Duals for SU(N) SUSY gauge theories with an antisymmetric tensor: five easy flavors

12 March 1998

Physics Letters B 422 Ž1998. 149–157

Duals for SU žN / SUSY gauge theories with an antisymmetric tensor: five easy flavors John Terning

1

Department of Physics, UniÕersity of California, Berkeley, CA 94720, USA Received 21 December 1997 Editor: H. Georgi

Abstract I consider N s 1 supersymmetric SU Ž Nc . gauge theories with matter fields consisting of one antisymmetric representation, five flavors, and enough anti-fundamental representations to cancel the gauge anomaly. Previous analyses are extended to the case of even Nc with no superpotential. Using holomorphy I show that the theory has an interacting infrared fixed point for sufficiently large Nc . These theories are interesting due to the fact that in going from five to four flavors the theory goes from a non-trivial infrared fixed point to confinement, in contradistinction to SUSY QCD, but in analogy to the behavior expected in non-SUSY QCD. q 1998 Published by Elsevier Science B.V.

1. Introduction In recent years our understanding of the infrared behavior of vector-like N s 1 supersymmetric ŽSUSY. gauge theories has increased dramatically, primarily due to the work of Seiberg w1,2x. In particular it is now known for SUSY QCD with a given number of flavors whether the theory has: an unstable vacuum, a confined description, a weakly coupled Žinfrared free. dual gauge description, a nontrivial infrared fixed point, or a trivial infrared fixed point. Some work has also been done on chiral SUSY gauge theories, but our understanding of these more complex theories is far from complete. Chiral SUSY gauge theories are of special interest since

1

E-mail: [email protected].

they can dynamically break SUSY, unlike most theories with vector matter 2 . Among the simplest chiral theories are those with an antisymmetric tensor. Consider SU Ž Nc . with one antisymmetric tensor, Ž Nc y 4. Nc’s and F flavors Ža flavor is one Nc and one Nc.; it is known that this theory is confining w4–6x for F s 3 or 4. Thus the simplest example of this type of chiral SUSY theory which admits a dual gauge description is F s 5. What is unknown is whether the theory has an infrared free dual gauge description or an interacting infrared fixed point. In this paper I show how to use the ‘‘deconfinement’’ method introduced by Berkooz in Ref. w7x and elaborated in Refs. w5,8,9x to construct simple duals for the

2 For a vector-like theory that dynamically breaks SUSY see Ref. w3x.

0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 0 7 4 - 4

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case F s 5 Žpointing out why this case is special. and Nc even and compare with the previously know dual for odd Nc . Using holomorphy I show that the dual Žand hence the original theory. has an interacting infrared fixed point at the origin of moduli space for sufficiently large Nc . Finally I present my conclusions, and discuss the analogous behavior in nonSUSY QCD.

Table 2 Field content of the ‘‘deconfined’’ theory, where D s 2 N y 2, and Cs 2 N q 1 SU Ž Nc . Sp Ž D . SU Ž2 . f SU Ž5 . SU Ž C . U Ž1 .R

U Ž1 . X

U Ž1 . Y

1

y1

y1

y1

2

q

I

1

1

I

1

q

I

1

1

1

I

x

I

I

1

1

1

p

I

1

I

1

1

1

r

1

I

I

1

1

1

s

1

1

1

1

1

0

Nq 3 2 Nq 3

Ny2

Nq3

2Ny 2 Ny2

2Ny2 Nq3

2 y N Ž N y 2.

2 yN Ž N q 3.

2Ny 2 N Ž N y 2.

2Ny2 N Ž N q 3.

Ny 1

Ny1

0

2. Duality for SU (2 N ) The theory I wish to study has gauge group SU Ž Nc . with 5 chiral superfields q in the Ždefining. Nc representation, one matter field A in the antisymmetric tensor representation, and Nc q 1 matter fields q in the Nc representation. This theory has the anomaly-free global symmetry SUŽ5. = SUŽ Nc q 1. = UŽ1.R = UŽ1. X = UŽ1. Y . The field content Žwith global charges. is given in Table 1. This theory has been considered previously by Berkooz w7x for even Nc Žs 2 N ., with the addition of a superpotential W s PfŽ A.. I will consider this theory with no superpotential. The case Nc odd with no superpotential has been discussed by Pouliot w5x Žsee also Ref. w10x.. I can replace the antisymmetric tensor by a composite ‘‘meson’’ operator of a confining Sp Ž2 N y 2. group: X

X

A a b ™ x a a x b b JaX bX ,

Ž 2.1 . X

where a,b are SU Ž2 N . indices and a ,bX are Sp Ž2 N y 2. indices and JaX bX is the invariant tensor. I must also introduce additional fields that transform under Sp Ž2 N y 2. and add terms to the superpotential in the deconfined description. The matter content of the model that accomplishes this is displayed in Table 2.

Table 1 Field content of the theory SUŽ Nc .

SU Ž5.

q

I

q

I

A

The superpotential in the ‘‘deconfined’’ description is W s xrp q rrs.

Ž 2.2 .

ŽI have set the coefficients of the superpotential to 1 by rescaling the fields.. The purpose of the superpotential is to remove the unwanted ‘‘meson’’ states Ž xr . and Ž rr . that appear when the Sp Ž2 N y 2. group confines. These two ‘‘mesons’’ get masses with p and s respectively. Note that, as discussed in Ref. w9x, gauge anomaly cancellation for Sp Ž2 N y 2. forces the fields r and p to have a fictitious global SU 2 f symmetry. This symmetry is fictitious in the sense that none of the physical low energy degrees of freedom transform under it: r is confined, and p is massive. This symmetry will be useful later in determining which of several dual descriptions might be useful. I can now use the known dual description of SU Ž2 N . gauge theory with fundamentals w1x to Table 3 Field content of the first dual description, where D s 2 N y 2, C s 2 N q1 SU Ž3 . Sp Ž D . SU Ž2 . f SU Ž5 . SU Ž C . U Ž1 .R q1

I

1

1

I

1

q1

I

1

1

1

I

x1

I

I

1

1

1

p1

I

1

I

1

1

r

1

I

I

1

s

1

1

1

U Ž1 . X

U Ž1 . Y

4

N

Nq1

3Ž N q 3. 6Nq2

3

3 5y N

N

3Ž N q 3. 10

y 3 N Ž2 N q 1.

3 Ž2 N q 1. Ž N y 5.

3Ž N q 3. 3Ny 1

3 Ž2 N y 2 . y 5 N Ž N y 1.

3 Ž2 N y 2 . y5Ž N q 1

3Ž N q 3.

3 Ž2 N y 2 .

6

1

1

y N Ž N y 2.

y N Ž N q 3.

2Ny 2

2Ny2

1

1

0

N Ž N y 2.

N Ž N q 3.

Ny 1

Ny1

4

0

y2

SUŽ Nc q1.

UŽ1.R

UŽ1. X

UŽ1. Y

I

1

4

1

y1

Ž qq . 1

1

1

I

I

1

y1

y1

Ž xp . 1

I

I

1

1

1

N Ž N y 2.

N Ž N q 3.

I

Nq5

2Ny 2 N

2Ny2 Nq1

Nq 3 2

2 yN

2 5y N

Nq 3

2Ny2

2Ny2

1

1

Nc q 6 4 Nc q 6

0

Nc y 4

Nc q 6

Nc y 2

Nc y 2

Ž qp . 1

1

I

I

1

Ž qx . 1

I

1

1

I

Nq 3

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Table 4 Field content of the second dual description, where C s 2 N q 1 SU Ž3.

SU Ž2.

SUŽ2. f

SU Ž5.

SUŽ C .

UŽ1.R

UŽ1. X

UŽ1. Y

4

N

Nq1

3Ž N q 3.

3

q1

I

1

1

I

1

q1

I

1

1

1

I

x2

I

I

1

1

1

p1

I

1

I

1

1

s

1

1

1

1

1

Ž qq .

1

1

1

I

I

Ž qp .

1

1

I

I

1

y1

1

I

1

1

I

z

I

1

1

1

1

Ž yy .

1

1

1

1

Ž x1 y.

I

1

1

1

I

write a dual description of this theory in terms of a theory with gauge group SUŽ3. = SpŽ2 N y 2.. The field content of this dual is given in Table 3. This dual has a superpotential W s Ž xp . r q rrs q Ž qq . q1 q1 q Ž xp . x 1 p 1 q Ž qp . q1 p 1 q Ž qx . q1 x 1 . Ž 2.3 . I have introduced some notation here to simplify the later exposition: qi refers to the field which is the ‘‘dual’’ of qiy1 , where q0 ' q, and I will denote a ‘‘meson’’ which is the mapping of qi pj Žand couples to qiq1 and pjq1 in the dual superpotential. by Ž qi pj .. For later convenience I will relabel the ‘‘meson’’ Žqx . by y. The massive fields Ž xp . and r can be integrated out, leaving the superpotential: W s x 12 p 12 s q Ž qq . q1 q1 q Ž qp . q1 p 1 q yq1 x 1 .

Ž 2.4 .

The anomaly matching is guaranteed to work by the anomaly matching of the SU duality used in its construction. The dual description obtained above is, unfortunately, almost completely useless, since it possesses a fictitious global SUŽ2. f symmetry: any field which transforms under this fictitious symmetry must either be massive or strongly coupled since it cannot appear in the physical spectrum. It is obvious that additional dual descriptions can be obtained by alternating the gauge group that duality is applied to

6Nq2

3 N

5y N

y 3Ž N q 3.

3

3

3Ny 1

y N Ž2 N q 1.

y Ž2 N q 1. Ž N y 5.

3Ž N q 3.

3 Ž2 N y 2 .

3 Ž2 N y 2 . y5Ž N q 1.

3Ny 1

y5 N Ž N y 1.

3Ž N q 3.

3 Ž2 N y 2 .

6

0

N Ž N y 2.

N Ž N q 3.

Ny 1

Ny1

4

0

y2

Nq5

N

Nq1

Nq 3

2

2

Nq1

N

Ny5

Nq 3

Nq 3

2Ny2

2Ny2

20

N Ž2 N q 1.

Ž2 N q 1. Ž N y 5.

3Ž N q 3.

3Ž N y 1.

3Ž N y 1.

4

yN

5y N

Nq 3

Ny1

Ny1

16

N

Ny5

3Ž N q 3.

3

3

w7,5,9x; what is surprising is that such an exercise turns out to be useful. Going through a repeated application of alternating dualities produces Žfor the case of five flavors. a dual with no fields transforming under SUŽ2. f . The remainder of this section is devoted to detailing this procedure, the reader who is interested in results rather than techniques is urged to skip ahead to Table 6 where the final dual is presented. The next step is to apply duality to the Sp Ž2 N y 2. gauge group. The field content of the resulting dual is given in Table 4, and the superpotential is W s zp 12 s q Ž qq . q1 q1 q Ž qp . q1 p 1 q Ž yx 1 . q1 q zx 2 x 2 q Ž yy . y 1 y 1 q Ž x 1 y . y 1 x 2 ,

Ž 2.5 .

where I have renamed Ž x 1 x 1 . to be z. After integrating out q1 and Ž x 1 y ., the superpotential is: W s zp 12 s y Ž qq . q1 y 1 x 2 q Ž qp . q1 p 1 q zx 2 x 2 q Ž yy . y 1 y 1 .

Ž 2.6 .

At this point it can be seen why the case of 5 flavors is so special. If the analysis so far had been done for F flavors, then the gauge group SU Ž3. would instead be SU Ž F y 2., and the field z would be an antisymmetric tensor 3. Then to further dualize the

3

The antisymmetric tensor for SU Ž3. is just a 3.

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Table 5 Field content of the third dual description, where C s 2 N q 1 SU Ž2.

SU Ž2.

SUŽ2. f

SU Ž5.

SUŽ C .

UŽ1.R

UŽ1. X

UŽ1. Y

2

N

Nq1

Nq 3

2

2

q2

I

1

1

I

1

x3

I

I

1

1

1

1

yN Ž N y 2.

yN Ž N q 3.

2Ny 2

2Ny2

p2 s

I 1

1 1

I 1

1 1

1 1

1 0

0

0

N Ž N y 2.

N Ž N q 3.

Ž qq .

1

1

1

I

I

Ž qp .

1

1

I

I

1

y1

1

I

1

1

I

z1

I

1

1

1

1

Ž yy .

1

1

1

Ny 1

Ny1

0

y2

N

Nq1

4 Nq 3 Nq5

1

Nq 3

2

Nq1

N

Ny5

Nq 3

2Ny2

2Ny2

2

2Ny 4

y N Ž3 N y 1 .

5q 6Ny 3N 2

Nq 3

2Ny2

2Ny2

4

yN

Nq 3

Ny1

Ny1

yN

3Nq 1 2Ny2

5y N

Ž q1 x 2 .

1

I

1

I

1

Nq1 Nq 3

2Ny2

Ž q1 p 1 .

1

1

I

I

1

Nq1

yN

Nq 3

2

2

1

8

N2

N 2y 3Ny 2

Nq 3

Ny1

Ny1

Ž q1 z .

1

1

1

I

SU Ž F y 2. would require the introduction of an additional ‘‘deconfinement’’ module, and hence an even more complicated description of the theory. I can now use the known dual of an SU Ž3. gauge theory with fundamental representations to find another dual, with the field content given in Table 5; the superpotential is

Note the ‘‘baryonic’’ operator mapping:

W s x 32 s y Ž qq . Ž q1 x 2 . y 1 q Ž qp . Ž q1 p 1 . q p 22 q Ž yy . y 1 y 1 q Ž q1 x 2 . q2 x 3 q Ž q1 p 1 . q2 p 2 q Ž q1 z . q 2 z 1 .

yN y 1

zp1 p 1

™

x 32 ,

zx 2 x 2 x 2 p1 p1 x 2 x 2 p1

™ ™ ™

p 22 , z1 x 3 , z1 p 2 .

Ž 2.8 .

After integrating out p 2 , Ž qp ., and Ž q1 p 1 . there are no longer any fields that transform under the ficti-

Ž 2.7 .

Table 6 Field content of the ‘‘final’’ dual description SUŽ2.1

SUŽ2. 2

SU Ž5.

SUŽ2 N q 1.

UŽ1.R

UŽ1. X

UŽ1. Y

yN

yN y 1

q3

I

1

I

1

Nq 1 Nq 3

2

2

x4

I

I

1

1

0

N Ž N y 2.

N Ž N q 3.

2Ny 2

2Ny2

y1

1

I

1

I

z2

I

1

1

Ž x 3 z1 .

1

I

Ž qq .

1

1

Ž q2 q2 .

1

1

Ž yy .

1

1

Nq1

N

Ny5

Nq 3

2Ny2

2Ny2

1

7y N

N Ž3 N y 1 .

3N 2y 6Ny 5

Nq 3

2Ny2

2Ny2

1

1

3Ny 1

y N Ž4 N y 3 .

5q 3Ny 4N 2

Nq 3

2Ny2

2Ny2

I

I

4

0

y2

N

Nq1

4

yN

5y N

Nq 3

Ny1

Ny1

1 1

Nq 3 4 Nq 3

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tious global SU Ž2. f symmetry, although the singlet field s still remains. The superpotential is given by

fields, and one component of x 4 remains uneaten. The operator mapping for four flavors is:

W s x 32 s y Ž qq . Ž q1 x 2 . y 1 q Ž yy . y 1 y 1

qq qAq

q Ž q1 x 2 . q 2 x 3 q Ž q1 p 1 . q 2 p 2 q Ž q1 z . q 2 z 1 . Ž 2.9 . To obtain a slightly simpler dual description, I can now apply duality one more time to the first SU Ž2. gauge group. After integrating out a number of massive fields I find the field content given in Table 6 with a superpotential given by: W s Ž qq . q3 x 4 y 1 q Ž yy . y 1 y 1 q Ž q2 q2 . q3 q3 q Ž x 3 z1 . x 4 z 2 .

Ž 2.10 .

Ny 1

™

Ž qq . ,

™

Ž yy . , Ž q2 q2 . ,

q

™

q 4A Ny2

™

AN

™

x4,

™

Ž x 3 z1 . ,

qA

2N

q

$

z2 ,

Ž 2.12 .

$

$

where the ˆ superscript indicates the remaining massless Žsinglet. component. Poppitz and Trivedi w4x showed that these theories can break SUSY with the addition of some singlet fields, some superpotential terms, and the gauging of a chiral UŽ1. symmetry. With corresponding manipulations of the dual, it can provide a weakly coupled description of their SUSY breaking models.

The operator mapping of the chiral ring is: qq

™

Ž qq . ,

qAq

™

qA Ny 1q

™

Ž yy . , Ž q2 q2 . ,

4

q A A

N

2N

q

Ny2

™

q3 z 2 ,

™

x 42 ,

™

y1Ž x 3 z1 . ,

3. Comparison with duality for SU (2 M I 1)

Ž 2.11 .

where y ' Žqx . and z ' Ž x 1 x 1 .. This dual has a simple relation to the spectrum of the confined description found by Pouliot w5x for the case of four flavors. Adding a mass term for one flavor in the original theory gives breaks the flavor symmetry to SU Ž4. = SU Ž2 N .. In the dual description the mass term maps to a linear term for Žqq ., which induces a vev for for the product q3 x 4 y 1. D-flatness ensures that, in an appropriate basis, each of these three fields has only one non-zero component. These vevs break the gauge symmetries completely and produce mass terms for extra components of Žqq ., Ž yy ., and Ž q2 q2 . with uneaten pieces of q3 and y 1. The vev of x 4 gives a mass to one component each of Ž x 3 z 1 . and z 2 , leaving two massless

The case of odd Nc has been studied previously by Pouliot w5x. In this Section I will briefly review his dual in order make comparisons with the even Nc case discussed above. The field content Žwith global charges. is given in Table 1, with Nc s 2 M y 1. Pouliot deconfined the antisymmetric tensor with Sp Ž2 M y 4. by introducing fields x, r, and p Žas discussed earlier for the even case. with a superpotential W s xrp. The odd Nc case is much simpler than the even case because no fictitious global symmetry is needed nor is a singlet field required. Pouliot then dualized SU Ž2 M y 1. to SU Ž2. in the usual fashion, and further dualized Sp Ž2 M y 4. to SU Ž2.. After integrating out massive fields, he arrived at a dual with a superpotential given by: W s Ž qq . q1 y 1 x 2 q Ž yy . y 1 y 1 q Ž qp . q1 p 1 qŽ x1 x1 . x 2 x 2 ,

Ž 3.1 .

and the field content shown in Table 7 Žusing the notation y ' Žqx ...

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Table 7 Field content of the dual description for the case Nc s 2 M y 1 SU Ž2.

SU Ž2.

SU Ž5.

SUŽ2 M .

q1

I

1

I

1

x2

I

I

1

1

y1

1

I

1

I

p1

I

1

1

1

Ž x1 x1 .

1

1

1

1

Ž qq .

1

1

I

I

Ž qp .

1

1

I

Ž yy .

1

1

1

1

The operator mapping is: qq qAq

™

Ž qq . ,

™

My 1

™

Ž yy . , Ž qp . ,

3 My2

™

q1 q1 ,

™

y1 x 2 p1 ,

qA

qA

2 My1

q

Ž 3.2 .

and for M G 3 q 5A My 3

™

Ž x1 x1 . .

Ž 3.3 .

Even though only three dualities were required in the derivation of the odd Nc case, as opposed to five dualities for even Nc , the resulting dual descriptions are quite similar. The five or six flavor models are special for odd Nc as well, since for a larger number of flavors the dual contains tensor representations.

4. The infrared fixed point I would now like to demonstrate that the dual of the SU Ž2 N . theory described above Žand the original theory itself. has a non-trivial infrared fixed point at the origin of moduli space. The situation is more difficult than in SUSY QCD since there are two gauge groups in the dual. The analysis can be simplified by using the fact that the ratio of the two intrinsic scales, L1 and L2 Žcorresponding to the SUŽ2.1 and SUŽ2. 2 gauge groups in the final dual description given in Table 6., can be varied arbitrarily. Holomorphy w11x requires that, aside from singu-

UŽ1.R

UŽ1. X

UŽ1. Y 2 M 2q y 5M y 1

6

Ž M y 1 .Ž 2 M y 1 .

2M q 5

4My 6

4My 6 y M Ž 2 M y 11 .

2My5

y M Ž2 M y 1.

2M q 5

4My 6

4My 6

2Mq1

2My1

2 M y 11

2M q 5

4My 6

4My 6

4M

y 6 M 2 q 13 M y 5

y6 M 2 q 3 M q 5

2M q 5

4My 6

4My 6

20

2 M Ž2 M y 1.

2 M Ž 2 M y 11 .

2M q 5

4My 6

4My 6

8

0

y2

4

4 M 2 y 10 M q 4

4M 2q 2 My 4

2M q 5

4My 6

4My 6

8

2y 4 M

22 y 4 M

2M q 5

4My 6

4My 6

2M q 5

lar points, there can be no phase transitions as this ratio is varied. There are two cases 4 to consider: for N ) 4 the SUŽ2. 2 gauge group is infrared free and L1 < L2 corresponds to the SUŽ2. 2 gauge coupling g Žrenormalized at a scale near L1 . becoming arbitrarily small, for N - 4 the SUŽ2. 2 gauge group is asymptotically free and the limit L1 4 L2 also corresponds to weak coupling for SUŽ2. 2 . In both cases the gauge coupling g ™ 0 as L1rL2 ™ 0 or `. Of course g cannot be simply set to zero for at least two reasons. Firstly, the massless spectrum is discontinuous in this limit, since setting g s 0 causes D-terms to vanish, thus enlarging the moduli space. More importantly non-perturbative effects from the SUŽ2.1 gauge interactions can affect the running of g in the infrared. Thus a careful study is required. Before proceeding with the details of the calculation I will sketch an outline of the analysis. I will analyze the dual at a renormalization scale somewhat below the interaction scale of SUŽ2.1 Ži.e. m - L1 . with an arbitrarily small Žbut non-zero. value for g Ž m .. At this scale the theory can be studied with perturbation theory in g Ž m ., and at lowest order in g Ž m . I will show that the SUŽ2.1 has a non-trivial infrared fixed point. I will then proceed to show that for sufficiently large N the SUŽ2. 2 interactions are infrared free at this scale, i.e. that coupling g has a trivial infrared stable fixed point. This is sufficient to prove that the theory with an arbitrary ratio L1rL2 4

For N s 4, the holomorphic gauge coupling does not run, and can be set to be arbitrarily small.

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D Ž y 1 . s 1 q gy 1Ž g s 0 . G 1

Ž 4.1 .

w12x fixed points 6 . the theory is at an infrared fixed point for 18r5 - NF - 6. Thus for g s 0 the bounds in Ž4.2.-Ž4.4. are not saturated. Recall that the case of five flavors in the original theory was special because it led to a simple dual without tensor gauge representations, and it is the absence of tensor representations that allows for a simple demonstration of an infrared fixed point for g s 0. A few more relations between anomalous dimensions are required to reach some definite conclusions for non-zero g. Recall that the exact b function for the SUŽ2.1 coupling is w13x:

D Ž Ž x 3 z1 . . s 1 q gŽ x 3 z1 . Ž g s 0 . G 1

Ž 4.2 .

b Ž g1 . s y

has the same infrared fixed point, since there cannot be a phase transition in the space of holomorphic couplings w11x. Thus the infrared limit can be understood simply through a perturbative analysis in the coupling g. It is instructive to first consider the theory at zero-th order in g Ži.e. with g set to zero.. With SUŽ2. 2 turned off, the fields y 1 and Ž x 3 z 1 . as well as the products q3 x 4 and x 4 z 2 become gauge invariant operators, so their scaling dimensions satisfy the bounds

D Ž q3 x 4 . s 2 q gq 3Ž g s 0 . q gx 4Ž g s 0 . G 1

Ž 4.3 .

=

g 13

16p 2 2 q 5gq 3Ž g s 0 . q 2gx 4Ž g s 0 . q gz 2Ž g s 0 . 1y

D Ž x 4 z 2 . s 2 q gx 4Ž g s 0 . q gz 2Ž g s 0 . G 1

g 12 4p 2

Ž 4.5 .

Ž 4.4 . Žwhere gf is the anomalous dimension of the field f . with equality holding if the operators are free. Thus for g s 0 the first two interaction terms in the superpotential Ž2.10. are the products of three gauge invariant operators, and are thus irrelevant operators 5. In other words, if the coefficients of the first two terms are labeled l1rm 0 and l2 then for g s 0, l1 and l2 run to zero in the infrared. Since the fields Ž qq ., Ž yy ., and y 1 only interact through these irrelevant terms, they are free fields and their anomalous dimensions vanish. The effective theory containing the remaining two operators and the SUŽ2.1 gauge interactions is in an interacting nonAbelian Coulomb phase Ži.e. it has a non-trivial infrared fixed point at the origin of moduli space.. This can be seen by noting that this is a special case of an SU Ž2. theory with NF flavors Žhere NF s 4. and trilinear superpotential terms which is dual to an Sp Ž2 NF y 6. theory with NF flavors and trilinear superpotential terms. The SU Ž2. theory is asymptotically free for NF - 6, while the Sp Ž2 NF y 6. theory is asymptotically free for NF ) 18r5, so Žassuming a la SUSY QCD that there is a conformal range of fixed point theories between the two Banks-Zaks 5

They can only be relevant if all three operators are dimension 1, in which case the operators are free, a contradiction.

,

thus at the fixed point: 0 s 2 q 5gq 3Ž g s 0 . q 2gx 4Ž g s 0 . q gz 2Ž g s 0 . .

Ž 4.6 . Since the last term in the superpotential Ž2.10. is a relevant operator Žfor g s 0. with R-charge 2 the anomalous dimensions satisfy gŽ x 3 z 1 . Ž g s 0 . q gx 4Ž g s 0 . q gz 1Ž g s 0 . s 0 ,

Ž 4.7 . which, with the bound Ž4.4., implies gŽ x 3 z1 . Ž g s 0. - 1 . Furthermore operators so

gq 3 ,gx 4 G y

q32 , 1

, 2 gq 3 q gz 1 G y1 ,

x 42 ,

Ž 4.8 .

and q3 z 1 are gauge invariant

Ž 4.9 . Ž 4.10 .

independent of g. Combining the fixed point condition Ž4.6. with the bound Ž4.10. gives 2gx 4Ž g s 0 . - y1 y 4gq 3Ž g s 0 . . Ž 4.11 . 6 If the superpotential couplings are taken to be arbitrarily small, a Banks-Zaks fixed point can be established in perturbation theory at the point where asymptotic freedom is almost lost, then by holomorphy w11x there is a fixed point for arbitrary superpotential couplings.

J. Terningr Physics Letters B 422 (1998) 149–157

156

This inequality with the bound Ž4.9. implies

gx 4Ž g s 0 . -

1 2

.

Ž 4.12 .

Returning to the theory with g Ž m . arbitrarily small, but non-zero, I note that the anomalous dimensions of Ž qq ., Ž yy ., and y 1 as well as the couplings l1 and l 2 vanish at g s 0. I proceed by making the plausible assumption that the anomalous dimensions and b functions of the theory with g Ž m . arbitrarily small can be reliably analyzed near the scale m with a perturbative expansion in g Ž m .. Thus I am assuming that the anomalous dimensions of the fields with SUŽ2.1 interactions have reliable perturbative expansions in g Ž m ., although I do not know the value first Ž g Ž m . independent. terms in these expansions since they are determined by the dynamics of the pure SUŽ2.1 fixed point discussed above. 7 Now consider the running of g Ž m . which is determined by w13x:

b Ž g . sy =

g3 16p 2 4 y N q Ž 2 N q 1 . g y 1 q 2 gx 4 q g Ž x 3 z 1 . 1y g3

sy

16p 2

g2 4p 2

Ž 4 y N q 2 gx Ž g s 0 . 4

qgŽ x 3 z 1 . Ž g s 0 . . q O Ž g 5 . .

Ž 4.13 .

The bounds Ž4.8. and Ž4.12. imply that the SUŽ2. 2 interactions are infrared free Ž b Ž g . ) 0. for N ) 6. Turning to the superpotential interactions, the b functions of the the first two terms in the superpotential Žexpanded to leading order in l1 , l 2 and g Ž m .. are:

b 1 s 1 q gŽ q q . q gq 3 q gx 4 q gy 1 s 1 q gq 3Ž g s 0 . q gx 4Ž g s 0 . q a l12 q bl22 2

y cg Ž m . ,

Ž 4.14 . 2

b 2 s gŽ y y. q 2gy 1 s d l12 q e l22 y fg Ž m . ,

7

Ž 4.15 .

Alternatively they are determined by the Žunknown. superconformal R-charge w1,2x.

where a... f are positive numbers. For the first two terms in the superpotential to be relevant these two b functions must vanish. The discussion above indicates that l1 and l2 must vanish with g Ž m .. Therefore for sufficiently small g Ž m ., b 1 cannot vanish since the bound in Eq. Ž4.3. is not saturated. These considerations suggest that it is however possible for b 2 to vanish. However the solution of b 2 s 0 with l1 s 0 is that l2 A g Ž m .. Thus in the infrared limit m ™ 0 the three couplings l1 , l2 , and g all run to zero. The conclusion of this analysis is that for N ) 6 the chiral operators qq ™ Žqq . and qAq™ Ž yy . Žas well as the SUŽ2. 2 gauge Žvector. multiplet and the dual quark y 1 ., correspond to free fields in the infrared, while the remaining fields are at an interacting fixed point. Thus these theories provide explicit examples of the phenomena suggested in Refs. w9,14x of a gauge theory splitting into a free sector and an interacting fixed point sector in the infrared. A similar analysis can be applied to the odd case, Nc s 2 M y 1, which can also be shown to have an interacting infrared fixed point for M ) 6. Although I have only proven that the theory with F s 5 flavors has an interacting infrared fixed point, I expect that the fixed point will persist up to the point where asymptotic freedom is lost: F s 2 Nc q 3. It should be noted that this analysis does not preclude a fixed point for N F 6; to obtain information about these theories would require more information about the anomalous dimensions gx 4Ž g s 0. and gŽ x 3 z 1 .Ž g s 0.. It is suggestive however that for N s 1 and M s 2 the original theory reduces to vector-like SU Ž2. and SU Ž3. theories both of which do indeed have a non-trivial infrared fixed points.

5. Conclusions I have displayed a new dual for SU Ž2 N . with an antisymmetric tensor, five flavors, and no superpotential. Using holomorphy to adjust the ratio of the scales of the two gauge groups in the dual description I have been able to show that in the five flavor case two composite ‘‘mesons’’ become free fields in the infrared, while other degrees of freedom are at an interacting infrared fixed point for N ) 6. Thus in going from five to four flavors Žfor sufficiently large

J. Terningr Physics Letters B 422 (1998) 149–157

N . the theory goes from a fixed point to confinement 8 without passing through an infrared free phase. Such behavior was seen previously w1,2x in the isolated case of vector-like SU Ž2., whereas in the generic case of vector-like SU Ž N . theories there is confinement for F s N q 1 flavors and an infrared free gauge description for N q 1 - F - 3 Nr2. ŽThe two bounds coalesce for N s 2.. The transition from a fixed point phase directly to a confining phase as the number of flavors is reduced has been argued to occur 9 in Žnon-SUSY. QCD w15x. Given that there is currently no non-perturbative understanding of non-SUSY SU Ž N . gauge theories with an arbitrary number of flavors, it is somewhat reassuring to find that the expected behavior of the confinement transition is actually realized in a large class of theories that are under non-perturbative control. However, there is no evidence to suggest that in QCD there are free, massless composites on the fixed point side of the transition. On the contrary the scalar and pseudoscalar Žpion. mesonic states are expected to be massive Žand broad. resonances on the fixed point side of the transition w15x. Thus while some of the qualitative behavior of the confinement transitions in QCD and the chiral SUSY theories discussed here is similar, the detailed physics of the two confinement transitions appears to be quite different.

Acknowledgements I would like to thank Csaba Csaki ´ and Yaron Oz for enlightening discussions as well as Martin

8

Without chiral symmetry breaking. However non-SUSY QCD has confinement with chiral symmetry breaking. 9

157

Schmaltz for enlightening discussions and a critical reading of the manuscript. This work was supported by the National Science Foundation under grant PHY-95-14797, and also partially supported by the Department of Energy under contract DE-AC0376SF00098.

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