New “electric” description of supersymmetric quantum chromodynamics

4 November 1999

Physics Letters B 466 Ž1999. 244–250

New ‘‘electric’’ description of supersymmetric quantum chromodynamics Yasuharu Honda a

a,1

, Masaki Yasue`

a,b,2

Department of Physics, Tokai UniÕersity, 1117 KitaKaname, Hiratsuka, Kanagawa 259-1292, Japan b Department of Natural Science, School of Marine Science and Technology, Tokai UniÕersity, 3-20-1 Orido, Shimizu, Shizuoka 424-8610, Japan Received 2 August 1999; accepted 21 September 1999 Editor: M. Cveticˇ

Abstract Responding to the recent claim that the origin of moduli space may be unstable in ‘‘magnetic’’ supersymmetric quantum chromodynamics ŽSQCD. with Nf F 3 Ncr2 Ž Nc ) 2. for Nf flavors and Nc colors of quarks, we explore the possibility of finding nonperturbative physics for ‘‘electric’’ SQCD. We present a recently discussed effective superpotential for ‘‘electric’’ SQCD with Nc q 2 F Nf F 3 Ncr2 Ž Nc ) 2. that generates chiral symmetry breaking with a residual nonabelian symmetry of SUŽ Nc .LqR = SUŽ Nf y Nc .L = SUŽ Nf y Nc .R . Holomorphic decoupling property is shown to be respected. For massive Nf y Nc quarks, our superpotential with instanton effect taken into account produces a consistent vacuum structure for SQCD with Nf s Nc compatible with the holomorphic decoupling. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 11.30.Pb; 11.15.Tk; 11.30.Rd; 11.38.Aw

It has been widely accepted that physics of N s 1 supersymmetric quantum chromodynamics ŽSQCD. at strong ‘‘electric’’ coupling is well described by the corresponding dynamics of SQCD at weak ‘‘magnetic’’ coupling w1,2x. This dynamical feature is referred to as Seiberg’s N s 1 duality w3x. In order to apply the N s 1 duality to the physics of SQCD, one has to adjust dynamics of ‘‘magnetic’’ quarks so that the anomaly-matching conditions w4,5x are satisfied. In SQCD with quarks carrying Nf flavors and Nc colors for Nf G Nc q 2, the N s 1 duality is

1 2

E-mail: [email protected] E-mail: [email protected]

respected as long as ‘‘magnetic’’ quarks have Nf y Nc colors w1,3x. Appropriate interactions of ‘‘magnetic’’ quarks can be derived in SQCD embedded in a softly broken N s 2 SQCD w6,7x that possesses the manifest N s 2 duality w8x. In SQCD with 3 Ncr2 - Nf 3 Nc , its phase is characterized by an interacting Coulomb phase w1x, where the N s 2 duality can be transmitted to N s 1 SQCD. On the other hand, in SQCD with Nf F 3 Ncr2, it is not clear that the N s 1 duality is supported by the similar description in terms of the N s 2 duality although it is believed that the result for 3 Ncr2 - Nf can be safely extended to apply to this case. Lately, several arguments w9–11x have been made to discuss other possibilities than the physics based

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 1 3 2 - 6

Y. Honda, M. Yasuer ` Physics Letters B 466 (1999) 244–250

on the N s 1 duality, especially for SQCD with Nf F 3 Ncr2. It is claimed in Ref. w9x that the origin of moduli space becomes unstable in ‘‘magnetic’’ SQCD and that the spontaneous breakdown of the vectorial SUŽ Nf .LqR symmetry is expected to occur. An idea of an anomalous UŽ1. symmetry, UŽ1.anom , taken as a background gauge symmetry has been employed. Their findings are essentially based on the analyses made in the slightly broken supersymmetric ŽSUSY. vacuum. On the other hand, emphasizing nonperturbative implementation of UŽ1.anom , the authors of Ref. w10x have derived a new type of an effective superpotential applicable to ‘‘electric’’ SQCD. However, the physical consequences based on their superpotential have not been clarified yet. Finally, extensive evaluation of formation of condensates has provided a signal due to spontaneous breaking of chiral symmetries although there is a question on the reliability of their dynamical gap equations w11x. These attempts suggest that, in order to make underlying property of SQCD more transparent, it is helpful to employ a composite chiral superfield composed of chiral gauge superfields w12–14x that is responsible for relevant expression for UŽ1.anom . In a recent article w16x, we have discussed what physics is suggested by SQCD with Nc q 2 F Nf F 3 Ncr2 and have found that, once SQCD triggers the formation of one condensate made of a quark-antiquark pair, the successive formation of other condensates is dynamically induced to generate spontaneous breakdown of the chiral SUŽ Nf . symmetry to SUŽ Nf y Nc . as a residual chiral nonabelian symmetry 3. The anomalies associated with original chiral symmetries are matched with those from the NambuGoldstone superfields. As in Ref. w9x, our suggested dynamics can also be made more visible by taking softly broken SQCD w17x in its supersymmetric limit. The derived effective superpotential has the common structure to the one discussed in Ref. w10x. It should be noted that the ‘‘magnetic’’ description should be selected by SQCD if SQCD favors the formation of no condensates. In this paper, we further study effects of SUSYpreserving masses. It is shown that our superpoten-

3

Another case with a chiral SUŽ Nf y Nc q1. symmetry has also been discussed.

245

tial is equipped with holomorphic decoupling property. In the case that quarks carrying flavors of SUŽ Nf y Nc . are massive, our superpotential supplemented by instanton contributions correctly reproduces consistent vacuum structure with the decoupling property. In SQCD with Nc q 2 F Nf F 3 Ncr2 Ž Nc ) 2., our superpotential w16x takes the form of S NcyN f det Ž T . f Ž Z .

½

Weff s S ln

L3 NcyN f

q Nf y Nc

5

Ž 1.

with an arbitrary function, f Ž Z ., to be determined, where L is the scale of SQCD. The composite superfields are specified by S and T : Ss

Nc

1 32p 2

Nc

WAB WBA ,

Ý

Tji s

A, Bs1

Ý

Q Ai Q jA ,

Ž 2.

As1

where chiral superfields of quarks and antiquarks are denoted by Q Ai and Q iA and gluons are by WAB with TrŽW . s 0 for i s 1 ; Nf and A, B s 1 ; Nc . The remaining field, Z, describes an effective field. Its explicit form can be given by B w i 1 i 2 PPP i Nc xTjiNNcqq11 PPP TjiNN f Bw j1 j 2 PPP j N

Ý Zs

c

i1 PPP i N f , j1 PPP j N f

f

c

x

det Ž T .

ž

'

BT

N fyN c

det Ž T .

B

/

Ž 3.

,

where B w i 1 i 2 . . . i Nc x s

1

Ý A1 . . . A N c

Bw i 1 i 2 . . . i N x s c

Nc ! 1

Ý A1 . . . A N c

Nc !

´ A 1 A 2 . . . A Nc Q Ai 11 . . . Q Ai NNc , c

´ A 1 A 2 . . . A N Q iA1 1 . . . Q iAN Nc . c

c

Ž 4. This superpotential is derived by requiring that not only it is invariant under SUŽ Nf .L = SUŽ Nf .R as well as under two additional UŽ1. symmetries but also it is equipped with the transformation property under UŽ1.anom broken by the instanton effect, namely, d L ; F mn F˜mn , where L represents the lagrangian of SQCD and F mn Ž F˜mn ; emnrs F rs . is a gluon’s field strength w12x. Note that Z is neutral under the entire chiral symmetries including UŽ1.anom and the Z-dependence of f Ž Z . cannot be determined by the symmetry principle.

Y. Honda, M. Yasuer ` Physics Letters B 466 (1999) 244–250

246

Although the origin of the moduli space, where T s B s B s 0, is allowed by Weff , the consistent SQCD must automatically show the anomaly-matching property with respect to unbroken chiral symmetries. Since the anomaly-matching property is not possessed by SQCD realized at T s B s B s 0, composite fields are expected to be dynamically reshuffled so that the anomaly-matching becomes a dynamical consequence. Usually, one accepts that SQCD is described by ‘‘magnetic’’ degrees of freedom instead of T, B and B. However, it is equally possible to occur that ‘‘electric’’ SQCD dynamically rearranges some of T, B and B to develop vacuum expectation values ŽVEV’s.. In this case, chiral symmetries are spontaneously broken and the presence of the anomalies can be ascribed to the NambuGoldstone bosons for the broken sector and to chiral fermions for the unbroken sector w5x. If the consistent SQCD with broken chiral symmetries is described by our superpotential, the anomaly-matching constraint must be automatically satisfied and is indeed shown to be satisfied by the Nambu-Goldstone superfields. In the ordinary QCD with two flavors, we know the similar situation, where QCD with massless proton and neutron theoretically allows the existence of the unbroken chiral SUŽ2.L = SUŽ2.R symmetry but real physics of QCD chooses its spontaneous breakdown into SUŽ2.Lq R w4x. However, since the SUSY-invariant theory possesses any SUSY vacua, which cannot be dynamically selected, both ‘‘magnetic’’ and ‘‘electric’’ vacua will correspond to a true vacuum of SQCD. In the classical limit, where the SQCD gauge coupling g vanishes, the behavior of Weff is readily found by applying the rescaling S ™ g 2 S and invoking the definition L ; m expŽy8p 2rŽ3 Nc y Nf . g 2 ., where m is a certain reference mass scale. The resulting Weff turns out to be WWr4, which is the tree superpotential for the gauge kinetic term. If S is integrated out w15x, one reaches the ADS-type superpotential w2x:

AD S Weff s

Ž Nf y Nc .

det Ž T . f Ž Z .

L3 NcyN f

1r Ž N fyNc .

.

Ž 5.

AD S In this case, Weff vanishes in the classical limit Ž . only if f Z s 0, where the constraint of BT N fyNc B

s det Ž T . , namely Z s 1, is satisfied. The simplest form of f Ž Z . that satisfies f Ž Z . s 0 can be given by f Ž Z . s Ž1yZ .

r

Ž r ) 0. ,

Ž 6.

where r is a free parameter. If one flavor becomes heavy, our superpotential exhibits a holomorphic decoupling property. Add a mass to the Nf flavor, then we have

½

Weff s S ln

S NcyN f det Ž T . f Ž Z .

L3 NcyN f

y mTNNf f .

q Nf y Nc

5 Ž 7.

Following a usual procedure, we divide T into T˜ with a light flavor Ž Nf y 1. = Ž Nf y 1. submatrix and TNNf f and also B and B into light flavored B˜ and B˜ and heavy flavored parts. At SUSY minimum, the off-diagonal elements of T and the heavy flavored B and B vanish and TNNf f s Srm is derived. This relation is referred to as Konishi anomaly relation w18x. Inserting this relation into Eq. Ž7., we obtain Weff

½

s S ln

S NcyN fq1 det Ž T˜ . f Ž Z˜ .

L˜ 3 NcyN fq1

5

q Nf y Nc y 1 ,

Ž 8.

˜ ˜ ˜ N fy N c y 1 Brdet Ž T˜ . from Z s where Z˜ s BT N N y N y 1 ˜ N f detŽ T˜ . and L˜ 3 N c y N fq 1 s ˜ N f f T˜ f c BrT BT Nf m L3 NcyN f . Thus, after the heavy flavor is decoupled at low energies, we are left with Eq. Ž1. with Nf y 1 flavors. We can also derive an effective superpotential for Nf s Nc y 1 by letting one flavor to be heavy in Eq. Ž1. for Nf s Nc . For Nf s Nc , at SUSY vacuum, we find that det Ž T . f Ž Z . s L 2 Nc ,

Ž 9.

which turns out to be the usual quantum constraint w1,13x of det Ž T . y BB s L2 Nc if r s 1 giving f Ž Z . s 1 y Z. The discussion goes through in the similar manner to the previous one. In this case, we find B s B s 0, leading to Z s 0, and t s Srm at SUSY minimum. As a result, Eq. Ž1. with Nf s Nc y 1 is derived if L2 Ncq1 is identified with m L2 Ncrf Ž0., where f Ž0. s 1 by Eq. Ž6.. The induced Weff is

Y. Honda, M. Yasuer ` Physics Letters B 466 (1999) 244–250

nothing but the famous ADS superpotential w2x after S is integrated out. It is thus proved that our superpotential with Eq. Ž6. exhibits the holomorphic decoupling property and provides the correct superpotential for Nc - Nf . Next, we proceed to discussing what physics is expected in SQCD especially with Nc q 2 F Nf F 3 Ncr2. It is known that keeping chiral symmetries unbroken requires the duality description using ‘‘magnetic’’ quarks. Therefore, another dynamics, if it is allowed, necessarily induces spontaneous breakdown of chiral symmetries. In our superpotential, Eq. Ž1., this dynamical feature is more visible when soft SUSY breaking effects are token into account. Although the elimination of S from Weff gives no effects on the SUSY vacuum, to evaluate soft SUSY breaking contributions can be well handled by Weff with S. Since physics very near the SUSY-invariant vacua is our main concern, all breaking masses are kept much smaller than L. The property of SQCD is then inferred from the one examined in the corresponding SUSY-broken vacuum, which is smoothly connected to the SUSY-preserving vacuum. Let us briefly review what was discussed in Ref. w16x in a slightly different manner. To see solely the SUSY breaking effect, we adopt the simplest term that is invariant under the chiral symmetries, which is given by the following mass term, Lmass , for the scalar quarks, fAi , and antiquarks, f iA : Lmass s Ý m2L < fAi < 2 q m2R < f iA < 2 . yL i, A

ž

Nf

žÝ /

< Weff ;i < 2 q GB

is1

ž

Ý

< Weff ;i < 2

isB , B

q GS < Weff ; l < 2 q Vsoft ,

/ Ž 11 .

Nf

Vsoft s Ž m2L q m2R . Ly2

which is assumed to be diagonal, E 2 KrE Tik )E Tj l s d i j d k l Gy1 with G T s G T ŽT † T ., and similarly for T † GB s GB Ž B B q B †B . and GS s GS Ž S †S .. Since the dynamics requires that some of the p acquire non-vanishing VEV’s, suppose that one of the p i Ž i s 1 ; Nf . develops a VEV, and let this be labeled by i s 1:

Ý

,

L

Ž 13 . for a s 1 ; Nc , where a B s zf Ž z .rf Ž z . and bB s z a BX with z s ²0 < Z <0:, and Nf

M 2 s m2L q m2R q GXT L2

2

Weff ;i ,

Ý

Ž 14 .

is1 Nc

X B s GT

pl

Ý Weff) ; a p

a

as1

y GB

Ý

Weff) ; x

xsB , B

pl px

.

Ž 15 .

The SUSY breaking effect is specified by Ž m 2L q m2R .

Ž M 2 r L2 . Ž p 1 2y pa 2 .

2

s1q

p1

GS Weff) ; l Ž 1y a B . q Ž M 2 r L2 . pa 2q bB X B

,

Ž 16 . which cannot be satisfied by pa/ 1 s 0. In fact, pa/ 1 s p 1 is a solution to this problem, leading to
p is1 ; Nc s LT2 ,

is1

q Ly2 Ž Ncy1. Ž m2L

2

pa

X

Ž 10 .

/

Together with the potential term arising from Weff , we find that Veff s GT

247

Ž 12 .

with the definition of Weff ;i ' E WeffrEp i , etc., where pl,i, B, B , respectively, represent the scalar components of S, Tii , B w12 PPP Nc x and Bw12 PPP Nc x. The coefficient G T comes from the Kahlar potential, K, ¨

p isNcq1 ; N f s j p is1 ; Nc ,

pl ; L3j

N fyNc

,

Ž 17 .

Y. Honda, M. Yasuer ` Physics Letters B 466 (1999) 244–250

248

in the leading order of j . Therefore, in softly broken SQCD, our superpotential indicates the breakdown of all chiral symmetries. This feature is in accord with the result of the dynamics of ordinary QCD w20x. Does the resulting SUSY-breaking vacuum structure persist in the SUSY limit? At the SUSY minimum with the suggested vacuum of
other hand, if all quarks with flavors of SUŽ Nf y Nc . are massive, we can further utilize instanton contributions w21x to prescribe vacuum structure. If our superpotential provides correct description of SQCD, both results must be consistent with each other. The instanton calculation for the gluino and Nf massless quarks and antiquarks concerns the following SUŽ Nc . – invariant amplitude:

SU Ž Nf . L = SU Ž Nf . R = U Ž 1 . V = U Ž 1 . A

where c Ž/ 0. is a coefficient to be fixed. At our SUSY minimum, the condition of E WeffrEpl s 0 together with E WeffrEp i s 0 for i s Nc q 1 ; Nf giving plrp i s m i reads

™ SU Ž Nc . LqR = SU Ž Nf y Nc . L = SU Ž Nf y Nc . R X

X

= U Ž 1. V = U Ž 1. A ,

Ž 18 .

where UŽ1.XV is associated with the number of Ž Nf y Nc .-plet superfields of SUŽ Nf y Nc . and UŽ1.XA is associated with the number of SUŽ Nc .LqR - adjoin and - singlet fermions and of scalars of the Ž Nf y Nc .-plet. The SUSY vacuum characterized by
N Ž ll . c det ž c i c j / ,

Ž 19 .

where c Ž c . is a spinor component of QŽ Q ., which can be converted into Nf

Nc

Ł pa s c L2 Nc

as1

Ł Ž m irL . ,

Nf

Nc

f Ž z. s

Ž 20 .

isNcq1

Ł Ž L2rpa . Ł Ž m irL . .

as1

Ž 21 .

isNcq1

By combining these two relations, we observe that the mass dependence in f Ž z . is completely cancelled and derive c s 1rf Ž z . giving f Ž z . / 0 instead of f Ž z . s 0 for the massless SQCD. Since f Ž z . s Ž1 y z . r , z / 1 is required and the classical constraint, corresponding to z s 1, is modified. These VEV’s are the solution to det Ž T˜ . f Ž Z˜ . s L˜ 2 Nc ,

Ž 22 .

where L˜ s L which is the quantum constraint Ž9. for Nf s Nc and which is also consistent with the successive use of the holomorphic decoupling. Therefore, our superpotential for Nc q 2 F Nf F 3 Ncr2 supplemented by the instanton contributions is shown to provide a consistent vacuum structure for SQCD with Nf s Nc compatible with the holomorphic decoupling. A comment is in order for the case with r s 1. The quantum constraint for Nf s Nc , Eq. Ž9., is rewritten as 2 Nc

det Ž T .

1y r

3 NcyN f

Nf Ł isNcq1 mi,

r

det Ž T . y BB

s L 2 Nc .

Ž 23 .

If r / 1, detŽT . / 0 is required and shows the spontaneous breakdown of chiral SUŽ Nf . symmetry. There is no room for detŽT . s 0. While, if r s 1

Y. Honda, M. Yasuer ` Physics Letters B 466 (1999) 244–250

as in the Seiberg’s choice w1x, there are two options: one for the spontaneous breakdown of chiral SUŽ Nf . symmetry and the other for that of vector UŽ1. symmetry of the baryon number. The latter case Nc corresponds to z s `, which means that z Ł as 1pa Nc cannot be separated into z and Ł as 1pa , and is possible to be realized by taking Nc

the vectorial SUŽ Nf .LqR symmetry, which has also been advocated in Ref. w9x. The dependence of the SUSY-breaking effect on various VEV’s can be summarized as

²0 < Tii <0 : s

½

LT2

Ž i s 1 ; Nc .

j LT2

Ž i s Nc q 1 ; Nf .

as1

rqN fyNc

Nf

p B p B s yL2 Nc

Ł Ž m irL . Ž s yL˜ 2 N . , Ž 24. c

isNcq1

as instanton contributions. In summary, we have demonstrated that dynamical symmetry breaking in the ‘‘electric’’ SQCD with Nc q 2 F Nf F 3 Ncr2 Ž Nc ) 2. can be described by

½

S

Nc yN f

det Ž T . Ž 1 y Z .

r

q Nf y Nc

L3 NcyN f

Ž r ) 0.

5 Ž 25 .

²0 < S <0 : ; LT3 j

N fyNc

,

Ž 28 .

and the classical constraint of 1 y Z s 0 is modified into 1yZsj ,

Ž 29 .

where j ™ 0 gives the SUSY limit. The parameter r will be fixed if we find ‘‘real’’ properties of SQCD beyond those inferred from arguments based on the symmetry principle only. The choice of

rs1

with Zs

,

²0 < B w12 PPP Nc x <0 : s ¦0 < Bw12 PPP Nc x <0; ; LTNc ,

Ł pa s 0,

Weff s S ln

249

BT N fyNc B det Ž T .

Ž 30 . AD S Weff

,

Ž 26 .

AD S which turns out to be Weff of the ADS-type:

seems natural since, in this case, with Nf s Nc q 1 correctly reproduces the Seiberg’s superpotential. Furthermore, the superpotential derived in Ref. w10x: `

AD S Weff s Ž Nf y Nc .

det Ž T . Ž 1 y Z .

L

3 Nc yN f

r

1r Ž N fyNc .

ž

Weff s S ln Z q Nf y Nc q

.

Ž 27 . This superpotential exhibits 1. holomorphic decoupling property, 2. spontaneously breakdown of chiral SUŽ Nc . symmetry and restoration of chiral SUŽ Nf y Nc . symmetry described by SUŽ Nf .L = SUŽ Nf .R = UŽ1. V =UŽ1. A ™ SUŽ Nc .LqR = SUŽ Nf y Nc .L = SUŽ Nf yNc .R = UŽ1.XV = UŽ1.XA , 3. consistent anomaly-matching property due to the emergence of the Nambu-Goldstone superfields, and 4. correct vacuum structure for Nf s Nc reproduced by instanton contributions when all quarks with flavors of SUŽ Nf y Nc . become massive. The breaking of chiral SUŽ Nf . symmetry to SUŽ Nc .LqR includes the spontaneous breakdown of

Ý c n Zyn ns1

/

Ž 31 .

with Zs

S NcyN f det Ž T . Ž 1 y Z .

L3 NcyN f

Ž 32 .

has the similar structure to Eq. Ž25.. This form implies that r s 1 although their additional terms may yield different physics from ours. It should be stressed that, in addition to the usually believed physics of ‘‘magnetic’’ SQCD, where chiral SUŽ Nf . symmetry is restored, our suggested physics of spontaneous chiral symmetry breakdown is expected to be realized in ‘‘electric’’ SQCD at least for Nc q 2 F Nf F 3 Ncr2. Therefore, we expect that there are two phases in SQCD: one with unbroken chiral symmetries realized in ‘‘magnetic’’ SQCD and the other with spontaneously broken chiral symmetries realized in ‘‘electric’’ SQCD.

250

Y. Honda, M. Yasuer ` Physics Letters B 466 (1999) 244–250

References w1x N. Seiberg, Phys. Rev. D 49 Ž1994. 6857; Nucl. Phys. B 435 Ž1995. 129. w2x I. Affleck, M. Dine, N. Seiberg, Phys. Rev. Lett. 51 Ž1983. 1026; Nucl. Phys. B 241 Ž1984. 493; B 256 Ž1985. 557. w3x K. Intriligator, N. Seiberg, Nucl. Phys. ŽProc. Suppl.. 45BC Ž1996. 1; M.E. Peskin, hep-thr9702094, in: Proc. 1996 TASI, Boulder, CO, USA, 1996; M. Shifman, Prog. Part. Nucl. Phys. 39 Ž1997. 1. w4x G. ’t Hooft, in: G.’t Hooft et al. ŽEds.., Recent Development in Gauge Theories, Proc. Cargese Summer Institute, Cargese, France, 1979, NATO Advanced Study Institute Series B: Physics, vol. 59, Plenum Press, New York, 1980. w5x T. Banks, I. Frishman, A. Shwimmer, S. Yankielowicz, Nucl. Phys. B 177 Ž1981. 157. w6x K. Intriligator, N. Seiberg, Nucl. Phys. B 431 Ž1994. 551; P.C. Argyres, M.R. Plesser, A.D. Shapere, Phys. Rev. Lett. 75 Ž1995. 1699; A. Hanany, Y. Oz, Nucl. Phys. B 452 Ž1995. 283. w7x R.G. Leigh, M.J. Strassler, Nucl. Phys. B 447 Ž1995. 95; P.C. Argyres, M.R. Plesser, N. Seiberg, Nucl. Phys. B 471 Ž1996. 159; M.J. Strassler, Prog. Theor. Phys. Suppl. 123 Ž1996. 373; N. Evans, S.D.H. Hsu, M. Schwetz, S.B. Selipsky, Nucl. Phys. ŽProc. Suppl.. 52A Ž1997. 223; P.C. Argyres, Nucl. Phys. ŽProc. Suppl.. 61A Ž1998. 149; T. Hirayama, N. Maekawa, S. Sugimoto, Prog. Theor. Phys. 99 Ž1998. 843. w8x N. Seiberg, E. Witten, Nucl. Phys. B 426 Ž1994. 19; B 431 Ž1994. 484. w9x N. Arkani-Hamed, R. Rattazzi, Phys. Lett. B 454 Ž1999. 290. w10x P.I. Pronin, K.V. Stepanyantz, Exact effective action for Ns1 supersymmetric theories, hep-thr9902163, Februari

w11x w12x

w13x w14x

w15x w16x

w17x

w18x w19x w20x w21x

1999; K.V. Stepanyantz, On the structure of exact effective action for Ns1 supersymmetric theories, hep-thr9902201, Februrari 1999. T. Appelquist, A. Nyffeler, S.B. Selipsky, Phys. Lett. B 425 Ž1998. 300. G. Veneziano, S. Yankielowicz, Phys. Lett. B 113 Ž1983. 321; T. Taylor, G. Veneziano, S. Yankielowicz, Nucl. Phys. B 218 Ž1983. 493. A. Masiero, R. Pettorino, M. Roncadelli, G. Veneziano, Nucl. Phys. B 261 Ž1985. 633. For recent studies on a SUSY Yang-Mils theory, M. Schlitz, M. Zabzine, Gaugino Condensate and Veneziano-Yankielowicz Effective Lagrangian, hep-thr9710125, October 1997; G.R. Farra, G. Gabadadze, M. Schwetz, Phys. Rev. D 58 Ž1998. 015009; V. Chernyak, Phys. Lett. B 450 Ž1999. 65. K. Intriligator, R. Leigh, N. Seiberg, Phys. Rev. D 50 Ž1994. 1092. Y. Honda, M. Yasue, ` Prog Theor. Phys. 101 Ž1999. 971; see also M. Yasue, ` Phys. Rev. D 35 Ž1987. 355; D 36 Ž1987. 932; Prog Theor. Phys. 78 Ž1987. 1437. O. Aharony, J. Sonnenschein, M.E. Peskin, S. Yankielowicz, Phys. Rev. D 52 Ž1995. 6157; E. D’Hoker, Y. Mimura, N. Sakai, Phys. Rev. D 54 Ž1996. 7724; N. Evans, S.D.H. Hsu, M. Schwets, Phys. Lett. B 404 Ž1997. 77; H.C. Cheng, Y. Shadmi, Nucl. Phys. B 531 Ž1998. 125; S.P. Martin, J. Wells. Phys. Rev. D 58 Ž1998. 115013; M. Chaichian, W.-F. Chen, T. Kobayashi, Phys. Lett. B 432 Ž1998. 120. K. Konishi, Phys. Lett. 135B Ž1984. 439; K. Konishi, K. Shizuya, Nuovo Cimento 90A Ž1985. 111. S. Dimopoulos, S. Raby, L. Susskind, Nucl. Phys. B 173 Ž1980. 208; T. Matsumoto, Phys. Lett. B 97 Ž1980. 131. D. Weingarten, Phys. Rev. D 51 Ž1983. 1830; C. Vafa, E. Witten, Nucl. Phys. B 234 Ž1984. 173. M.G. Schmidt, Phys. Lett. B 141 Ž1984. 236.