K.G.Klimenko ON NECESSARY AND SUFFICIENT CONDITIONS FOR SOME HIGGS POTENTIALS TO BE BOUNDED FROM BELOW

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1 I N S T I T U T E F O R H I G H E N E R G Y P H Y S I C S И Ф В Э ОТФ K.G.Klimenko ON NECESSARY AND SUFFICIENT CONDITIONS FOR SOME HIGGS POTENTIALS TO BE BOUNDED FROM BELOW Serpukhov 2984

2 K.G.Kllmenko ON NECESSARY AND SUFFICIENT CONDITIONS FOR SOME HIGGS POTENTIALS TO HE BOUNDED FROM BELOW Submitted to TMPh

3 УДК М - 24 Abstract КИнепко K.G. On Necessary and Sufficient Conditions for Some Biggs Potentials to be Bounded From Below. Serpukhov, p. 14. (IH» 84-43). Refs. 8. The necessary and sufficient (ИЗ) conditions have been obtained to make the Higgs potentials be bounded from below. Here these potentials are constructed from: (i) two doublets, as well as two doublets and a singlet of SU(2)-group; (ii) adjoint and vector representations of SO(n). For the potential constructed from the adjoint and fundamental SU(n) multiplets, the problem of NS conditions is solved partially. Клименко К.Г. О необходимых и достаточных условиях ограниченности снизу некоторых потенциалов Хиггса. Серпухов, стр. (ИФВЭ ОТФ 84-43). Библвогр. 8. Получены необходимые я достаточные (НД) условия ограниченности снизу потенциалов Хиггса, построенных из: 1) двух дублетов, а также двух дублетов и сннглета SU(2) группы; 2) присоединенного в векторного представлений SO(n). Для потенциала из присоединенного и фундаментального мультиплетов SU(n ) вопрос о НД условиях решен лшпь частично. Институт фвзики высоких энергий, 1984.

4 INTRODUCTION In constructing the unified models for elementary particles interaction one should spontaneously break the initial group of symmetry. For this a mechanism based on the introduction of one or several multiplets of the scalar Higgs fields is usually applied. According to the general principles of the theory, the Higgs potential must satisfy the conditions for renormalizability and boundedness from below, ie it must be a polynomial of not higher than the fourth power and be positive at large values of fields. In the simplest cases with a small number of coupling constants (for example the Salam-Weinberg model with one Higgs doublet), one may easily point out only such values of the constants, for which the Higgs potential is bounded from below. In the models for weak and electromagnetic interactions, containing several scalar doublets as well as in the Grand Unification theories, which are based on groups of a higher rank and have several multiplets of scalar particles, the Higgs potentials have a much more complicated structure. As a consequence, the problem of the necessary and sufficient (NS) conditions for the potential parameters, under which it is bounded from below, becomes non-trivial. One should note, that in a number of refs.' ' for some potentials, only sufficient conditions were given, which do not at all cover the whole parameter region, where the demand for the positivity at large values of scalar fields is satisfied. In the present paper we consider a number of interesting (from the point of view of physical applications) Higgs potentials in a tree approximation and also obtain NS conditions to bound them from below. Here we additionally demand that this property is fulfilled for any values of the dimensional constants of scalar particles interactions, which is equivalent to the positive definiteness of that part of the potential, which contains only the fourth powers of fields. Methodically the problem of NS conditions for the boundedness from below is a generalization of the problem of the search for the

5 residual invariance group under spontaneous symmetry ^ breakdown (see, eg' '). As a matter of fact,all information on possible types of symmetry in the theory may be obtained, if study the invariant properties of the minimum of the potential, which is denoted as V] throughout the paper. On the other hand, it is the knowledge of the minimum of the function Vj that turns out to be decisive in search for NS conditions for the positivity of the Higgs potential. In section 1 after the example of the Salam-Weinberg model with two doublets we display the method which helps to obtain the required estimates. These results were used in section 2, where the Higgs potential composed of two doublets and a singlet of SU(2) group is examined. (SV.CJ a structure of the scalar sector may be used in SU(2)x U(l)xU(i) theories and, as was noted in' 2/ ; in order to suppress the monopoles production). In the Grand Unification models based on SO(n) groups, one of the chains of the spontaneous symmetry breaking contains adjoint and vector multiplets of scalar fields. The Higgs polynomial from these representations of the SO(n) group is discussed in section 3. In the concluding fourth section some NS conditions for the boundedness from below are given for the SU(n)- invariant potential, which is formed by the adjoint and fundamental representations. Here we managed to obtain the estimations only for the case when the coupling constant a and /8 (see formulae (20) and (21)) have the same sign. But if a,/3 are opposite in their signs, then the methods proposed in this paper allow to obtain only sufficient conditions. 1. SALAM-WEINBERG MODEL WITH TWO DOUBLETS In this section we are going to consider the Higgs potential in the Salam-Weinberg model with two scalar doublets ф^ and ф 2, whose homogeneous part of the fourth order has the form V=V +V, where V o = А (^+ 1 1^1 )2 +Л 2 (^+^2) 2 +^1)(^+^2) (1) +2у(^1 V o 1 22 ф 1 )+г ) {ф+ф 2 )2 + г,*{ф + 2 ф 1 )2. (2) (There must not be other combinations of ф х and ф 2 in the expression for V, since there is assumed an invariance under the change of the sign at fields.) Evidently, the boundedness of the Higgs potential from below entails with necessity the requirement V>0 for all ф х and ф 2. Here along the ray where V=0, the principal role goes over to the relations between dimensional parameters, including the coefficients at the powers of fields smaller than 4. Thus, in order to *'HI8 concerns only the potentials without ф 3 terms.

6 simplify the problem here and henceforth, let us assume that the potential is bounded from below for all values of the dimensional constants. For this the relation V>0 will be necessary and sufficient for any <^1 and ф 2 - Let V( ф, ф ) = minv(<*, ф ) under the condition that Ф + 1Ф 1 =Ф Х, Ф + 2Ф 2 =* Г _ (3) In this case VC^j, ф 2 ) > 0 if and only if V( Ф 1, Ф 2 )>0 for all ф 1, Ф 2^.О, which are not simultaneously zeros. Apparently, V o depends only on Ф! and Ф 21 then to find V( Ф 1, Ф ) it is enough to minimize Vj ( ф у, Ф 2 ) under condition (3). Since the model contains an SU(2) invariance, then without the loss of generality one may put ф- = (0, V Ф 2 ). Introduce the new notations: ф = (x^, x 2 ), rr^j - x 2 sin«, Rex 2 = coscuv^j"! x f I Then Vj takes the form: ) \, (4) where ctgs = Jmr//Rer/. The minimum value of function (4) over the variables a and x-j, which lie within the intervals 0 ^ ш < 2n, xj ^ Ф 1, is fo, at p -2 IT/I >0 Ф Ф (p -2 \T) ), at ф-2\т)\<0 Thus, we have obtained the function V( Ф-, Ф 2 ) = V ( Ф-, Ф 2 ) + + У 1 (Ф 1, Ф 2 ). If Ф г = 0, then V(0^2)>0for all Ф 2 >0 when A 2 >0. Let Ф, 0. The function V( Ф^, Ф 2 ) may be represented in the form: Ф 2 (А + Ah 2 + 2yh) at p-2 i/ >0 1 1 «+ p >> at p-2 i77 ] < 0 where 0< h = Ф 2 / ф^ <. It follows from the lemma of Appendix 1 that function (6) will be positive over the variable h under the condition: \ A,, A_> 0; у >-/ГГАТ; p>2 7j iu U{ Aj.Aa >B : ; 2у +р -2 т7 I >-2y/l^; p < 2\-q\ i. Set (7) is right the NS condition for the boundedness of the Higgs potential from below in the Salam-Weinberg model with two doublets. (5) (7)

7 2. POTENTIAL WITH THREE MULTIPLETS The results of the previous section may straightforward used to find the NS conditions for the boundedness from below of the potential, consisting of two SU(2) doublets and one singlet (we give only the ф 4 -terms): 2а(ф* Хф 2 Щф г )+ r,^2) 2 + г,*{ф*ф х ) 2 (8) Here ф х, ф 2 are doublets, ф is a singlet with respect to the SU(2) group, V was given in (2), and potential (8) is also invariant under the change of sign of any multiplet. As was pointed out(ref./ 2^, in order that the Salam-Weinberg model may predict an insignificant number of magnetic monopoles at the present temperature of the Universe, it is necessary to break spontaneously the U(l) symmetry, corresponding to the electrical charge conservation law, at higher temperatures. For this both the potentials with three doublets and the Higgs potentials (8) are permissible. Besides, in some SU(2)xU(l)xU(l) expansions of the standard model of electroweak interactions structure (8) of the Higgs sector is possible. Consider, as in section 1, the function У(Ф 1, Ф 2,Ф) = minv(<^1, ф 2,ф ) under the condition *T*l"*l' «^^V Ф*Ф=Ъ- (9) Since in our case V Q depends only on Ф 1, Ф 2, Ф, and the minimum of V 1 under condition (9) is reduced to formula (5), one has: where Ф 2, Ф) = У о (Ф 1 ф at 2' Ф)+ at P<2\ v \ (10) V (11) O<*1»*2 Let p > 2 г/1. Here we must find the positivity condition of a homogeneous polynomial of the second order (11) at Ф :, Ф 2, Ф>0.ТМз may be easily done using Appendix 2, which shows that V o > 0 only for the parameters found in the domain S^: Q 1 =fx,a 1,A 2 >0; a >- - >ДТ 2 ; у> - where

8 In the case of p < 2\i)\ the maximum parameter range, where function (10) i-> positive, is п% : Thus, V(ф, ф, ф ) >0 for all values of the fields (which is equivalent to the boundedness from below of the whole Higgs potential in the sense explained in section 1), when and only when the coupling constants belong to the domain Q 1 with p > 2\rj\ and to the set fl 2 with p< 2\ v \. 3. S0(n)-INVARIANT POTENTIAL a) even n Consider a homogeneous polynomial of the fourth order formed by the vector h^ and the adjoint ф^^ representations of the S0(n) group, where n = 21: V = аф 2 + ЬН 2 + снф + У г (14). Here n n # V 1= -,.*., «V a H \ViV n n 2 ф.. ф ф ф. 4-fi 2 h ф.. ф..h, (15) Find the conditions at which function (14) is positively defined. We shall proceed according to section 1, ie find first the minimum of the function V 1 at constant values of the quantities Ф and H. The corresponding calculations were performed in ref./ 5 / and are presented in Table 1. In order to find V( Ф, H), substitute the expressions for У..(ф,Н) from Table 1 into formula (14). With the help of the lemma from Appendix 1 one may easily obtain the NS conditions (see Table 2) for the parameters of potential (14), under which it is positive for all ф ±, h ± in the case of a< 0. With a > 0 and P>0, we have to use two different expressions for У(ф, H), depending on the relation between H and 2aФ/( -1). If /SHSi.2 аф/(2-1), then the formulae (A1-A5) give the following positivity conditions for the function V ( Ф, H): )} (16) 2( -l) J

9 Table 1 The minima of the potential V-j at Ф and H constant a a, /3 $ 0, 0 > 0 Vj(Ф, Н) аф 2 /2 a 0, /3 < 0 (аф 2 4-0НФ)/2 a>0 /ЗН>2аФ/( -1) аф 2 /2( -1) /ЗН<2 а ф/( -1) оф 2 0НФ J3 2 H 2 ( - 1) 21 8ia a> 0 2аФ 4- j8h>0 аф 2 21 )8НФ j8 2 H 2 ( - 1) 2 8 а /3<0 2аФ + /3H< 0 ( аф 2 + /8НФ)/2 Table 2 Necessary and sufficient conditions for the boundedness from below of a S0(2I)-invariant potential, constructed from the vector and adjoint representations of the Higgs fields а > 0 /3 2 ( -1) 2 4а а<0 {а 4- >о; Ь> 0; /3>0 с > -2V Ь(а4-2)1 V < - 1>2 <a, «.. 2 2(1 3 ) / '"" о "" JD «/ а р vt * / л с 4- >-2V(a+ ^)(b - )j... 1 а (5 г а Р а off * У о? ор Or ^ 4а ^ ^ 8 а ) + о>5 с + "5" > -2\/b(a 4- ^)! 2 ' 4а 2 ( 3 + fа4- - >0; b >0; 2 С4- J-> -2л/ь(а4-2){ 2 2

10 If /Зн<2аФ/( -3), then the function V( Ф, H) will be positive, if the parameters satisfy the condition: {a + тг >0 > b - i> + 8 a ' 2 0(0-1) Sla 2a (17) The values of the parameters at which function (3 4) is positively defined for all Ф, H> 0 is the crossing of sets (36) and (37). The result is presented in the corresponding square of Table 2.The case a > 0, j8<0 is treated analogously (see Table 2). b) n = 2E+1 With the odd n = 2t+1 the potential still has the form of (14), the function V, at Ф and H fixed achieves the minima of those given in Table 1, excluding the case of a>0, /3>0. Here in the whole range a> 0, j8>0 the minimum of the potential V.. equals аф^/2, and the function V(Ф, H) is П fy УЧФ, H) = (a + 2?) Ф f ЬН + снф. (18) Let Ф = 0, then expression (38) is positive, provided b > 0. When Ф Ф- 0, one may extract from (38) the factor Ф and apply the lemma from Appendix 1. As a result, we obtain the conditions for the positive definiteness of potential (34) (a > 0, /3>0): { a + ~ > 0; b > 0; с > -2 Vb(a + i) (19) For other relations between the signs of the coupling constants a and /S the results coincide with the case of even n and are given in Table HIGGS POTENTIAL WITH TWO SU(n) MULTIPLETS Consider finally the Higgs potential constructed from the adjoint Ф- г and fundamental h. representations of the SU(n) group. The corresponding homogeneous polynomial of the fourth order of the fields ф and h has the form: V = a Ф 2 + ЬН 2 + снф + V, (20)

11 where Ф= 2 Ф* Ф 1 ; Н = 2 h.h 1 ; ( ф к )*= й 1 ; h* = h 1. (21) i,k=3 Х к i=l х 1 к Х п к с V = а 2 * </> 1 The minimum У 3 (Ф,H) of potential (23) at constant Ф and H was studied in' 6 ' 7 ' so as to find the residual groups in the spontaneous breaking of the initial SU(n). Here \f (Ф, H) has such a complicated form, that we managed to obtain the 5 conditions for the positive definiteness of potential (20) for only two regions of the constants a and j8. The case of a < 0, )6 < 0. With such signs of the parameters a and /8, the expression for ^-(Ф, Н) may be easily found with the help of paper/ 6 ''. After all operations У(Ф,Н) takes the form: o(n 2-3n+3) о 9 в(р-\) У(Ф, H) = (а + -^ ; --) Ф* + ЬН 2 + (с + Р К П } ) ФН (22) п(п-1) п If Ф= 0, then function (22) is positive only when b>0. Assuming Ф ^ 0 and extracting the factor Ф^ from (22), we may reduce the problem of the positivity conditions of the function ^(Ф, H) at Ф, H>0 to the study of the corresponding secojid order polynomial (see Appendix 1). Thus, in order that at a< 0, /S < 0 function (20) would be positively defined, the following conditions should be necessarily and sufficiently satisfied:! &+ n(n-l) > 0 ; b > 0! The case of a> 0, /6>0. We shall consider the possibility of odd n> 3 only when the minimum of function (23) at constant Ф and H equals (see ref./ 6 /) 2 *) 2 Н)= min i, g., 4 [sin 4 fl(n+l)(n-3)+sin 2 fl(e-2n+y)-t-n]} (24) 0 n(n " 3) 4 в4 п/2 where у =/ЗН(п-1) /аф. The expression in square brackets in (24) is a second order polynomial with respect to sin 2 0, whose minimum is achieved at (sin 2 <9) =(2n-6-y)/2(n+3)(n-3). Depending on the location of this point with respect to the range O^sin 2 0<3, one obtains different expressions for У}(ф,Н) and hence, also for У(Ф,Н) Therefore, if 0 Ф < /3H(n-3) 2 /2a(n-3), then 2*) In the ease of even n V (Ф, H) is of a more complicated form, and we failed to ir vestigate function (20) for positive definiteneas. 10

12 У(Ф,Н) = (a + ) Ф 2 -н ЬН 2 + снф < 25 > п-1 п-1 Provided Ф > Н(п-1) 2 /2а(п-3), н<1 - (26 > Let M 1 and M? be sets of the parameters of potential (20), which make functions (25) and (26) positive in their own definition regions (the explicit form of these sets may be obtained, if use the lemma of Appendix 1). The crossing M ПМ, leads to the following NS conditions for the positive definiteness of potential (20): a(n 2 +3) a fi2 (n-l) 4 0<b( 0<b<(a* )P ) n(n -1) > 0 ; П 2 4a 2 (n-3) 2 { + > 0 LJ! - ; (27) a(n 2 +3) js(n-l) / а(п 2 +3) д 2 (п-1) 3 a+ - >0; c+ >-2v/(a+ )(b- ^ )i n(n 2-3) n <n+d n(n 2 -!) 4an(n+l)(n-3) In the cases when a>0, /3<0 and a<0, /3> 0 we could not find the NS conditions for the boundedness from below of the Higgs potential because of a more complicated expression for the minimum of the function Vj. Here one may obtain a sufficient condition if one uses some lower boundary of function (23), with Ф and H constant, instead of the minimum V ;1 ( Ф, H). CONCLUSION In the present paper at the examples of some Higgs potentials the method for obtaining the NS conditions for the boundedness from below has been elaborated. In order to escape additional calculations, we require that the potential should contain a 0^-asymptotics in infinity. Thus we have reduced the problem to the search for the conditions of the positive definiteness of that part of the notential, which consists of the fourth powers of fields (see section 1). Further analysis is based on the search for the minimum value of the function, defined throughout the article as V (the form of this function is defined by the initial potential). The symmetry of the minimum point of V^ defines the residual invariance group of the 11

13 theory (see, eg,' ~ '). Thus, in order to find the NS conditions for the boundedness from below, it is necessary to solve the problem of the spontaneous breakdown first. Hitherto this has been done for simplest potentials consisting of one or two multiplets. We have not considered one-multiplet potentials,which are relatively simple to investigate, and turned to some non-trivial cases of two and three multiplets. Here, as becomes clear from section 4,the knowlegde of the minimum point of the function V-, as well as of the expression V, * does not always lead to the successful solution of the problem on NS conditions for the boundedness of the Higgs potential from below. Under these circumstances, however, one may obtain sufficient conditions, if instead of V-j min one uses some lower estimation of Vj. The author is sincerely grateful to A.I.Oksak and G.L.Rcheulishvili for the interest to his work and useful discussions. REFERENCES 1. Mohapatra R.N., Senjanovic' G. - Phys. Rev., 1979, D20, N12, pp Langacker P., Pi S.-Y. -Phys. Rev. Lett., 1980, 45, N.I, pp Parke S,, 91 S.-Y..-Phys. Lett., 1983, B3O7, N.3, pp Huffel H... Pocsik G. - Z. Phys., 1981, C8, N.I, pp Klimenko K.G. - TMPfc, 1983, 55, N1, pp Buccella F., Ruegg H., Savoy C.A. - Nucl. Phys., 1980, B169, N. 3, pp Ruegg H. - Phys. Rev., 3S80, D22, N. 8, pp Enqvist K. and Maalampi J. - Preprint HU-TFT-80-10, university of Helsinki, Received 2 February,

14 Let us prove the following lemma: Appendix 1 2 Lemma. Let f(x) = ax + bx + c. The inequality f(x) >0 is satisfied for all k < x <oo (k > 0), if and only if the parameters a,b,c lie in the region: f a >0; с >ak 2 ; b> -2V"a<?! Uf a > 0; c< ak 2 ; b > - - ak. (A.I) Proof. With a< 0 the function f(x), evidently, cannot be positive in the infinite interval k«x<o. Therefore, a>0. First consider the case when the discriminant is negative, ie D = b -4ac<0. In this situation {a>0; c>b 2 /4a; - <*> < b <«, { (A. 2) and f (x) > 0 for all xf[k, ). Now, let D > 0. The conditions of the lemma will be satisfied, if the greater root of f(x) is smaller than k, which is equivalent to the restrictions a>0, Ь>-2ак; Ь 2 /4а > с >-k(b+ak)!. (A.3) Unifying regions (A2) and (A3), one obtains set (Al), which means that the lemma is proved. In the case of k=0 one may easily see that f(x) > 0 in the interval 0 < x <oo only for the parameters a,b,c from the region ja>0; c> 0; b >-2/асЧ (A.4) In order that f(x) is positive at the segment 0 ^ x<: k, the following conditions should be necessarily and sufficiently satisfied: Ic>0; k 2 a>c; b>-2/ac IUj c>0; ak 2 <:c; b > -ak- } (А.5) 1С Formula (A.5) is proved in the same way as (A.I). One should only carry out the substitution x-» 1/x, к * 1/k. Appendix 2 Let us consider the function: f(x,y,z) =Ax 2 + Л 2 у 2 + * 2 z 2 + 2axy + 2j8xz+2yyz (A.6) Prove, that the maximal possible set of the parameters of this function, which make f(x,y,z)>0 for all x,y,z>0 (and are not simultaneously zeros), coincide with (12). Assume that z^). Then it follows from (A.4) that f(x,y,0)> 0 only when { Л >0; X 1 > 0, a> - хдл*!! (A.7) ч ;

15 Now take that z*0. Extracting from (A.6) the factor z 2 and replacing the variables x/z-> x, y/z-> y, one comes to the conclusion that the function f(x,y) =Ax 2 + A y 2 + 2axy + 2 /3x -i- 2yy + A o (A.8) must be positive for all x,y >0. The NS conditions for this are represented by the inequalities, which are valid for all y^o: \ 2 + 2yy + A 1 y 2 >0. (А.9) ay + >- [A(A 2 +2yy+ Ajy 2 )] 1/2 (A. 10) and besides for A>0. (A.9) and (A. 10) follow with evidence from (A.4), if we treat (A.8) as a function of only one variable x. From inequalities (A.9) and (A.4) one obtains again the condition А 2 >0, А 2 > 0; y>-vx^t 2 } - (A. 11) Turn again to inequality (A.10). For the y-values, when the LHS is positive, the inequality does work. Relation (A. 10) must also be valid for the y, with which the left-hand side becomes negative. This leads to the following three variants of the lemma of Appendix 1: the inequality (here Д а =а 2 -АА, Ар = fi 2 - АА 2,А= а/s-ay) у 2 Д а + 2yA +A^< 0 (А.12) must be valid for the following values for y: 1) a4 0, P40: 0<y <oo 2) a < 0, /S>0: у >-/8/а 3) о > 0, /S<0: 04y<-@/a Let in the first case the set of the parameters of function (A.6), which satisfy inequality (A. 12), be <uj, whereas in the second case it will be ujg, and &> 3 in the third (not to overdo we do not write these sets in their explicit forms, since they may be easily obtained from Appendix 1). Next put «4 = fa>0, )8>0L Now, in order to make the proof of our assertion complete, it would be enough to unify all UX (i = 1,2,3,4) and to cross them with sets (A.7) and (A. 11). As a result, one obtains region (12). 14

16 Цена 17 коп. Индекс 3624 К.Г.Клименко О необходимых в достаточных условиях ограниченноств снизу некоторых потенциалов Хиггса. Редактор А.А. Аятшюва. Технический редактор Л.П.Тимкина. Корректор Е.Н.Горина. Подписано к печати Т-0Г7262. Формат 60x90/16. Офсетная печать. Печ. л. 0,87. Уч.-изд.л. 1,11. Тираж 250. Заказ 308. Индекс Шна 17 коп. Институт физики высоких энергий, , Серпухов Московской обл.