Revised Criteria for Stability in the General Two-Higgs Doublet Model
aa r X i v : . [ h e p - ph ] A p r Revised Criteria for Stability in the General Two-Higgs Doublet Model
Yithsbey Giraldo and Larry Burbano
Departamento de F´ısica, Universidad de Nari˜no, A.A. 1175, San Juan de Pasto, Colombia ∗ We will revise one of the methods given in the literature to determine the necessary and sufficientconditions that the parameters must satisfy to have a stable scalar potential in the general two-Higgs doublet model. We will give a procedure that facilitates finding the conditions for stability ofa scalar potential. The stability guarantees that the scalar potential has a global minimum, that is,the potential is bounded from below, which is a necessary condition to implement the spontaneousgauge-symmetry breaking in the models.
I. REVISED CRITERIA FOR STABILITY
We obtain the stationary points of J ( k ) using Eq. (46)of Ref. [1] as follows:( E − u ) k = − η , with | k | = 1 , (1.1)where | k | < u = 0. Now, suppose wefind two solutions p and q with their respective Lagrangemultipliers u p and u q such that( E − u p ) p = − η , ( E − u q ) q = − η , (1.2)where | p | = 1 , | q | = 1 and u p = u q . (1.3)Let us evaluate the function J ( k ) at these stationarypoints J ( p ) = u p + η + η T p ,J ( q ) = u q + η + η T q . (1.4)Given that the matrix E is symmetric, from Eq. (1.2),we have ( u q − u p ) p T q = η T ( p − q ) , (1.5)and taking into account (1.4), we obtain J ( p ) − J ( q ) = ( u p − u q )(1 − p T q ) . (1.6)The product p T q = | p || q | cos θ = cos θ <
1. Accord-ing to (1.3), the case θ = 0 implies that p = q , andfrom (1.5), we deduce u p = u q , which contradicts theassumed in (1.3). Further, the inequality p T q < | p | <
1. Therefore, in any case, itis true that the factor (1 − p T q ) >
0, and consequently,from (1.6), we conclude that u p < u q ⇐⇒ J ( p ) < J ( q ) . (1.7) ∗ E-mail:[email protected]
The result (1.7) is quite useful because it makes it easierto find the conditions of the parameters to have a stablescalar potential. The process would be as follows: com-pute all the “regular” Lagrange multipliers { u i } ( i ≤ { µ j } ( j ≤ E − u ) = 0, omitting the values µ j for whichthe corresponding η j = 0, on the basis that E is diago-nal (as you can see from (1.1)). Finally, consider u = 0for solutions within the sphere | k | <
1, and with them,form the set S = { u i , µ j , } , which has at most ten ele-ments.The result (1.7) suggests taking the smallest value of S to establish a stable scalar potential. Since the valuesof S are in general free parameters, let us assume thateach one of them is the lowest value.If the smallest value is a regular solution { u i } , immedi-ately impose the condition J ( p i ) >
0, that is, f ( u i ) > S .If the smallest value is an exceptional solution, { µ j } ,you must first verify that it gives a valid stationary point,that is, f ′ ( µ j ) ≥
0. If this is not right, you can discardthis value from the set S . In the case of being satisfied,impose the condition f ( µ j ) >
0, which would guaranteethe stability of the potential according to the result (1.7).The conditions arising from the exceptional solutionsmay not be necessary since the inequality f ′ ( µ j ) ≥ S is 0, you should check first that f ′ (0) >
0; if not, dis-card this value from S . If it is satisfied, set the condition f (0) > S that, given their structure, cannot be thesmallest, are discarded if the lowest value gave a validstationary point (according to the result (1.7)). Other-wise, they should be analyzed.So far, the conditions above give stability in a “strong”sense. If for one of the cases above we have f ( u ) = 0,proceed as indicated in Ref. [1, 2], considering, in thiscase, J ( k ), which would guarantee the stability of thescalar potential in the weak or marginal sense. For theremaining stationary points, it follows that J ( k ) >
0, asstated in (1.7).Finally, we build the set I = { values not discarded from S } , (1.8)from which we obtain the sufficient conditions to guar-antee the stability of the scalar potential. Let us applythe results above to a particular model. II. EXAMPLE: STABILITY FOR THDM
Let us analyze the two-Higgs-doublet model (THDM)of Gunion et al., with the Higgs potential given in Eq.(79)of Ref. [1]. After examining the potential, the corre-sponding Lagrange multipliers, including 0, which couldresult in possible stability conditions, give the followingset: S = (cid:26) u = 14 (2 λ − λ ) , u = 14 (2 λ − λ ) , u = 0 , µ = 14 ( κ − λ ) ,µ = 18 (cid:18) − λ + λ + λ + q ( λ − λ ) + λ (cid:19)(cid:27) , (2.1) where κ = (cid:16) λ + λ − p ( λ − λ ) + λ (cid:17) . The firsttwo parameters are the regular Lagrange multipliers, and the last two are the appropriate exceptional solutions in S . Note that µ < µ , but we still cannot discard µ sincewe must first check if f ′ ( µ ) ≥
0. The global minimumof J ( k ) occurs where the minimum valid value of S is.(i) If u is the smallest value of S in (2.1), then f ( u ) > ⇒ λ + λ > . (2.2)(ii) If u is the smallest value of S in (2.1), then f ( u ) > ⇒ λ + λ > . (2.3)Since u and u are regular solutions, the inequali-ties (2.2) and (2.3) are necessary.(iii) If u = 0 < u , u , µ , µ , (2.4)we can observe that f ′ ( u ) = 4 u u ( u + u ) > , (2.5)so u is not discarded. Taking into account the in-equalities (2.2) and (2.3) in f ( u ), we have f ( u ) = h − λ − λ + 2 p ( λ + λ )( λ + λ ) i h λ + 2 λ + 2 p ( λ + λ )( λ + λ ) i u + u ) > , (2.6)and from Eq. (2.4) we can show that the factors u + u > − λ − λ +2 p ( λ + λ )( λ + λ ) >
0; therefore λ > − λ − p ( λ + λ )( λ + λ ) . (2.7) (iv) If µ < u , u , u , µ , (2.8)then f ′ ( µ ) = (2 λ − κ )(2 λ − κ )( λ + λ − κ ) > λ − κ ) >
0, (2 λ − κ ) > λ + λ − κ ) > µ must be included inthe set I . Besides, f ( µ ) = h − κ − λ + 2 p ( λ + λ )( λ + λ ) i h κ + 2 λ + 2 p ( λ + λ )( λ + λ ) i λ + λ − κ ) > , (2.10)and using (2.8), we can show that the factors ( λ + λ − κ ) > − κ − λ +2 p ( λ + λ )( λ + λ ) > ; therefore κ > − λ − p ( λ + λ )( λ + λ ) . (2.11)So the Lagrange multiplier µ is not considered since µ > µ . In short, for the THDM to be stable, the followingconditions on the parameters are sufficient λ + λ > , λ + λ > , λ , κ > − λ − p ( λ + λ )( λ + λ ) . (2.12) III. CONCLUSIONS
We can see that the application of the result (1.7) isessential to get a consistent model and be able to derivesufficient conditions to have a stable scalar potential. Itallows us to identify either necessary conditions (for reg-ular solutions) or conditions that may not be necessary,coming from exceptional solutions (including 0). Bothconditions generate sufficient inequalities that guaran-tee the stability of a scalar potential. As an example,we can appreciate it, in the expression (152) of Ref. [1], where u < u , u , so for stability conditions, only u is considered. In this sense, it may happen that someLagrange multipliers, although not being the smallestvalues, must be taken into account for stability condi-tions. You can appreciate it from Gunion’s potential inSect. II (Eq. (79) of Ref. [1]), since if µ were not a validstationary point, we would have had to analyze µ . Inthat way, we can reduce the number of sufficient con-ditions arising from exceptional solutions (including 0)provided that f ′ ( µ j ) < f ′ (0) ≤ [1] Maniatis M, von Manteuffel A, Nachtmann O andNagel F 2006 Eur. Phys. J.
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