 350 MeV wic increases to δm = GeV for m / = TeV. Te corresponding numbers for te uncertainty in α s M Z read 80 MeV and 600 MeV. A furter uncertainty is connected to te unknown supersymmetric parameters wic may also be of te order of a few undred MeV. Assuming, e.g., an uncertainty of 0% for m in te range between 00 and 800 GeV and adopting te remaining parameters from Eq. leads to an uncertainty of at most 500 MeV for M. Note tat te parametric uncertainty is of te same order of magnitude as te teoretical error due to missing iger order corrections as estimated in te previous paragrap. 6 Conclusions In tis paper we ave discussed te ligtest Higgs boson mass of te MSSM to tree-loop accuracy. At tis order an exact calculation is out of range and tus we eavily exploit te metods of asymptotic expansion in order to provide precise approximations. Tis procedure as been successfully tested at two-loop order against te full result. Te result 8

29 is expressed in terms of DR parameters for te quark and squark masses for wic we ave found very good convergence of te perturbative expansion. We provide a userfriendly Matematica program H3m wic allows te computation of M in a simple way. In particular, it is possible to apply various SUSY breaking scanarios, invoke a spectrum generator, and use te output in order to compute M. As already mentioned, wit te elp of te asymptotic expansion we ave implemented analytical results valid for various ierarcies in te supersymmetric masses. H3m is set up in suc a way tat it is straigtforward to include furter ierarcies in case tey are needed for future investigations. We ave performed several studies wit te new tree-loop corrections. In particular, we ave considered teir renormalization sceme and scale dependence, and teir numerical effect in te various SPS scenarios. Furtermore, assuming msugra, we ave considered te m 0 -m / parameter space and computed two and tree-loop corrections. On te basis of tese investigations we estimate te remaining teory uncertainty of M to about 00 MeV for m / = 00 GeV and to about GeV for m / = TeV. Acknowledgements We tank Steven Martin for providing us wit C++ routines for te relation between te DR and on-sell top quark mass, and Pietro Slavic for providing us wit te analytic result for te two-loop SQCD corrections. We are grateful to Sven Heinemeyer for carefully reading te manuscript and many valuable comments, and to Tomas Hermann for useful comments on Appendix A. Tis work was supported by te DFG troug SFB/TR 9 Computational Particle Pysics and contract HA 990/3-, and by te Helmoltz Alliance Pysics at te Terascale. Appendix A: DR counterterms In tis appendix we provide details about te computation of te counterterms for te gluino, top quark and squark masses and te mixing angle in te top squark system. To define our framework for te computation of te top squark masses and mixing angle counterterms, we start from te bare Lagrangian containing te kinetic energy and mass terms L 0 = µ L, R 0 µ L R 0 L, R 0 M 0 L R 0, 4 were te superscript 0 labels te bare quantities, L and R denote te interaction eigenstates and te top squark mass matrix was defined in Eq. 8. 9