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1 t ь * *j G * * * e * я у iioufct»! e д к & с г ft з у о *? «a CEv4ue«&C0ftO x : r- >.i сз г.-j -г ^ t " и л -л :\ #? с ь с ( {it«;'"?!?cff*ps»'eessc '( e f И е- с ft в INSTITUTE OF THEORETICAL AND EXPERIMENTAL PHYSICS 6 i A.Yu.Morozov, M.A.OIshanetsky, M.A.Shitman GLUINO CONDENSATE IN SUPERSYMMETRIC GLUODYNAMICS Preprint 105 Moscow - ATOMINFORM «limit;

2 УДК M-I6 OLUIHO СО1ТОЕ 9Д?8 ЯГ ЗПРЕНЗТШВТН1С GLUODYfTAMICS: Preprint ITEP 87-Ю5/ A.Yu.lorozov, «.A.Olshanetslcy, M.A.Shiftman - X.: ATOMIHPORM, 1987] 29 We extend the method for calculation of the gluino condensate In SUSY Yang-Mills theories with no matter proposed in the previous work to cover the cases of SP(H) and exceptional groups (0,, P., B 6, B? ). In all these cases the condensate Tz ЛКу is non-vanishing and takes I( <J{ ) different values where T( Q ) is one half of the Dynkin index for the adjoint representation of the gauge group Q. fig, - f ref. - 12

3 1» Introduction The question of existence of the gluino condensate,- in supersyimnetric Yang-Milla theories with no mutter became eepecially accute in connection with the proposal' ' to use the condensate (1) in the shadow sector for a dynamical SUSY breaking within the superstring approach. The literature /2/ devoted to this issue is rather rich (зее ref/ ' for a critical analysis). Speaking briefly we may say that there are several arguments in favor of the gluino condensation none of which, however, is fully conclusive and - what is more important - does not give the possibility of reliable calculation of the condensate (1). The main difficulty is due to the fact that the condensate (1) is a dynamical parameter emerging in the theory with the strong coupling regime. In this work we continue to develop the construction put /2/ forward in ref.' ' which allows one to fix the condensate (1) In an indirect way* The basic virtue of the construction is that all calculations are carried out in the weak coupling regime and are undar complete theoretical control. Then, using sonte general properties of the supersynmetrie theories, we are able to propagate the result to the strong coupling regime.

4 /2/ The method of ref. ' bears the inductive character. It first,- the condensate <CfLU^>for SU(2) group is found. Then it is assumed that^ ^jni* 0 for Stf(H-1), and with this input <СТгДХ> for SU(N) is obtained. The analogous chain Ofc) s /iv^j) -^ ОГ=?-) ->... -^ O/0i/) ha-s been constructed in /2/ for the orthogonal groups as well. The strategy of this work based on two fundamental observations of refа У^**' ia as follows. (i) For the given group G introduce auxilary matter superfields in a specially chosen representation of the group G (let us call the model with the auxilary matter fields "intermediate**). (11) If the mass term of the matter fields is small the gauge group G is spontaneously broken down to smaller group G, G <CG. The breaking takes place thanks to large vacuum expectation values of the scalar fields provided that in the smaller group G' / ^ O. (ill) Calculate ^TiJU^in the intermediate model in terms of ОГгЛЛ^/ (the result will depend on <TiX b and the mass parameter m). (iv) Tend т-9<у> во that the auxilary matter disappears from the spectrum} use the exact result for the m dependence of n the inteteediate model in order to «.х<тг.лду in the 4 limit in -* o». She aim at the present work is to extend the method to cover symplectic and exceptional groups. The condensate (1) can be calculated in all the eases with the only

5 3 For this group the program sketched above ie not applicable by technical reasons. Hotice that the approach close in spirit has been proposed recently in ret. devoted to the proof of the spontaneous breaking of the discrete chiral symmetry "^дтлгу"*' д. ( -tlr>-r,s\- invariance is a remnaut of the classical U(l) symmetry destroyed by the quantum anomaly» T(G) stands for one half of the Dynkin index for the adjoint representation of the gauge group G). The authors of ret, considered the groups 0(N) and G and managed to Introduce auxilary fields in such a way as to break down the gauge group completely and reduce the situation to that with the weak coupling regime. In this regime they found T(G) degenerate vacuum states. According to Witten the nujriber of vacuum states in eupersymmetric theories is an invariant which does not change with ю (in particular, ^ ), 2. Generalitiea Below we will need the following notations. The auxilary matter fields are.denoted by S and I. There is no need to introduce more than two chiral superfields. For real representations of the group we will need only one superfleld, or in other words, S may coincide with T. We will use the letters T and T for one half of the Dynkin index in the adjoint representations of the groups G and G. Formally

6 4 (Certain oare should be taken her» in order to encore consistency of normalizations of the Q and e'generatore). The corresponding quantity for the matter fields is denoted where the *um runs over all matter supernultiplete <3 and I), Thm calculation of the gluino condensate is based on the following chain of arguments'. (1).Xbe theory without the mass teim possesses a non-aaomalous t-eyametry under wwhieh both euperfields S and T tranefor» in the earn war. So guarantee that this R-eymmetry is anonaly- -free we must ensure certain relation between the chiral R charges of gluino Д and matter fermions ^ and ^т * Spe ~ cifically, if Х-*е и Л,' «Т/т (Cf, in the instanton field there are J2T gluino sero modes and natter sero modes.) The istariance of the Yukawa coupling *(u, j\ requires the fennion transformations to be supplement (2).The dependence of the gluino condensate on the mass parameter a can be exactly determined in the whole interval of variation of s«bev the mass «era b* DtSTJ ^мщ Piret of all»- notice that the condensate depends on я» analytically, the partial derivative ~-<ТгМ\*> </ 4/» Indeed, this partial derivative is equal to the connected pert of the correlation function

7 -0. С») The correlation function of two lowest components of the superfields of one and the same chirality must vanish. The dependence on the phase of m can be found by using the same R transformations. Notice that the lagrangian with the mass term switched on, does not change if the E transformations of the fields are accompanied by the additional phase rotation. ~- /, n m. > hie м,»т_»- rn ы. Since under such a transformation <CTiJA">-**e "^Тг-L^we see that (3) where the numerical constant С is now m-independent. 54irther consideration is necessary for proving the fact that Actually we will be able to calculate C. (3).The identity^ ± 5 CfQ. allows us to connect the gluino condensate with the vacuum expectation value V^<4^>=.^ebAin the whole interval of variation of i: If m->0 the value of Iris determined by the form of the superpotential of the matter fields S and T. The superpotentiai consists of two parts: the classical term mstplus a non-perturbative correction А/гГ^Л^У"*.( Д stands for the scale parameter of the intermediate model.) The functional form of the noii-perturbative term is unembigously fixed by dimensional arguments (2x+y«3) and the R-invariance (зс»~г^/ т-т н ) ). кв а

8 result and 1 'к (4) The coincidence of the mass dependence in eqs,(3) and (4) shows that any other contributions can not appear in the supexpotential. Thus, the coefficient С in eq.(3) - the aim of our analysis - is expressed in terms of the coefficient Л. emerging in the non-perturbative correction to the superpotential. (4).In order to find the m-independent parameter A the authors of ref. proposed to investigate the theory in the limit fa-9>0. In this limit the original gauge group G breaks down to G by V, At the scale below г к»е have SUSY G 7 Yang-Milla theory with additional light ohiral superfields. Under the appropriate choice of S and T all additional light superfields are singlets with respect to G, To ensure this fact It is necessary to eat up all G 7 non-singlet components of S and X in the super-higgs mechanism (they must b«paired with the G/G' gauge bosons and make them heavy). In this case 3>T'+ T M automatically* Ihe dynamics of the light chiral euperfields is described by the enperpotential АД (S T). She coefficient A can be expressed in terms of the gluon condensate in the theory with the gauge group G. Using the inductive line of reasoning we reduce the question of tha gluino condensate for all theories (except E^, see /2/ below) to the SU(2) case, which has been analysed in ref.' '.

9 In this way we calculate the condensate. (5).To establish the relation between A and <ftijla)> et us 7 trace the evolution of tfas effective lagrangian with decreasing i*-, the normalization point It M»-iTthe gauge symmetry G ia not broken, and the super-yang-mille lagrangian hae the form At М«тГоп1у the gauge group G survives. The coupling constant tk r t in the corresponding legrangiex differs from Я -. These two constants coincide at h- Therefore» / ^ A- 2. "Ill A 2. /_-X Thus, at A<<vtha effectix 7 lagrangieii acquii-вв the terra that can be written as follows зт~ьт'-т н L (%T) If the G theory possesses a non-vaniehing condensate _( the interaction (6) generates at scales у ч<<^л the non-perturbative superpotential л.($т) for light chiral singlet superfields - the only superfields below A,.». U The lsat thing to do is to express the condensate i U^i-KjA, 1 terms of the original parameter Д (the

10 a scale parameter of the intermediate theory) and V. To this end use eq.(5) plus the standard definitions Then > and the non-perturbative superpotential in the effective actio below A i takes the form This expression completes our efforts to determine the coefficient i. in terms of the gluino condensate in the theory with the gauge group G'. Returning to eq.(4) with m-^e-^w see that <;' (8) where Л is the scale parameter for G gluodynamica. The T- fold degeneracy of the vacuum state in the super-yang-mills theory with the gauge group G becomes evident from eq.(8). Hot to obscure the simple essence of the construction by the lengthy chain of formulae let us formulate the result in somewhat different terme. Ae explained is/^ the expressit terms ot the bare quantities ftt 0, q o, M Q.

11 9 is ;Jiist exact irrespectively of th.e relation between РП О and H c * Moreover» varying K c we continuously pass from the liffl.lt of G gluodji;&--ics { m c ~ Si. ) to G' gluodynamics 1 Ч С > i : j i j \ о i only those terms survive "that correspond to ihe unbroken subgroup G j. and the expression in eq f,(9) reduces to \ If ise know the proportionality ooefi"3 - this is our original assumption - and,. Einraltsaeously, if we know the connection between the combinetion Т н /Т у 3.-4JH/T 4 ) ъ Sn- ^ -85чУт^ ^ and [\ r i at PV**O - we immediately establish the content i t 8 Ц. (9)» Tending М в -тн в and using we get the expression fop тт- г <ТгЛА]> in terms o.f /v^ с Tb.ua, the question of the gluino condensates for all groups is reduced to determination of the chain G *P G* ^... > SU(2) for each group and the choise of the matter fields at each stage. The superfields S and T must possess the folio* «ing properties: (i) their expectation values along the valleys in the general situation must break Q down to G' ; (li) ell components non-singlet with respect to G' must be atm up by the super-higgs mechanism. The reservation made above, about the generality of the situation G ^ G, is essential. If in the general case G la broken by down to G * -ф specific choice of parameters G-*-G G i, and only with some IT / it would be necessary to prove the stability of such vacuum solutions with respect

12 to perturbative and non-perturbstive corrections. One more 10 condition on the fields S and T is the requirement of existence of the mass term (n S T \. In other words, the product of representations S*T must contain a singlet in G. G > G The general pattern of the gauge symmetry breaking can be described as follows. A general vector from the matter multiplet H^^ K.; is transformed to some standard с form by using the gauge freedom in G ( this can be viewed as the transition to the unitary gauge). By definition the group G does not act on this standard vector. G 7 -singlet vectors form a subspace H in M : H \ yv~ v, For instance, if H i s *я в adjoint representation of SU(IT) and H is the space of the diagonal matrices. (There exists an extension jj(h) of the subgroup G' which consists of transformations conserving the subspace H as a whole. In the example considered H(H) is the extension of the subgroup G by the transformations consisting of A// permutations of the diagonal elements. In certain cases the factor-group includes continuous components, and its Lie algebra is nontivial. For example, for the N-plet representation of SU(II) group the space H is one-dimensional, H» (1X, О. - - О ); G / - ^u(h/-i) ; J/THVG'- l/(i) Let us denote the invariants of the gauge group G built from chiral superfields by 1. _. T* ЗЪв invariants the

13 11 may be bilinear, trllinear, etc. The corresponding degrees of freedom remain light after the symmetry breaking (see below). It is clear that the invariants depend not on all coordinates in the multiplet H but on the coordinates on the subspace H'. More exact mathematical assertion bears the name of the Luna-Richardson theorem' ' and essentially reduces to the following. The polynomials -L-... -i-^, considered on H, coincide with the polynomials invariant under the transformation from the factor-group In the first example discussed above (the adjoint representation) the invariants are IJL«^A ;...X^^TiA, where A ia a matrix from the adjoint representation. It is evident that Xj depend on the eigenvalues of A; symmetric polyniraials of the eigenvalues give all the invariants. Leas trivial situation emerges in the theories with the gauge groups St. (2^1 E^ and S,, to be dealt with below. The matter mass term is proportional to the bilinear invariant, denoted by X A. In the limit Ki-ч- 0 in the intermediate model there exist the so called valley, the flot directions along with the D-terms vanish. As follows from the general theorjr ', the number of degrees of freedom parametrizing the bottom of the valleys is equal to the number of independent chiral invariants. Choosing certain vector from the multiplet II we actually fix the gauge condition (the analogue of the unitary gauge). Then the variables of the angle type ara eliminated, and we are left with the variables of the"moduli" type, corresponding to the invariant combinations of the coordinates in H. Let us notice that the complete relation between the

14 12 condensates <~*глл. and <«XV>./( with all coefficients explicitly written out) can be found only provided there exists only one G-singlet chiral invariant, X L. With more than one invariant their combination appear in eq.(7). This does not effect the analysis as far as the order of magnitude is concerned, but the exact relation between Д s and T; will require a concrete computation. /2/ SU(ff) and 0(Ю theories have been discussed in r e f/. Here we will concentrate on all other simple Lie groups, except Eg : -> Sp (Jf-1 )»... =»- St. (1) -=i SU(2), SUO)-*... 30(8) S0(8) -* -- S0(8) In the cased of F., Б. and E_, we have to deal with several invariants, and the exact numerical value of the condensates remains unspecified (only order-of-magnitude estimates are available). The lowest dimension representation of the group Eg, the most interesting from the point of view of applications, is the adjoint representation, and it is impossible to find G', S and T in such a way that Т»т' + T M. Thus. /2/ for 8g the method of ref, is inapplicable. Mathematically all relevant results can be extracted from the tables where different symmetry breaking patterns for different representations are collected. Such tables allowing us to select the appropriate chains ere compiled in ref.. A part of them la reproduced in the Appendix. Below we give

15 necessary comments * 13 3«Sr,(2S) The Himplactit groups are direct generalization of SU(N) with complex numbers substituted by quaternions. Let us denote quaternions by q * q + q^^» (a»1 r 2,3) where the "imaginary units" e^ can be represented by the Pauli matrices, e a -» t6^. The Lie algebra of Sp» (2N) consists of qusternionic anti-hermiteanif *JT matrices Q г where the bar marks the quaternionic conjugation Q = ^""Я?" 6^* If, instead of quaternions we used the Pauli matrices we could rewrite the I by H matrices C^ in terms of complex Matrices 2U «?]?. The matrices act on the column rector consisting of H quaternions 5c^-=DC^«o -f-с^бг «, The norm of the vector, tf is conserved under the action of the group o If» instead of quaternions» complex numbers ar used, we have two 2N-dimensional ^representations R, and R_ with the following «tructure» It la e«7 to that froa two vectors, froa R. and 1Ц, one can built a. bilinear iavarient, th» nor» of the vector

16 ч In terme of the components (%" i ), oc»i e 2, the invariant involves the antisymmetric rank-two tensor. In other words, the Ш representation is pseudoreal, and one needs two such multiplets to write down the mass term. With this matter sector there are no other chiral invariants. The standard form of the vectors S and T is Given arbitrary vector we can alwaya reduce it to eq.(12) by a combination of transformations from G. Eq.(12) defines the subspace H in the space М^\?, ф К г» Their common stationary subgroup is Q =%>(A/V~0) With respect to O*' the generators of the original group and two fundamental 2K-plet representations can be decomposed as follows: {2.n/}!in eq.(13) denotes the adjoint representation of the So (2R) group. (Below we «ill us* the same notation for other groups as well.) The dimension of $>^(2W) is equal to N(2H+1). She pairings indicated by th* arrows correspond to the masses of the gauge bosons belonging to 2{5-1)-pl*ta of Sh (2Я-2). From four singlet matter fields in eq.(i4) three

17 are paired with the singlets from eq<,(13) - the corresponding gauge bosons acquire тазвез. Иге unbroken gauge group is G* a Sp(2N-2), and only one light singlet survives, in full agreement with the counting of invariants. (Let us parenthetically note that one can reduce S and T to a more special form» (Ho this end one should act on the vectors (12) by the transformat ione from the factor-group j/(b()l t xs(a($~ Then the variable V corresponding to the mass invariant, is the only Invariant of the factor-group 4», The group Gy. The fundamental 7-plet of the group V \tu. is the real representation» which allows us to write the mass term using only this representation. By choslng an arbitrary vector from the 7-plet as the vacuum configuration we break G, down to SU(3). Before explanatory remarks let us write down the decompositions of the adjoint and fundamental representations of G^ respect to SD"(3)? - i +Ъ+Ъ щ^ "к/ тяг ~* with She singlet appearing in the decomposition of 7. corresponds to the standard vector» In other words, the standard form of the 7-plet Is as follows* the 3 and 3 (with respect to Stf(3)) components vanish, and the only non-vanishing component is SU(3) singlrt. Shan, the stationary subgroup Is evidently Sn(3). d%e mass tent has the usual formwv Y J +h.c, and wv Y J

18 16 no other chiral 6^ invariants can Ъз built from one 7- plet,. Acting on the standard vector by elements of G«we get the orbit of the group* She latter is known to be e sphere» S я *"fsu$. Ems, in the general ease G^""* SU(3) ( and six gauge bosons (3+3) get masses. The intermediate model is SU(3) gluodynamics plus one light singlet corresponding to 5. The <$roup She dimension of the algebra P^ is 52 while its fundamental representations is 26. It is real and, hence, the mass term can be constructed from one 26-plet. The vectors in this multiplet ere representable as traceleas hermitean octonionic matrices' ' " '" S \ (16) Xhe algebra F^ sets in this representation in the following way. 24 &«merators not belonging to the SC(8) subalgebra have the form of antihersxitean octonionic matrices / ft *h\ end «ct on 3 ее the noraal cojamxtator. 39» r ainjng 28 generators of?^ form the S0(8) eubalgebre..

19 17 It annihilates the diagonal elements of 3, and each matrix element k: ia eq,(16) is transformed according to one of its t three 8-dimensional representations { &y & -rectors, 3$ 3 L -leftis is handed spinors, 8^ -right-handed apinora). This relation implies the following decomposition of the 26-plet: Z6= <l + i t$*и3 [в] for the adjoint representation we hare where the last three terms correspond to those in eq.(19). Hotice that apart from the bilinear invariant (the ниве tent) <v,3y T>i{v f v} (22) there exist* a trilinear Рл singlet, the determinant of the matrix 3 It can Ъе written in a symmetric form with the aid of a new operation called the Preundenthal prodtict where /З^ЭЛ denotes *he anticommutator -M *'i + 4 м this notation the trilinear invariant reduces to. With (г4) One can check that it is symmetric with respect to interchange of the indices 1,2,3. The arbitrary matrix from 26 can be reduced to the dlago-

20 18 nal form by i 1^ transformations (the Treundenthal theorem )s?/л} (25) The presence of two independent parameters ie correlated with the existence of two invariants, (22) and (24). In the general case with allecfi in eq.(25) different, PA is broken down to SO(8), which should be clear from the description given above. The SO(8) algebra annihilates all vectors lying in the eubspace H» On the ether hand, in P^ there exists a subgroup, conserving the apace Я as a whole - this is the extension of S0(8) by the group of permutations of «Cj. The invariants (22) and (24) can be expressed as polynomials of Л!. invariant under the permutations, - {j and «^W»*^j. In a special case of equality two of the *j the symmetry breaking pattern is Rj - $0(9) ш one can check, however, that this regime is unstable. The requirement of vanishing P-terms for the vacuum configuration leads to ^f^ existence of <"ГгДА> р follows from the fact that Determination of the constant in the relation < ТЪД»- rcohst Л Fjf turns out to be a more complicated problem than previously due to the presence of two independent parameters J i and «(^. She relation between Лc^^and the bare quantities depends on the behaviour of the theory in the transitional domain, in particular, on the masses of the gauge bosons from fy /S0(W. If previously, the masses of all heavy bosons were equal (up to known coefficients), now they depend on the dimeneionlees ra-

21 19 tio ^i/^,. Heedless to say that with e ^ e ^ «rffj the constant in eq.(26) is known by an order of magnitude. For exact calculation of the constant, however, one will have to fix the relation between «i-± and oc-^. Thus far, the question is not completely investigated. o r The grmip K» There are 78 generators in the total set; 52 of them form the aubalgebra F4 plus 26 additional generators that can be represented as traceless hermitean octonionic m trices given in eq.(16): The fundamental representation of E^ is 27. It is described by the same matrices (16) with no additional condition on the trace. The action of the algebra Eg on the 27-plat is as follows: (28) Prom this we see that for matrices proportional to unity, c the algebra E 6 is broken down to the aubalgebra P*, and, therefore. The 27-pIet forms the complex representation. Hence, to introduce the maaa term we need We will describe the corresponding representations as octonionic matrices with complex coefficients. If the complex conjugation of the coefficients

22 is denoted by bare. I Е * 5= h*j-±{t } 3]. (30) Both, for 27 and 27 (separately) there exists a trilinear 20 invariant quite analogous to (24): Besides that, we can write the mass' term (bilinear invariant) This is not the end of the story, though,, From 27 and 27 we can build one more, quartic, invariant» Denote by y, ( M ' ) the symmetric tensor appearing in the trilinear invariant J3 С Хг J * Then the invariant tensor of the fourth rank is proportional to *fr}ft, <* "^. In terms of '- / ^ the quartic invariant mentioned above can be written in the following way How, let us show that in the general case the algebra Eg is broked dovm tojo{8). ELrst, we check this formally by inspecting the decompositions (27) and (29). We use eqs.(20), (2i) obtained previously for F4, * tj + * in * z Ц], (33) Th» octet* la 040*(33)» (34) are connected with the matrix elementя J? in eq.(28), off-diagonal matrix element? Tin (28) and off-diagonal matrix elements of the matrix from 27-plet, Two aingleta

23 in eq.(33) are diagonal elements of the matrix T while three singlets in 27 (or 27) are the diagonal elements of the 21 matrix О. Six octets from 7S are paired with six octets from giving masses to the corresponding gauge Ъозопз. Besides that, the diagonal elements of the matrix T (зее the term2"[ jin eq.o3)) do not conserve the trace of the matrix J (or 3 ), and hence, these bosons also acquire masses. The light sector includes only gauge bosons from 0(8), two singlets from 27 and two from 27. (This number is in agreement with the existence of four chiral invariants in the case at hand). In the space of the representation 27 there is the subspace H to which one can reduce the arbitrary- matrix from the multiplet. More concretely, H is the subspace built from the diagonal matrices H= { M*«i(4 it Ji,^3)} (35) Almost every vector from H has$0(8) as its stationary subgroup. If some fi-s coincide the stationary subgroup becomes larger. For example, if two/л are equal to each other the stationary subgroup is SO (10) while the equality ecj- =J.2.~*^3 leads to Рл as the stationary subgroup. As in the previous section we can show that the special regimes (with 5 0 (10) and PA ) are unstable. The factor group JV(H)/(S'sSOGfJji.a generated by permutations of «^, c^j,, U$ and the group of the diagonal matrices of the form (36)

24 22 The polynomials on Ы (see eq.(35)) invarient with reepeet to the factor group correspond to the four invariants built above. Here we write down tlie expressions for the invariants in terrae of the chirel superfields Instead of these four invariants one can ohose four light dynamical variables as follows! 7. The group E? 133 generators of the algebra B» can be divided in four parts:? 6 (37) Here W< denotes the matrices from the 27-plet of the subalgebra E. Correspondingly, OS) The fundamental representation of E? has dimension 56. It is pseudcreal. In other word* we will need two 56-plets to be able to introduce the mass term. The arbitrary vector from 56, \jf, can be decomposed as (with respect to E^ ) (39)

25 23 Here and b are aingleta while X and X 27-pl*te of the aubalgebr» B^ * She transformations from 2$ act is this baeie according to the following rules«jx - i< з г,г>, -i As wee already mentioned, we need two 56. The mass term can be written aa l40) For each of the two 56-pleta there exieta the quartie invariant X (42) The pattern of the gauge symmetry breaking induced by two 56-plet«±m 7 -+ $0(8) (43) Thie pattern, according to ' ' bears the general character (see Sable 2 in the Appendix). To see this let us write down the decompositions of the adjoint and fundamental representations of Sj with respect to$q(s}t st*6-l*util 2 * * Ш * Ш (44)

26 24 All fields in the octets are paired and become massive. Apart from that, nine SO(8) singlet fields are also paired and become massive. In the low-energy limit we are left with SO(8) gluodynamics plus seven light singlet superfields, corresponding to 7 invariants that can be built from two 56-plets, Three of them are indicated explicitly (see eqs.uo and (42)). Prom the decomposition (44) we see that dim Я = 16. Besides that, the ccntiauous part of the factor algebra М(Н)/ coinsides with SU(Z)@SU(Z)*>SU(Z). It is worth emphasizing the fact of coincidence of the Lie algebras -V/")/ff and 6' with those parametrizing /12/ the magic square' '. With the help of the magic square one can get a unified description of the gauge symmetry breaking for all groups except Eg. We are grateful to Yen Kogan and Victor Hovikov for valuable remarks. Appendix le present here a part of the Tables from ref.' 1. They answer the following question. Assume we have the group ^ and the matter fields in the representation И What is the gauge symmetry breaking pattern? In other words, which subgroup О survives from 6 in the presence of the vacuum expectation values of the general form? We list in th* Sable all representations H of the groups ь( ло, &, E and Bj that lead to non-trivial 6, i.e. T M <7(ty. In the case of the unitary and orthogonal group* the number of different possibilities is r«rj large.

27 25 the eoaplete catalogue can be found in vmt,' lvf. Tar all rep» resentations in the fables the corresponding dimensions as well as the weights in the Synkls graphs are indicated. fable 1 includes the irreducible representations, the reducible* oaes are collected in Sable 2. We also give (Cable 1) the decoapositlone of И and $ with respect to S» the form of the invariant apace H (whet, it is known), the values of Тц and Тц, «ad, finally, the lie algebra of the factor group Botice that the Dynkin indices for various non-trivial representationa can be aaeiljr found from the instanton calculus. Zbe Dynldn index Tjbf is equal to one half of the number of the fermion zero modes, provided the imbedding. is organized in each a way that the instanton topological charge is ftv**»fl (unity). One should select a chain Stf(Z)-*&"*. 5*6 and decompose M in the sum of irreducible representations of 3U(2) (they are numbered by the spin which can take integer and half integer values )s laob representation C^jHj ааа»ф45ф*фвго modes in the lnataaton Held. Saerefore for inetanoe, feartitleadjoint representation of S» we have 1 - «.early, it is not necessary to go all the way down to SU(2). One can stop at the group «here the Dyateln indices are already known, Por example, fo» 9л :

28 26 E 6 : 78 = * T 78 =9+3= 27 = 26 + i r> Х 27~ 26 3; E 7 : 133 = Д. 1 I Tl T 1 In ^^^ > = 12+2, = Д, + Д Е» T 56 = 2» 3 «6; E 8 : 248 = L *^"^ r*^d ^>^ f^ *\/ ' ^ **^ Г7 T 248 = = 30.

29 Table 1 Irreducible Representations 1) M Rank Representation' U G» Decomposition p) G H Я(Н) "JfT 1. GnSp(N); dim 9p(N)aN(2N+i); T Q =N+1 N 2?J Sp(N-1) gjf^il -traceless aymmetrical qua- 4 ternion matrioies 0 l О oo се f>f..o _^ 2. G 2 t 0 3. P 4 { И SU(3) dim G 2 «14 SU(3) 1) Ш-1) +2 2) ffr^?j 2) G'+(2,2,1,...1)+...+(1,...1,2,2) 1J ^ +^" + 1 * 1 2) G-?6 + П1 1) 2)..- dim P 4 «52; T Q -9,-4 X+ I + д. сч з" + f diagonal matricies 1/2 SU(2!..0) (seeo6)) 4. Eg; S0(0) ^ dim Eg- 78; T G «*12 2)"G» + 8 V +51 +sf зее(25) (see(28)) ^ ' 56^ {о-о-л-е-о-о (see(39)) 6 : P 4 " 1) ) G^ + 26 ' 5. E? ) -j dim B^» 133; T G «= 18 E 6 1) (aee(35)) 3 0 UO)

30 Reducible Representations Table 2 Rank Representation И I Sp (N) m.25 SptN-l-j-J) SU(2) SU(2) SU(2) G /^ ^/ SU(2) SU(2) P Д6 SU(3) SU(3) SO (8) UCD U(1)!' SO (8) U(1) ff>u(1) 1 i 6 ra2j + ra'u SU(3) SU(3) Q SU(3) j *7 7 SO (8) SU(2)$SU(2)$SU(2)

31 REFERENCES 1. Dine M.,Rohra R.,3eiber«N.,Witten E,.-Phys.Lett./1985, 156B Shifman M.A.,-Vainshtein A.I. Preprint ITEP~78,« Affleck I.,Line M. n Seiberg N e Nuol.Phys,,1984iB2 J. t 493} 1985.B » Aaiati D»,fiossi G./Veneziano G,- Hucl.Phys., 1985,3242/1 e 5. Cordes S.P.,Dine M.-Nucl.Phya > B273.5e2 <, 6. Witten E.-Nucl.Phy3.,J1982,'B202, Clark T.E./Pignet 0.,Sibold K. Hucl.Phys.,1979>B159.*1. Konishi K.-Phys.Lett..t B/439o 8. Luna D./Richardson R.W. Dtike Math.Journ.,^979, : 4J5»487, 9. Buocella P.,'Derendinger J.-P.,Savoy C,Ai/Perrara S, : - Pays.bett.51982,1151,375. Gatto R./Sartori G r Phys.Lett.,1982,»riOT,>79j1985;! l5j2i Proceesi g^ Phy3,Lett».1985»'161B/117» Girardi. Phya,Lett«;И B O.Elashvili A.G. Punk.Anal. i Ego Priloz./I972,'6,< Frettndenthal H» Oktaven, ; Ausnahnugi > uppen und Oktaven Geometrle. Utrecht,>1951» Jacobson N. Exceptional Lie Algebras.-Maroel,Dekkeri* 1971, No.4. 12«0lshanetsky M.A.,Rogov V-B.K"-^eor.Mat.Fiz..И967^72.До. 1. А.Ю.Морозов, М.А.Ольшанецкий, ЫЛ.Шяфтн Конденсат глгоино в супереизилвтричной глшдинамике. Работа поступила в ОНТИ Подписано к печати TI5972 Формат 60x90 I/I6 Офсетн.печ. Усл.-печ.л.1,75. Уч.-изд.л.1,3. Тираж 290 экз. Заказ 105 Индекс 3624 Цена 19 коп.

32 ИНДЕКС 3624 М.,ПРЕПРИНТ ИТЭФ, 1987, 105, с.1-29