Spontaneous breakdown of internal symmetry in internal symmetry ⊗ supersymmetry

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  • PHYSICS LETTERS 31 March 1975 Volume 56B, number 1



    L. O'RAIFEARTAIGH* Dublin Institute for Advanced Studies, Dublin 4, Ireland

    Received 15 November 1974

    It is shown that the addition of supersymmetry to a (real irreducible, self-coupling) internal symmetry produces a Goldstone potential for the spontaneous breakdown of the internal symmetry along the (discrete) canonical direc- tions, the supersymmetry remaining intact.

    In this note we wish to show that supersymmetry [1-3] has the following property: Let

    F) = + + lf00) + ... - (1 )

    be a set of chiral superfields [4] belonging to a real, irreducible, self-coupling representation* 1 of a compact semi- simple internal symmetry group and in self-interaction via the most general, renormalizable, parity-conserving Lagrangian which is invariant with respect to internal and supersymmetry, namely, the Wess-Zumino [2] ,u type Lagrangian

    .~ highest t _ep~a_ m a ~ a a g a ~ I = weight of [ +2-- (~+~+ + ~-~- ) +-~ d~'r(~+cI~+4~+ + cba ~ q~L) j (2)

    =a.A'*a.A ~ -a.Baa.Bc~-~aatp ,~ +(F~F u +ao~G~)

    + m (FaAo~ + G,,,B a - ~a qs.) + gdut~. Y .[Fa(AoA~, = Bt~B.), ) + 2 Gcr4B.y + ~J~(At3 - 75B3) ~ 1, (3)

    where A + iB = .v/2A +, F - + iG = v~-F~, and let

    _~.. _ *_(F 2 + G2~ - *-m2(A 2 mgd.tpyA.(AoA.r+ BaB.y ) -2~ ~ ~- '~=0-2 ~ ~+B2) + (4 )

    + g2 d,.g.rda#o ((At~A.r _ BgB~) (AuA o - B#Bo) + 4A3AvB~Ba},

    be the potential for this Lagrangian obtained in the usual way by eliminating the dummy fields F and G and omitt- ing the kinetic and fermion terms.

    Then the parity conserving (B a = 0) minima of this potential are at the points A s = - (m/g) f~ where f~ are the solutions of the fixed point equation*3

    d a, la l , = I . ,

    for the bilinear map f~ ~f~ = d~faf~ of the representation onto itself, and only at these points. A shift to one of these minima breaks the internal symmetry yielding a one-parameter mass formula

    m,, a = m(6 ~ - 2d~o~f~) , (6)

    * Work supported in part by the U.S. National Science Foundation under Grant NSF GP 40354X. ,1 We are considering here the direct product of internal and supersymmetry. See also refs. [4, 5]. ,2 We shall refer to the coefficients of the powers of 0 in the expansion of a superfield as the weights of the superfield. ,3 This type of equation, considered as a starting point for symmetry breaking has been studied by others [6].


  • Volume 56B, number 1 PHYSICS LETTERS 31 March 1975

    but leaves the supersymmetry intact. Thus, paradoxically, the addition of supersymmetry to internal symmetry results in the breakdown of the inter-

    nal symmetry. The above result generalizes results found earlier for the case of no internal symmetry in refs. [7, 8] and for the case of SU(2) SU(2) in ref. [4]. Note that the potential in (4) is non-negative and that it is much more restrictive than that allowed by internal symmetry alone. (For example, a potential of the form -p2(Aw4,,) + o2(ha.Aa)2 which is group-invariant, and would allow breaking in any arbitrary direction, is not allowed). For SU(3) there are actually three solutions of (5), namely, the directions orthogonal to U-, V- and/-spin respec- tively, and if we choose the Y-direction, the one parameter Gell-Mann-Okubo mass formula reads M() = 0, M(1)2/M(O) 2 = 9. For SU(2) SU(2) a minimum which has been found [8] corre:-,onds to the solution f a = 6in of (5) with da~ / = eiikeabc/3! and yields the mass formula M(1) = 0, M(2)2/M(O/~ = ~'. In both these examples the argument in M refers to isospin, and we note the existence of massless Goldstone particles.

    It should be noted, of course, that since supersymmetry is not broken by the above mechanism, the usual pro- blems (strict equality of the masses etc.) persist after breaking. It should also be noted that since the mechanism (at least in the above simple form) requires a D-coupling, the only simple compact groups to which it can apply [9] for the adjoint representation are SU (N) for N/> 3. However it also applies to the adjoint adjoint representations of semi-simple groups of the form G G where G is any simple compact group since the direct product of the anti- symmetric structure constants forms a symmetric D-coupling.

    To establish the result stated let us suppose that the potential (4) has a minimum at Aa = %. Then defining t - - " t " A s - A a e a we return to the Lagrangian A? in the form (3), that is, before the elimination of the dummy fields

    F and G, and note that when expressed in terms of A~ this form becomes

    m "/2 = "/2 kinetic + "/~fermion +kaFa +~--(FaA'3 + GaB#) g--g-"~ daf~,{Fa(A'~A'7 - B3B 7) + 2Ga'B3A'3,} , (7)

    z 3X/3


    ~,, = me~, + gd,,~.le~e, r , m,~a = mS,,a + 2gda~/e. r .

    The Euler-Lagrange equations for the dummy fields in (7) are clearly

    ' + d " " " F a = ha+ Xam~A a g otoT[A3A 7 -- Bt~B~t )


    (9) Gu= ma#B ~ + 2gd~,yA'oBT ,

    i and hence the expansion of the potential up to second order in the fieldsA,, and Ba is easily seen to be

    ' _ *~2+ , + + , , V2(ho ,not ) = 2 ,,ot ma#~,aAfj [(m2)a# gdaf~/X~]A'w4f + [(m2)aa - gd,,f~hT]BaB # (10)

    The results now hinges on the following lemma. Lemma: The necessary and sufficient condition for the potential (10) to have a minimum at A',, = B a = 0 is X~.=0. Proof: It is clear from (10) that the necessary and sufficient condition is

    maa?~a=O, fim2)aa>~ ++-gdaa.rX 7 , (11)

    where the inequality is to be understood in the sense of matrices. It is clear also from (11) that ~'a = 0 is a sufficient condition. To show that X~ = 0 is necessary we first take the expectation value of the inequality with respect to ;~. We obtain

    0 >>- +gda#q, XaXak. r =o daasXaXOh= O. (12)

    Next we let/a a be any vector orthogonal to ),,, and with the same norm as X a and define u a = ;% cos ~ +/a a sin ~. Then taking the vacuum expectation value of the inequality with respect to v,. we obtain

    ( m 2),~ ~a ~# sin2~ i> +-gd,,3.r~, r {I.tal.t~ sin2~ + 2/.t,~ ~'3 sin ~ cos ~}. (13 )


  • Volume 56B, number 1 PHYSICS LETTERS 31 March 1975

    But from (13) we see that ifd~,a~gaX#h ~ is not zero we can obtain a contradiction by taking ~ sufficiently small. Hence (13) implies that

    dat~.r/aa ht~ h ~ = 0 . (14)

    Combining (12) and (14) we have finally

    dag~,Xah~, = 0 . (15)

    But now using the definition (8) of maa in the equality in (11) we have

    mha+ 2ga#.rh~e ~ = 0, (16)

    and taking the inner product of this equation with h a and using (15) we obtain

    mh~,ha = 0 =~ X,, = 0 . (17)

    This establishes the lemma. The result stated above now follows almost immediately. First using the lemma and the definition (8) of h a

    P we see that thenecessary and sufficient condition for a potential minimum at A a = B a 0 or equivalently A,, = e a, B,~ = 0 is

    me a + gda~/e~e ~ = 0 , (18)

    and setting e a = -(rn/g)f~, in this result we obtain (5) which is the main part of our result. The mass-formula fol- lows by making the same substitution e a = - (m/g)f~, in (8). Finally we see that the shift to the potential minimum at A a = - (m/g) f a does not disturb the supersymmetry since the shifted Lagrangian .6? in (7) is still supersymmetric First for any shift A a ~ A 'a = A,~ + e a, 27 is just

    1 a (~a - - a /~__ higheStweight of -~ ~ - +~/~ (~+ + ~a_) + (~cI~+ + cI, a & ) + usual interaction (19)

    which is manifestly supersymmetric at the formal level. Secondly the shifts we are considering are by definition to a potential minimum, so the vacuum is stable. Thirdly it may be noted from (9) that the condition h,~ = 0 for a minimum is just the condition for the dummy field F a to have zero vacuum expectation value, and this, inturn, is the condition for the fermions and bosons to have the same masses. Finally, since V = 0 at this minimum, and V ~> 0 everywhere, any other minima that may occur, namely that at the origin, or parity non-conserving minima, will not have a lower value of V. All the above results are established, of course, only in the tree approximation.

    Note that for our result we have had to use the potential only up to second order in the fields. This is because in (8) we have absorbed the shift in the field in the coupling constants h a and maa before eliminating F and G. Such an absorption is possible because .~ in (8) satisfies the identity

    .12 (A a + ea, m~, ~a) = Z? (Aa, ma~ + 2g da~e~, h a + m a~e ~ + g da~e~en, ) . (20)

    This identity holds for both the Lagrangian and the potential seperately, as one can easily see by inspection before F and G are eliminated (it is not so obvious afterwards). However, the fermion part of the Lagrangian, re- presenting the full supersymmetry, is necessary to maintain the identity beyond the tree approximation even for the potential alone. Note that the identity (20) not only simplifies the proof that h a = 0 but shows immediately why the supersymmetry is preserved, namely by the absorption of the shift in A a in the constants.

    For the original WZ-Lagrangian the identity (20) reads

    Z?(A + e, rn, h) = Z?(A, m + 2ge, h + me + ge2) , (21)

    and its existence has been noted and exploited in ref. [7], though in somewhat less explicit form. One direct con- sequence of (21) for example is


  • Volume 56B, number 1 PHYSICS LETTERS 31 March 1975

    .67(A, m, O) = .12(A + m/2g, O, -m2/4g) , (22)

    which shows that the WZ-Lagrangian o f mass m is the potential minimum of a WZ-Lagrangian o f mass zero and 'scale' parameter - m 2/4g.

    Identities of the type (20)-(22) are a general feature of supersymmetry. This we can see by starting with the trivial observation that i fA is the lowest weight (coefficient of unity) of a superfield av(A, ~k .... ) (chiral or other- wise) then by definition

    (A+e,~ .... ) =q~(A,~ .... )+e . (23)

    Hence ifP(q~) is a polynomial in cb

    p(c~(A + e)) = P(c~(A ) + e) = P'(~b(A)) , (24)

    where P'(q~) is a polynomial of the same degree but different coefficients. Furthermore (24)holds even ifp(qb) contains other superfields, and holds for the coefficient of each power of 0 in the expansion of P(&) separately. Since the Lagrangians are just the coefficients of the highest powers in such expansions, we therefore have in ge- neral

    ~(a +e) = ~'(A), (25)

    where ~ ' is a Lagrangian of similiar form but different coefficients.

    The author would like to thank the organisers of the Aspen Center for Physics and the Strobl Workshop in Weak Interactions for their hospitality during the period of this work. He should also like to thank J - L . Gervais, Z. Horvath, H. Stremnitzer and B. Zumino for discussions on some technical points and A. Jaffe and A. Pais for general ac~vice and encouragement.


    [1] J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39. [2] J. Wess and B. Zumino, Phys. Letters 49B (1974) 52. [3] B. Zumino, Prec. XVIIth Intern. High energy conference, London, 1974. [4] A. Salam and J. Strathdee, ICTP Trieste preprint IC/74/42. [5] R. Frith and J. Jenkins, Durham Univ., preprint 1974. [6] R. Brout, Nuovo Cim. 47A (1967) 932;

    L. Michel and L. Radicati, Prec. 1968 Coral Gables Conf.; A. Pals and N. Cabbibo, 1968 Nobel Symposium (Svartholm, Almqvist and Wiksell Stockholm 1968). See also references therein.

    [7] J. Iliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310. [8] A. Salam and J. Strathdee, Phys. Letters B49 (1974) 465. [9] G. Racah, Rend. Lintel 8 (1950) 108;

    B. Gruber and L. O'Raifeartaigh, J. Math. Phys. 5 (1964) 1796.



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