JHEP12(2017)040 Springer June 9, 2017 : October 8, 2017 metry, Renor- pproach to the : December 7, 2017 November 21, 2017 : : Received Revised gies. The stability of the Published Accepted al values are discussed and lues at the unification scale the pure standard model. Published for SISSA by https://doi.org/10.1007/JHEP12(2017)040 . 3 1705.01605 The Authors. c Beyond Standard Model, Higgs Physics, Non-Commutative Geo

We consider a complex singlet scalar in the spectral action a , [email protected] Physics Department, American University ofBeirut, Beirut, Lebanon E-mail: Open Access Article funded by SCOAP Abstract: Hosein Karimi noncommutative geometry model Implications of the complex singlet field for compared at the two-loop level with a real scalarKeywords: singlet and standard model. It iswhich shown is that able there to is producevacuum a and Higgs range and the top of deviation quark initial of masses va gauge at couplings low from ener experiment ArXiv ePrint: malization Group JHEP12(2017)040 8 1 3 13 10 13 15 15 Riemannian D us 4 commutative version of the Dirac operator ace-time has been able erwise consistent with the the defining ingredients of information like gauge field niquely, by using very weak als the fermionic content of nd, in addition to the Higgs, dditional constrain called the rum of this operator. In the . It is shown that this field model are not totally arbitrary ion of the space, and just like c operator in the commutative rtant features distinguishing it . These relations are consistent rect fermionic content and also ]. We will see in this letter that the results 6 – 1 – ] and the papers that followed, it was shown that 3 ]. In this model, space-time is taken to consist of a continuo 3 The Lagrangian of this model comes from the most general form part of the space and can be considered as the generalized non 2.1 Running of the renormalization group equations 3.1 comparing the complex and the real cases D and there are relations between themwith at grand the unified unification scale theories suchthere as SU(5) is unified a theory. singletcan Seco help scalar the field situation with presentunification the of in low spectral the Higgs action mass in spectral which high action is energies not [ oth consistent with axioms of noncommutative geometryfirst plus order an a condition. Thisfrom Lagrangian the possesses minimal standard three model. impo First, the couplings of the the simplest possible noncommutative structureleads has to the the cor gauge symmetry of the standard model. to produce the standardconstraints model [ coupled with gravity, almost u 1 Introduction The noncommutative geometry approach to the structure of sp 5 Conclusion A 2-loop RGEs for complex singlet extended standard model 3 Top quark and Higgs masses at low energies 4 Implications on the vacuum instability Contents 1 Introduction 2 The model of it. This operatorthe has Dirac all operator the inthe useful model. the geometrical standard Moreover, informat in model,interactions the its and noncommutative geometry, structure the other scalar reve work sector of are Chamseddine embedded and in Connes the in spect [ manifold tensored with athis finite hyperspace noncommutative is space. an operator One4 which of coincides with the Dira JHEP12(2017)040 ]). 15 -loop renormal- re added to the plex field in this work. ]. Finally, right-handed three-loop equations to y condition on the sin- ompared with the pure consistent with the cur- on the negativity of the singlet can accommodate rticle masses. The proper 14 hat we can do is to follow uplings, although vacuum with neutrino mixing. We we also discuss the effects reliable, even at high en- ino by see-saw mechanism urthermore, we use 2-loop acy. This agrees with the , oward current experimental fication scale when we start 1 rs started to study the insta- wa coupling of the top quark real singlet field. s coupling also plays a role in cuum stability of the models o be between 0.411 and 0.455 ases: when the added singlet x. It is also seen that such an ts Yukawa interaction. These e seriously (For example [ htly better than the other two s, it still could have dramatic e [ in the experimentally observed unknown right-handed neutrino e presence of the complex scalar e reason we cannot predict what gs are derivable in this scale from ]. It is interesting to check the effect of any . The resulting value for this coupling at Z- 3 16 , – 2 – 2 ], the singlet scalar field was assumed to be real. Then using 1 6 We will show in this letter that even though a few extra terms a In [ Since the discovery of the Higgs particle in 2012, researche Yukawa coupling contributes in the RGdetermining equations as the well. Higgs Thi and topits quark effect masses by at following low RGvalue energies. of equations right-handed down W neutrino and Yukawa looking couplingat - at unification turns the scale out pa as t we will see in section RG equations due toHiggs self-coupling the at complex high energies singletexactly can happens field, be for substantial. yet the coupling Th their is that effect the experimentally is about 0.995. Besides, theRG values equations. of scalar We sector argue couplin instability that is in not this cured, acceptable butfield. the range situation of is We the improved use co byassess th two-loop the equations loop and correction effects near in to presence the of leading a complex order or boson mass region is also between 0.517 and 0.530, while Yuka standard model. Although this instability cannotergies, because make of the the standardconsequences long during model the lifetime un inflation of periodmodification the [ of tunneling the proces modelcoming on this from situation. noncommutative Therefore geometry the will va be addressed and c neutrino appears into theterms picture are automatically needed as to welland give as usually a i are small added mass to to the the standard left-handed model neutr by hand. extra scalar field can be responsible for dark matter particl improve if the extra singlet scalar field is taken to be comple separations of the standardfrom model experimental gauge values couplings and atof run the three-loop them uni corrections. upward. Subsequently is a complex field, whenshow it that is while real, running and RGenergies, the equations the pure from model standard unification with model scale addedcases. t complex Yet, singlet like behaves the slig standardgauge model couplings itself, at one can low only energies atta within some percent of accur ization group equations, ita was Higgs shown field that the withglet model the field with is mass the not ofOur necessary order consideration and 125GeV. shows we therent In assume model experimental fact, the values in singlet the of its to realit renormalization the most be Higgs group general and a equations top form to com quark compare is masses. the following F c bility problem of the standard model effective potential mor JHEP12(2017)040 he y introduced tive algebra. us and studied find applicable set of . These are shown to hod to asymptotically emannian manifolds in ned based on three con- y assuming extra dimen- ], the authors discovered osmological constant and 3 m the standard model plus c operator and were able to re is to look at the spectrum and Alain Connes found an o a much broader range of l cases with complex singlet ses. The form of potential is we use the initial conditions ructures leading to the stan- her orders are higher powers ace which possesses no space- he geometrical notions to the quely from these assumptions. bility are slightly better (e.g. with its faithful representation nsor. o many developments in various ) (2.1) C ( Dirac Operator 3 . M ) ⊕ ]. They assumed the space-time is a direct ,D 3 H H ]. Moreover here a neutrino coupling is present , – 3 – 8 ]. This expansion is controlled by a scale called ⊕ A C 22 ( , = 18 F ]. A . The Hilbert space is then the space of 1 by 4 spinors. In 13 µ , ∂ 5 µ ]). iγ spinorial space-time one can consider Dirac operator to be t 20 = , D 17 D , 14 We have therefore a so called spectral triple which is shown b , 1 Riemannian manifold with a noncommutative space. They also 11 D is the algebra of 4 by 4 complex matrices which is a noncommuta ]. He also used the new geometry to define a noncommutative tor A 12 Geometrical structures in noncommutative geometry are defi Unlike Kaluza-Klein type theories which enlarge geometry b We stress that the above results are not merely derivable fro For precise definitions refer to [ 1 be enough to defineexpected a limits. rich geometry and can yield features of Ri on the Hilbert space, and a special operator called the spectral action which isshow that based the on standard the model spectrum arises naturally of and the almost Dira uni cepts; a Hilbert space, an algebra of a given set of operators time dimensions. In earlydard models, model finding noncommutative was st thethat matter a of noncommutative space trial with and the error. algebra Eventually, in [ Λ. Doing so, thethe first term second of term the gives expansionof the turns the total out geometrical curvature to invariants of be such the space-time. as c curvature Hig and Riccisions, te here the added structure is a finite noncommutative sp product of 4 areas of mathematics,axioms finally and Alain definitions Connesspaces to was [ generalize able geometrical inits concepts 1980 geometrical t properties. to application Later of in this 1996, new Ali geometry in Chamseddine physics [ compare with [ 2 The model After years of investigations byspaces with mathematicians less to constraints expand than t metric spaces, which led t a complex singlet. Thepredicted by reason the is spectral that actionin approach in RG [ our equations considerations, and contributesalso to restricted the and values of isdescribed particle different mas in from the extended literature. standard mode In our case, the results for sta Now one way to seeof the geometrical the invariants such Dirac as curvatu operator.expand the trace One of can Dirac operator for [ example use heat kernel met As an example, for a 4 familiar 4 by 4 matrix this case JHEP12(2017)040 ) ]. ), 4 C ( 3 2.1 (2.5) (2.2) (2.3) (2.4) M isenberg pectral action spin manifold. D space-time. ilbert space and is actly the same inter- D es. Next, we can triple ut of D. Function f is a cles to the Hilbert space. m of its eigenvalues. The nonlinear fluctuations, the the 4 nite spaces consistent with o this fermionic part of the the spin manifold of fermions. Dirac operator based on the algebra ( e above settings of the non- s: . quely to the same algebra [ e the chirality operator called is the algebra of quaternions Algebra of the whole space can nd left handed leptons and 12 )) . action consistently, another operator . C H F i ( and we assume its validity in the , D 3 ) ]. is the algebra of complex numbers. 4 ⊗ M C ( 5 C ψ, Dψ ⊕ 4 γ h and is required to be positive and even. r to [ H M + ]. As an interesting breakthrough in 2014 N ⊕ ⊕ 96 10 G × C Λ)) + H ( 96 – 4 – ⊕ 1 D/ ⊗ ( ) H ⊗ f M = M ( first order condition with the algebra of functions on the 4 D F ∞ F A = Tr ( 4 = 16 by 16 matrices and members of the Hilbert space = C A S × D = ) that happens when the perturbations of Dirac operator is 2 A 2.2 × ] that this algebra is dictated by a generalized version of He are 2 9 F A The vector and scalar parts of the model are described by the s 2 We have then 16 spinors and it turns out later that they have ex Members of To be able to introduce an inner product and define this part of 2 to enrich the algebra by grading mechanism and add antiparti which are represented usingLater 2 on, by the 2 samethe matrices, authors noncommutative and showed geometry requirements that leads the almost classification uni of fi is the algebra of 3 by 3 matrices on complex numbers, Lambda is an energysource cutoff to needed generate to physical make constants dimensionless such as term o which leads to the Pati-Salam unified model [ which is the traceaction of is: Dirac operator and depends only on the su which is a specialrequired case to of be ( linear. This is called consistent algebra is it was discovered in [ commutation relations. In this letter we consider the model They also observed that by letting the Dirac operator to have called reality operator is needed. For exact definitions refe is able to produce the standard model when it is tensored with be written as direct product of consist of 16 spinors, which means they possess 64 elements. actions as fermions incolored one right generation and including left four handedγ quarks. right a Next, one can introduc The latter is the commutative algebra of smooth functions on current work. of the whole spacedefined as is the then tensorial a sum 384 of the by operators 384 on matrix different which part acts on the H Therefore members of thethis Hilbert space space by hand are to now take 1 into by account 128 the matric three generations The particle content of thecommutative geometry. model Then is Dirac thereforemodel. action coming provides from dynamic th t JHEP12(2017)040 . 2 D (2.7) (2.6) ish for d by Λ , appears such that ll be then 0 0, roximation F F n > n of the trace. − . der inner automor- n udu. ) that any mass in the ts of the spectrum of a e the trace. Existence n u ]: ( − to show up in the mass are the spectral function 4 F 19 tion group equations and 0 or in form of connections. The trace is then reduced [ F n se a function cal constants. For example s the root of unification in F n tative space are responsible ∞ 3 − e Dirac operator under such ions associated with the au- 0 elevant for our purposes. In − 4 4 Z . F e square of the Dirac operator Λ c interactions, just like the stan- = n embedded in the Dirac operator 4 =0 ∞ hat the expansion is valid up to energies X n =  2 du, F ) Λ) u ( D/ – 5 – F (  ∞ 0 F depend only on the geometrical invariants such as Seeley deWitt coefficients Z  manifold form Riemannian aspects of the curved 4 n = a D 2 0 and momenta of spectral function for 4 Λ)) = Tr n < ,F D/ − ( ]. (0) 7 f F ) is over even numbers, therefore the forth term is suppresse = Tr ( 2.6 0 ) is expected to work well. The unification of the couplings wi , is related to the cosmological constant and the third one, is supposed to be positive. The odd terms in the expansion van F 4 2.6 f F ). The coefficients ]. Therefore we trust the model on high energies where the app α 5 is expected to be a cutoff function which can control expansio ( f F ) = Next, one can use heat kernel asymptotic expansion to comput The sum in ( To start, first we need to make the fermionic part covariant un 2 For the special case of Robertson-Walker metric it is shown t α 3 ( the first one, These coefficients along with Yukawa couplings make the physi derivatives at zero for 4 addition, We expect Λ tomodel. be right Therefore below it Plank is energy logical which to is assume much higher higher terms are irr of Λ in the action is crucial so one can rely on the expansion. term of fermions asthis well model as [ all the coupling constants which i dard model when allOn the top vector fields of are that,Dirac added here operator. to we Dirac get operat the Yukawa terms and the Higgs as par space-time. The Dirac action then contains all the fermioni what is expected fromrunning GUT them down theories. to experimental Writing energies the is renormaliza also feasible of expansion ( curvature and therefore reveal the geometricalup informatio to the order defined by powers of Λ. Taylor coefficients to a series with coefficients known as phisms of the Hilbert spaceautomorphisms. by The adding fluctuations inner associated fluctuationsfor to of the the th existence noncommu oftomorphisms gauge fields of and the the continuous Higgs. 4 Inner fluctuat in the Higgs kinetic term. Normalization of this term causes manifolds without boundaries. Ithas is important equivalent to geometrical saying informationF th in its spectrum and u close to the Planck order [ The function JHEP12(2017)040 m ]]:                     5 9 (2.8) 9 × 3 × 1 18 . In the ) 18 ⊗ ) is [[ u ) a,b ∗ d 18 ∗ k 0                   H 2.3 k 3 ⊗ × ⊗ ], and are not j i 3 18 δ 4 j i I b i δ × mion interactions a 9 σ H ) i µ ¯ in [ d H ab ( k ǫ W gebra in ( 2 is a 192 by 192 matrix ⊗ g i 2 i j δ 9 D 0 ( a − block diagonal and contains 2 H s. ( × er their coincidences with the y to exclude interactions not 2 t same form of standard model I a 18 λ g with a matrix algebra in the noncom- a × and Higgs scalar fields µ contains gauge fields as it does in 9 0 i V ) a 3 k u λ D g k a i µ 2 V counterpart gauge theory. ⊗ 3 − a i j 9 g 0 0 0 0 0 0 0 0 0 λ 6 δ i 2 b a × µ 6 ¯ V H − I 3 3 ) 3 ab g 3 µ × i – 6 – ǫ × i 2 6 × σ 3 B ) 6 i I µ 1 ν ) − ) g ∗ e µ W 3 i 6 ∗ 6 k stand for the U(1), SU(2), and SU(3) gauge fields 2 0 × k B g 3 1 ⊗ − i 2 ) and what follows, we will study the scalar sector of I g ⊗ b ) µ W i 3 6 a µ − H . The first part of D × ¯ ) justifies chosen names of fields and their coefficients as B ( 2 + H 3 2.11 ab 1 ( ) × ǫ µ AB g e 2 2.5 i k I 3 D 2 D ) , and µ ⊗ 3 V − 0 ( B a ) on the other hand, there is no fermionic field and the spectru µ 1 , H g µ ( D B i 2 ( 2.5 B 1 + 6 ig µ × 3 + D ) 0 ( ( ν µ The second part is responsible for all the other fermion-fer k D 3 4 0 0 0 ( 0 0 0 0 0 0 0 0 0 0 0 ⊗ 0 ⊗ b µ ¯ γ H µ ab = ⊗ ǫ 5 ( D The forms of these matrices come from very few axioms, listed The Fermionic part at ( The Dirac operator for the noncommutative space defined by al γ what we see here is an special case of a general theme, startin                                       AB 4 + D respectively. the standard model; generates bosonic and scalar potentialspotential which terms. have the exac Inthe equation action. ( for nonzero components of which justifies the choice of names, Yukawa couplings arbitrary. The zeros appearexperimentally observed. automatically Nonzero and components are arefields named necessar and aft constants in theall standard the model. vector The bosons. firstand matrix The is acts second on matrix all contains 48 Higgs known term fermions. trace part of action ( mutative geometry, the spectral action principle leads to a JHEP12(2017)040 ent is a (2.9) (2.13) (2.11) H ndent parts odel. . 2 interaction. After ]: H AB 2 5 rt action along with | H = 0 (2.12) . D σ sequently the Hilbert 0 | − i 0 2 H = i 1 etry allows us to gauge ) is [ σ hσ ′ urs when σ h m mionic and anti-fermionic ) are well defined. The λ h h B rator of the whole space is the following choices of the ′ 2.5 1 2 + 2.5 A = = vable from this operator and 2 | + ed fermion which is coinciding rmion-Higgs interactions, plus σ for the geometry as an axiom. 4 ! is natural to call it right-handed | r of ( .. .. H ,D J 0 0 hσ ′ h λ λ .. σ AB 0 1 4 ) to identify the bosonic part of the +

D 2 + 2.6 H = = 4 | h ′ σ B | ′ , which presents scalar-fermionic interactions self-interaction, and σ AB A has only one nonzero element, only one of the λ D , λ σ D – 7 – ′ 1 4 . = 0 + AB + fermionic and Yukawa interactions (2.10) J , w 2 2 σ ,D D | 2 σ m | ! iσ , v ′ ′ C.C. 2 σ + B + ′ ! m + 2 AB v A 1 1 2 σ R H D D 0 + ν + + hσ R h B 2 ′ λ ν

w AB A c H + D 2 h ] that only one off-diagonal element can be nonzero. This elem D = = 2 = 5 | m

σ i σ has the role here as the charge conjugate has in the standard m H 1 2 | = σ J = λ D to be a complex singlet with two degrees of freedom. Although causes all the other operators to be the direct sum of two depe V ψ, Dψ h 5 σ ]. ). Calculations up to first three terms yield Einstein-Hilbe 6 2.5 Physically important geometrical information is also deri The Dirac operator is however not simply the direct sum of fer To include antiparticles, we can double the algebra, and con Having the above operator, both parts of the action ( We take It is evident that 5 we need only to find coefficients introduced in ( absent in the standard model. Since away three of them. The potential has local minimum which occ complex doublet with four degrees of freedom, the gauge symm action ( terms coming from off-diagonal elements of fermionic part of action is containing fermion-gauge and fe therefore indicates a singlet thatwith gives a mass right to handed a neutrinotherefore right in a hand the 384 standard by 384 model. matrix Dirac as ope we noted before which can be exchanged by the act of and proposes the symmetry breaking,vacuum which expectation we values: formulate with fermions is involved with this newneutrino sigma-interaction [ and it proper redefinition of the fields, the scalar potential secto parts. It is shown in [ Gauss-Bonnet terms, plus Higgs potential, space, by assuming theThis existence operator of a reality operator JHEP12(2017)040 ) 2 σ σ λ 2 hσ h and λ λ − 2.13 v (2.14) (2.15) 1 − σ 1 other two q vary all the .   2 1 op RG equations ! 2 ) hσ d to the picture which lore high energy scales. . 2 λ mechanism is appearing ) ( σ σ 2 λ 2 hσ nism which determines the λ r. After substituting ( awa coupling and the Higgs n be fitted in the standard w h ew scalar which is supposed s self-coupling is determined λ 2 2 λ 4 wa couplings and the unified lso relates, in the unification ed by the factor of oncommutative geometry not w mix and due to the ). That is because the scalar w icle is only able to gain mass en when the mass is measured. v . It is clear from RG equations r the Higgs mass. ar only in two-loop corrections . to be much greater than 1 tions and therefore is not going t in two-loop corrections of the − A − + w 1 σ h λ 1299 λ s . 2 h h v λ λ ( 2 2 = 0 w 2 2 p 2 v v 2) 5) . . = − 1 h – 8 –

(125 2(246 , one can find the effective potential and renormal- ± 2 1 ) = , m   z σ 2  M λ ( σ h λ r λ 2 4 w w = + h M λ 2 v ), and diagonalizing the mass matrix of the square terms, the  ]. Having those relations, we will start from unification and 5 = is a complex field. We therefore suppose 2.11 2 ± σ m We used SARAH which is a Mathematica package to derive two-lo ]) for this model and presented the results in appendix 21 symmetry, one massless Goldstone boson is expected to appea scale, the scalar couplingsgauge to coupling other [ parameters such as Yuka due to the presence ofonly the scalar predicts field. the It singlet is field remarkable that and n its potential terms, but a that the extra field cannot correctto the help gauge the couplings evolu couplingsfield to potential meet terms in are quadratic exactlyof and Yukawa one couplings their point evolutions, couplings which (figure appe themselves enter jus Having the model described inization section group equations inThere some are however loop two free order parametersself-coupling. and here. Knowing run The the them neutrino Higgs Yuk in mass to now, the exp the pure value standard of model Higg as free parameters to probe the implications of this2.1 formula fo Running of the renormalization group equations into potential ( to be highlyappears massive. since Here we have another massless field adde It is believed thatmodel a to highly explain massive neutrinofrom right oscillations. a handed Such neutrino singlet a ca naturally. scalar neutral The field. part price of In this treatment the is model of described course to above have this a n The smaller one is responsible for the Higgs mass and is modifi It is obvious from the above setting that the three scalars no ([ In models with extraHiggs scalars mass and though, the value there of is this coupling a is not see-saw determined mecha ev we get scalar masses turn out to be JHEP12(2017)040 , x 2(a) ters. Yet no remarkable impli- shed lines are indicating m when we add the imag- kawa couplings. However, it is clear from figure of the particles with square high energies and follow them ceable consequences for the ollow their evolutions to very odel. The black and yellow lines nning RG equations, including alar field added to the standard tly at very high energies. These ) which can be sensitive to small p RGEs with two-loop equations for rections. Yellow solid lines show the creates meaningful changes. ions of the complex scalar field, in the n this diagram. This difference is from higher orders are not going to make the eate when we run the equations upward. 2.15 also taken into account. The black dotted – 9 – , and include three-loop corrections of all the other parame 2 also shows that the added singlet field, no matter is it comple 1 . The behavior of gauge couplings at the unification scale. Da Though replacing the real scalar field with a complex field has inary component. Choosing acceptabletwo-loop initial values effects, and show ru that this difference is meaningful. As cations on the gaugefield couplings couplings as evolutions, there it are new can Feynman cause diagrams between noti the starting from the same points, thecouplings couplings indicate behave differen Higgs massvariations through of the the relation couplings. ( two-loop corrections shift themdifferent for energy about scales. ten percent if we f gauge couplings. The latterof is the due singlet. to theor Figure Yukawa real, interaction does not cause meaningful changes in evolution of Yu again it matches with two-loopsituation corrections any suggesting better. that Thegauge graph and on Yukawa couplings thetoward when left experimental we compares values. start one-loo Again at adding the a same singlet doesn’t points at Figure 1 model is negligible.model Red dotted described lines in have section two-loop correct the evolution of standardsituation model is couplings slightly up better when tolines two-loop one-loop are corrections cor for are RGare equations so up close to thatthe three their same loops separation order cannot for of be errors theIn that distinguished standard all experimental i m uncertainties cases, cr the corrections coming from a real or complex sc JHEP12(2017)040 g is one n ). 2.15 ning this along with (b) eals itself when we com- tion scale. The advantage al singlet is added to the ts of higher orders of loop ons, toward unification (a) or ]. We choose the approach orth investigating. Another 8 ercent at low energies and as the values of the couplings at hey are determined by both d the value of gauge couplings lts of equation ( gh energies, and relates all the ities at the experimental arena. lex. In case of SM, except for the hen the real scalar singlet is present, m experiments. In all the cases, the at the unification scale. 2 1 scussed in the next section; when we run ) t ν . Interestingly, the scalar couplings are k k g = ( shows the evolution of all the parameters in n – 10 – 2(b) ] the effects of the scalar field couplings were shown to be able 6 (a) . Running of the couplings incorporating two-loop correcti ] and, for simplicity, define the ratio The other observation which justifies our consideration rev 6 also not free parametersand at the the unification ratio energy, of instead neutrino t and top quark Yukawa couplings [ 3 Top quark and HiggsIn masses our approach, at we low run RG energies is equations downward that from the the unifica spectralYukawa couplings model to predicts the initial unified conditions gauge at coupling hi and dashed lines areneutrino for Yukawa the coupling, case theinitial that initial values the for values singlet experimentally are unknown fieldfrom coming couplings unification is fro are and comp di look for the best fits. from unification (b). Solid lines are for the SM, dots are for w pare two-loop and one-loopLagrangian, equations. the scalar couplings Whether getnoted complex modified before or for this re about can ten in p principle dramatically modify resu at this scale is not predicted. Figure of the freeother parameters parameters of causes predictions the forAs model the discussed which physical before quant can however, the be unification fixed, scale then itself an run to save the model after the Higgs small mass discovery. different scenarios. There islow about energies ten between percent real differencecorrections and in are complex models. negligible, theencouraging Since fact difference the is effec we that in see [ here does w of [ Figure 2 JHEP12(2017)040 . To t the n on scale, GeV and and 16 g 7 for the real 10 . that everything gies. Then it is × g for three different nce it has a Yukawa h lead to retrieving -loop and one-loop n and ctral action, it is not tence of a new scalar sses simultaneously in at unification energy in n hod is straight forward; and n added scalar field coming or top quark coupling. rgies. The fact that these lar fields. Up to one-loop, ses the importance of loop g Trivially the reason is that eal scalars and use two-loop cale and find the experimen- he real scalar with a complex r a reasonable g, the correct as it depends directly to the ter particles. As it was noted ark mass in these two cases. uark mass. two-loop corrections. As we s very important and we will are consistent with the known minimizing the errors between and than the current observational imposes no restrictions on the e is depending on trips. lings which does not happen for g ]. Then we run the equations supposing is obtained to be around 2 8 n – 11 – , unification scale, varying between 2 U ] however, the authors showed initial conditions and RG 6 )). the lines indicate what initial values are acceptable to mee 3 2.15 5 for the complex scalar case which means that at the unificati . are free parameters. It hands us couplings values at low ener however, the neutrino is assumed to play a significant role si U 2 GeV. In figure . The colored strips in two diagrams show all the choices whic 4 18 Up to two loop corrections, the suitable To illustrate even more, we show possible choices for The other important aspect of the situation is to compare two Our considerations shows that there is a rather short range f Our main result in this section is that the assumption of exis It is notable that if we use initial values coming from the spe , and 10 g × , Yukawa coupling of neutrino is around 6 times bigger than the scalar and around 2 5 fits together. This happens for a scalar couplings (eq. ( corrections, as we arethere is comparing no remarkable real change scalar inone. top and quark mass complex However, if the sca we replaceThis two-loop t differentiation corrections is differentiate still topuncertainties. one qu order The of situation magnitude is smaller different for the Higgs mass unification energies are indicated by lighter colors on the s choices for n andlow U energies always within exist the to experimentally fit acceptable the values. Higgs and top quark ma particle masses atnoted low before energies, however, incorporating the correct one-loop choice or of unification scal correct particle masses at law energies. It turns out that fo masses of top quark andin Higgs. section For simplicity we neglectcoupling ligh and its masswe comes assume from a the see-saw initialn mechanism. conditions The predicted met in [ from spectral model issinglet real. field. In Nevertheless, this spectralRG section, action equations we to consider incorporatecorrections. both complex higher We also and order study r corrections prediction of and the as theory forfield and top the q predictions of the spectral model at unification possible to run the minimaltally standard acceptable model from mass unification ofthese s initial the conditions Higgs imply particle unificationminimal of at standard the low gauge model. coup energies. equations In are [ consistent with the low Higgs mass for when the possible to find the bestthe values for result these three masses parameters and by errors experimentally exist known and values arediscuss at more it low than in ene the experimental next uncertainties section. i give some examples, the small window of correct choices of figure JHEP12(2017)040 can be U and n or unified gauge coupling, , which lead to consistent n value at this scale, in order to o, n top quark masses. This is true for eft hand side diagram incorporates nd green lines respectively. one-loop corrections. It can be inferred t. Therefore suitable oop corrections, left diagram, make the es. Each set of three lines are for a specific , the lines associated with top quark, solid brown g – 12 – . At any unification scale, there is a small window of choices f . Each line shows suitable choices of unification scale and value and are illustrated with a particular thickness. The l , and the root of neutrino and top quark Yukawa couplings rati Figure 4 Figure 3 lines, and Higgs, dashed lines, always have a collision poin g two-loop corrections while the diagram onfrom the diagrams right that has within only a reasonable range of always found to assureboth the low real energy and values complex for cases the which Higgs are distinguished and by blue a g revive experimental values of particle masses at low energi low energy particle masses withchoices experimental a values. little Two-l more restricted. JHEP12(2017)040 is n situation illustrates 2, and . 39 When the 6(b) ∼ r when we add the u is about 0 ctions at high energy ). As noted before, it 35 and up to two-loop, U happens for the effective ∼ r. Figure 6(a) 49 and . u n be seen, in the tree level, ark imply effective potential et, the situation is slightly e 0 ome energy scale below the rgies. It is however useful to calar quadratic term and its nification and run RG equa- is possible to find acceptable range of values for couplings; ental data. One result is that auge values deviations at low wn initial conditions and one d model, and this pushes the hese two couplings along with normalization group equations ections are not being expected have no clue about their mag- f the Higgs and the supposedly ∼ y significance. Thus, we do not , we take into account the Yukawa of stigation shows that the addition g t does not affect the Higgs mass e modify the equations as much as 52 and . 0 at unification, ∼ changes its sign. It turns out that this goes to much higher energies. g g h λ which is much smaller than the unification 6 – 13 – GeV . For any 6 2 g and 3 g 5. It shifts the instability to lower energies, however late . 32 end to the best results. shows two-loop corrections have an effective role to make the ∼ 5 u 28 higher in the real case compared with the complex one. . 53 and . As we saw in the previous sections, there are a number of predi With an additional scalar field, it is interesting to see what For the standard model alone, best quantities are: 0 To find this result, in addition of all the known couplings of SM 6 ∼ instability of the effectivebecause potential of to the higher see-saw energies. mechanism I between Higgs and new scala scales in spectral approachtions which downward. suggest Doing to this start givesparticularly from ideas this the about is u the useful acceptable nitudes for as the they extra are couplings notthe which constrained Higgs we with self-coupling the current is experim stronger than in the pure standar investigate whether this additional fieldneeded could in in order principl to cureof instability. A a straightforward complex inve field could in principle cure the equations (figur potential. There are twointeraction new with couplings the associated Higgs with inthe the the Higgs self-coupling s model are described only earlier.heavily constrained massive by T the singlet. masses Therefore o cannot there run are the not renormalization enough group kno equations from low ene happens at the energy scale of order 10 is especially important due toto the save fact the that potential. higher loop corr scale. Figure better while three-loop corrections areexpect too higher small order to corrections have to an resolve this issue. g neutrino which is around 0 In the standard model, the observedof masses the of Higgs Higgs and field top qu toby become the unstable fact at that highunification. the energies. For the Higgs This standard self-coupling ca modelup changes itself, to one its can some sign use order the at re and s find the point at which 4 Implications on the vacuum instability We saw that for bothinitial scenarios conditions. (complex On or the realenergies other singlets), do hand, it again not inbetter fit both in within cases, the g the complex case experimental for uncertainties. Y 3.1 comparing the complex and the real cases singlet and all the parameters of the model, the instability scalar (complex or real) is added, up to one-loop, about 0 JHEP12(2017)040 different scenarios. wn for the standard model e includes three-loop corrections and uation. ts are added to RG equations, and the upling. The lower line has only one-loop (b) In scalar extended standardis model, not the determined coupling with Higgsgreater mass initial and value could which havefrom a might being save instable. the potential – 14 – . Comparison between the behavior of Higgs self-coupling in . After the Higgs discovery, all of the initial values are kno Figure 6 initial values are less than ten percent of loop effects. (a) The initial values usednot realistic. to draw However this it diagramfields shows are that can addition in of principle new loop have effects corrections. more than Theenergies higher due errors to of the experimental these uncertainties lines of the at high Figure 5 corrections. The line shiftstunneling to time the increases right when consequently.suggests two-loop that effec The going blue to higher dashed orders lin will not improve the sit parameters and one can follow the evolution of Higgs self-co JHEP12(2017)040 ]. 20 , 14 , 11 ]. The model , ,   10 2 2 3 3 g g ading order, for three- + 440 + 360 e geometry point of view, , 2 otential is expected to be 2 re. Further investigations dard model at unification  2 plings in the potential are 2 alar is added; however, the tion and will be the subject alized models derived from ribute in a significant way. 2 g erimental arena which is of g ge couplings at low energies. atic potential, higher powers 3 s are derived using SARAH inglet field makes the theory on. Equivalently in noncom- ehave desirably. asses. We however witnessed he complex singlet extended his approach [ ce in the simple version that g unification point predicted by o the issue of gauge couplings al change in the settings of the berg uncertainty relations leads gs unification at high energies. al theory. The Pati-Salam model 260 + 135 + 175 2 2 − 1 1 adds some familiar extra features to 2 g g 2 2 g + 27 + 199 2 + 45 t 2 t 2 1 K K g – 15 – 45 85 − − + 11 2 2 2 ν t ν K K K 15 20 15 − ]. The equations are consistent with the literature [ − −    21 4 4 4 π π 3 π 3 2 3 3 g 1 g g 7680 2560 12800 + + 3 + 2 2 3 2 π 3 2 π g 3 g 1 π g 96 7 16 19 41 160 − − = = = The spectral action approach coming from the noncommutativ Comparing the results of two-loop corrections and near to le 2 1 3 dt dt dt dg dg dg to Pati-Salam model as the most general possible outcome of t however, does not uniquelyshowed lead in 2014 to that the imposing model generalized versions we of considered Heisen he Yet, the little change towardstandard better model results with might this urgenoncommutative minim us geometry to principles. investigate other more gener loop, shows that thereWe is believe the no root hope ofnot for all meeting loop of at such corrections one inconsistencies point tomutative goes and geometry cont back therefore approach t lack there ofwe is a considered unification, true here unificati but was the that one pri could not fully revive the gau A 2-loop RGEs forHere complex we singlet present extended 2-loopstandard standard renormalization model model group with equations right-handed ofpackage for neutrino. t Mathematica [ These equation we considered here is the simplesthas special a case of rich that content gener of beyondof SM our fields further that investigations. might help the situa the same order ofscale. deviation Comparing of these errors the weslightly gauge conclude better couplings that than the in the complex thefull pure s treatment standard stan is model not or possible. when a real sc the standard model, for exampleof a new geometrical singlet invariants, field with andOur quadr prediction considerations in of this gauge paperthe couplin show theory, that it starting is from possible the tothat revive there both is top quark a and deviation Higgs for m the gauge coupling values at exp now positive all alongstable. the way In up this to respect unification both scale real and and the complex5 scalar p models b Conclusion The noncommutative model introduced in section what happens when we use such initial conditions. All the cou JHEP12(2017)040 2 3 2 ν t t sh K K K
λ  , 4  sh
ν 32  h λ 5 2 K − λ t h 3 h 2 96 3 g λ λ K  90 2 2 4 sh − 2 2 sh 2 2 2 1 t +85 h λ h λ g / − g g , 2 λ λ K h 2 4 200 2  , t λ
64 g  4 2 + 3 s  sh ν K 3 40 288 6 − 63 3 λ 1887 λ 2 g t 28800 ν K 2 − − 2 + 5400 sh K + 3 2 − K 1 270 + 9600 sh 2 3 λ h , h g 6 λ 3  g h + 80 λ λ − 2 2 2 g . λ 2 10  g 2  + 160 2 ν 2 2 1 sh 4  / g 1 h + 10 ν 16 2 2 + 1800 g 2 + 17600 ν 312 + 80 K λ g λ 2 g 6 ν 4 K t +160 g 2 4 h 305 K t − +1 sh 2
h / K λ 2 2 K 2 2 2 λ K 2 9 + 48 2 λ h h + 960 5760 300 2 g 2  + 80 h λ λ − 2 g sh 4 2 g − − + 760 2 + 45 − λ − h 2 ν λ 4 4 2 9 sh  4 t 2 2 g 2 λ 4 + 30 2 g t 4 2 144 +45 λ 1 2 2 K 2 g + 5  4 ν 16 g K − 2 1 2 g g + 240 g 5 K 4 2 − ν 1 1 + 5400 2 80 g + 6 h 4 K ν 1 h g − g 2 2 17 K λ 2 g 27 λ 2 t ν − g K 2 +108 + 120 h t 2 17 48 3450 1 289 + 57600 2 2 2 3 K K λ 2 − g +3625 + 675 ν K h + 150 / g 2  1350
2 2 − 2 sh 5 − h 230 λ 2 g 4 −
2 K / / 2 2 1 λ λ sh − h g + 75 2 sh 100 π 4 + 15 2 1 h 2 9 g − + 5 5 2 t 2 λ 45 λ – 16 – 4 λ 1 2 g 2 g 2 λ 1 − t g 2 2 + 80 12 + 9 2 t g − t K g 4 2 2 sh 1 − 32 20 256 2 K 48 1 h g h K g 4 − λ K )) 3 2 g 2 108 λ 2 + 4 t λ
1 g − 2 − 1 400 − 8 2 450 + ( g + 279 2 3 g 2 g 3 2 2 8 2400 2 h 3 s t + 3 g 9 2  − 270 g 9 + 3 2 4050 t g λ λ 2 4 − )) g h − − K 54 1677 2 t 1 − + 45 − K 4 )) 2 λ − 4 ( 2 h g 96 t t + 1200 4 4 g − sh 2 2 2 s − λ 2 ( + 16 2 1 + 8 g 2 ν 2 λ 2 4 4 K ν λ − s g g 6 g h 2 1 540 1 2 4 2 / 4 + 160 1 K λ 2 2 1 K 3 g g 2 2 λ 9 / 1671 20 2 g 1 2 g g 73 g g −
9 g 2 g + 12 60 ( 4 2 − − 16 2 21 g )+16000 ( 2 9 2 1187 / − 32 − ν 2000 2   1 t − g 9 + 20 480  + 15 135 ( g sh 2 h sh 3411 K ) + 20 ( + − 2 t s 4 1 t λ + 15 40 t − λ λ − 2 + 3600 1 ( + 4 g 4 λ 2 + 45 / 20 K − 4 2 t + 5 60 600 20 2 g 4 1 K s 2 4 1 / 1 2 sh 2 π 1 h g 2 3 g sh λ 9 K 17 1 g 1    + 4 λ − g 1 2 λ g 2 2 18 200 g λ 60 g 4 4 4 20 3 g 1 2 1 8 − 27 −   1 100  80 π π π g sh h g  2 3  2 17 171 1 1 1   λ λ 5 ν ν 223 2 π 2 2 2 / 80 − t sh ν 200 π 102400 256 256 256 K K 117 π π π 100 80 8 64800  1  1 λ K K 16 160 16 16 16 + 5 + − + + 24000 − − + 38400 + 24 − + + + − − + + + 225 = = = = = t s ν h sh dt dt dt dt dt dλ dλ dK dK dλ JHEP12(2017)040 A ]. , ]. ] ]. ommons SPIRE (2013) 089 (2012) 019 (2008) 38 , IN SPIRE ][ 12 04 58 SPIRE IN , IN ][ ][ JHEP JHEP , , , es. ]. arXiv:1411.4048 [ , esting the problem and also redited. J. Geom. Phys. arXiv:1304.8050 SPIRE , [ IN ity. Part I ]. hep-th/0610241 ld like to thank Jad El Hajj and [ hep-ph/0011335 ][ Beyond the Spectral Standard Model: [ Two-loop stability of a complex SPIRE (2015) 025024 Geometry and the Quantum: Basics ]. ]. ]. IN Compt. Rend. Hebd. Seances Acad. Sci. , Gravity and the standard model with (2013) 132 ][ The Minimal model of nonbaryonic dark (2007) 991 11 (2001) 709 D 92 SPIRE SPIRE SPIRE 11 IN IN IN – 17 – ][ ][ ][ hep-th/9606001 [ JHEP , Academic Press, San Diego U.S.A. (1994). B 619 , Why the Standard Model Noncommutative Geometry as a Framework for Resilience of the Spectral Standard Model Spectral Action for Robertson-Walker metrics The Spectral action principle Phys. Rev. ]. , arXiv:1004.0464 ), which permits any use, distribution and reproduction in [ ]. ]. ]. (1997) 731 Vacuum stability, neutrinos and dark matter Nucl. Phys. , 186 SPIRE SPIRE SPIRE arXiv:1208.1030 arXiv:1105.4637 arXiv:1411.0977 [ [ [ IN IN IN Investigating the near-criticality of the Higgs boson ][ ][ ][ (2010) 553 CC-BY 4.0 Adv. Theor. Math. Phys. hep-th/0101093 , 58 This article is distributed under the terms of the Creative C C* algebras and differential geometry Noncommutative geometry (2012) 104 (2012) 101 (2014) 098 ]. 09 10 12 (1980) 599 [ SPIRE IN arXiv:1202.5717 arXiv:0706.3688 arXiv:1307.3536 290 matter: A Singlet scalar singlet extended Standard Model [ Emergence of Pati-Salam Unification [ JHEP neutrino mixing JHEP Unification of all FundamentalFortsch. Interactions Phys. including Grav JHEP Commun. Math. Phys. [ [ [1] C.P. Burgess, M. Pospelov and T. ter Veldhuis, [8] A.H. Chamseddine, A. Connes and M. Marcolli, [9] A.H. Chamseddine, A. Connes and V. Mukhanov, [6] A.H. Chamseddine and A. Connes, [7] A.H. Chamseddine and A. Connes, [3] A.H. Chamseddine and A. Connes, [4] A.H. Chamseddine and A. Connes, [5] A.H. Chamseddine and A. Connes, [2] D. Buttazzo et al., [13] A. Connes, [14] R. Costa, A.P. Morais, M.O.P. Sampaio and R. Santos, [10] A.H. Chamseddine, A. Connes and W.D. van Suijlekom, [11] C.-S. Chen and Y. Tang, [12] A. Connes, References any medium, provided the original author(s) and source are c Open Access. Acknowledgments The author is gratefulall to the Professor insightful Ali discussions Chamseddine andHasan for suggestions. Hammoud sugg He for also their wou help on working with the Linux packag Attribution License ( JHEP12(2017)040 ] ] , , second Higgs mass (1975) 601 10 (2012) 222 (2003) 279 arXiv:1203.0237 [ arXiv:1202.1316 [ Stabilization of the 388 r index theorem B 709 J. Diff. Geom. , (2012) 031 06 Phys. Rept. (2012) 043511 d A. Strumia, Phys. Lett. , , , A. Riotto and A. Strumia, Complex Scalar Singlet Dark Matter: JHEP ]. D 86 , ]. SPIRE IN SPIRE – 18 – ][ Phys. Rev. IN [ , ]. ]. SPIRE arXiv:1205.6497 [ SPIRE IN arXiv:0806.0538 IN Higgs mass and vacuum stability in the Standard Model at NNLO ][ , Heat kernel expansion: User’s manual ][ The Spectral geometry of a Riemannian manifold Invariance theory, the heat equation, and the Atiyah-Singe (2012) 098 SARAH ]. ]. ]. 08 SPIRE SPIRE SPIRE hep-th/0306138 IN IN IN arXiv:1112.3022 [ edition, CRC Press, Boca Raton U.S.A. (1995). Vacuum Stability and Phenomenology [ [ [ JHEP [ Electroweak Vacuum by a Scalar Threshold Effect implications on the stability of the electroweak vacuum [19] P.B. Gilkey, [20] M. Gonderinger, H. Lim and M.J. Ramsey-Musolf, [21] F. Staub, [18] P. Gilkey, [22] D.V. Vassilevich, [16] J. Elias-Miro, J.R. Espinosa, G.F. Giudice, G. Isidori [17] J. Elias-Miro, J.R. Espinosa, G.F. Giudice, H.M. Lee an [15] G. Degrassi et al.,