Dynamical Dirac Mass Generation in the Supersymmetric Nambu--Jona-Lasinio Model with the Seesaw Mechanism of Neutrinos
aa r X i v : . [ h e p - ph ] O c t Dynamical Dirac Mass Generation in the Supersymmetric Nambu − Jona-LasinioModel with the Seesaw Mechanism of Neutrinos
Tadafumi Ohsaku
Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, 50937 K¨oln, Germany ∗ (Dated: October 27, 2018)The dynamical generation of Dirac mass in the supersymmetric Nambu − Jona-Lasinio ( SNJL )model with the seesaw mechanism of neutrino is investigeted. The right and left handed Majoranamass parameters are introduced into the SNJL model; we regard them as external model parameters.The question on the origin of these Majorana masses are set aside, and we concentrate on theexamination of the effect of the Majorana mass parameters on the dynamical generation of Diracmass. The effective potential of the model and the gap equation for the self-consistent determinationof Dirac mass are derived and solved. We use both the four-dimensional covariant and three-dimensional non-covariant cutoff schemes for the regularizations of the effective potential. We findthere are cases of the first and second order phase transitions with respect to variation of the couplingconstant of the Nambu − Jona-Lasinio-type four-body interaction of the SNJL model. In the case ofsecond-order phase transition, the dynamically generated Dirac mass | φ S | can arbitrarily be smallcompared with the right-handed Majorana mass parameter | M | and thus the seesaw condition0 < | φ S | ≪ | M | can be satisfied by a fine tuning of the coupling constant, while at the first-ordercase it seems very difficult and/or ”unnatural” to satisfy the condition. The numerical results donot depend on the difference of the cutoff schemes qualitatively. PACS numbers: 11.30.Pb,12.60.Jv,12.60.Rc,14.60.St
Recent experimental observations of flavor oscillations confirmed that neutrinos have small masses ( the mass oftau neutrino, <
18 MeV ) compared with other particles inside their same generations. This fact motivates us toconsider the seesaw mechanism of neutrinos [1-3] more seriously, because the mechanism seems to provide the uniqueexplanation on tiny masses of neutrinos. The seesaw mechanism states that Dirac masses ( for example, ∼
300 GeV )of neutrinos will be suppressed by very heavy Majorana masses ( for example, 2 × GeV ). The electroweak gaugetheory of the standard model is the most important success of modern particle physics, which gives very accurateagreements with various experimental results, and still it will give us the horizon of particle phenomenology. Inthe electroweak symmetry breaking, the top condensation model [4,5] obtains much attractions from us until now,because it is one of possible candidates toward beyond the standard model. The top condensation scenario uses ageneralization of the Nambu − Jona-Lasinio ( NJL ) model [6] to include the electroweak gauge symmetry, and itconsiders a fermion-antifermion condensate. Recently, Antusch et al. gave a theory of an NJL-type four-body contactinteraction model with introducing a right-handed Majorana mass parameter [7]. In that work, the authors derived agap equation for the dynamical generation of a Dirac mass term of neutrino, and solve them numerically. Moreover,they argued the standard model with right-handed neutrinos as a low-energy effective theory of their model underits condensation scale Λ, and performed a renormalization group analysis of model parameters of the renormalizableeffective theory. It is an interesting problem for us to examine a supersymmetric version of a four-fermion modelcombined with a right-handed Majorana mass term for neutrino. This is the motivation of this paper. We employ thesupersymmetric Nambu − Jona-Lasinio ( SNJL ) model as our model Lagrangian. Historically, the SNJL model wasintroduced for the investigation of dynamical chiral symmetry breaking in supersymmetric quantum field theory [8],and it is extended to construct a supersymmetric version of the top condensation model [9]. The SNJL model wasalso used to investigate the SUSY BCS-type superconductivity [10,11].We introduce the following one-flavor SNJL model with including the Majorana mass parameters:
L ≡ L
SNJL + L M + L m , (1) L SNJL ≡ h (1 − ∆ θ ¯ θ )(Φ † + Φ + + Φ †− Φ − ) + G Φ † + Φ †− Φ + Φ − i θθ ¯ θ ¯ θ , (2) L M ≡ M † h Φ + Φ + i θθ + M h Φ † + Φ † + i ¯ θ ¯ θ , (3) L m ≡ m † h Φ − Φ − i θθ + m h Φ †− Φ †− i ¯ θ ¯ θ . (4)Here, Φ + and Φ − are right and left handed chiral superfields of neutrino, respectively. We take the convention ofspinor algebra by the book of Wess and Bagger [12]. M and m are right- and left-handed Majorana mass parameters( in general, given as complex numbers ), respectively. The term of the coupling constant G in (2) will be used as afour-body contact interaction similar to the case of the ordinary Nambu − Jona-Lasinio model [6]. In four-dimensionalspacetime, G has mass dimension [mass] − and the theory is non-renormalizable. In this paper, we perform thefour-dimensional covariant and three-dimensional non-covariant cutoff regularizations, and a choice of a numericalvalue of the cutoff will be regarded as an introduction of the limit of the applicability of the theory by our hands. Theelectroweak gauge symmetry is not considered in this model for the sake of simplicity, hence it is a drastically simplifiedmodel prepared for our examination on the dynamical generation of a Dirac mass under including the Majorana massparameters. In the seesaw mechanism of neutrino masses, a right-handed Majorana mass of neutrino should take ahuge value compared with its Dirac mass. The mechanism of the generation of Majorana mass might be given in thedifferent framework from the dynamical generation of Dirac mass ( for example, by a vacuum expectation value of aHiggs field ), hence we regard M ( and also m ) as a model parameter. Of course, a simultaneous consideration of thedynamical generations of both the Dirac and the Majorana mass terms can be possible, by using the similar frameworkof the supersymmetric theory of superconductivity [10,11]. Here, we assume the generation of the Majorana massterms happens outside, a possible underlying theory, of our model. In this paper, we just focus on the examinationon some effects of the existence of the Majorana mass parameters in the dynamical generation of a Dirac mass term.Our attitude to the problem is similar to the work of Ref. [7], and we can say that our theory is a SUSY versionof Ref. [7], though we do not consider the electroweak symmetry breaking. Usually, it is believed that the existenceof the SUSY is favorable to solve the problem of gauge hierarchy. Because our model is determined as a one-flavormodel, we do not consider the Kobayashi-Maskawa-like mixing matrix in our theory. By using this model, we derivethe effective potential and the gap equation for a self-consistent Dirac mass.By using the method of SUSY auxiliary fields of composites [8-11], the model Lagrangian will be converted intothe following form: L = h (1 − ∆ θ ¯ θ )(Φ † + Φ + + Φ †− Φ − ) + 1 G H † H i θθ ¯ θ ¯ θ + h S (cid:16) HG − Φ + Φ − (cid:17) + M † Φ + Φ + + m † Φ − Φ − i θθ + h S † (cid:16) H † G − Φ † + Φ †− (cid:17) + M Φ † + Φ † + + m Φ †− Φ †− i ¯ θ ¯ θ . (5)Here, S and S † are Lagrange multiplier multiplets to keep the following definition of the composite collective chiral andantichiral superfields H = G Φ + Φ − , H † = G Φ † + Φ †− , respectively. Due to the non-renormalization theorem of super-symmetry, the ordinary SNJL model with keeping N = 1 SUSY cannot break the chiral symmetry dynamically [8,9].We consider it is also the case in our model. In order to generate a Dirac mass term in our theory, we have introduceda SUSY breaking mass ∆ in (5). Expanding L in terms of component fields, eliminating the auxiliary fields of chiralmultiplets through their Euler-Lagrange equations, and assuming that the collective fields are spacetime-independent,one finds the Lagrangian in the following form: L = − | φ S | G − ∂ ν φ † + ∂ ν φ + − ∂ ν φ †− ∂ ν φ − − ( | φ S | + 4 | M | + ∆ ) φ † + φ + − ( | φ S | + 4 | m | + ∆ ) φ †− φ − +(2 M φ S + 2 m † φ † S ) φ † + φ − + (2 M † φ † S + 2 mφ S ) φ †− φ + − i ¯ ψ + ¯ σ ν ∂ ν ψ + − i ¯ ψ − ¯ σ ν ∂ ν ψ − + φ S ψ + ψ − + φ † S ¯ ψ + ¯ ψ − − M † ψ + ψ + − M ¯ ψ + ¯ ψ + − m † ψ − ψ − − m ¯ ψ − ¯ ψ − . (6)At the case | M | 6 = | m | , the Lagrangian has a skew structure in the right-left symmetry space, and thus the contributionsof the right and left handed particles in quantum corrections of a loop expansion will have differences. The generatingfunctional is given as Z = Z D ψ + D ¯ ψ + D ψ − D ¯ ψ − D φ + D φ † + D φ − D φ †− D φ S D φ † S exp h i Z d x ( L + (source)) i = Z D ψ + D ¯ ψ + D ψ − D ¯ ψ − D φ + D φ † + D φ − D φ †− D φ S D φ † S × exp h i Z d x (cid:16) − | φ S | G + Π † Ω B Π + 12 ΨΩ F Ψ + (source) (cid:17)i , (7)where, the matrices Ω B and Ω F are given as follows:Ω B ≡ (cid:18) k − | φ S | − | M | − ∆ M φ S + m † φ † S )2( M † φ † S + mφ S ) k − | φ S | − | m | − ∆ (cid:19) , (8)Ω F ≡ i / ∂ − M † iγ − M − iγ − φ S iγ − φ † S − iγ − φ S iγ − φ † S − iγ i / ∂ − m † iγ − m − iγ ! , (9)( γ = γ γ γ γ ) while the definitions of several fields in our Lagrangian areΨ ≡ ( ψ MR , ψ ML ) T , ψ MR = ( ψ + , ¯ ψ + ) T , ψ ML = ( ψ − , ¯ ψ − ) T , Ψ ≡ ( ψ MR , ψ ML ) , ψ MR = ( − ψ + , − ¯ ψ + ) , ψ ML = ( − ψ − , − ¯ ψ − ) , Π ≡ ( φ + , φ − ) T . (10)Here, ψ MR and ψ ML are right- and left- handed Majorana fields, respectively. T denotes transposition. Ψ can becalled as an eight-component Nambu-notation-field [13] defined in terms of the Majorana fields. ( Of course, it isequivalent to the formalism of eight-component Nambu-notation given by Dirac fields which was used in relativistictheory of superconductivity [14,15]. ) Because we have three complex mass parameters φ S = | φ S | e iθ S , M = | M | e iθ M , m = | m | e iθ m , (11)we can absorb only two of these phases θ S , θ M and θ m by a redefinition of fields ψ + and ψ − , and one phase degreeof freedom remains.We employ the steepest descent approximation for the integration of the collective fields φ S and φ † S . Therefore, weobtain the effective action in the following form:Γ eff = Z d x (cid:16) − | φ S | G (cid:17) + 2 i ln DetΩ B − i ln DetΩ F . (12)The effective potential will be obtained after the diagonalizations of the matrices Ω B and Ω F as V eff = | φ S | G − i Z d k (2 π ) ln ( k − E B + ( k ))( k + E B + ( k ))( k − E B − ( k ))( k + E B − ( k ))( k − E F + ( k ))( k + E F + ( k ))( k − E F − ( k ))( k + E F − ( k )) , (13)( the diagonalization of Ω F has been done by the same method given in Ref. [14] ) where, the ”quasiparticle” excitationenergy spectra become E B ± ( k ) = q k + | φ S | + 2 | M | + 2 | m | + ∆ ∓ p ( | M | − | m | ) + | φ S | ( | M | + | m | + 2 | M || m | cos Θ) , (14) E F ± ( k ) = q k + | φ S | + 2 | M | + 2 | m | ∓ p ( | M | − | m | ) + | φ S | ( | M | + | m | + 2 | M || m | cos Θ) , (15)Θ ≡ θ S + θ M + θ m . (16)The masses appear in these spectra show complicated structures, though the Lorentz symmetry is still kept in ourtheory. ( These spectra have some similarities with the mass spectra of top/bottom-stop/sbottom appear in theMinimal Supersymmetric Standard Model (MSSM) [16-18]. ) We show several limiting cases of these spectra. Under | M | = | m | , these spectra become like E B ± ( k ) = q k + ( | φ S | ∓ | M | ) + ∆ , (17) E F ± ( k ) = q k + ( | φ S | ∓ | M | ) (18)at Θ = 0. On the other hand, our E B ± and E F ± become E B ± ( k ) = q k + | φ S | + 4 | M | + ∆ , (19) E F ± ( k ) = q k + | φ S | + 4 | M | (20)at Θ = π and they coincide with the dispersion relations of the zero-chemical-potential case in the SUSY BCSsuperconductivity [10,11]. In fact, the BCS-type superconductivity is a dynamical generation of a specific choice ofthe right- and left- handed Majorana masses. At | M | 6 = 0, | φ S | 6 = 0, | m | = 0, E B ± ( k ) = q k + ( | M | ∓ p | M | + | φ S | ) + ∆ , (21) E F ± ( k ) = q k + ( | M | ∓ p | M | + | φ S | ) , (22)and thus, in the case | M | ≫ | φ S | > | m | = 0 ( the seesaw condition ), the spectra become E B + ( k ) = s k + | φ S | | M | + ∆ , (23) E B − ( k ) = q k + 2 | φ S | + 4 | M | + ∆ , (24) E F + ( k ) = s k + | φ S | | M | , (25) E F − ( k ) = q k + 2 | φ S | + 4 | M | . (26)Hence, the seesaw mechanism suppresses the mass of the branch of E F + ( k ) just the same with the ordinary seesawtheory [1-3]. In our theory, the seesaw mechanism occurs also in the scalar sector. The choice | m | = 0 reduces thenumber of mass parameters in our theory, and thus the phase degree Θ disappears from our theory.After performing the Wick rotation in the momentum integration R d k of V eff and introducing the covariantfour-momentum cutoff, the effective potential is found to be V eff = | φ S | G + 116 π " ∆ + Λ ln (1 + α + / Λ )(1 + α − / Λ )(1 + β + / Λ )(1 + β − / Λ ) − α ln(1 + Λ /α + ) − α − ln(1 + Λ /α − ) + β ln(1 + Λ /β + ) + β − ln(1 + Λ /β − ) ,α ± ≡ ( E B ± ( k = 0)) , β ± ≡ ( E F ± ( k = 0)) . (27)Here, Λ denotes the four-momentum cutoff. α ± and β ± correspond to the squares of the masses appear in E B ± , E F ± .At Λ ≫ α ± , β ± , one obtains V eff = | φ S | G + 116 π " ∆ − α ln Λ α + − α − ln Λ α − + β ln Λ β + + β − ln Λ β − . (28)We also examine our V eff in the three-dimensional non-covariant cutoff scheme: V eff = | φ S | G + 18 π " + α + ) / − Λ α + p Λ + α + + 2Λ(Λ + α − ) / − Λ α − p Λ + α − − + β + ) / + Λ β + p Λ + β + − + β − ) / + Λ β − p Λ + β − − α ln p Λ + α + + Λ √ α + − α − ln p Λ + α − + Λ √ α − + β ln p Λ + β + + Λ p β + + β − ln p Λ + β − + Λ p β − . (29)The energy spectra of particles we have obtained are Lorentz symmetric, while the three-dimensional cutoff scheme (usually be employed in a finite-temperature Matsubara formalism ) breaks the Lorentz symmetry explicitly. Therefore,the self-consistent gap equation for | φ S | is found from the stationary condition of the effective potential ∂V eff ∂ | φ S | = 0 tobe ∂V eff ∂ | φ S | = 2 | φ S | G − π " ∂α + ∂ | φ S | n α + ln (cid:16) α + (cid:17) − β + ln (cid:16) β + (cid:17)o + ∂α − ∂ | φ S | n α − ln (cid:16) α − (cid:17) − β − ln (cid:16) β − (cid:17)o (30)in the covariant cutoff scheme, while ∂V eff ∂ | φ S | = 2 | φ S | G + 14 π " ∂α + ∂ | φ S | n Λ p Λ + α + − Λ p Λ + β + − α + ln p Λ + α + + Λ √ α + + β + ln p Λ + β + + Λ p β + o + ∂α − ∂ | φ S | n Λ p Λ + α − − Λ p Λ + β − − α − ln p Λ + α − + Λ √ α − + β − ln p Λ + β − + Λ p β − o (31)in the three-dimensional cutoff regularization. Here, the derivatives appear in the gap equations become ∂α ± ∂ | φ S | = ∂β ± ∂ | φ S | = 2 | φ S | (cid:16) ∓ | M | + | m | + 2 | M || m | cos Θ p ( | M | − | m | ) + | φ S | ( | M | + | m | + 2 | M || m | cos Θ) (cid:17) . (32)Hereafter, we examine the case | m | = 0 and thus the phase Θ disappears from V eff and the gap equations. Weused the numerical package Mathematica ver. 6 for our calculation. At the case | m | = 0, the determination equationfor the critical coupling in the covariant cutoff scheme is obtained as follows: G cr = 4 π (∆ + 4 | M | ) ln (cid:16) Λ ∆ +4 | M | (cid:17) − | M | ln (cid:16) Λ | M | (cid:17) . (33)From this results, we obtain lim | M |→ G cr = 4 π / { ∆ ln(1 + Λ ∆ ) } , coinsides with that of the ordinary SNJL model [8].Figure 1 shows the critical coupling G cr as a function of cutoff Λ under the unit ∆ = 1. G cr shows the well-knowndependence on Λ, and becomes small at a large Λ. G cr depends on | M | quite sensitively, and when | M | / ∆ becomeslarge, the critical coupling also becomes large and the dynamical generation of a finite VEV of | φ S | will be suppressed.This stems from the fact that the denominator of Eq. (33) approaches to zero under the limit | M | / ∆ → + ∞ . Thecritical coupling in the three-dimensional cutoff scheme is found to be G cr = 2 π " Λ p Λ + 4 | M | − Λ p Λ + 4 | M | + ∆ +(4 | M | + ∆ ) ln p Λ + 4 | M | + ∆ + Λ p | M | + ∆ − | M | ln p Λ + 4 | M | + Λ2 | M | − . (34)This expression of G cr of the three-dimensional non-covariant cutoff with | M | = 0 coinsides with the results ofRefs. [10,11] of the zero-density case. Figure 2 shows G cr of the non-covariant cutoff scheme as a function of Λ. Theresult of Fig. 2 is qualitatively the same with that of Fig. 1. In the covariant cutoff scheme, the second derivative of V eff with respect to | φ S | at the origin will be given by G cr of Eq. (33) as follows: ∂ V eff ∂ | φ S | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | φ S | =0 = 2 (cid:16) G − G cr (cid:17) . (35)The numerical results of V eff of Eq. (27) by the covariant cutoff scheme are shown in Figs. 3, 4, 5 and 6, while thatof Eq. (29) of the non-covariant cutoff scheme are given in Figs. 7, 8 and 9. All of our numerical calculations of V eff are performed under a fixed SUSY breaking scale ∆: Λ / ∆ is fixed to 100, and we use the energy unit ∆ = 1. Theglobal minimum of V eff is quite sensitive to both G and | M | , and thus we regard them as the relevant parametersto control a numerical value of self-consistent mass | φ S | . A too large G gives an unphysical solution like | φ S | > Λ.Hence we will choose a numerical value of G close to G cr . In our numerical calculations, we find there are cases offirst and second order phase transitions they take place under the variation with respect to G . In the case of firstorder transition shown in Figs. 3, 4, 5, 7 and 8, they are depicted by several numerical examples of G , the globalminimum of V eff usually locates at | φ S | ∼ O ( | M | ) and such a | φ S | is too large for satisfying the seesaw condition0 < | φ S | ≪ | M | in our model. In this case, when we take a small value of G to try to obtain a | φ S | ≪ | M | , V eff suddenly changes the global minimum from | φ S | 6 = 0 into its origin | φ S | = 0. Hence it is difficult to obtain a small andfinite VEV of | φ S | which satisfies the seesaw condition. When | M | is large, G cr also becomes large, and the energyscale of the tree level part | φ S | /G becomes large for obtaining a finite VEV of | φ S | . See the differences of energyscales of Figs. 3, 4, 5 and 6. The reason of | φ S | ∼ O ( | M | ) stems from this fact under the first-order transition.On the other hand, in the case of second order transitions given in Figs. 6 and 9, a global minimum of V eff cantake an arbitrary small value of | φ S | compared with the right-handed Majorana mass | M | by choosing an appropriatenumerical value of G , and thus the seesaw condition 0 < | φ S | ≪ | M | can be satisfied. In this case, the theoryapproaches to a critical point with a continuously vanishing | φ S | .Our model becomes the ordinary SNJL model which has no Majorana mass term, at the limit | M | → / Λ → ∞ . The ordinary SNJL shows second-ordertransition under the variation of G . Therefore, there is a boundary of the first and the second order phase transitions inthe parameter space of the ratio | M | / ∆. For example, we found in our numerical calculation, the boundary betweenthe second and the first order transitions locates at the region 0 . < | M | / ∆ < .
55 under the parameter choice | m | = 0, ∆ = 1, and Λ = 100 in the covariant cutoff scheme: In this case, the phase transition becomes second-orderat | M | / ∆ ≤ .
5, while it becomes first order at | M | / ∆ ≥ .
55. ( Similarly, we found the boundary in the region0 . < | M | / ∆ < .
55 under the choice | m | = 0, ∆ = 1, and Λ = 100 in the case of non-covariant cutoff scheme. )The boundary could be understood as 2 | M | / ∆ ∼ . (36)Therefore, we obtain the conclusion that the right-handed Majorana mass | M | should take a value much smaller thanthe SUSY breaking mass scale ∆ ( in the case Λ ≫ ∆ ), which is considered to have a TeV energy scale mass inmodern particle phenomenology, to satisfy the seesaw condition | M | ≫ | φ S | > G close to G cr ( a fine-tuning procedure [5] ), while in the first-order case itis impossible to do the procedure to obtain a finite VEV of | φ S | which satisfies the seesaw condition. At the origionof the effective potential, the second derivative ∂ V eff ∂ | φ S | of (35) is always negative under the second-order transition,while it will take positive values under the first-order transition. This means G > G cr is always satisfied for obtaininga finite VEV of | φ S | in the case of second-order transition, while V eff can have a finite VEV | φ S | 6 = 0 even if G < G cr under the first-order transition. ( Therefore, G cr lost the precise meaning under the first-order transition case. )These results of V eff we have obtained were also confirmed in our numerical evaluation of the gap equations (30) and(31), and also by the behavior of second derivative ∂ V eff ∂ | φ S | at the origin | φ S | = 0 of the cases of Figs. 3-9. The authoralso examined the Λ-dependence of V eff , by choosing Λ / ∆ = 500. Still, the case | M | / ∆ = 0 . | M | / ∆ = 1 gives the first-order transition. Hence the order of transition is determined bythe relation between | M | and ∆ under | M | , ∆ ≪ Λ, as indicated in (36). Figure 10 shows the first-order transition ofthe example ∆ = 1, | M | = 1000 and Λ = 1000000.Our conclusions are summarized as follows: • We have shown that the dynamical Dirac mass with the seesaw mechanism of neutrino can be described by oursimplified model. • When | M | > ∆ or | M | ∼ ∆ with Λ ≫ ∆, a self-consistent solution of | φ S | will take a numerical value of theorder O ( | M | ). In this case, it is difficult to satisfy the seesaw condition 0 < | φ S | ≪ | M | . • When | M | is much smaller than ∆ with Λ ≫ ∆, we can obtain an arbitrarily small | φ S | by variation with respectto G . In this case, it is possible to satisfy the seesaw condition.The final comment. Recently, the Nambu − Jona-Lasinio and Gross-Neveu models have been derived from stringtheory with intersecting D-branes [19-21]. It is intersting for us whether the Majorana mass parameters we haveconsidered here can be derived from such an attempt, and they ultimately have their origins in string theory, ornot. In another point of view, effective actions obtained from compactifications of superstring theory has a commonfeature, no matter what method of compactification one uses [22,23,24]. While, the MSSM can be regarded as thelow-energy effective theory of the top-condensation model. Hence, our ultimate goal is summarized in the followingscheme ( see, also [25] ): Superstrings → SNJL + Majorana → MSSM . (37) ∗ Present address: Department of Physics, University of Texas at Austin,[email protected] M. Gell-Mann, P. Ramond and R. Slansky,
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FIG. 1: The critical coupling constant G cr of the covariant cutoff scheme as a function of cutoff Λ. We choose ∆ = 1.FIG. 2: The critical coupling constant G cr of the non-covariant cutoff scheme as a function of cutoff Λ. We choose ∆ = 1.FIG. 3: The effective potential V eff ( obtained by the covariant cutoff scheme ) of the first-order phase transition as a functionof the dynamical Dirac mass parameter | φ S | . We set the parameters as ∆ = 1, | M | = 10, Λ = 100, | m | = 0. In this case, weobtain G cr ∆ = 17 . V eff at the origin is negative at thecase G ∆ = 17 .
2, while it is positive at the cases G ∆ = 16 . , . , . | φ S | . The parameters ∆, | M | , Λ and | m | are set as the same with Fig.3. The positiveness of curvature of V eff at the origin isclear under the energy scale of this figure.FIG. 5: The effective potential ( obtained by the covariant cutoff scheme ) of the first-order phase transition as a function of | φ S | . We set the parameters as ∆ = 1, | M | = 1, Λ = 100, | m | = 0.FIG. 6: The effective potential ( obtained by the covariant cutoff scheme ) of the second-order phase transition as a functionof | φ S | . We set the parameters as ∆ = 1, | M | = 0 .
1, Λ = 100, | m | = 0.FIG. 7: The effective potential ( obtained by the non-covariant cutoff scheme ) of the first-order phase transition as a functionof | φ S | . We set the parameters as ∆ = 1, | M | = 10, Λ = 100, | m | = 0.FIG. 8: The effective potential ( obtained by the non-covariant cutoff scheme ) of the first-order phase transition as a functionof | φ S | . We set the parameters as ∆ = 1, | M | = 1, Λ = 100, | m | = 0.FIG. 9: The effective potential ( obtained by the non-covariant cutoff scheme ) of the second-order phase transition as a functionof | φ S | . We set the parameters as ∆ = 1, | M | = 0 .
1, Λ = 100, | m | = 0.FIG. 10: The effective potential ( obtained by the covariant cutoff scheme ) of the first-order phase transition as a function of | φ S | . We set the parameters as ∆ = 1, | M | = 1000, Λ = 1000000 and | m | = 0.= 0.