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Vacuum stability and at high scales with two Higgs doublets Emanuele Bagnaschi, Felix Brümmer, Wilfried Buchmüller, Alexander Voigt, Georg Weiglein

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Emanuele Bagnaschi, Felix Brümmer, Wilfried Buchmüller, Alexander Voigt, Georg Weiglein. Vacuum stability and supersymmetry at high scales with two Higgs doublets. Journal of High Energy Physics, Springer, 2016, 2016 (3), pp.158. ￿10.1007/JHEP03(2016)158￿. ￿hal-01444206￿

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Distributed under a Creative Commons Attribution| 4.0 International License JHEP03(2016)158 Springer , March 2, 2016 a March 22, 2016 : : January 27, 2016 : Accepted Published 10.1007/JHEP03(2016)158 Received , doi: and G. Weiglein a Published for SISSA by A. Voigt a 2 and pseudoscalar masses of at least about . [email protected] β , [email protected] , W. Buchm¨uller, b . 3 1512.07761 F. Br¨ummer, The Authors. a c Supersymmetry Phenomenology

We investigate the stability of the electroweak vacuum for two-Higgs-doublet , [email protected] Laboratoire Univers et Particules dePlace Montpellier, Eug`eneBataillon, 34095 UMR5299, Montpellier, Universit´ede Montpellier, France E-mail: [email protected] [email protected] Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany b a Open Access Article funded by SCOAP ArXiv ePrint: model cannot be matched toHiggs. supersymmetry Light at neutral very and highsupersymmetric charged scales when UV requiring therefore completion a emerge of 125 as the GeV a Standard promising Model signature at of theKeywords: a grand unification scale. scale model contains twoconstraints Higgs point doublets. towards low In values of thisa tan TeV. case If vacuum the stability andare experimental similar of and the Higgs essentiallyalso fields independent given are of also electroweak-scale the kept masses light, higgsino (split the mass. conclusions supersymmetry Finally, with if two all Higgs doublets), are the Abstract: models with ataken supersymmetric to UV be of completion.superpartners the decouple order at The of this the supersymmetryone scale. grand Higgs breaking We unification doublet, show scale scale. matching that to is contrary We the to first supersymmetric the study UV Standard the completion Model case is with where possible all if the low- E. Bagnaschi, Vacuum stability and supersymmetrywith at two high Higgs scales doublets JHEP03(2016)158 5 6 8 ]. From the point of 2 , 1 ]. 7 ], which has been realised in string 5 , 14 4 3 S 11 M – 1 – GeV. Example scenarios include universal high- 17 − 4 7 11 15 t 7 17 M 3 13 8 ] and [ 3 1 ] and higher-dimensional field theory with flux [ 6 3.1 The pure3.2 THDM The THDM3.3 with higgsinos The THDM with split supersymmetry 2.1 Preliminary remarks 2.2 Conventions for2.3 the THDM Matching to2.4 the MSSM at Running the to2.5 scale the scale Matching at2.6 the weak scale Higgs-mass predictions states, would then bewith the spontaneously non-supersymmetric broken effective supersymmetry.very field high This theory energy of UV scalescale a completion of supersymmetry UV would about [ completion take 10 effecttheory [ at a embedding of the Standardtheory, the Model leading into candidate a fortry a more unified to fundamental theory guarantee theory. of a all perturbativelyview In interactions, of controlled relies particular superstring stable on theory, string vacuum supersymme- the state genericis expectation [ at for or the close scale to ofthan the supersymmetry string the breaking scale, electroweak which is scale. of course The usually many Standard orders Model, of magnitude possibly larger supplemented by other light The structure of thecrease of electroweak symmetry and and strong toshorter a interactions distances. unification seems It of is the to thenspace-time fundamental point natural forces symmetries to towards as of expect we an that the probe symmetriestry, in- Standard shorter grand larger Model and unification than of the and internal additional and space-time physics, dimensions, including will supersymme- play a crucial role for the 1 Introduction 4 Summary and outlook A Details on the matching atB the weak Vacuum scale (meta)stability 3 Results Contents 1 Introduction 2 THDM models as effective field theories JHEP03(2016)158 ] and references 10 ]. Recently, a detailed analysis of the ], but without imposing constraints from 11 16 – 13 – 2 – ], however, without taking vacuum stability constraints into 12 ]. Therefore, to allow for a supersymmetric UV completion at scales 9 , 8 ]. This is because at higher energies the running Higgs quartic coupling in 8 GeV, more states need to be kept light in the low-energy theory, besides the 17 − 15 GeV [ In the present paper we show that several kinds of two-Higgs-doublet models can in- A particularly interesting class of models retains both MSSM Higgs doublets as light Our ability to extrapolate some non-supersymmetric low-energy effective theory to Admitting a supersymmetric UV completion at high scales is a nontrivial constraint on In the past, the main motivation to consider supersymmetric extensions of the Stan- 11 near the Planck scalehigh-energy was supersymmetry. studied in refs. [ deed be matched tofrom GUT-scale vacuum or instability. even to We study string-scale supersymmetry three without exemplary suffering models using the spectrum gener- states at low energies,symmetry. The with matching or of withoutenergies the two-Higgs-doublet the has model light previously (THDM) higgsinos beenmatching to for and the discussed a MSSM gauginos in variety at of ofhas ref. high THDM been split [ models performed super- as in functionaccount. ref. of [ With the regards supersymmetry to breaking vacuum scale stability, the extrapolation of a THDM to high energies be positive definite at thestill UV be completion formally scale, unbounded theof from RG-improved the below tree-level running potential at couplings. may intermediategeneral energies This deeper when would than expressed signal the in the realistic terms presence electroweak of vacuum. additional vacua which are in potential becomes unbounded from below (althoughhas the been lifetime estimated of the to electroweaktherein). be vacuum longer More than generally, theimposes demanding age additional of a constraints the stable onsupersymmetric universe, any or UV see low-energy at completion. ref. theory, least Although [ even supersymmetry sufficiently if ensures long-lived it that vacuum can the be potential will matched to a of 10 Higgs doublet and possibly gauginos and higgsinos. high energies may alsothe Standard be Model limited itself: by as vacuum a stability. result of This, the quartic again, coupling is turning already negative, the seen Higgs in 10 the Standard Model becomespositive definite. negative, The while maximal matching thethe scale D-term is electroweak-scale even potential spectrum lower for in consistsand split supersymmetry of supersymmetry, higgsinos where the is [ Standard Model and the MSSM gauginos fixed by some unknown ultravioletto dynamics us. at Our a hypothesis value for whichscale the presently of present appears paper its unnatural is breaking that is supersymmetry high, does that exist but, it since playsthe the no low-energy role effective theory. in For stabilisingitself example, the cannot it electroweak is be hierarchy. well matched known to that the its Standard minimal Model supersymmetric by extension (MSSM) above about dard Model used to bestabilise the a hierarchy large problem: hierarchyscale electroweak-scale between against supersymmetry the radiative allows electroweak corrections. to scale However,try. and so Should far a the no much data evidence higher showshave in to fundamental no conclude its sign that of favour the supersymme- surface electroweak scale during is not the actually second protected by run supersymmetry, of but the LHC, one may JHEP03(2016)158 DR), the parame- 1 TeV, is in agreement and that the remaining S & M A M 2 and of the non-standard Higgs is . A GeV and the electroweak scale. While β M 17 − 15 – 3 – ]. The conclusions are similar but somewhat more = 10 21 S – , namely tan -loop renormalisation group equations in between the M A n 19 ]: a pure type-II THDM, the THDM with additional [ M 17 ττ MS scheme (or one of its cousins such as → H,A ] and the limits from the searches for additional Higgs bosons, in par- 18 )[ and relatively large sγ GeV), and the THDM with the full and higgsino field content of split β 14 → b 1)-loop precision. If the masses of two heavy states are comparable to each other, In particular, we take all the eigenvalues of the Higgs mass matrix to be comparable to For the present study we will always use precisely one effective field theory between − n each other, and therefore theto running the parameters measured of pole the massesthe of THDM often the must Standard be considered Model matched case .much directly where higher Thus, the our than study the mass differs electroweak scale from scale. In this case the appropriate procedure would be intermediate thresholds certainly offer interestingwe possibilities will always to assume generalise that one our setstates work, of will here particles decouples obtain close masses to include at the most of Standard the Modelinvestigate order the particles cases of where and a they TeV.all a also MSSM These second include gauginos. a “light” Higgs pair states doublet; of will higgsinos, we always or will a pair furthermore of higgsinos and two heavy states arewhich widely the separated, logarithms then shouldan they be intermediate define resummed, effective two theory. using distinct the thresholds renormalisation between group equationsthe of scale thresholds, and at eachparameters threshold of crossing the the resulting heavy( effective states theory are are decoupledthey matched by to should hand. those be The of decoupledappropriate the simultaneously threshold full correction and theory at their with leading-log mass order. difference If accounted on for the by other an hand the masses of 2.1 Preliminary remarks The standard procedure for treatingto theories “run with and several match” hierarchically the separatedand effective scales field renormalised is theory using parameters. the Thatters is, are the evolved theory according is to regularised their Unification because the predictedtoo mass large of in the the Standard parameter Model-like regions Higgs allowed by the is2 other always constraints. THDM models as effective field theories at low tan with all these constraints asof well BR( as with theticular in experimental the bounds channel fromrestrictive the for measurement the THDM withon light the Higgsinos. other For hand, the we THDM find with that split supersymmetry, the model cannot be extrapolated to the scale of Grand electroweak-scale higgsinos (which hastion the at appealing 10 propertysupersymmetry of at gauge the coupling electroweak scale. unifica- supersymmetric It UV turns completion, out a thattive stable the on combined vacuum, the requirements and of low-energy a a spectrum. 125 GeV For Higgs the are pure quite THDM restric- we find that the parameter region ator framework FlexibleSUSY [ JHEP03(2016)158 (2.1) (2.2) , and are 17 . − u 15  ˜ h . u c ˜ 10 . B ˜ h † 1 ˜ 2 B ∼ | † H 2 2 2 0 S u H ) + h H ˜ γ 2 √ † 2 1 0 u M ˜ g , to match the THDM to H √ H + † | 2 A 4 u + H , λ M ˜ h u 4 )( . Moreover, if there are light ˜ ˜ h 2 V W 2 ) + † 1 ˜ H 2 + W H † 1 † H 2 H  TeV is a strong assumption, which 2 . H † u 2 H ( c ˜ γ . . √ 7 2 H u λ ˜ g √ )( + 1 + h d + itself. Here we postulate that the various H ) + 2 ˜ h † d 1 1 ) in the spectrum, they are coupled to the ˜ S H B ˜ h u H † H 2 1 ˜ ˜ h † ( M 1 B – 4 – 3 H H 1 , H λ 2 d 0 d H 2 12 )( ˜ ˜ h γ √ 2 + 2 0 d m ˜ g 2 √ H  + ) † 1 2 d + − H ˜ h H d 2 ( † 2 6 ˜ ˜ h and the other involving W H H λ 2 † 2 ˜ ( 1 W 2 + H 1 .. H , and then to evolve the Standard Model running parameters 2 H λ c 2 2 2 2 d A . H ) ˜ γ √ 2 2 m + d M ˜ g 2 H √ + h + + ) † ) and higgsinos ( 1 1 a 1 = H ˜ H G ( H † 1 † 5 1 , 2 i λ Yuk H H ( ˜  2 1 W 1 2 −L λ , m + ˜ B = = 4 V V If all the couplings allowed by gauge symmetry were actually present (and sizeable) in Imposing that all Higgs bosons acquire masses Yukawa couplings between the higgsinos, right-handed and Higgsthe bosons. THDM, this would leadneutral to currents. phenomenologically However, unacceptable matching rates toparameter of supersymmetry space flavour leads as changing to we strong will restrictions now on describe the in detail. The gauge symmetries of the general THDM with higgsinos and gauginos further allow for For each Yukawa term allowedsuch in terms, the one Standard involving gauginos Model, ( the generalHiggs THDM doublets contains with two the Yukawa terms argument for a second light Higgs doublet. 2.2 Conventions for theWe use THDM the following conventions for parameterising the scalar potential of the THDM as the UV theory byscale some might means need to unknown be tolight low scalar us. for doublet. anthropic It reasons, We do andon has not that rather been subscribe this to frail argued would these predict assumptions, thatelectroweak arguments; precisely hierarchy, the and it this one would seems electroweak that not to even necessarily us preclude if that a they anthropics (presently rest unknown) should anthropic indeed be related to the scale physics. The Higgsmatching mass parameters conditions of to the the low-energygenerically theory unknown expected are supersymmetric to determined theory be bycontributions the to at of the the Higgs mass order matrixsuch of cancel that each all other of to a itsabout very entries the high are degree reasons of of — precision, the in order our of approach at we most assume a that TeV. the We hierarchy refrain problem from is speculating solved by the Standard Model at down to the electroweak scale. as discussed above125 is GeV also technically the masses unnatural of since further relatively as light Higgs for bosons should the be discovered affected by Higgs high- boson at to decouple the non-standard Higgs bosons at the high scale JHEP03(2016)158 and (2.4) (2.3) S ]. The 11 µ/M , are the Higgs S d U(1) theory. H × A/M and u H in our analysis. h ) are not specific to the UV m 2.3 , where , , . S 0 0 g g , g , g , , , M .    = = 0 = = = 2 2 2 S 0 0 0 0 0 d d d 0 u u , and that all other soft parameters are g g g = 0 ˜ ˜ g g ˜ ˜ γ g g M S 7 + + − = ˜ , λ M 2 2 2 2 0 u = 1 supersymmetric SU(2) g g g g = γ 1 2    – 5 – 6 N = ˜ 1 4 1 4 1 4 at the scale − λ = 0 in our analysis, and since the Yukawa terms d u = = = = = 7 γ are also absent at the matching scale (up to small is of course unknown, but we use the GUT model H λ 1 2 3 4 5 . = ˜ S λ λ λ λ λ c = . = u M 2 ˜ γ 6 H λ + h L = ` . and 5 read at the tree-level 1 R λ g ¯ e ∗ d S † 5 3 u H M H 2 q -term potential of an iσ + ≡ − 0 D L g q = R ¯ d 1 † u and H H ) in the pure THDM case, and the electroweak gaugino threshold corrections in 2 g ] as a guidance. It predicts that the squark and slepton soft masses are degen- + 2.3 7 ≡ L q g R If there are winos or binos in the spectrum, the matching conditions for their Yukawa Note that the tree level matching conditions eqs. ( Since we are setting In the following we set these threshold corrections to zero forFollowing the definiteness, same line with of the reasoning, we also neglect the higgsino threshold corrections The one-loop threshold corrections to these couplings are e.g. listed in ref. [ gives ¯ u S † d H threshold corrections which we neglect), our model becomes ancouplings effective at type-II the THDM. scale the case of both the pure THDM and thecompletion THDM being with the light MSSM, higgsinos. butemerges apply from in the any model in which the quartic scalar potential correspondingly reduced. understanding that this iseffects, a we will source assume of a model conservative 3 GeV dependence. uncertainty on to To eqs. account ( for the neglected of ref. [ erate to leading ordergenerated at at the subleading matching order.are scale In suppressed this not case onlyby the by the squark a near-degeneracy and loop of slepton factor the threshold but corrections squarks also and by the sleptons, small and ratios their impact on our results is Here exact spectrum at We identify doublets of the minimal supersymmetricM Standard Model. Tree-level matching at the scale 2.3 Matching to the MSSM at the scale JHEP03(2016)158 , 10 r µ 10 (2.9) (2.5) (2.6) (2.7) (2.8) (2.10) & ) may be 2.8 ) distinguishing L DR conversion terms The latter is used − 1 MS– B ]. 26 – . ) are satisfied, the condi- ,  24 2 2.7 / 1 ) meta -parity or )–( 2 λ , , , . 2 λ R / 1 0 0 0 0 2.5 ≡ 1 λ ) > > > > 2 r λ 1 2 2 2 + ( µ / / 1 λ λ GeV 1 1 4 λ ) ) λ 2 2 . 10 λ λ + 82 1 1 B . + 2 ( 3 λ λ 2 , as well as the “wrong Higgs” and λ 2 – 6 – ] λ + ( + ( 0 1 + log u,d 2 . 23 γ 3 4 + / ], finding complete agreement. 1 , ˜ λ λ 1 41 ) 2 λ 28 + − , u,d λ γ 3 1 t 27 & , ˜ λ λ 7 ) , M r 6 4 ( , ), as described in the following as well as in more detail in t µ 5 ( λ = M λ λ down to the electroweak scale according to their renormalisation ] in combination with SARAH 4.6.0 [ ) is given in appendix S 17 2.9 M . A ) is replaced by an inequality which should hold at all renormalisation scales evolve from i 2.8 λ FlexibleSUSY makes use of 2-loop renormalisation group equations and provides an In order to numerically study the running of the parameters in the presence of the The stability conditions can be relaxed if one allows for additional vacua besides the To obtain a scalar potential that is bounded from below, a set of sufficient conditions The SARAH version we use contains an additional bug-fix, which corrects the 1 in the left- and right-handed one-loop self-energies. automatic matching of the THDM tothe input parameters matching at the at electroweak the scaleappendix (we scale perform boundary and vacuumFlexibleSUSY stability 1.2.1 conditions, [ weto use compute the the 2-loop renormalisation spectrumpreliminary group safety-check, generator equation we for framework have thethe compared effective ones the field provided expressions theories. by obtained As PYR@TE from a [ SARAH with where A derivation of eq. ( electroweak one, and merely imposes thatyears. the lifetime of In the electroweaktion that vacuum ( be case, assuming that the conditions ( Numerically it will turn outquence that of the the first supersymmetric threeviolated matching conditions at conditions, are intermediate while always scales. the satisfied fourth as one a eq. conse- ( on the running scalar couplings is ref. [ group equations. Note that Yukawa couplings, are protectedwill by not the be symmetries generated ofassume during the is the effective the running theory case. if and they We therefore are therefore work zero with at all the these matching couplings scale, set which to we zero henceforth. that there is some conservedthe quantum higgsino number from (such the as the lepton Higgs, doublets, the such higgsino that and there the right-handed are2.4 leptons. no Yukawa couplings Running between toThe the scale We will again neglect possible effects from small threshold corrections. We also assume JHEP03(2016)158 (2.13) (2.11) (2.12) . , and the β , where we A t = 125 GeV. , the quartic M h 2 d MS parameters v M + 2 u , , v   174 GeV implies that, q β β . 2 ≈ 2 ) t = v . This leads to correlations M µ v ) sin ) cos 4 4 = λ λ r + + , µ 3 3 . In practice, however, the theory λ . Setting λ A 2 h,H S m + ( M . + ( M β 2 1 β and = v v 2 is very mild. This allows us, in principle, 2 2 and A p S ≡ cos ( sin m β MS renormalised CP-even Higgs one-loop self- h 1 β 2 M – 7 – λ λ 2 tan 2 Re Σ β , v v − − − ) is the cos 2 h r β β β M , µ 2 tan cot sin p = ( 2 A 2 12 2 12 MS Higgs mass h 1L m m m 2 h, = = = M 2 1 2 2 and Σ 2 12 t m m MS gauge and Yukawa couplings as well as the VEVs of the THDM m , one of which can (in principle) be fixed by requiring M β = r µ denotes the CP-even Higgs mass matrix expressed in terms of the and tan 2 h A of the one-loop-corrected mass matrix M m We calculate the CP-even Higgs pole masses by numerically finding the two eigenvalues We note that our models have the appealing feature that there are very few parameters , 2 h,H S Here, at the scale energy matrix, where the Higgs fields at the external legs are taken to be the Higgs gauge between the Higgs andcolliders. and In the THDM masseschargino with which and higgsinos are neutralino and in masses, gauginos, principlefrom and the testable the may Higgs . at in masses This addition depend will be on also all affected become the by evident two-loopM in corrections the next section. room for significant variation, as we will2.6 detail in the next Higgs-mass section. predictions In the THDM with higgsinos, the and Higgs hence masses depend receive loop on the corrections higgsino from mass and parameter M Moreover, requiring vacuum stability forcessensitivity us of into the the region low-energy ofto spectrum rather predict low to a tan sharpuncertainty on correlation the between calculation tan of the lightest Higgs mass is still so large that there is still left in the low-energy theory.gauge Since the couplings quartic via couplings arewhich the essentially directly determined supersymmetric affects by them boundary the in is conditions, the the pure the matching THDM, scale the only Higgs free mass spectrum parameter is completely determined by the parameters allow us to expresscouplings, the the pseudoscalar entire scalar potential in terms of More details on the matching procedure at the loop level are given in appendix level, the well-known THDM relations By integrating the 2-loop renormalisationters group of the equations THDM we (potentially obtainmatch including the the higgsinos running and THDM parame- gauginos) to at experimentallyby the known scale calculating input parameters. the Thefrom matching known is input performed parameters at the 1- and leading 2-loop level. In particular, at the tree 2.5 Matching at the weak scale JHEP03(2016)158 . · , , 8, β . 1 2 h,H = 2 = 30. Planck . M values, S β . Thus, β M = β , and this M of the order 2 ττ β p S M → GeV. The vacuum . It is metastable region, tan β H,A 14 β 10 · ), which together with the = 200 GeV and somewhat lowers the upper = 2 GeV for two values of tan sγ A S S 14 m M → M ], = 125 GeV and the central value 10 200 GeV at large tan 1 · b . h 12 . M A = 0 = 2 ) is solved iteratively. S M µ M 2.13 ) (since the charged Higgs is similarly light we show contours of the lightest Higgs mass ] and the limits from the LHC searches for sγ 1 2 for 18 – 8 – 2 , eq. ( → )[ . This scale, an order of magnitude below 1L b sγ 2 h, 1 M → b for a SUSY breaking scale = 125 GeV enforces β h M should be close to 125 GeV still points to somewhat small tan and tan h region. For example, choosing ]. In the top row of figure comes from the measurement of BR( . A M β β 21 A M – M 19 are the eigenvalues of [ 2 ττ h,H 5. M . By contrast, when allowing for the vacuum to be metastable, the most severe < ∼ → t GeV in the bottom row of figure In order to understand why the THDM allows a matching to the supersymmetric For comparison, we also show the case of a higher SUSY breaking scale Note that absolute vacuum stability forces one into the low tan We remark that including high-scale one-loop threshold corrections from heavy hig- β Note that for part of the parameter space considered in ref. [ M 2 17 standard model at very highquartic scales couplings. one This has is to shown study in the figure renormalisation group flowof of the the grand unification scale, the electroweak vacuum is either metastable or unstable. is about the highestsymmetric for field which theory the matching canthe to be same a justified. as weakly for coupledformerly While the four-dimensional metastable the lower super- region qualitative SUSY is behaviour breakingcorrections, now scale similar in unstable. remarks case, the as we plane Concerning above observe is the apply. that higgsino a one-loop large threshold part of the constraint on requirement that tan 10 as the pseudoscalar) andincluding or by neglecting the these limitsis threshold phenomenologically from corrections disfavoured the only anyway. LHC affects a searches parameter for regionwith which pseudoscalar Higgs masses exceedingof a TeV for on the large tan boundary of the unstableHowever, the region constraint and opensparameter up region a is newTHDM excluded stable from by the region the measurement around of constraint tan BR( on the charged mass in a is absolutely stablein only the in bulk the ofintermediate white tan the unshaded parameter region space at (orange low regions), tan andgsinos, unstable which we in have the neglected red in region generating of these plots, can have a significant impact The low-energy parameterrequiring space the is lightest strongly Higgsfrom constrained boson the by mass measurement to vacuum be ofH,A (meta)stability, 125 by BR( GeV, and byas the a function experimental of bounds where 3 Results 3.1 The pure THDM eigenstates. Since the Higgs self-energy has to be evaluated at the momenta JHEP03(2016)158 5000 5000

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125 = 2 = 2 µ µ 3000 3000 58 GeV . ) [GeV] ) [GeV] t t

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plane in the pure THDM 125 M M 128 128 125 ( ( β GeV, GeV, = 172 A A t m m 2000 2000 14 17 M 10 10 · · ). Unshaded regions are allowed ) – tan t t 122 M

122 M 1000 1000 ( = 2 = 2 122 (

122 A A S S m m M M 0 5 0 5 0 5 0 5 0 5 0 5

......

1 GeV

. 1 2 1 4 3 3 2 2 4 3 3 2 β tan β tan in the , large GeV (bottom row). The Higgs mass prediction is h β = 174 17 t M – 9 – 10 M · 500 500 = 2 GeV GeV S 450 450 13 16 M 10 10 · · 400 400 76 GeV (solid black, dashed green and dotted blue). Left: full range . 34 GeV = 2 = 2 0 . 350 350 µ µ  ) [GeV] ) [GeV] t t = 173 34 300 300 . t M M ( ( ); right: region of low tan M GeV, GeV, A A t 250 250 m m M 17 14 = 173 ( GeV (top row) and t A 10 10 · · 200 200 14 M m 122

122 10 · = 2 = 2 150 150 125 128 125 128 . Contours of the lightest Higgs mass S S , low = 2 β M M S 100 100

M

20 10 50 40 30 20 10 50 40 30 β tan β tan by vacuum stability. Inlarger the than the orange age region, ofuncalculable the the because universe. electroweak perturbative Red vacuum control regions is is are unstable lost. excluded but by vacuum its stability. lifetime Grey regions is are Figure 1 for computed for of tan JHEP03(2016)158 14 14 2 / 10 10 1 ) , all 13 13 2 S λ 10 10 1 ). shows, 12 12 λ M 15 . 10 10 2 + ( 11 11 = 5 = 1 2.10 4 10 10 β β λ . Hence the 10 10 S + 10 10 3 meta M 9 9 λ λ λ ) and ( 10 10 8 8 [GeV] [GeV] 2.9 can compensate the GeV, for two different 10 10 r r 2 7 7 µ µ 14 λ 10 10 = 2000 GeV, tan = 2000 GeV, tan 1 10 A A 6 6 λ · M M 10 10 2 / √ 1 5 5 ) 2 = 2 10 10 Vacuum stability conditions Vacuum stability conditions λ is negative at 4 4 1 S λ 4 10 10 M λ 3 3 + ( 10 10 1 2 3 2 2 λ λ λ are defined in eqs. ( 10 10 0 4 2 0 4 3 2 1 0 8 6 1 2 2 ...... 0 0 0 0 0 0 0 0 1 0 0 0 0 0

− − −

opigvalue Coupling opigvalue Coupling meta λ and 14 14 – 10 – t 3 10 10 g y λ 13 13 10 10 also to large values in the IR. In the UV, at 12 12 15 . 2 10 10 λ 11 11 is sufficiently large such that = 5 = 1 10 10 β β 2 1 2 10 10 λ g g 0 is the most stringent stability constraint. As figure 10 10 9 9 > 10 10 the absolute value of the top-quark Yukawa coupling is large in 2 8 8 = 5 this is no longer the case, and only the weaker metastability λ β [GeV] [GeV] 10 10 1 r r β 7 7 µ µ λ 10 10 = 2000 GeV, tan = 2000 GeV, tan √ 3 4 A A 6 6 λ λ + M M 10 10 4 5 5 λ 10 10 15 the coupling . + 4 4 Renormalisation group running Renormalisation group running . For tan 3 10 10 4 λ 3 3 = 1 . Renormalisation group running of dimensionless parameters (left column) and the λ denotes the renormalisation scale. 10 10 1 2 β r λ λ 2 2 µ 10 10 5 0 5 0 0 5 0 5 5 0 5 5 5 0 ...... 0 0 1 1 1 0 0 1 0 1 1 1 0 1

− − − − − − opigvalue Coupling opigvalue Coupling THDM. Due tocondition the boundary conditions thefor tan coupling negative condition is satisfied. For small values of tan the IR. This drivesquartic the couplings are coupling determined by the gauge couplings, which approximately unify in the Figure 2 vacuum stability conditions (right column),points in characterised the by THDM a for stablerow). electroweak vacuum (top row) and metastable behaviour (bottom JHEP03(2016)158 . µ which shows , which imposes A ) in this 3 β M , while the sγ GUT S , however, a . = 2000 GeV. M M → 4 µ should be close as well as on b GeV. The best- GUT 3 h . Figure , 2 16 M µ , for the electroweak M 1 10 = µ M .) The Higgs sector is · S values than in the pure µ = M A 2 = 2 GeV the parameter space is M M 17 ). An absolutely stable region = GUT 10 equal to sγ · 1 M 2 values is now excluded because the M . As shown in figure → M = 2 β β b S M values than for the pure THDM, in accordance – 11 – A = 2000 GeV to avoid experimental limits from LHC M 3 M . Hence, a scenario where at the weak scale the particle µ , favouring somewhat higher β GeV a wide range of tan 14 , given that the squarks are decoupled. We can therefore assume 3 10 M · = 200 GeV; the picture is qualitatively very similar for = 2 µ S M We find that in the case of light gauginos the vacuum stability conditions are always The low-energy spectrum now depends on the gaugino masses It is important to notice that the existence of a stable region at small tan Higgs mass consistent with observationare can only essentially be excluded obtained by forscenario. the small constraint values Hence, of from the theto measurement extrapolation the of of grand BR( theboson unification THDM mass. scale with is light not higgsinos compatible and with gauginos the up measured value of the Higgs affected by the gluino onlythe through precise two-loop value effects, of andthat therefore the is gluino not is very sufficiently sensitive heavy to to have escaped detectionsatisfied at and the therefore LHC so imply far. no constraint on tan For simplicity we choosesuperpartner a masses, while common keeping low-scaleRun value 1. (Alternativelyleads we could to have very imposed similar results gaugino for mass a unification low-scale at value of 3.3 The THDM with splitWhen supersymmetry retaining thesector full as gaugino the spectrum light degreesthe of of gauge the freedom, couplings this MSSM approximately particlemotivated as content choice unify has for well at the the as the appealing matching its scale feature scale in that complete this Higgs case is therefore to a supersymmetric UVhiggsinos completion at the at LHC the could thereforeUV grand be completion unification interpreted at as scale. the a possible grand A hint unification for discovery scale. a of supersymmetric light even more constrained. no constraints on the parameter content of the Standard Modelabout is a supplemented by TeV and the light Higgs neutral bosons and of charged a higgsinos second is doublet fully at compatible with the matching vacuum is unstable.to For a 125 GeV metastable favours vacuum somewhatwith the higher the requirement constraint that fromremains the at measurement small of values BR( THDM of case. tan For a higher SUSY breaking scale In the case that the gauginos,Higgs squarks bosons and sleptons of are the decoupled THDMthe at the and low-energy scale their mass superpartners spectrum havethe masses depends results at for on the the electroweakAlready scale, additional at parameter 3.2 The THDM with higgsinos JHEP03(2016)158 5000 5000

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128

122 122

128 125 µ µ 3000 3000 58 GeV ) [GeV] ) [GeV] . t t plane for the case where GeV, GeV, M M 122

( ( 125

128 125 β A A 17 14 = 172 t m m 2000 2000 10 10 M · · )–tan t

122 ). Unshaded regions are allowed t = 2 = 2 122 M ( 1000 1000 M S S ( A A M M m m 0 5 0 5 0 5 0 5 0 5 0 5

......

1 2 1 4 3 3 2 2 4 3 3 2 1 GeV β tan β tan . in the h , large β GeV (bottom row). The Higgs mass prediction is = 174 M t 17 – 12 – M 10 · 500 500 = 2 450 450 S M 400 400 76 GeV (solid black, dashed green and dotted blue). Left: full range = 200 GeV = 200 GeV . 34 GeV 0 . µ µ 350 350  ) [GeV] ) [GeV] = 173 t t 34 300 300 . t GeV, GeV, M M ( ( ); right: region of low tan M t A A 17 14 250 250 m m M = 173 ( 10 10 t · · A 200 200 GeV (top row) and M 122 m 122 14 = 2 = 2 125 S S . Contours of the lightest Higgs mass 150 150 128 128 125 10 · , low M M β 100 100 = 2

20 10 50 40 30 30 20 10 40 50 S β tan β tan by vacuum stability. Inlarger the than the orange age region, ofuncalculable the the because universe. electroweak perturbative Red vacuum control regions is is are unstable lost. excluded but by vacuum its stability. lifetime Grey regions is are Figure 3 the spectrum at the electroweakM scale consists of thecomputed THDM for with higgsinos, with of tan JHEP03(2016)158 5000 5000 4000 4000 = 2000 GeV 3 , = 2000 GeV 2 , 3 1 GeV = 200 GeV (bottom M 3000 3000 M 16 µ ) [GeV] ) [GeV] t t

= 140 plane for the case where 10 . Unshaded white regions M M · ( ( µ A β A A M m m 2000 2000 = 2

140 S )–tan

GeV, 145 t

145 = 200 GeV, M 16 , large M ( 2 1000 1000 β , A 10 1 · m M 130110 = 0 5 0 5 0 5 0 5 0 5 0 5 = 2 ......

4 3 3 2 2 1 µ 4 3 3 2 2 1

S β tan β tan in the M h = 2000 GeV (top row) and M µ – 13 – 500 500 450 450 ; right: region of low tan A

140 GeV, with = 2000 GeV 400 400 M 3 , 16 = 2000 GeV 2 , 3 10 1 350 GeV 350

140 · , low M M β 16

150 ) [GeV] ) [GeV] t t 300 300 = = 2 10 M

152 150 M · ( ( µ S A A 250 M 250 m m = 2

155 S GeV, 200

200 153 = 200 GeV, M 16 2 , 150 10 1 150

· 130 125 . Contours of the lightest Higgs mass

M 110 100 100

=

= 2

40 30 20 10 50

β

20 10 tan 50 40 30 µ β S tan M 4 Summary and outlook We have studied thedoublet, matching with of or the without Standarddard additional model Model, higgsinos at supplemented high and scales by gauginos, closeof a to the to the Standard second the GUT Model Higgs scale. supersymmetric is strongly A stan- motivated supersymmetric by ultraviolet unified completion theories, in particular . supersymmetry) for row). Left: full range ofare tan allowed by vacuum stability. Grey regions are uncalculable because perturbative control is lost. Figure 4 the spectrum at the electroweak scale consists of the THDM with gauginos and higgsinos (split- JHEP03(2016)158 , A M and relatively high values of β – 14 – . t M are already essentially excluded by the constraints from A M the large top-quark Yukawa coupling drives one of the quartic Higgs β that are incompatible with low energy measurements. A The Standard Model requires fine-tuning of the and the Higgs It is remarkable that the extrapolation of two-Higgs-doublet models to the GUT scale The matching of the pure THDM to its supersymmetric extension at high scales implies The extrapolation of the Standard Model to high scales is severely constrained by M A Details on the matchingIn at the following the the weak appliedmatching procedure scale is for the performed matching at at the the scale weak scale is described. The This work has been supportedlaborative Research by Center the 676 German “Particles, Sciencewas Strings Foundation also and (DFG) supported the within in Early the partTraining Universe”. Col- Network by PITN-GA-2012-316704. This the research European We thank Commissionuseful J. through Bernon, discussions. the S. “HiggsTools” FB Kraml Initial thanks and the S. Shirai DESY for theory group for hospitality. be fine-tuned. This situation ispuzzle unsatisfactory. will It eventually is be conceivable provided that by an the explanation UV of completion. this Acknowledgments superpartners of the twoLHC Higgs doublets. could be Hence, interpreted agrand as discovery unification of a scale. just possible light hint higgsinos for at a the supersymmetricmass. UV completion In at two-Higgs-doublet models the also the mass term of the second Higgs doublet has to rare processes. Finally,vacuum in stability the conditions are case always fulfilled, ofof but both a Higgs higgsinos mass and of 125 gauginos GeV implies at values the TeVimplies scale essentially no the constraints on the masses of light neutral and charged higgsinos, the of light higgsinos theTHDM. lower bound Because of is this slightly preferencethe more for discovery stringent low of values than of additional for tan HiggsIn the bosons principle, case smaller at of pseudoscalar the the massesvacuum. LHC pure can appears But be challenging these consistent in values with this of a scenario. metastable electroweak couplings to large values inbe the satisfied. IR. As a consequence, all vacuum stabilitya conditions lower can bound on thesignificant additional sensitivity on Higgs the boson remaining masses theoreticaltal of uncertainties error induced about of by a the the TeV. mass experimen- This of bound the top shows a quark and from unknown higher-order corrections. In case the necessary requirementthe of Standard stability Model or apossible matching metastability for to of the its measured the supersymmetricmatching mass extension electroweak of consistent at vacuum. the with the Higgs vacuum GUTsmall boson. stability In scale values is On is of possible the not tan contrary, for as two-Higgs-doublet we models. have shown, For a JHEP03(2016)158 = ) = t Z (A.6) (A.7) (A.8) (A.1) (A.2) (A.3) (A.4) (A.5) ) and M t , from M ( M VB ( δ MS , MS , , # SM(5) s . SM(5) em  α r , α H µ m ] and log 34 1 3 − , , , 940 [ )  i . ) ) r ) t ˜ χ t t ) t µ t , , M m M M 127 M ( ) ) M ( ( ( t . / t ( s W  log em M M MS MS θ α ] using the iterative approach described , , ( ˜ g ( r α 2 ∆ is an up-type fermion is a down-type fermion THDM em µ =1 using the 1-loop QED and 3-loop QCD 34 m ) = 1 ∆ i W X cos i i θ Z t − 4 3 πα − SM(5) em SM(5) s 1 THDM em 4 M THDM s M α 1 α sin ( − 2 log p t ) if πα ) if πα r – 15 – t t MS − 5 3 4 ]. The vertex and box contributions, MS weak mixing angle. The coupling constants 4 , µ m ) = ) = t t t r M ) read M p r q ( r ( µ 37 1876 GeV [ m M M u d , . µ log ( ( in the THDM is determined from the Fermi constant ) of the THDM are determined from the corresponding ( SM(5) em t i ) of the THDM are calculated as /v /v 29 ) = ) = ) = α 9 is the t [ log ) ) α t t t 16 t t W M = 91 2 3 ( θ r M − M M M i W ( M M ( ( ( THDM THDM em s i y " Z θ − ( ( 2 3 1 i i g α α  g g g M π denote the electromagnetic and strong coupling constants of s m m em π 2 α α 2 ( and ∆ˆ ρ using the relations ] and ) = ) = ) = i t THDM s r r 34 [ α µ µ m M ( ( 5 ( s i − MS mass in the THDM is calculated from the top pole mass α em y ] using the full 1-loop self-energy plus 2-loop Standard Model QCD cor- and α 10 ∆ · 36 ), via the relation ∆ t M ], which are evolved to the scale ( MS weak mixing angle MS gauge couplings MS Yukawa couplings THDM em 16638 35 MS masses . α MS , ] taking into account the full 1-loop THDM corrections and leading 2-loop Standard The The The 34 GeV [ = 1 . 29 F 1184 [ SM(5) s -functions in the Standard Model with 5 active quark flavours. . The 173 in [ Model corrections to ∆ˆ potential non-Standard Model particles are neglected here. THDM effective theory. As input,0 we use β G The terms involving theparticles masses have of not been the integrated charginos out and at the the high-scale gluino and are are thus only part present of if the low-energy these where the threshold corrections ∆ α where the THDM, respectively, and of the THDM are related to the corresponding Standard Model ones, JHEP03(2016)158 ) τ τ M , ( (A.9) , with i (A.12) (A.13) (A.14) (A.10) (A.11) ) τ t , M MS mass τ M SM(5) τ M m = (3) ) t ζ r , M 48 , µ ) are 1- and 2-loop t 2 τ MS mass of the . = − M m # r 2 t pole mass, qcd 2 r , = = µ , µ m τ 2 (2) t t r 2 p m ( M , µ 2 b R τ = m 2 (1 + log 4) 2 p 2028 log , = and ∆ ( π using the 1-loop QED RGE and i 2 − R t ) p t t 2 t ( qcd 2 r ), is obtained from the ) + Re Σ , . t ) (A.15) t M R b µ M m t ( τ (1) t M M 2 M ( M m b qcd , = = ) + Re Σ m ) t r b (2) t t r ) = + 2821 + 16 , M τ , µ ) + Re Σ m 396 log M , µ t 2 τ /m " 2 τ ) M =  t = M ( 4 m – 16 – r m 2 t 2 r π r M = 4 = 3 µ ) + ∆ m , µ = t ) is evolved to g , µ r 2 2 t = τ 2 2  SM(5) τ t p M ], 4608 p r ( , µ M ( M ( m ) as M 2 b L τ ) as ( t 30 S τ t , µ − = m 2 b qcd = 3 log , 2 using the 1-loop QED and 3-loop QCD RGE. Afterwards, M M 2 , ( 2 (  m t = p − Re Σ τ b (1) t ) p ( SM(5) τ ), is calculated by first identifying the t ( 2 h 4 = t M b L t qcd m m m p ) S t , m  t M ( 2 m M ( 2 ) + Re Σ p L b ( (1) t t ( M 2 ∆ 3 π τ ( + ∆ Re Σ S b g m M m − h 12 ( ∆ t SM(5) b MS mass in the THDM, + Re Σ 1 −  t M m SM(5) τ + Re Σ M m SM(5) τ + 18 GeV in the Standard Model with 5 active quark flavours by first evolv- = Re Σ ) = ) = . r r m b ) = ) to the scale + t denote the scalar, left- and right-handed parts of the top self-energy in the are the scalar, left- and right-handed parts of the 1-loop self- MS scheme in which all Standard Model particles, except the bottom quark, b µ µ ) = ( ( m t M m ( ∆ ) is converted to ) is converted to ) = ( ) = 4 M t t t b qcd qcd b , , ( t S,L,R t S,L,R b m M m M M t (2) t (1) ( ( ( ( m τ SM(5) b m m MS mass in the Standard Model with 5 active quark flavours at the scale m m ∆ ∆ b SM(5) b SM(5) SM(5) τ are neglected. Afterwards, m the In this identification, the 1-loop Standard Model electroweak corrections to where Σ energy in the the top quark and thelepton W, in Z the and THDM, Higgs bosons, are omitted. Finally, the The bottom quark m ing m where Σ MS scheme without the gluongluon contribution, corrections and taken ∆ from ref. [ rections, JHEP03(2016)158 ) τ and (B.1) S (A.16) (A.17) (A.18) M potential 4 at φ i ] for the case λ 32 self-energy in the τ , ) , , t ) ) t t M M M = ( lepton, the top quark and the ( β r β τ ) are obtained from the running t , µ 2 Z M ) cos ) sin ( , ) ) t t t d t M v M M M M ( ( = ( ( 2 2 2 2 2 bounce Z Z g g p S ( m m − ) and is the characteristic size of the bubble. Note 2 t 2 e ) + ) + T ZZ t t 4 √ R must be performed until a convergent solution √ M , which are fixed by boundary conditions at the  ( i t M M (or more precisely, the tunnelling rate times – 17 – u ( ( λ τ R v 2 2 1 1 M τ  g g 5 5 + Re Σ / / ∼ 2 3 3 Z p p p M ) = ) = ) = t t t M M M MS gauge couplings via ( ( ( d u 2 Z ] v v ] for the MSSM, adapted to the THDM case, potentially including m is undetermined for a classically scale invariant potential. 31 29 R ) and the t . For this reason, an iteration between the matching of the is the euclidean action of the “bounce” instanton solution which interpolates are the scalar, left- and right-handed parts of the 1-loop S M ( M Z is the transverse part of the 1-loop Z self-energy in the THDM including higgsinos MS vacuum expectation values m S,L,R τ bounce S T ZZ A more precise estimate in quantum theory was discussed e.g. in ref. [ If a consistent solution to this boundary value problem has been found, the pole mass As shown above, the matching at the weak scale at the 1- and 2-loop level introduces The where between the false and thethat, true vacuum, at and this level, of the Standard Model. Following their analysis, for a single scalar field with a nomenological point of view, itlifetime might larger be than the more age reasonablethe of to true the demand universe. vacuum metastability during Semiclassically, withcan the a a be tunnelling cosmic probability estimated time into as [ higgsinos and gauginos, if present in the theory. B Vacuum (meta)stability Absolute stability of the electroweak vacuum is a strong requirement. From the phe- the matching to the Standardto Model this at boundary value problem has been found. spectrum is calculated atas described the in 1-loop ref. level. [ This calculation follows a similar procedure a dependency of the gaugeon and the Yukawa couplings particle as spectrum well of asgauge the the and THDM vacuum (possibly Yukawa expectation couplings values including enter higgsinosrameters, the including and the gauginos). renormalisation quartic group These couplings high equations scale, for all model pa- and Σ and gauginos if present in the theory. where the running Z mass is given by MS scheme where all StandardW, Model Z particles, except and the Higgs bosons, are omitted. Z mass, where Σ JHEP03(2016)158 (B.8) (B.4) (B.5) (B.6) (B.7) (B.2) (B.3) ), the yr. The 2.8 , 10 )–( ! 3 . . iξ 2.5 2 ) e 2 = 10 ) H d † 1 θ τ + H 2 cos c 2 ( χ ˜ λ iξ = Im e + , ) to zero. The remaining , which, for the Standard 2 | θ ρ ) + √ λ ! 2 | B.5 ) S d sin − 05 during its entire RG evolution , d φ 2.8 . 0 ) are satisfied: 2 2 + ∆ 0 , − c ( , | 4 . H > − 2 2.8 ) † 1 4 φ − 1 2 1 2 √ 1 ) 2 are one-loop corrections from particles R φ λ λ H λ θ ( π )–( 4 ab λ 1 S

8 λ (2 r | λ ] and set )( 2 2.5 3 − 2 = p = 33 2 λ = Re 2 − ) 1 cos ) ) + φ

λ is the age of the universe, θ θ ˜ 4 λ µ 4 – 18 – p ∂ τ λ can be estimated as ( growing large. being permissible at low scales). exp 1 2 + + λ ) = sin(2 2 | 4 , c ,H 3 3 d φ λ  2 | = ( − λ λ ! τ is defined by H R + 1 + sin(2 1 eff † L 1 2 ≡ 2  θ be larger than about + ( 1 when V iξ c H ˜ λ e 2 λ ) direction is R   = θ b max sets the first two terms in eq. ( 2 φ p with λ ≈ sin φ 2 p p d χ = , b 0 − + + χ 1 2 are real and a θ c 1 H i † λ 1 ξ = ] (somewhat larger cos H p 32 φ  ab [ = ( and . We require is negative, then the potential is unbounded from below along the direction , 1 2 2 b φ a 1 = 0 and √ 1 ˜ λ ) is the running quartic coupling, and ∆ = ρ r

4 /λ µ Planck 2 ( V φ, χ, ρ, = λ λ M 1 To map this onto a one-dimensional problem, we choose a gauge and a field basis In our case the model is somewhat more complicated as it involves several scalar degrees p H = Choosing effective potential along the where If however a such that This allows us to write thepositive quartic definite potential if as the the sum stability of conditions three eqs. terms ( which are manifestly may be violated, whichunstable. corresponds To see to this one explicitly, we particular follow direction ref. [ in field space becoming up to of freedom. However, outfirst of three the turn four out always conditions toconditions for be on absolute satisfied the as stability quartics. a eqs. consequence ( The of remaining the supersymmetric condition boundary eq. ( where coupling to tunnelling probability is dominatedModel, leads by to the a condition largest that value of the tunnelling probability for negative (neglecting the Higgs mass term), JHEP03(2016)158 B (B.9) B 750 (B.10) (B.11) Erratum [ ]. Nucl. Phys. . Phys. Lett. , ]. ) (2004) 65 , 2 λ SPIRE 1 065 at the electroweak IN (2005) 073 2 λ . SPIRE λ 0 √ 1 B 699 IN 06 − λ + ][ √ 4 λ between the electroweak scale r + 2 JHEP + , µ , 2 3 ]. Split symmetries the RG-improved one-dimensional λ r λ ( µ Nucl. Phys. , GeV + 2 , | ]. λ 1 ) 10 1 SPIRE r λ 2 bounce λ varies between µ 82 IN π . ( S √ λ 8 2 ][ λ arXiv:1108.6077 | SPIRE 4 [ 3 IN ]. 1 + log . . Vol. 2: Loop Amplitudes, = ][ = ]. – 19 – λ 41 Supersymmetric unification without low energy − SPIRE IN & (2012) 63 SPIRE bounce Splitting supersymmetry in string theory ) ][ S where IN . 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Schweizer, G.F. Giudice and A. Strumia, G.F. Giudice and A. Romanino, I. Antoniadis and S. Dimopoulos, L.E. Ibanez and A.M. Uranga, L.J. Hall and Y. Nomura, N. Arkani-Hamed and S. Dimopoulos, M.B. Green, J.H. Schwarz and E. Witten, [7] [8] [5] [6] [2] [3] [4] [1] any medium, provided the original author(s) and source are credited. References scale and beyond our simple one-dimensional approximation. Open Access. Attribution License ( and with expression without further loop corrections, we obtain a reasonablyvacuum accurate from eq. necessary ( condition for the longevity of the electroweak The tunnelling rate willproblem is be effectively one-dimensional. dominated Using by for bounces along this line in field space, so the or equivalently JHEP03(2016)158 , ]. ]. ¯ B A ]. ]. − Phys. , B SPIRE , ]. Phys. SPIRE ]. SPIRE (2014) IN , SPIRE IN [ (2013) 089 IN ][ IN 11 ]. ][ ][ Averages of SPIRE 12 SPIRE IN IN ][ ][ SPIRE JHEP IN , JHEP (2014) 1130 , ][ (2015) 139 . ]. 185 Comput. Phys. 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