Confining N=1 SUSY Gauge Theories from Seiberg Duality

Trieste Meeting of the TMR Network on Physics beyond the SM

PROCEEDINGS

Conﬁning N=1 SUSY gauge theories from Seiberg duality

Matthias Klein Departamento de F´ısica Teórica C-XI and Instituto de F´ısica Teórica C-XVI Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain E-mail: [email protected]

Abstract: In this talk I review and generalize an idea of Seiberg that an N = 1 supersymmetric gauge theory shows confinement without breaking of chiral symmetry when the gauge symmetry of its magnetic dual is completely broken by the Higgs effect. It is shown how the confining spectrum of a supersymmetric gauge theory can easily be derived when a magnetic dual is known and this method is applied to many models containing fields in second rank tensor representations and an appropriate tree-level superpotential.

1. Introduction For all of the models considered in [5] a dual description in terms of magnetic variables is known Due to holomorphicity properties and non-renor- [6, 8, 9, 10] and the authors of [5] used the fact N malization theorems valid in =1supersym- that the electric gauge theory confines when its metric theories it has become possible to argue magnetic dual is completely higgsed. that some supersymmetric gauge theories with special matter content confine at low energies. Seiberg already used this idea as an addi- The first example is due to Seiberg [1] who found tional consistency check in his original paper es- that supersymmetric quantum chromodynamics tablishing electric-magnetic duality for non-Abel- ian N = 1 supersymmetric gauge theories [11]. (SQCD) with gauge group SU(Nc)andNf quark He showed how the confining superpotential of flavors shows confinement when Nf = Nc or Nf = SQCD with Nf = Nc + 1 could be obtained by Nc + 1. This has been generalized to more complicated models. All N = 1 supersymmetric a perturbative calculation in the completely bro- gauge theories with vanishing tree-level super- ken magnetic gauge theory. Under duality the potential which confine at low energies could be fields of the magnetic theory (which are gauge classified [2, 3, 4] because they are constrained singlets as the gauge symmetry is completely bro- by an index argument. When a tree-level su- ken) are mapped to the mesons and baryons of perpotential is present the index argument is no the electric theory and the confining superpo- longer valid. Because of the lower symmetry tential is easily shown to be the image of the the non-perturbative superpotential is less con- magnetic superpotential under this mapping [11]. N strained and one expects more confining models This a realization in = 1 supersymmetric gauge to exist. Indeed, Csáki and Murayama [5] showed theories of an old idea of ’t Hooft and Mandel- that many of the Kutasov-like [6] models exhibit stam [12] that confinement is driven by conden- confinement for special values of the number of sation of magnetic monopoles. 1 quark flavors Nf . These models contain fields in Now, many gauge theory models have been tensor representations of the gauge group and an found that possess a dual description in terms of appropriate superpotential for these tensor fields. magnetic variables in the infrared. This allows 1Some further confining models with non-vanishing us to predict many new examples of confining tree-level superpotential are discussed in [7]. gauge theories. The idea described in the previ- Trieste Meeting of the TMR Network on Physics beyond the SM Matthias Klein

ous paragraph was first used by the authors of Nf = Nc: In this case the superpotential [13] to determine the confining spectrum of the vanishes even at the non-perturbative level [1]. model proposed by Kutasov [6] and has been ap- As a consequence the flat directions that param- plied by Csáki and Murayama [5] to six further etrize the moduli space of vacua are not lifted in models that confine in the presence of an appro- the quantum theory. The low-energy spectrum priate superpotential. is given by the mesons M ij defined above and In this talk I review the original example of the baryons B =detQ,B¯=detQ¯(the quarks Seiberg [11] and explain how electric-magnetic Q, Q¯ are viewed as (Nf × Nc)-matrices). The duality is used to obtain the low-energy spectrum physical degrees of freedom at low energies being and the form of the non-perturbative superpo- gauge invariant means that the theory confines. tential of confining gauge theories [14]. One finds The classical constraint det M = BB¯ is modified that all of the gauge theory models based on sim- in the quantum theory [1] to ple gauge groups considered in [10, 15] confine det M − BB¯ =Λ2Nc . (2.2) when the gauge groups of their magnetic duals are completely broken by the Higgs effect. For The expectation values of the mesons and baryons nine of these theories the confining phase has not that satisfy this constraint span the quantum been discussed before. moduli space. The observation that the expectation values of det M and BB¯ cannot vanish 2. Phase structure of SQCD simultaneously tells us that the chiral symmetry is spontaneously broken. Let us briefly review the well-known phase struc- Nf = Nc + 1: There is again a quantum N ture of SQCD [16]. By this we mean an =1 moduli space, but now the classical constraints SU N N supersymmetric ( c) gauge theory with f are not modified by quantum effects. In the low- N quark flavors, i.e. f chiral matter supermulti- energy theory they can be derived from the non- Q plets transforming in the fundamental repre- perturbative superpotential [1] sentation of the gauge group and the same amount ¯ BMB¯ − det M of matter multiplets Q transforming in the anti- W = , (2.3) np 2Nf −3 fundamental representation. Consider first the Λ case of vanishing tree-level superpotential. De- where the baryons Bi are defined as the deter- pending on the relative values of Nf and Nc the minant of the quark matrix Q with the i-th line the low-energy theory resides in different phases. omitted. This describes confinement without Nf = 0: This is pure super Yang-Mills the- breaking of the chiral symmetry. N ≤N ≤ 3N ory. It is believed to show confinement. Accord- c +2 f 2 c: The low-energy energy ing to an index argument by Witten [17] there theory is rather complicated and more appropri- are Nc distinct supersymmetric vacua. ately described in terms of dual magnetic vari- 0 renormalization group [11] and resides in a non-

1 Abelian Coulomb phase. Seiberg found a dual N −N Λ3 c f Nc−Nf W N −N , description of this model [11] by an SU(Nf −Nc) np =( c f) M (2.1) det gauge theory with Nf (magnetic) quark flavors q, q N 2 M ij where Λ is the dynamically generated scale of the ¯ and f additional singlets mag which couple theory and the meson matrix M is defined by to the magnetic quarks via the superpotential ij αi j M = Q Q¯α, i, j =1,...,Nf, α =1,...,Nc. Wmag = Mmagqq.¯ (2.4) The minimum of the potential lies at infinite field expectation values and therefore the theory has This magnetic theory flows to the same infrared no stable vacuum for this range of parameters. fixed point. The gauge invariant operators of

2 Trieste Meeting of the TMR Network on Physics beyond the SM Matthias Klein both theories are in one-to-one correspondence: are denoted byq ˆ, ˆq¯. They couple to the meson singlets Mˆ mag via the tree-level superpoten- M ←−→ µM , (2.5) q mag tial (2.4). Due to non-renormalization theorems N c − N f 3 N c − N f this is not corrected in perturbation theory. But B ←−→ − ( − µ ) Λ B mag . there are non-perturbative corrections generated The mass scale µ had to be introduced by dimen- by instantons. The full superpotential of the low- energy magnetic theory reads [11] sional analysis. The three scales Λ, Λmag and µ are related by 3−Nf Wˆ mag = Mˆ magqˆˆq¯+Λmag det Mˆ mag. (3.1) N −N −N 3Nc−Nf 3( f c) f Nf −Nc Nf Λ Λmag =(−1) µ . From the fact that all physical degrees of free- (2.6) dom of the effective magnetic theory are gauge Nf > 3Nc: In the infrared the theory flows invariant one expects that, as a consequence of to the trivial fixed point of free quarks and glu- the duality mapping, the degrees of freedom of ons. the electric theory are gauge singlets as well. We will see that this intuition is right. The mapping (2.5) gives 3. Confinement from duality M ←−→ µM , p mag (3.2) Let us consider deformations [11] of the theory B ←−→ µ − 1 Λ 2 N f − 3 q,ˆ described in the previous section by mass terms W =Tr(mM), where m is an (Nf × Nf )-matrix and the scale matching (2.6) now reads p −N of rank . By the duality mapping (2.5) this cor- 2Nf −3 3 f Nf Λ Λmag = −µ . (3.3) responds to adding a term µ Tr(mMmag)tothe superpotential (2.4) in the magnetic theory (cf The effective electric theory is described by the figure 1). As our treatment of SQCD is restricted mesons M and the baryons B, B¯. One can check to the (Wilsonian) low-energy effective action we that the ’t Hooft anomaly matching conditions have to integrate out the massive modes from [19] are satisfied for this low-energy spectrum. the deformed model. In the electric theory this These conditions require that if one gauges the just leads to a reduction of the number of quark global symmetries of the theory then the values of flavors by p. Therefore the low-energy theory is the various triangle anomalies (which in general an SU(Nc) gauge theory with Nf − p quark fla- will not vanish) must coincide for the microscopic ˆ vors and vanishing superpotential W =0.In description in terms of quarks and gluons and the the magnetic theory, integrating out the massive macroscopic description in terms of mesons and M components of mag leads to non-vanishing ex- baryons. Performing this calculation one finds pectation values for q,¯qand thus the gauge sym- that the mesons and baryons obtained from the metry is broken spontaneously. The low-energy duality mapping (3.2) are just the right degrees theory is an SU(Nf −Nc −p) gauge theory with of freedom to match the global anomalies of the N − p Wˆ f quark flavors and superpotential mag = microscopic theory. This means that the theory Mˆ qq mag ˆˆ¯, where hats denote the low-energy fields. is in the confining phase [1], in agreement with One finds that the two effective theories are again the result for Nf = Nc+1 of the previous section. dual to each other [11], as shown in figure 1. It is easy to determine the full superpotential of It is interesting to consider the special case the effective electric theory by applying the dual- p = Nf − Nc − 1. Then one has an effective ity mapping (3.2) to the magnetic superpotential SU(Nc) gauge theory with Nc + 1 quark flavors (3.1). Using the scale relation (3.3) one finds on the electric side. The magnetic gauge sym- BMB¯ − det M metry is completely broken by the Higgs effect. Wˆ = , (3.4) 2Nf −3 However, one color component of each of the Λ Nc +1 quark flavors stays massless after the sym- which coincides with the superpotential (2.3) found metry breaking. These 2(Nc + 1) gauge singlets in the previous section for Nf = Nc +1.

3 Trieste Meeting of the TMR Network on Physics beyond the SM Matthias Klein

duality SU(Nc),Nf flavors ←−−−−−−→ SU(Nf − Nc),Nf flavors W =Tr( mM) Wmag = Mmagqq¯+ µ Tr(mMmag)       y y duality SU(Nc),Nf −p flavors ←−−−−−−→ SU(Nf − Nc − p),Nf −p flavors

Wˆ =0 Wˆmag = Mˆ magqˆˆq¯

Figure 1: The electric theory is deformed by adding mass terms for some of the quarks and the magnetic theory is deformed correspondingly. After having integrated out the massive modes one ﬁnds two eﬀective theories that are again dual to each other. Displayed are the tree-level contributions to the superpotentials.

4. Generalizations An Sp(2Nc) gauge theory with 2Nf quarks Q transforming in the fundamental representa- The results on SQCD described in the previous tion and vanishing tree-level superpotential can sections have been generalized to other gauge equivalently be described [22] by an Sp(2(Nf − theory models involving different gauge groups 2 − Nc)) gauge theory with 2Nf quarks q and and/or matter fields transforming in representa- Nf (2Nf −1) meson singlets Mmag that couple to tions other than the fundamental. For each of the magnetic quarks via Wmag = Mmagqq.The these models the electric theory shows confine- duality mapping between the gauge invariant op- ment without breaking of the chiral symmetry erators of the two theories is simply given by when the gauge symmetry of its magnetic dual M ↔ µMmag; there are no baryons in symplec- is completely broken. tic gauge theories. For Nf = Nc +2 the magnetic theory is completely higgsed, and one finds that 4.1 other gauge groups the electric theory confines. The confining spectrum (mesons M) can be obtained from the ef- The simplest extension of the results on SQCD fective magnetic theory via the duality mapping. described above consists in gauge theories with To obtain the correct confining superpotential orthogonal or symplectic gauge groups. Let us M first consider an SO(Nc) gauge theory with Nf W Pf = N − (4.2) matter fields Q (quarks) transforming in the vec- Λ2 f 3 tor representation of the gauge group and vanish- more care is needed, because it is due to instan- ing tree-level superpotential. This has a dual de- ton corrections in the effective magnetic gauge scription [11, 20] in terms of a magnetic SO(Nf + theory. 4−Nc) gauge theory with Nf quarks q transform- 1 N N 4.2 gauge theories containing tensor fields ing in the vector representation and 2 f ( f +1) meson singlets Mmag that couple to the magnetic For any N = 1 supersymmetric model with van- quarks via Wmag = Mmagqq. The gauge invari- ishing tree-level superpotential the form of the ant operators of both theories are in one-to-one most general superpotential that can possibly be correspondence to each other by a mapping very generated by non-perturbative effects is complete- similar to (2.5). For Nf = Nc − 3 the magnetic ly fixed by the requirement that it be invariant theory is completely higgsed, and one finds that under all symmetries of the considered model the electric theory confines. The confining spec- [18, 23, 2]. For a theory with gauge group G trum (mesons M and baryons B)aswellasthe and chiral matter fields φl in representations rl correct confining superpotential of G and dynamically generated scale Λ one finds

Q 2 µ MBB (φl) l ∆ W = (4.1) W ∝ l , (4.3) Λ2Nf+3 Λb can be obtained from the eﬀective magnetic the- where µl is the (quadratic) Dynkin index of the ory via the duality mapping. representation rl, µG denotes the index of the

4 Trieste Meeting of the TMR Network on Physics beyond the SM Matthias Klein

P µ − µ b N adjoint representation,P ∆ = l l G and = silon tensor of rank c for the baryons. In ad- 1 µ − µ 2 (3 G l l) is the coefficient of the 1-loop dition the flavor indices of the baryons are con- β-function. In general the complete non-pertur- tractedwithanepsilontensorofrankkNf leav- bative superpotential consists of a sum of terms ing 2Nf independent baryons. It is easy to see of the form (4.3) with different possible contrac- [14] that these are exactly the degrees of free- tions of all gauge and flavor indices. The rela- dom that are mapped under duality on the mag- tive coefficients of these terms cannot be fixed by netic singlets Mˆ mag,j,ˆq,ˆq¯that stay massless after symmetry arguments but must be inferred from the Higgs effect. The matching of the ’t Hooft a different reasoning. anomalies between the microscopic (i.e. quarks If the theory is confining at every point of and gluons) and the macroscopic (i.e. confined) the moduli space then the superpotential must description of the electric theory can thus be seen either vanish or be a smooth function of the con- as a consequence of the anomaly matching be- fined degrees of freedom [2]. All such models with tween the electric and the magnetic theory. In a smooth confining superpotential could be clas- this sense confinement can be derived from dual- sified [2] as they have to verify the constraint ity. ∆ = 2. But only for some of these smoothly It is straightforward to apply this idea to all confining gauge theories containing tensor fields gauge theory models of [10, 15] based on simple a dual description in terms of magnetic variables gauge groups. One first builds the gauge invari- is known. ant composite operators whose expectation val- On the other hand many dualities for mod- ues span the moduli space, then finds the duality els including tensor fields have been found once mapping between the electric and the magnetic an appropriate tree-level superpotential for the theory for these operators and finally applies this tensors is added. Let us review one example mapping to the completely higgsed effective mag- first studied by the authors of [6]. They con- netic theory to obtain the confined spectrum of sidered an SU(Nc) gauge theory with Nf quark the electric theory. In addition, in most cases flavors Q, Q¯, an additional matter field X in the at least some of the terms of the confining su- adjoint representation and a tree-level superpo- perpotential can be determined from the mag- k+1 tential Wtree =TrX ,wherek>1issome netic tree-level superpotential. To constrain the integer. This model has a dual description in possible form of the confining superpotential we terms of an SU(kNf − Nc) gauge theory with would like to generalize the formula (4.3) to the 2 Nf quark flavors q,¯q, an adjoint tensor Y , kNf case of a non-vanishing tree-level superpotential. singlets Mmag,j, j =0,...,k −1 and tree-level Therefore divide the matter fields into two sub- ¯ ˆ ¯ ˆ superpotential sets {φl} = {φ¯l}∪{φˆl},with{φ¯l}∩{φˆl} =∅, Xk−1 and add a tree-level term for the hatted fields: k j Y n W Y +1 M qY q. lˆ mag =Tr + mag,k−1−j ¯ (4.4) ˆ Wtree = h φˆl , (4.6) j=0 ˆl For Nc = kNf − 1 the magnetic theory is com- where h is a dimensionful coupling parameter pletely higgsed and one expects the electric the- and the n are positive integers. To be invariant ory to confine. Indeed, one finds that the ’t Hooft ˆl under all global symmetries the full superpoten- anomaly matching conditions are satisfied if the tial must be of the form [14] confined spectrum of the electric theory is given Q µ α φ¯ l¯ Y βˆ by [13] W ∝ l¯( ¯l) φˆ l hγ , b ˆl (4.7) j Λ Mj = QX Q,¯ j =0,...,k−1, (4.5) lˆ

Nf k−1 Nf k Nf−1 B =(Q) ···(X Q) (X Q) , where the powers α, βˆl, γ must verify the follow- B¯ =(Q¯)Nf ···(Xk−1Q¯)Nf (XkQ¯)Nf−1 , ing relations: γ −1α , where the gauge indices are contracted with a =1 2 ∆

Kronecker delta for the mesons and with an ep- βˆl =µˆlα+γnˆl . (4.8)

5 Trieste Meeting of the TMR Network on Physics beyond the SM Matthias Klein

SU(Nc) tensors adj + + + k+1 k+1 k+1 2(k+1) Wtree X (XX¯) (XX¯) (XX¯)

Nc kNf − 1 (2k +1)Nf −4k−1 (2k +1)Nf +4k−1 (4k +3)(Nf +4)−1 α k 1 2(k +1) 2(k +1)

βX (k−1)Nc k(Nf − 1) (k + 1)(2k(Nf +2)−1) 2(k+1)((2k+1)(Nf +4)−3)

βX¯ k(Nf − 1) (k + 1)(2k(Nf +2)−1) 2(k+1)((2k+1)(Nf +4)+1)

γ −Nc 3 − Nf −(Nc + Nf +4) −2(k +1)(Nf +4)

Sp(2Nc) SO(Nc) tensors 2(k+1) k+1 2(k+1) k+1 Wtree X X X X

Nc (2k +1)Nf −2 k(Nf −2) (2k +1)Nf +3 k(Nf +4)−1 α 2k+1 1 1 k

β 2k(Nc +1) (k−1)(Nf − 1) 2k(Nf +1)+4 (k−1)(Nc − 2k)

γ −(Nc +1) 3−Nf 1−Nf −(Nc +2k)

Table 1: Gauge theories that confine in the presence of a tree-level superpotential. The microscopic spectrum consists of Nf quark flavors and additional fields transforming in tensor representations represented by their Young tableaux. The coefficients α, β, γ refer to the powers in the non-perturbative superpotential (4.7).

The calculation of the confining spectra and 2Xk+1 X2k n n N − , the confining superpotentials for all simple group with j + ¯j = c 4 (5.1) j=0 j=0 models of [10, 15] is performed in [14]. The re- N n Nc−2n c sults are displayed in tables 1 and 2. Bn = X Q ,n=0,..., , 2 n¯ Nc−n¯ Nc−n¯ B¯n¯ =X¯ Q¯ Q¯ , n¯ =0,...,Nc , 5. A new confining model i Ti =Tr(XX¯) ,i=1,...,2k+1, To illustrate the ideas presented above I would where the gauge indices are contracted with one like to treat one of the models of [14] in more (···) epsilon tensor for the B¯ , Bn and with two ep- detail. Consider an SU(Nc) gauge theory with silon tensors for the B¯n¯ . Nf + 8 quarks Q, Nf antiquarks Q¯,ananti- The authors of [10] found a dual description symmetric tensor X and a conjugate symmetric of this model in terms of a magnetic tensor X¯ and tree-level superpotential Wtree = k h Tr(X X¯ )2( +1). This is a chiral theory and the SU((4k +3)(Nf +4)−Nc) difference between the number of quarks and an- gauge theory, with Nf + 8 quarks q, Nf anti- tiquarks is required by anomaly freedom. The quarksq ¯, an antisymmetric tensor Y , a conjugate gauge invariant composite operators are given by symmetric tensor Y¯ and singlets Mmag,j, Pmag,r, ¯ ¯ ¯ ¯ ¯ ¯ Pmag,r and tree-level superpotential Mj = QQ(j) ,Pr=QXQ(r) , Pr = QXQ(r) , ¯ j ¯ ¯ j ¯ W −h X X¯ 2(k+1) ... , with Q(j) =(XX) Q, Q(j) =(XX) Q, mag = Tr( ) + j =0,...,2k+1,r=0,...,2k, where the dots indicate terms involving Mmag, (¯n0,...,n¯2k,n0,...,n2k+1) k 2 B¯ =(X¯(XX¯) Wα) Pmag, P¯mag. The duality mapping for the gauge ¯ n¯0 ¯ n¯1 ¯ n¯2k invariant operators is given by ·(XQ) (XQ(1)) ···(XQ(2k)) n ·Q¯n0 Q¯n1 ···Q¯ 2k+1 , M ,P,P¯ ↔ M ,P , P¯ , (1) (2k+1) j r r mag,j mag,r mag,r

6 Trieste Meeting of the TMR Network on Physics beyond the SM Matthias Klein

SU(Nc) tensors 2 adj adj + + adj + + adj + + k+1 2 k+1 k+1 k+1 Wtree X + XY X + XYY¯ X + XYY¯ X + XYY¯

Nc 3kNf − 1 3kNf − 5 3kNf +3 3k(Nf +4)−1 α 3k k 4k 2k

Sp(2Nc) SO(Nc) tensors 2 + 2 + k+1 2 k+1 2 k+1 2 k+1 2 Wtree X + XY X + XY X + XY X + XY

Nc 3kNf − 4k − 2 3kNf − 4k +2 3kNf +8k+3 3kNf +8k−5 α (k) (3k) 3k k

Table 2: Gauge theories that confine in the presence of a tree-level superpotential. The microscopic spectrum consists of Nf quark flavors and additional fields transforming in tensor representations represented by their Young tableaux. The coefficient α refers to the power in the non-perturbative superpotential (4.7).

B¯(¯ni,nj ) ↔ B¯(¯mi,mj ) , B ¯b B mag the baryons , . A large VEV of breaks the gauge symmetry to Sp(2((2k +1)(Nf +4)−2)) with mj = Nf − n2k+1−j , [10]. The low-energy theory contains 2(Nf +4) m¯ j = Nf +8−n¯ k−j , (5.2) 2 quarks Q, a symmetric tensor X and tree-level B ↔ B , n mag,m superpotential Tr X2(k+1). This model is known with m =(2k+1)(Nf +4)−2−n, to show confinement [5]. A large VEV of ¯b breaks SO k N B¯n¯ ↔ B¯mag,m¯ , the gauge symmetry to (2(2 +1)( f +4)+3) [10]. The low-energy theory contains 2(Nf +4) withm ¯ =2(2k+1)(Nf +4)+4−n.¯ quarks Q, an antisymmetric tensor X and tree- The dependence on the mass scale µ has been level superpotential Tr X2(k+1). This model is suppressed. known to show confinement [5]. For Nc =(4k+3)(Nf +4)−1 the magnetic The effective low-energy superpotential of the theory is completely higgsed and the electric the- magnetic theory contains the terms M2k+1qq¯ + ory confines with low-energy spectrum given by P2kqYq¯ . We thus expect that the confining su- the composite fields perpotential of the electric theory has terms pro- portional to BM¯ k B, BBP k¯b. The detailed M ,P,P¯, 2 +1 2 j r r analysis gives [14] j=0,...,2k+1,r=0,...,2k, BM¯ B B ≡B , 2k+1 (2k+1)(Nf +4)−2 W = h2(k+1)(Nf +4) Λ2(k+1)((8k+5)(Nf +4)−2) B¯ ≡ B¯(Nf +8,...,Nf +8,Nf ,...,Nf ,Nf −1) , (5.3) BBP k¯b + 2 (5.4) ¯b ≡ B¯ , 2(Nf +4)−1 2((8k+5)(Nf +4)−2) 2(2k+1)(Nf +4)+3 h Λ + ... , of eqs. (5.1). From (5.2) one finds the mappings B,B¯ ↔ q, q¯ and ¯b ↔ Y¯ . One color component of where the dots stand for possible further terms each of the fields q,¯q,Y¯together with the meson that could be generated by instanton effects in singlets are exactly the degrees of freedom that the completely broken magnetic gauge group. stay massless after breaking the magnetic gauge group. 6. Conclusion As a further consistency check let us consider deformations of the theory along the flat direc- I have shown how the non-Abelian duality of tions corresponding to large expectation values of N = 1 supersymmetric gauge theories discov-

7 Trieste Meeting of the TMR Network on Physics beyond the SM Matthias Klein

ered by Seiberg can be used to find new mod- D. Kutasov, A. Schwimmer, Phys. Lett. B 354 els that confine in the presence of an appropri- (1995) 315, hep-th/9505004. ate superpotential. This is a very interesting ap- [7] T. Hirayama, K. Yoshioka, Phys. Rev. D59 plication of the proposed duality because it en- (1999) 105005, hep-th/9811119. ables us to obtain non-perturbative results for [8] K. Intriligator, Nucl. Phys. B 448 (1995) 187, the electric theory by a perturbative calculation hep-th/9505051. in its magnetic dual. Confinement in the electric [9] R. G. Leigh, M. J. Strassler, Phys. Lett. B 356 theory can be understood from the Higgs phase (1995) 492, hep-th/9505088. of the magnetic theory. The confining spectrum [10] K. Intriligator, R. G. Leigh, M. J. Strassler, can easily be derived from the duality mappings Nucl. Phys. B 456 (1995) 567, hep-th/9506148. of gauge invariant operators. For SU and SO gauge groups one also obtains the form of the [11] N. Seiberg, Nucl. Phys. B 435 (1995) 129, hep-th/9411149. confining superpotential by applying these mappings to the magnetic tree-level superpotential. [12] G. ’t Hooft, in “High Energy Physics”,Proc. To determine the full confining superpotential of the EPS International Conference, edited one needs to include instanton corrections in the by A. Zichichi, Editrici Compositori (1976); S. Mandelstam, Phys. Rep. 23 (1976) 245. completely broken magnetic gauge group. For Sp gauge groups the tree-level superpotential of [13] D. Kutasov, A. Schwimmer, N. Seiberg, Nucl. hep-th/9510222 the completely higgsed magnetic theory vanishes. Phys. B 459 (1996) 455, . In this case the whole magnetic superpotential is [14] M. Klein, to appear in Nucl. Phys. B, non-perturbative and therefore more difficult to hep-th/9812155. obtain. [15] J. H. Brodie, M. J. Strassler, Nucl. Phys. B 524 (1998) 224, hep-th/9611197. Acknowledgments [16] K. Intriligator, N. Seiberg, Nucl. Phys. 45 (Proc. Suppl.) (1996) 1, hep-th/9509066. I would like to thank the organizers for this very [17] E. Witten, Nucl. Phys. B 202 (1982) 253. interesting workshop and for their warm hospi- [18] I. Affleck, M. Dine, N. Seiberg, Nucl. Phys. B tality. I would also like to thank L. Ibáñez, 241 (1984) 493; Nucl. Phys. B 256 (1985) 557. C. Csáki and H. Murayama for helpful discus- sions and e-mail correspondence. This work is [19] G. ’t Hooft, in “Recent Developments in Gauge supported by the European Union under grant Theories”, Proc. of Cargèse Summer School 1979, edited by G. ’t Hooft et al., Plenum Press, FMRX-CT96-0090. New York (1980). References [20] K. Intriligator, N. Seiberg, Nucl. Phys. B 444 (1995) 125, hep-th/9503179. [1] N. Seiberg, Phys. Rev. D49(1994) 6857, [21] P. Pouliot, Phys. Lett. B 367 (1996) 151, hep-th/9402044. hep-th/9510148. [2] C. Csáki, M. Schmaltz, W. Skiba, Phys. Rev. [22] K. Intriligator, P. Pouliot, Phys. Lett. B 353 Lett. 78 (1997) 799, hep-th/9610139; Phys. (1995) 471, hep-th/9505006. Rev. D55(1997) 7840, hep-th/9612207. [23] K. Intriligator, R. G. Leigh, N. Seiberg, Phys. [3] B. Grinstein, D.R. Nolte, Phys. Rev. D57 Rev. D50(1994) 1092, hep-th/9403198. (1998) 6471, hep-th/9710001; Phys. Rev. D58 (1998) 045012, hep-th/9803139. [4] G. Dotti, A. V. Manohar, Phys. Rev. Lett. 80 (1998) 2758, hep-th/9712010. [5] C. Csáki, H. Murayama, Phys. Rev. D59(1999) 065001, hep-th/9810014. [6] D. Kutasov, Phys. Lett. B 351 (1995) 230, hep-th/9503086;