We review various effective potential methods which have been useful to
compute the Higgs mass spectrum and couplings of the minimal supersymmetric
standard model. We compare results where all-loop next-to-leading-log
corrections are resummed by the renormalization group, with those where just
the leading-log corrections are kept. Pole masses are obtained from running
masses by addition of convenient self-energy diagrams. Approximate analytical
expressions are worked out, providing an excellent approximation to the
numerical results which include all next-to-leading-log terms. An appropriate
treatment of squark decoupling allows to consider large values of the stop
and/or sbottom mixing parameters and thus fix a reliable upper bound on the
mass of the lightest CP-even Higgs boson mass.Comment: 10 pages, latex + psfig.sty, 6 figures uuencoded. Based on talk given
  at the SUSY-95 International Workshop on Supersymmetry and Unification of
  Fundamental Interactions, 15-19 May 199

Quirós, M.

English

arXiv

We review various effective potential methods which have been useful to compute the Higgs mass spectrum and couplings of the minimal supersymmetric standard model. We compare results where all-loop next-to-leading-log corrections are resummed by the renormalization group, with those where just the leading-log corrections are kept. Pole masses are obtained from running masses by addition of convenient self-energy diagrams. Approximate analytical expressions are worked out, providing an excellent approximation to the numerical results which include all next-to-leading-log terms. An appropriate treatment of squark decoupling allows to consider large values of the stop and/or sbottom mixing parameters and thus fix a reliable upper bound on the mass of the lightest CP-even Higgs boson mass.We review various effective potential methods which have been useful to compute the Higgs mass spectrum and couplings of the minimal supersymmetric standard model. We compare results where all-loop next-to-leading-log corrections are resummed by the renormalization group, with those where just the leading-log corrections are kept. Pole masses are obtained from running masses by addition of convenient self-energy diagrams. Approximate analytical expressions are worked out, providing an excellent approximation to the numerical results which include all next-to-leading-log terms. An appropriate treatment of squark decoupling allows to consider large values of the stop and/or sbottom mixing parameters and thus fix a reliable upper bound on the mass of the lightest CP-even Higgs boson mass

Quiros, M.

CERN Document Server

arXiv:hep-ph/9509385v1  25 Sep 1995IEM-FT-114/95hep–ph/9509385STATUS OF EFFECTIVE POTENTIAL CALCULATIONS ∗M. Quiro´s †CERN, TH Division, CH–1211 Geneva 23, SwitzerlandAbstractWe review various effective potential methods which have been useful to compute the Higgs massspectrum and couplings of the minimal supersymmetric standard model. We compare resultswhere all-loop next-to-leading-log corrections are resummed by the renormalization group, withthose where just the leading-log corrections are kept. Pole masses are obtained from runningmasses by addition of convenient self-energy diagrams. Approximate analytical expressions areworked out, providing an excellent approximation to the numerical results which include allnext-to-leading-log terms. An appropriate treatment of squark decoupling allows to considerlarge values of the stop and/or sbottom mixing parameters and thus fix a reliable upper boundon the mass of the lightest CP-even Higgs boson mass.IEM-FT-114/95September 1995∗Based on talk given at the SUSY-95 International Workshop on Supersymmetry and Unification of Funda-mental Interactions, Palaiseau, 15-19 May 1995.†On leave of absence from Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006-Madrid, Spain.Work supported in part by the European Union (contract CHRX-CT92-0004) and CICYT of Spain (contractAEN94-0928).1 IntroductionThe effective potential methods to compute the (radiatively corrected) Higgs mass spec-trum in the Minimal Supersymmetric Standard Model (MSSM) are useful since they allow toresum (using Renormalization Group (RG) techniques) leading-log (LL), next-to-leading-log(NTLL),..., corrections to all orders in perturbation theory. These methods [1, 2], as well asthe diagrammatic methods [3] to compute the Higgs mass spectrum in the MSSM, were firstdeveloped in the early nineties.Effective potential methods are based on the run-and-match procedure by which alldimensionful and dimensionless couplings are running with the RG scale, for scales greaterthan the masses involved in the theory. When the RG scale equals a particular mass threshold,heavy fields decouple, eventually leaving threshold effects in order to match the effective theorybelow and above the mass threshold. For instance, assuming a common soft supersymmetrybreaking mass for left-handed and right-handed stops and sbottoms, MS ∼ mQ ∼ mU ∼ mD,and assuming for the top-quark mass, mt, and for the CP-odd Higgs mass, mA, the rangemt ≤ mA ≤ MS, we have: for scales Q ≥ MS, the MSSM, for mA ≤ Q ≤ MS the two-Higgsdoublet model (2HDM), and for mt ≤ Q ≤ mA the Standard Model (SM). Of course there arethresholds effects at Q = MS to match the MSSM with the 2HDM, and at Q = mA to matchthe 2HDM with the SM.The neutral Higgs sector of the MSSM contains, on top of the CP-odd Higgs A, twoCP-even Higgs mass eigenstates, Hh (the heaviest one) and H (the lightest one). It turns outthat the larger mA the heavier the lightest Higgs H . Therefore the case mA ∼MS is, not onlya great simplification since the effective theory below MS is the SM, but also of great interest,since it provides the upper bound on the mass of the lightest Higgs (which is interesting forphenomenological purposes, e.g. at LEP 2). In this case the threshold correction atMS for theSM quartic coupling λ is:∆thλ =316pi2h4tX2tM2S(2− 16X2tM2S)(1)where ht is the SM top Yukawa coupling and Xt = (At−µ/ tanβ) is the mixing in the stop massmatrix, the parameters At and µ being the trilinear soft-breaking coupling in the stop sectorand the supersymmetric Higgs mixing mass, respectively. The maximum of (1) corresponds toX2t = 6M2S which provides the maximum value of the lightest Higgs mass: this case will bereferred to as the case of maximal mixing.2 Leading-log vs Next-to-leading-log resultsRecent effective potential studies [4] have proved that the L-loop improved effective potential,with (L+1)-loop RG equations is exact to Lth-to-leading-log order. In particular the caseL=0 (tree level potential improved with one-loop RG) describes the LL approximation, andthe case L=1 (one-loop effective potential improved with two-loop RG) describes the NTLLapproximation. In particular, the effective potential in the NTLL approximation is expectedto be highly scale independent. But, can we quantify the scale independence of it?We have minimized the effective potential at the scale t∗ such that [5]ddtΦmin(t)ξ(t)∣∣∣∣∣t=t∗= 0 (2)2where ξ(t) is the anomalous dimension of the Higgs field. In fact the scale dependence isFigure 1: Left panel: Plot of φmin(t)/ξ(t) as a function of µ(t) in the LL (dashed line) andNTLL (solid line) approximation. The dotted line shows the would-be scale independent result.Right panel: Plot of mH,der(t)/mH(t) as a function of µ(t). In both cases mt = 160 GeV andthe supersymmetric parameters are MS = 1 TeV, Xt = 0 and tanβ ≫ 1.measured by the quantity Φmin(t)/ξ(t), which is constant for a scale invariant theory. Wehave plotted it versus the RG scale µ(t) in Fig. 1 (left panel) where we can see that the bestminimization scale µ(t∗) is around the value of the running top-quark mass and that minimizingthe effective potential at the high scale MS produces a departure of the scale invariance ofΦmin(t)/ξ(t) of order 10%. This measure of scale invariance is associated with the minimumof the potential, i.e. with the first derivative. An independent measure, associated with thesecond derivative of the effective potential, is given by the ratio mH,der/mH , where mH,der isthe second derivative of the potential at the scale µ(t) and mH the second derivative at theminimization scale µ(t∗), run to the scale µ(t) with the anomalous dimension of the Higgs field.The corresponding plot is shown in Fig. 1 (right panel) where we can see a similar tendency inthe sense that the ratio (mH,der −mH)/mH is O(10)% for large values of the scale µ(t). Thisfeature of µ(t∗) being close to the top-quark running mass on shell (i.e. mt(Q = mt) = mt)remains for all values of the top-quark mass and supersymmetric parameters, as in Fig. 2.We also have defined physical (pole) masses for the top quark and the Higgs boson. Forthe top-quark we have included the usual one-loop QCD corrections, and for the Higgs bosonone-loop electroweak corrections asM2H = m2H(t) + Re[ΠHH(M2H)−ΠHH(0)](3)where mH(t) is the running Higgs mass and ΠHH the Higgs self-energies which can be foundin Ref. [5]. We have plotted in Fig. 3 MH as a function of tan β (left panel) and as a functionof Mt (right panel). From Fig. 3 we can see that the present experimental band from CDF/D03Figure 2: Plot of µ(t)∗ as a function of mt for tanβ ≫ 1 (solid lines), tan β = 1 (dashed lines),and supersymmetric parameters MS = 1 TeV, Xt = 0 (left panel) and X2t = 6M2S (right panel).The dotted lines correspond to µ(t∗) = mt.for the top-quark mass requires MH <∼ 140 GeV, while if we fix Mt = 170 GeV, then the upperbound MH <∼ 125 GeV follows. It goes without saying that these figures are extremely relevantfor MSSM Higgs searches at LEP 2.3 Comparison with other approachesIt is worth at this point to compare our NTLL calculation with other similar calculations aim-ing to evaluate two-loop corrections to the lightest Higgs mass in the MSSM. For this purposewe have plotted in Fig. 4 MH as a function of the minimization scale µ(t∗) for different valuesof supersymmetric parameters, and for the LL (dashed lines) and NTLL (solid lines) approxi-mations. We can see that our definition of pole mass (3) provides a highly scale independentHiggs mass in the NTLL approximation. Moreover, as was the case in Fig. 1 (left panel), theLL approximation exhibits a strong dependence with the minimization scale.✷ Kodaira, Yasui and Sasaki [6] have used an effective potential method neglecting gaugecouplings in the one-loop correction to the effective potential (g = g′ = 0) and neglecting thecontribution of the stop mixing to the threshold corrections (Xt = 0). Furthermore they havenot introduced any wave function renormalization for the Higgs and thus have worked withrunning Higgs masses. Finally they have chosen as minimization scale µ(t∗) = v = 246.22GeV and claimed that the difference between the NTLL and the LL approximations is positiveand sizeable, unlike a previous result in Ref. [2] where it was claimed it to be negative andsmall. A quick glance at Fig. 4 allows to understand these seemingly contradictory results.We have plotted with a ✷ the minimization scale chosen in Ref. [6] and with a ✸ our choice.Had we chosen as minimization scale the ✷ we had found (even if our analysis includes gauge4Figure 3: Left panel: Plot of MH as a function of tan β for Mt = 170 GeV and MS = 1 TeV.The solid (dashed) curve corresponds to the X2t = 6M2S (Xt = 0) case. Right panel: Plot ofMH as a function of Mt for tanβ ≫ 1 (solid lines), tan β = 1 (dashed lines), and X2t = 6M2S(upper set), Xt = 0 (lower set). The experimental band from the CDF/D0 detection is alsoindicated.coupling effects and considers pole masses) that (NTLL–LL) is positive and sizeable. Howeverour choice of the minimization scale at the ✸ position suggests (NTLL–LL) negative andtiny. Since we made our choice of the minimization scale such that the effective potential isas scale independent as possible we could say that the latter statement is the correct one.However, since we have defined pole (scale independent) masses for the Higgs boson (unlike inRef. [6]) in the NTLL approximation, and the LL approximation is strongly scale dependent,the quantity (NTLL–LL) is scale dependent and then meaningless to qualify the goodness ofthe approximation.△ Langacker and Polonsky [7] work in the LL approximation and fix as minimizationscale µ(t∗) =MZ . They find larger values than those in Fig. 3. We have plotted in Fig. 4 theirminimization scale with a △. We can easily see that for that scale the two-loop correctionsthat they disregard are large and negative. Had they evaluated them they should have foundsimilar results to ours.✸ Finally, Hempfling and Hoang [8] use diagrammatic and effective potential approachesto evaluate the lightest Higgs mass at two-loop order. They use various approximations, as e.g.g = g′ = 0 in the effective potential and only deal with the case of zero stop mixing, Xt = 0.Their results contain LL and NTLL corrections to one- and two-loop orders, while our LL andNTLL corrections are resummed to all-loop by the RG. However, in the case of no mixing,where we can compare both approaches, our results agree within ∼2-3 GeV, which we consideras fully satisfactory, given all their simplifications and the fact that this corresponds to theuncertainty of our own calculation, due to the tiny scale dependence of the effective potentialand pole masses, and other small effects as, e.g., the possible presence of light charginos and5Figure 4: Plot of MH as a function of µ(t∗) for Mt = 170 GeV, MS = 1 TeV and Xt = 0. Thesolid (dashed) curves correspond to the NTLL (LL) approximation, and the upper (lower) setto tan β ≫ 1 (tanβ = 1).neutralinos.4 An analytical approximationWe have seen from Figs. 1 and 2, and previous considerations that, since radiative correctionsare minimized for scales Q ∼ mt, when the LL RG improved Higgs mass expressions [9] areevaluated at the top-quark mass scale, they reproduce the NTLL value with a high level ofaccuracy, for any value of tanβ and the stop mixing parameters [10]mH,LL(Q2 ∼ m2t ) ∼ mH,NTLL. (4)Based on the above observation, we can work out a very accurate analytical approximationto mH,NTLL by just keeping two-loop LL corrections at Q2 = m2t , i.e. corrections of order t2,where t = log(M2S/m2t ).Again the casemA ∼MS is the simplest, and very illustrative, one. We have found [10, 11]that, in the absence of mixing (the case Xt = 0) two-loop corrections resum in the one-loopresult shifting the energy scale from MS (the tree-level scale) to√MS mt. More explicitly,m2H =M2Z cos2 2β(1− 38pi2h2t t)+32pi2v2m4t (√MSmt)t (5)where v = 246.22 GeV.In the presence of mixing (Xt 6= 0), the run-and-match procedure yields an extra piece inthe SM effective potential ∆Vth[φ(MS)] whose second derivative gives an extra contribution tothe Higgs mass, as∆thm2H =∂2∂φ2(t)∆Vth[φ(MS)] =1ξ2(t)∂2∂φ2(t)∆Vth[φ(MS)] (6)6Figure 5: The neutral (Hh, H) and charged (H+) Higgs mass spectrum as a function of theCP-odd Higgs mass mA for a physical top-quark mass Mt = 175 GeV and MS = 1 TeV, asobtained from the one-loop improved RG evolution (solid lines) and the analytical formulae(dashed lines). All sets of curves correspond to tanβ = 15 and large squark mixing, X2t = 6M2S(µ = 0).which, in our case, reduces to∆thm2H =34pi2m4t (MS)v2(mt)X2tM2S(2− 16X2tM2S)(7)We have compared our analytical approximation [10] with the numerical NTLL result [5] andfound a difference <∼ 2 GeV for all values of supersymmetric parameters.The case mA < MS is a bit more complicated since the effective theory below the su-persymmetric scale MS is the 2HDM. However since radiative corrections in the 2HDM areequally dominated by the top-quark, we can compute analytical expressions based upon the LLapproximation at the scale Q2 ∼ m2t . This has been done in Ref. [10] where LL are resummedto two-loop. Our approximation differs from the LL all-loop numerical resummation by <∼ 3GeV, which we consider the uncertainty inherent in the theoretical calculation, provided themixing is moderate and, in particular, bounded by the condition,∣∣∣∣∣∣m2t˜1−m2t˜2m2t˜1+m2t˜2∣∣∣∣∣∣ <∼ 0.5 (8)7where t˜1,2 are the two stop mass eigenstates. In Fig. 5 the Higgs mass spectrum is plottedversus mA.5 Threshold effectsThere are two possible caveats in the approximation we have just presented: i) Our expan-sion parameter log(M2S/m2t ) does not behave properly in the supersymmetric limit MS → 0,where we should recover the tree-level result. ii) We have expanded the threshold function∆Vth[φ(MS)] to order X4t . In fact keeping the whole threshold function ∆Vth[φ(MS)] we wouldbe able to go to larger values of Xt and to evaluate the accuracy of the approximation (1) and(7). Only then we will be able to check the reliability of the maximum value of the lightestHiggs mass (which corresponds to the maximal mixing) as provided in the previous sections.This procedure has been properly followed in Refs. [10] and [12], where the most general caseFigure 6: Plot of the exact (solid line) and approximated (dashed line) effective mixing Xefft asa function of At, for MS = 1 TeV and µ = 0.mQ 6= mU 6= mD has been considered. We have proved that keeping the exact thresholdfunction ∆Vth[φ(MS)], and properly running its value from the high scale to mt with the cor-responding anomalous dimensions as in (6), produces two effects: i) It makes a resummationfrom M2S to M2S +m2t and generates as (physical) expansion parameter log[(M2S +m2t )/m2t ]. ii)8It generates a whole threshold function Xefft such that (7) becomes∆thm2H =34pi2m4t [M2S +m2t ]v2(mt)Xefft (9)andXefft =X2tM2S +m2t(2− 16X2tM2S +m2t)+ · · · (10)In fact we have plottedXefft as a function of At (solid line) and compared with the approximationwhere we keep only terms up to X4t (dashed line), as we did in the previous sections. The resultshows that the maximum of both curves are very close to each other, what justifies the reliabilityof previous upper bounds on the lightest Higgs mass as, e.g., in Fig. 3.6 ConclusionsWe have seen that effective potential methods, when decoupling is properly accounted for, areuseful and powerful techniques to analyze the MSSM Higgs sector and provide easy-to-use ana-lytical approximations (with radiative corrections RG resummed) to the Higgs mass spectrumand couplings. In particular, an appropriate treatment of stop and sbottom decoupling allowsto consider large mixing parameters and put reliable upper bounds on the lightest Higgs bosonmass in the MSSM.References[1] Y. Okada, M. Yamaguchi and T. Yanagida, Prog. Theor. Phys. 85 (1991) 1; Phys. Lett.B262 (1991) 54; J. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett.B257 (1991) 83; Phys. Lett.B262 (1991) 477; R. Barbieri and M. Frigeni, Phys. Lett. B258 (1991) 395; R. Barbieri,M. Frigeni and F. Caravaglios, Phys. Lett. B258 (1991) 167[2] J.R. Espinosa and M. Quiro´s, Phys. Lett. B266 (1991) 389[3] H.E. Haber and R. Hempfling, Phys. Rev. Lett. 66 (1991) 1815; A. Yamada, Phys. Lett.B263 (1991) 233[4] B. Kastening, Phys. Lett. B283 (1992) 287; C. Ford, D.R.T. Jones, P.W. Stephensonand M.B. Einhorn, Nucl. Phys. B395 (1993) 17; M. Bando, T. Kugo, N. Maekawa andH. Nakano, Phys. Lett. B301 (1993) 83; Prog. Theor. Phys. 90 (1993) 405; C. Ford,Phys. Rev. D50 (1994) 7531[5] J.A. Casas, J.R. Espinosa, M. Quiro´s and A. Riotto, Nucl. Phys. B436 (1995) 3; (E) B439(1995) 466[6] J. Kodaira, Y. Yasui and K. Kasaki, Phys. Rev. D50 (1994) 7035[7] P. Langacker and N. Polonsky, Phys. Rev. D50 (1994) 2199[8] R. Hempfling and A.H. Hoang, Phys. Lett. B331 (1994) 999[9] M. Carena, K. Sasaki and C.E.M. Wagner, Nucl. Phys. B381 (1992) 66; H.E. Haber andR. Hempfling, Phys. Rev. D48 (1993) 4280[10] M. Carena, J.R. Espinosa, M. Quiro´s and C.E.M. Wagner, Phys. Lett. B355 (1995) 209[11] H.E. Haber, R. Hempfling and A.H. Hoang, CERN preprint, in preparation[12] M. Carena, M. Quiro´s and C.E.M. Wagner, preprint CERN-TH/95-15710

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