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Volume 135B, number 5,6 PHYSICS LETTERS 16 February 1984

SUPERPOTENTIAL SYMMETRIES AND PSEUDO NAMBU-GOLDSTONE SUPERMULTIPLETS

T. KUGO 1 and I. OJIMA 2

Max-Planck-lnstiut far Physik und Astrophysik, Werner-Heisenberg-Institut fiir Physik, Munich, Fed. Rep. Germany

and

T. YANAGIDA

Department of Physics, College of General Education, Tohoku University, Sendai 980, Japan

Received 1 November 1983

It is pointed out that the (bosonic) symmetries of the superpotential term is an important notion in supersymmetric theories. Nambu-Goldstone like theorems are proved for the spontaneous breaking of the superpotential symmetries, under the assumption that supersymmetry itself remains unbroken.

It is well known that quite often in supersymmetric theories many unexpected pseudo Nambu-Goldstone particles appear when internal symmetries break down spontaneously [ 1 ]. The origin of such phenomena, however, has never been understood clearly.

We show in this letter that the notion of symmetries ofsuperpotential term has much importance in this context: The spontaneous breaking of superpotential symmetries explains the appearance of such unexpected massless particles in almost all the cases. Indeed the superpotential term has necessarily a much larger symmetry than the full lagrangian since the former is a function of chiral multiplets ¢ alone while the latter contains anti-chiral multiplets Cas well as ¢. So, for instance, when the lagrangian is invariant under a real Lie group GO, the superpotential term is necessarily invariant under the complex form G~ of G O at least. This has been known probably to many authors [2 -5 ] . The purpose of this note is to prove that N a m b u - Goldstone like theorems hold for such superpotential symmetries. Actually, Lee and Sharatchandra [4] and Lerche [5] have mentioned such a type of theorem for the first time correctly referring to the super-potential

1 On leave of absence from Dept. of Physics, Kyoto Universi- ty, Kyoto 606, Japan.

2 On leave of absence from Research Institute for Mathematic- al Sciences, Kyoto University, Kyoto 606, Japan.

symmetries. Unfortunately, since they did not recog- nize sufficiently the importance of the notion of superpotential symmetry, however, their "theorem" was not so clearly written and further was restricted to the simple systems consisting of chiral supermultiplets alone. Therefore we here generalize their result to arbitrary systems with global (N = 1) supersymmetry and present the theorems in a clear and complete form.

Before going into the most general cases, we first concentrate on simpler systems consisting of chiral multiplets ~i(x, O) alone. The general form of the lagrangian for such a system is given by ,1

£= f d 2 O d 2 O f ( ~ , ¢ ) + ( f d 2 O g ( ¢ ) + h . c . ) . (1)

We assume that the superpotential g(¢) is invariant under transformations of a symmetry group G(C):

(e)~i ~ [~)i, 6A XA ] = eA A/A (¢ ) , (2)

8 (e )g(¢) = e A (~g/~¢i ) ~A (~) = 0 . (3)

Here X A's are the generators of G(C) and AA(¢) take the simple forms

At" A (4) = ( TA)/¢], TA: repr. matrix of X A , (4)

+1 For notations and conventions we follow the book of Wess and Bagger [3.]

402 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


when X A -transformations are linear but may be non- linear functions of q) in general. Notice that the symmetry group G(C) of superpotential is necessarily a o ) mp lex group as mentioned above; i.e., the transformation parameters e A are complex, e A @ C. Indeed, if the invariance eq. (3) holds for real eA, it does also for complex CA, since (Og/~q)i) AA(q)) = 0 follow from the fl)rmer. We use the notation G(C) to emphasize this point.

We confine ourselves to the cases where the supersymmetry is not broken spontaneously, which in particular implies that the vacuum expectation value (VEV) of the superfields q)i

%(x , 0))= v;, (5)

is 0-independent (and also x-independent by the trans- lational invariance). We assume that the system (1) is non-singular and has positive metric, at least on the vacuum realizing (5)

det f / ¢ 0 , (*i)*f/ ' .P/>~ O. (6)

Here and hereafter we use the abbreviations such as

f i -- f i ( v ' v*) = 0f(q), ~)/Sq)il(~i=oi,~i:v.i ,

f / E fi](O, O*) = ~}2f(q), q))/~¢tOq)ji¢i=vi,~i=o,i ,

etc. (7)

The generators X A of G(C) that leave the vacuum invariant, i.e.,

[q)i, XA]lea:o = A/A(°) = 0 , (8)

define the unbroken symmetry subgroup H(C) of G(C), and the other generators corresponding to G(C)/H(C) change the VEV

[¢i, xA ] [~=o = AA(°) :/: O, (9)

and are called broken generators.

Theorem 1. Consider the system ( I ) satisfying (6) and assume that the supersymmetry remains unbroken. If the vacuum breaks the superpotential symmetry G(C) down to H(C), there must appear massless (pseudo Nambu-Goldstone) supermultiplets, in one to one correspondence to the broken generators X A . Namely, the number of the appearing massless supermultiplets is the same as the number of broken generators, dim(G(C)/H(C)) = dim G(O) - dim H(C).

Proof. The equations of motion read as

- ~DB~f/oq)i + og/oq)i + o . (lO)

By multiplying Aft(q)) to this and using the symmetry (3) of superpotential g(q)), we immediately obtain from (lO)

0 = (DD~f/Oq)i)A A (q))

= [(02f/Oq)ioCk)DD~ k

+ (O3f/Oq)iO~kOCt)(D~k)(D&~l)] A/A (q)). (11)

Now recall that the VEV v i = (q)i(x, 0)) is independent o fx and 0 by the assumption of unbroken supersymmetry; i.e.,

D , v i = D & o i = O . (12)

Therefore, by the substitution of

q)i(x, O) = q)~(x, O) + o i , (13)

into (11), eqs. (11) reduce to the following simple form in the linear approximation in q)~:

.fj A A (V) DD~'] = 0 . (14)

These imply that the fields

CA)(x, 0) - (A~(o))*~/q)~(x, 0) -----(~xA(o), ¢ ' ) , (IS)

are massless (chiral) supermultiplets at the tree level unless they vanish identically. Since the "metric" f / i s non-singular, eqs. (15) clearly show that for each one of broken generators X A (for which A A ( v ) ¢ 0) there appear a massless supermultiplet q)(A), say (pseudo-) Nambu-Goldstone supermultiplet. So the number of independent Nambu-Goldstone supermultiplets q)(A) is identical with the number of broken generators, dim G(C) - dim H(C). The eqs. (14) say the masslessness of q)(A) only at the tree level approximation. But, owing to the assumption of unbroken supersymmetry, their masslessness is guaranteed by the nonrenormaliza- tion theorem [6] at any order of perturbation theory. q.e.d.

Remarks . (i) In this theorem dim G(C) (dimH(C)) means dimcG(12)(dimcH(C)) , i.e., the dimension of G(C)(H(C)) regarded as a complex Lie group. This should be so because the chiral Nambu-Goldstone superfields q)(A) are complex and hence the number of independent modes of(15) has to be counted by the number of linear independent vectors among AA(o)

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regarded as complex vectors. (ii) The Nambu-Golds tone superfields ~b (A) contain 2 × [dim G(C) - dim H(C)] real bosons as their first components. Clearly these real bosons are Nambu-Golds tone bosons directly associated with the broken generators X A and iX A (i: imaginary unit) of a real Lie group (G(C)) R which is equal to G(C) as a set but regarded as a group with real coordinates. The more precise correspondence of real bosons Re q~(A)(x, 0 = 0) and Im ~)(A)(x, 0 = 0) and broken generators X A and iX A would be easily seen if we examine the field-independent terms in the transformation laws of q~(A)

6 (e)(9 (A) = (AA (o), eBAB((~'+ o)) (16)

and separate their real and imaginary parts. Theorem 1, in its appearance, might be regarded as

a supersymmetric extension of Weinberg's pseudo- Goldstone theorem [7] in which the symmetry of the potential term V(¢) is relevant. One should, however, notice that the superpotential symmetry neither implies nor is implied by the potential symmetry in general. Indeed the relation between the potential V(~o) and superpotential g(~b) is not so simple:

_ Og(~0)* [~g(~0)'~, g ( t p ) - ( - ~ i ) [f-l(~0, tp*)] I. (17)

where ~0(x) = ~(x, 0 = 0) and ( f -1)~ is the inverse matrix of b2f(~0, ~0*)/O(~i ~*/" Fur the / the assumption that the supersymmetry is unbroken is essential in the present theorem.

This theorem 1 is in essence identical with the one which was given by Lerche in ref. [5]. A similar state- ment was also made by Lee and Sharatchandra [4]. But here we have given it in a rather general form and made much clearer the proof and the conditions under which it holds.

Next we proceed to the general cases in which the gauge-vector multiplets Va(x, O) are also present. The lagrangian now takes the following form in general ,2.

g = fd2Od2Oy(¢, ~eg v)

+ ( f d20~hab(¢)e~ aw~lV~ + h.c. )

+ ( f d2Og(¢~) + h.c.), (18)

where ( V ) / = Va(Ta ) / i s the gauge vector multiplets of a gauge group Gloc. with Ta's being representation matrices of the generators X a of Gloc., and Wa

a - W~T a = -(4g~-lDfi(e-g~Dc~eg'V ) is the field- strength spinor-superfield ,3. The Gloc.-gauge invariance of the lagrangian requires that the functions f ( ¢ , ~) (real) and g(40 satisfy the conditions

(Of/O~i)(Ta)/(~i - ~ i (ra) / (Of /~i ) = 0 , (19)

(Og/Oc~i)(Ta)Jd~ / = 0 , (20)

and that hab(¢ ) transforms as a symmetric product of two adjoint representations of Gloc..

The symmetry group G(C) of the superpotential g(~) is defined by (2) and (3) also here and necessarily contains Gloc. as a (linearly realized) subgroup as is seen from (20): G(C) D Gloc.. [Here one should notice that Gloc. is in fact a complex Liegroup since the gauge transformation parameters in supersymmetric theory are chiral superfields which are complex.] We denote the generators of G(C) generally by X A with capital superfix A and the generators of Gloc. in particular by X a with lower case a. We use a symbolic notation like X A E G(X A ~ G) to indicate that the generator X A is (is not) contained in the Lie algebra of G.

Theorem 2. Consider the general system (18) with non-singular and positive matr icesf / (v , v*) and Re hab(O), and assume that the supersymmetry remains unbroken. I f the superpotential symmetry G(C) is spontaneously broken down to H(C), a massless Nambu-Golds tone supermultiplet appears corresponding to each one of broken generators X A c G(C) such that

4:2 We have assumed in (18) that the field-strength superfields Wg appear only quadratically. But the inclusion of their higher order terms give no changes to our discussion below.

4:3 The gauge group Gloc. is not necessarily simple. Then the gauge coupling constant g"is a diagonal matrix being constant only within each simple (or abelian) subgroup of Gloc. But we write it for notational simplicity as if it were a constant over the whole group Gloc.

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[ePi, XAl[•= o =AA(o)~0 a n d X A ~ G l o e . . (21)

So the number of the independent Nambu-Golds tone multiplets is given by

dim G ( C ) - dim H ( C ) - [dim Gloc . - dim(Glo c. ~ n ( e ) ) ] .

(22) Here dim Glue. - dim(Glue, f) H(C)) is the number of "would-be" Nambu-Golds tone multiplets which are actually eaten by the gauge-vector multiplets to form massive vector supermultiplets.

Proof. This theorem is just the naturally expected one is the previous theorem and the Higgs phenomenon are known. So the proof goes quite similarly to the usual proof of the Higgs theorem [8].

First, in order to obtain the linearized equations of motion, let us expand the lagrangian (18) up to qua- dratic terms in dp~ = ~i - vi and va :

£ = f d 2 0 d 2 0 [(q~', ¢ ' ) +g(Vu, ¢;') +g(ck', Vv)

+ ~ 2 ( V o , Vo) + }(Re hab ) VaDD2DV b ] (23)

+ (superpotential term) + total derivative t e rms .

Here we have used the property (19) o f f and the notations hab = hab (o) and

((p, o~) - ~if / tI ,} , (Vo)i = Va(Ta)/o i " (24)

The linearized equations of motion following from (23) are:

- ~DDf/(~b' + ~,Vo)/

+ (3g((J + o)/OdPi)linear terms in ¢' = 0 , (25)

¼(Re hab)DD2DVb +~,(Tao, ~')

+g,(ck', Tao) +g2(Tao, rbo) V b = 0 . (26)

Just as before, eq. (25) and the superpotential symmetry (3) lead to

DD~b (A) +~(AA(o), Tav)DDV a = 0 , (27)

where ~b(A)'s are the superfields defined in (15) and, previously in the absence of gauge coupling oY = 0, de- noted the (dim G(C) - dim H(C)) Nambu-Golds tone modes.

In the present case of~4= 0, eq. (27) indicates that

those modes ¢(A) mix with the vector modes V a. As is seen further from (26), it is only the modes

ck (a) = (Tar, ok') = ( AA=a(v), ok') (28)

among 4~(A)'s that actually mix with V a. These fields represent the (dim G l o c . - dim(H fh Gloc.)) independent modes corresponding to the broken generators X a contained in the gauge group Gloc.. They can be gauged away. We can simply impose ,4

(Tar, q~') = 0 , (29)

as gauge fixing conditions. Indeed, since (29) has the same form as the "unitary gauge" conditions in the usual (non-supersymmetric) Higgs case [8], the usual reasoning is already sufficient to show that (29) can be imposed. [Or equivalently, one could show that those chiral modes q~(a) of (28) are absorbed into the vector multiplets V a by the gauge transformation with chiral parameters dependent on ~(a)'s.] In the gauge (29), eq. (26) reduces to ,s

~-DD2DV a + 2(M2) ab V b = 0 , (30a)

(M2) ab = ~g2(Tao, Teo)(Re h)e 1 . (30b)

These eqs. (30) show that actually (dim Gloc. - dim(H A Gloc.)) vector multiplets V a corresponding to broken generators X a E Gloc. with Tau 4~ 0 acquire non-zero masses in coincidence with the "absorpt ion" of the chiral modes q~(a) in (28), while the other V a corresponding to the unbroken generators X a E (Gloc.

H) remain massless. On the other hand, the other chiral modes q~(A)

of (15) corresponding to broken generators X A not contained in Gloc. still represent physical N a m b u - Goldstone supermultiplets as before. In fact, since D2D = 0, it follows from eqs. (30) in this gauge that

(Tao, Tbu)DDV b = 0 ,

or equivalently, owing to the positivity (6) o f f / ,

+4 This "supersymmetric unitary gauge" in fact fixes the gauge only in the sector of broken generators. One can still impose, for instance, DD V a = 0 for the indices a of unbroken generators.

,S Notice that, when the coefficient hab of the vector multiplet kinetic term takes the canonical value ~ab, the mass matrix M 2 of (30b) actually coincides with the well-known gauge boson mass matrix of the usual nonsupersymmetric Higgs case [8].

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(Tav)i DD V a = 0 . (31)

So eqs. (27) reduce simply to the massless equations

DD¢(A) = 0 , (32)

as is desired, q.e.d.

Remarks . (i) Theorem 2 as well as theorem 1 are valid even for such "trivial cases" that the superpotential g(¢) is identically zero. When g(O) = O, any large group can be said to be the symmetry group G(C) of the superpotential term. But, as far as linear symmetry transformations are concerned, it is sufficient to take G(C) = GL(N; C) for the system containingN chiral supermultiplets ¢i (i = 1 ,2 . . . . . N ) since GL(N; C) is the largest group to be realized linearly and faithful- ly on ¢i.

(i) In this theorem 2, we have assumed the existence of non-singular kinetic term for the gauge vector multiplets V a. But the number counting of the N a m b u - Goldstone modes in theorem 2 itself remains true with- out such an assumption (e.g., even for the cases with no kinetic term of va) .

To conclude this letter, we discuss two simple examples which are probably instructive. We consider hereafter only symmetries of linear transformations.

As an example of theorem 1, we consider the lagrangian

£ = f d2Od20(VI'i'I'i + ~i(Oli + qg~gP2i + XX)

+ ( f d20(g~i¢liX + m~idP2i + #(X-V)2)+h.c. ) (33)

where the index i runs over 1 -n . This is the model dis- cussed first by Buchmialler et al. [9] and later by other authors [4,5] for the case o f n = 2. The symmetry of the lagrangian is G O = SU(n) X U(1), under which the superfields 4 , q~l,2 and X transforms as (n*, -1 ) , (n, +1) and (1, 0). But the symmetry of the superpotential is G(C) = GL(n;C). Under this GL(n; C), xI ti and (q~l,2)i are contravariant and covariant n-vectors, respectively, and X is scalar.

The vacuum candidates at the tree level correspond to the minima of the potential V(¢) (17). As is easily seen, a supersymmetric minimum is realized at

(~oi} = 0 , (¢1i) = mC/ , (~b2i> = - g o C i , (X) = o , (34a)

where C/is an arbitrary constant n-vector but can be generally brought into the form

C i = ~ C , (34b)

by the SU(n) transformation belonging to the lagrangian symmetry G O The n 2 generators X(', (1 ~< i',

• 1

j --~ n) o f GL(n ; C) have a representation by n × n matrices (Ti.,')J = 8!'8./, . which act on the vectors ~)ai(a I t . l 1 = l o r 2 ) a n d 4 1 a s

• F . i •

8 = ,! q%,

8(e)xI *i= -* / (T i / , " ) : , f = - b / c : . (35)

For the vacuum realizing the VEV (34), the superpotential symmetry GL(n; C) is broken clown to the subgroup H(C) consisting of the n(n - 1) generators X/t with/" :/= n (i.e., 1 <~i<~n, 1 <~/<~n - 1), and the other n generators X i are broken ones [4,5]. Corre-

n

spondingly, the n Nambu-Goldstone modes, say q)(i,n), are given by the general formula (15) and now read explicitly

~)(i,n) = c * (m * ~l i _ g * o * ~2i) . (36)

They transform by (35) as

8(e)¢(i ,n) = 1cl2(1ml2 + Igvl2)e n

+ (terms linear in ¢1,2) (37)

and so the inhomogeneous term shows that Re ¢(~ n)(O = 0) and Im ¢(i,n)(o = 0) are just the Nambu-Goldstone bosons corresponding to broken generators X / and iX respectively. But we notice that the same fields Re •(i,n)(o = 0) and Im ~p(~n)(o = 0) for i 4: n can also be regarded as Nambu-Goldstone bosons of the generators

E t i ' n ) = X / - X / n , i = 1 , 2 . . . . . n - 1, (38a)

g~ i'n) = i.Xin + i X f , i = 1, 2 . . . . . n - 1 , (38b)

instead o f X / and iX/, respectively, since in the choice o f broken generators there always exists an arbitrariness o f adding unbroken generators like X n terms here in (38). The generators E~I'2 n) in (38), which are anti- hermitian, are those of SU(n) in G O and hence the 2(n - 1) bosons of both real and imaginary parts of ¢(i,n)(o = 0)(1 <~ i ~< n - 1) are actually true N a m b u -

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Goldstone bosons corresponding to the breaking of the lagrangian symmetry SU(n) down to SU(n - 1). On the other hand, for the diagonal part i = n, iX n is a U(1) generator of G O but X n (or equivalently the trace part 2n= 1 X[) is a generator of scale transformation not belonging to G 0. So Im ¢(n,n)(o = 0) is a true N a m b u - Goldstone boson but Re (a(n,n)(o = 0) is a pseudo Nambu-Golds tone boson associated with a breakdown of scale transformation symmetry of superpotential

14,5]. As an example of theorem 2, we discuss the follow-

ing model with zero superpotential g(~b) = 0:

2 = f d Z O d 2 b [ ~ i ( e g V ) / ¢ ~ - g-'V/l , (39)

where V ( n X n matrix) is a gauge vector multiplet of gauge group Gloc. = U(n) ,6 and q~a(/= 1 -n , a = l - m ) belongs to a representation n of Gloc. and m* of a global U(m). The lagrangian symmetry is Gloc. X global U(m). We assume n < m. This model (39) was construct- ed by Aoyama [10] as a straightforward generalization of the supersymmetric CP n - 1 model considered by D'Adda, Di Vecchia and Ltischer [ 11 ] in two-dimensional space-t ime. As was remarked above, it is sufficient to take G(C) = GL(nm;C) as the superpotential symmetry since ~b a has nm components. If we define the generators Y / a n d Z a b of GL(n; C) and GL(m; C) acting on the indices i and a of Ct .a, respectively, in a similar way to the Xj t in the preceding example, then, the (nm) 2 generators of GL(nm; C) may be given by Yj ® zab :

[e) a, Yj; ® Zab, ' ] =(8f@)~bS.b (8b 'sa , ) = 61'<3a,@ ' .(40)

The generators Za b may be identified with 1 @Za b = Zf= 1 Y[ @ Zba of the GL(m ', C) subgroup of GL(nm; C).

As a gauge fixing of Gloc., we can take the following n 2 chiral conditions

oa(x, 0 ) = 5 a for 1 ~ a , i < . n , (41)

and so the "vacuum expectation value" becomes

(~ba(x, 0))= 6 a . (42)

Under this VEV, the superpotential symmetry GL(nm;

,6 More precisely, we should write Gloc. = GL(n; C) since the transformation parameter is chiral superfield and hence complex.

C) breaks down to a n m ( n m - 1) dimensional subgroup H(C) spanned by the generators

Y( ® Z b - z c ~ i ~ b. (43) - a -c~1 " ]

The nm broken generators may be given by Za b with b ~< n and i f ~ = 0 the original nm fields q~a themselves would be the corresponding Nambu-Goldstone superfields since [~a, Z[] = 6951. In place of the first n 2 generators Za b with 1 ~< a, b ~< n, however, we can equivalently take

Yi=l ZII +6g(Yl(®zb-_ z~c c8 jb ) , (44) since the second terms of the RHS are unbroken generators ell(C). Therefore the first n 2 fields ¢ / co r re - spond to local symmetry generators Y! and hence are

1 unphysical. They were in fact gauged away as was seen in (41). The remaining n(m - n) superfields ~a with n + 1 <~ a ~< m are physical Nambu-Goldstone modes. Further, as is easily seen in the same way as in the previous example, all of the 2 × n (m - n) real bosons of real and imaginary parts of ~a(0 = 0) are true Nambu-Goldstone bosons associated with the breaking of the lagrangian symmetry U(m) down to U(m - n) X U(n).

An interesting point of this model [10] is that it provides us with a concrete way to construct the invariant lagrangian for the supersymmetric non-linear realization corresponding to the coset U(m)/U(n) X U(m - n) (which is a K~hler manifold). After the gauge fixing (41), the remaining fields O7(n + 1 ~< a ~< m) just give the complex coordinates of this manifold. The invariant lagrangian £ = f d20d20 K(O, ~) or the KZlller potential K ~ - ) is simply obtained from (39) by eliminating the gauge vector multiplets by the use of the equation of motion 8£/6 V = 0. [This is possible because V has no kinetic term in (39)]. This yields a result K(~b, ~) = tr In M with M{: : (8i. + ]~m=n+l a - j ¢i Cd ) as the K~ihler potential [12,13]. Prob- ably similar methods would exist to obtain the K~ihler coordinates and potential explicitly also for other ho- mogeneous K~hler manifolds G/H.

The authors would like to thank D. F6rster, L. Girardello, W. Lerche and H.S. Sharatchandra for dis- cussions and especially R.D. Peccei for helpful discus- sions and encouragements.

407


Note added in proof. In some explicit examples, Kubo and Sakakibara [14] counted the numbers of Nambu-Golds tone supermultiplets in the same way as our theorems 1 and 2. We thank them for calling our at tention to their work.

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