Theoretical Physics Institute
University of Minnesota
TPIMINN95/18T
UMNTH135095
OUTP9525P
hepth/9507170
(revised)
Matching Conditions and Duality in SUSY Gauge Theories in the Conformal Window
Ian I. Kogan, Mikhail Shifman, and Arkady Vainshtein
Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP , UK ^{1}^{1}1permanent address
Theoretical Physics Institute, Univ. of Minnesota, Minneapolis, MN 55455, USA
Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia
Abstract
We discuss duality in SUSY gauge theories in Seiberg’s conformal window, . The ’t Hooft consistency conditions – the basic tool for establishing the infrared duality – are considered taking into account higher order corrections. The conserved (anomaly free) current is built to all orders in . Although this current contains all orders in the ’t Hooft consistency conditions for this current are shown to be oneloop. This observation thus justifies Seiberg’s matching procedure. We also briefly discuss the inequivalence of the “electric” and “magnetic” theories at short distances.
1 Introduction
In this paper we discuss the infrared duality between different (“electric” and “magnetic”) SUSY gauge models observed in Ref. [1]. Supersymmetric (SUSY) gauge theories are unique examples of nontrivial fourdimensional theories where some dynamical aspects are exactly tractable. The first results of this type – calculation of the gluino condensate and the GellMannLow function – were obtained in the early eighties [2, 3]. The interest to the miraculous features of the supersymmetric theories was revived after the recent discovery [1], [4][6] of a rich spectrum of various dynamical scenarios that may be realized with a special choice of the matter sector (for a review see Ref. [7]). The basic tools in unraveling these scenarios are:

instantongenerated superpotentials which may or may not lift degeneracies along classically flat directions [8];

various general symmetry properties, i.e. the superconformal invariance at the infrared fixed points and its consequences [1].
A beautiful phenomenon revealed in this way is the existence of a generalized “electricmagnetic” duality in [12] and some versions of theories [1].
In Ref. [1] it was argued that and gauge theories with flavors (and a specific Yukawa interaction in the “magnetic” theory) flow to one and the same limit in the infrared asymptotics. If both theories are conformal – this is the so called conformal window. In other words, for these values of the GellMannLow functions of both theories vanish at critical values of the coupling constants. In particular, for the “electric” gauge theory with flavors in the fundamental representation the function corresponding to the gauge coupling has the following form [2, 3]:
(1) 
where is the anomalous dimension of the matter field,
(2) 
The critical value of the coupling constant is determined by the zero of the function,
(3) 
According to Ref. [1] two dual theories, to be discussed below, have the following content: the first one (“electric”) has gauge group and massless flavors while its dual theory (“magnetic”) has gauge group, the same number of massless flavors (but with different quantum numbers) plus colorless massless “meson” fields [1]. The electric and magnetic theories are supposed to have one and the same infrared limit (although their behavior at short distances is distinct; see below). Moreover, the electric theory is weakly coupled near the right edge of the window where the magnetic one is strongly coupled, and vice versa.
The main tool used in [1] for establishing the infrared equivalence is the ’t Hooft consistency condition [13]. As was first noted in [14] the chiral anomaly implies the existence of the infrared singularities in the matrix elements of the axial current (and the energymomentum tensor) which are fixed unambiguously. Therefore, even if we do not know how to calculate in the infrared regime the infrared limit of the theory should be arranged in such a way as to match these singularities.
The standard consideration is applicable only to the so called external anomalies. One considers the currents (corresponding to global symmetries) which are nonanomalous in the theory per se, but acquire anomalies in weak external backgrounds. For instance, in QCD with several flavors the singlet axial current is internally anomalous – its divergence is proportional to where is the gluon field strength tensor. The nonsinglet currents are nonanomalous in QCD but become anomalous if one includes the photon field, external with respect to QCD. The anomaly in the singlet current does not lead to the statement of the infrared singularities in the current while the anomaly in the nonsinglet currents does. Thus, for the ’t Hooft matching one usually considers only the set of external anomalies.
At first sight in the conformal window the set of the external anomalies includes extra currents due to the vanishing of the function at the conformal point, . In the framework of supersymmetry it means that the trace of the stress tensor vanishes as well as the divergence of some axial current entering the same supermultiplet as the stress tensor . Therefore, the idea that immediately comes to one’s mind is that the standard matching conditions should be supplemented by the new singlet axial current. Actually, this was the starting point in the first version of this paper. The point is false, however.
Our analysis shows that:
(i) The number of the matching conditions at the conformal point is not expanded and is the same as in Ref. [1].
(ii) However, what changes is the form of the conserved current, to be used in the matching conditions; coefficients in the definition of the conserved current are dependent, i.e. are affected by higher loops.
(iii) Although the current is different from the naive one (where is set equal to zero) consequences for the matching conditions and superpotentials remain intact provided one takes into account higher order corrections consistently everywhere, together with the specific form of the NSVZ function. The crucial observation is the fact that the conserved current which includes all orders in coupling constants still yields the ’t Hooft consistency conditions with no higher loop corrections. The fact that higher orders in have no impact in some relations is due to the existence of a new type of holomorphy in the effective Lagrangian for the anomalous triangles in external fields.
The paper is organized as follows. In Sec. 2 we briefly review those results of Ref. [1] which are relevant for our analysis, introducing notations to be used throughout the paper. In Sec. 3 we discuss the construction of the conserved (anomaly free) current to all orders in the coupling constants. Sec. 4 treats the anomaly matching conditions at the multiloop level. It is shown here that the higher order corrections present in the current are canceled in the ’t Hooft consistency conditions for the external backgrounds. Sec. 5 explains the cancellation of the higher order corrections in the triangles for the current to the baryon currents. In Sec. 6 we discuss the selection rules for the superpotentials. In Sec. 7 we comment on inequivalence of the electric and magnetic theories at short distances. Sec. 8 is devoted to incorporating the Yukawa couplings in the analysis of the infrared fixed points. The anomalous dimension of the field is derived here from the requirement of the conformal symmetry in the infrared limit.
2 Oneloop anomaly matching condition
The action of the electric theory is
(4) 
where and are the matter chiral superfields in the and color representations, respectively. The subscript is the flavor index running from 1 to . The theory has the following global symmetries free from the internal anomalies:
(5) 
where the quantum numbers of the matter multiplets with respect to these symmetries are as follows
0  1  
0  1 
Table 1
The transformations act only on the matter fields in an obvious way, and do not affect the superspace coordinate . As for the extra global symmetry it is defined in such a way that it acts nontrivially on the supercoordinate and, therefore, acts differently on the spinor and the scalar or vector components of superfields. The charges in the Table 1 are given for the lowest component of the chiral superfields.
The notion of the symmetries was originally introduced in Ref. [15]. The current considered in Ref. [1] is a conserved current that is free from the triangle anomaly at the oneloop level. At the classical level there are two conserved axial currents. One of them – sometimes the corresponding symmetry is called – is the axial current entering the same supermultiplet as the energymomentum tensor and the supercurrent. The current is classically conserved if all matter fields are massless; at the quantum level, generically, it acquires the internal anomaly proportional to , see Refs. [17, 16, 9]. Another one is the flavor singlet current of the matter field. The anomaly of the latter current is purely oneloop. First it was shown in Ref. [17], later an independent analysis of this anomaly was carried out by Konishi et al [18, 19]. The corresponding expression is usually called the Konishi relation.
Seiberg’s charge refers to a combination of these two currents chosen in such a way as to ensure cancellation of the internal anomaly at one loop.(Let us parenthetically note the simplest example of conserved current in the Abelian gauge theory was found long ago in Ref. [17].) This nonanomalous symmetry transforms superfields in the following way:
(6) 
In the component form the transformations are
(7) 
where and are the scalar and fermion components of chiral superfields and .
The conserved current is defined in [1] as
(8) 
where and are the standard spinor indices and . Here and below the contribution of the scalars in the currents is consistently omitted.
It was suggested [1] that for there is another (magnetic) theory with the same number of flavors but different color group, , in which one has an additional “meson” supermultiplets . (Below, to distinguish the quark and gluon fields of the magnetic theory from those of the electric one the former will be marked by tilde.)
The quantum numbers of the new chiral quark superfields and the meson superfield with respect to the global symmetries are as follows:
Table 2
where the quantum numbers for meson field are defined from the superpotential
(9) 
and the conserved current must be defined as
(10) 
where and are dual gluino and quarks and are the fermions from the supermultiplet .
Let us also note that if the number of the dual colors is the same as , i.e. , then the charge of is zero. At this point, , the electric and magnetic theories look selfdual. As we will see shortly, the actual situation is more complicated. The fermions do not decouple from the current defined beyond one loop.
The electric and magnetic theories described above are equivalent in their respective infrared (IR) conformal fixed points – with the choice of the quantum numbers above the highly nontrivial ’t Hooft anomaly matching conditions for the currents corresponding to are satisfied. If the theory is weakly coupled at the conformal point the dual theory will be strongly coupled, and it is only the anomaly relations which can be compared, because they can be reliably calculated in both theories. The presence of fermions from the meson multiplet is absolutely crucial for this matching. Specifically, one finds for the oneloop anomalies in both theories [1]:
(11) 
where the constants and were introduced in [1] and are related to the traces of three and two generators.
For example, in the anomaly in the electric theory the gluino contribution is proportional to and that of quarks to ; altogether as in (11). In the dual theory one gets from gluino and quarks another contribution, . Then the fermions from the meson multiplet (see Eq. (10)) add extra , which is precisely the difference.
The last line in Eq. (11) corresponds to the anomaly of the current in the background gravitational field,
(12) 
In the electric theory the corresponding coefficient is
while in the magnetic theory it is from quarks and gluinos and from the fermions, i.e. in the sum again .
As was discussed above, in accordance with the standard logic, the set of the matching conditions above includes only those currents that do not have internal anomalies. The number of the matching conditions is rather large and the fact they are satisfied with the given field content is highly nontrivial.
3 Conserved currents
In the consideration above it was crucial that there exists a singlet axial current ( current) whose conservation is preserved at the quantum level. The particular form of the current (8) assumes that the coefficients are independent numbers. We will show below that this is not the case if higher loops are included and we will determine the coefficients in terms of the anomalous dimensions of the matter fields. This definition is consistent with the fact that the anomaly in the divergence of the current is multiloop.
Let us consider first the electric theory. At the classical level there exist two conserved singlet currents. The first one is the member of the supermultiplet containing the stress tensor and the supercurrent [20], it has the following universal form:
(13) 
The current is the lowest component of the superfield (see Ref. [16]),
(14) 
The current (13) corresponds to the transformation of the superfields
(15) 
In components this means
(16) 
this symmetry exists in the presence of the Yukawa couplings of the form .
The second classically conserved current, , which we will refer to as the Konishi current, is built from the matter fields only:
(17) 
Note, that although the Konishi current (17) superficially looks identical to the second term in equation (13), actually they are different – the (omitted) contributions of the scalars in (13) and (17) are different. The current (17) is the lowest component of the superfield
(18) 
The Konishi current corresponds to the transformation of the superfields:
(19) 
Both currents and have anomalies at the quantum level. The anomaly in the current is multiloop and in the current is oneloop. In the operator form the anomalies can be written as follows [9]:
(20) 
and
(21) 
Here the anomalous dimension is defined as
and in one loop is given by equation (2). Equation (21) is the Konishi anomaly [18][19]. The second term in the righthand side of equation (20) is due to higherloop effects and represents the violation of the holomorphy of the effective Lagrangian.
By virtue of the Konishi anomaly (21) the second term in the righthand side of equation (20) is transformed into the same gauge anomaly. The corresponding divergence in the current looks as follows:
(22) 
where the coefficient in the square bracket is the numerator of the NSVZ function (1). The denominator will appear after taking the matrix element of the operator .
Let us write down in parallel the anomaly in the matter current
(23) 
From these anomalies we easily recover the form of the only conserved current in the theory
(24) 
One could give an alternative derivation of the very same conserved current considering the mixing matrix we will between and currents which arise already at the oneloop level. Diagonalizing the mixing we can find two renormalization group (RG) invariant currents, one of which coincides with the conserved and the second one, which is not conserved, is (see Ref. [9] for details).
Consider now how the conserved current looks like in two limits. In the conformal point, when the coefficient in front of vanishes (see Eq. (3)). Thus in the infrared limit the current flows to . On the other hand in the extreme ultraviolet () limit (i.e. ) the current flows to the Seiberg current (8). Therefore the genuine current interpolates between the and currents.
Keeping in mind that in the magnetic theory there are two distinct matter fields with a superpotential, let us generalize the procedure of construction of the conserved current to the case with some number of matter superfields and a nonvanishing superpotential .
The definition of the current is general since it has a geometrical nature,
(25) 
where is the fermionic component of the chiral superfield . In the presence of the superpotential there are two sources of the current nonconservation: the first source is possible classical nonconservation due to , the second one is the quantum anomaly. In the superfield notations we have the following generalization of the equation (20):
(26) 
where is the first coefficient of the function and invariants characterize the gauge group representation of the field (they are defined as where are the matrices of the group generators).
We can also construct the Konishi current
(27) 
for each superfield . The divergence of this current is given by the generalized Konishi relation:
(28) 
(A comment on the literature: the anomaly in the current (26) was expressed in terms of anomalous dimensions in Ref. [16, 9]; for a recent instructive discussion which includes the classical nonconservation [20], see Ref. [21]. )
Let us look for the conserved current as a linear combination of the currents and :
(29) 
The divergence of this current can be immediately found from Eqs. (26) and (28),
(30) 
where the subscript marks the component of the chiral superfield, in particular, . All terms proportional to occur by virtue of the substitution of the Konishi relation (28) into (26).
For the conserved current to exist both terms in Eq. (30), the superpotential term and the one with , must vanish. All higher order corrections reside in the anomalous dimensions which depend on the gauge coupling constant and the constants in the superpotential .
Let us first omit these higher order corrections, i.e. put . For a generic superpotential there may be no conserved currents at all. This situation is of no interest to us, so we assume that one (or more) conserved currents exist. Let us denote by the set of the coefficients ensuring the vanishing of at . (These coefficients are rational numbers which in many cases were found by Seiberg et al.)
Now let us switch on the higher order corrections, . It is crucial that the anomalous dimensions appear only in the combination . This means that the coefficients ensuring the current conservation at are different from only by a shift by ,
(31) 
Equation (24) given above is a particular example of this general result with . As another illustration let us consider the magnetic theory. In this case we have two Konishi currents,
(32) 
where the notations have been introduced in Sec. 2. Equation (10) gives the values of in this case:
(33) 
Higher order corrections will change these coefficients to
(34) 
where and are the anomalous dimensions of the fields and respectively. Thus, the extra terms in the current as compared to one (see Eq. (10)) are
(35) 
One more comment concluding this section. The conserved current is in the same supermultiplet with the stressenergy tensor only in the infrared limit. Thus the relation between the dimension and the chiral charge of the chiral superfield is valid only for the IR fixed point, but not for the UV one.
4 Cancellation of higherloop corrections for the external anomalies
Now, when the conserved current is constructed we can proceed to discussing the anomalies of this current in the background of weak external fields. As was mentioned in Sec. 1, these anomalies, via the ’t Hooft consistency conditions, constraint the infrared behavior of the theory, and, thus, crucial in establishing the electricmagnetic duality. In Ref. [1] the anomaly relations were analyzed at the oneloop level. Since the duality can take place only in the conformal points where the coupling constants are not small a consideration of higherloop corrections is absolutely crucial. As we see, for instance, from Eq. (26), generally speaking, higherloop corrections are present. Below we will demonstrate that in the external anomalies for the current constructed above all higher order contributions cancel out.
To warm up we begin our consideration of the multiloop effects starting from the example of triangle. The definition of the baryon charges for the electric theory is given in section 2. We have corrections both in the definition of the current (see Eq. (24)) and in the anomalous triangle.
If we introduce an external field coupled to the baryon current then the anomaly for the current in the electric theory takes the form similar to equation (22),
(36) 
where . Let us emphasize that corrections to the anomalous triangle do not vanish and enter through the anomalous dimension which is, in turn, related to the second term in the square brackets in Eq. (20). As far as the matter current is concerned here corrections to the bare triangle vanish,
(37) 
Assembling these two pieces together we find
(38) 
One can see that the external anomaly in the current has no corrections – those coming from the definition of the current are canceled by the corrections in the triangle containing . Effectively the naive oneloop anomaly is preserved in the full multiloop calculation in the case at hand.
As a matter of fact it is not difficult to generalize the assertion above to the case of arbitrary number of matter fields with a superpotential (e.g. in the magnetic theory we have two distinct sets of the matter fields – the magnetic quarks in the fundamental representation of the gauge group and color singlets).
Since the background field can be treated as an additional gauge field what is to be done, Eq. (30) must be supplemented by the extra term proportional to . Namely,
(39) 
where is equal to the trace of the square of the generators of the baryon charge, i.e. ( is the baryon charge of the field ). The analogy with the second term in Eq. (30) is quite transparent. The color generators are substituted by those of the baryon charge, and the coefficient in the square brackets is substituted by .
The coefficients in Eq. (30) are chosen in such a way that the righthand side of Eq. (30) vanishes. This vanishing is achieved provided , see Eq. (31). It is important that the very same combination, , enters Eq. (39). This means that the external baryonic anomaly reduces to
(40) 
Thus, the higher order corrections obviously drop out.
If the gravitational field is considered as external the calculation of the corresponding triangle is very similar to that in the external field. The only difference compared to the case of the baryon current is the substitution of by unity,
(41) 
In the electric theory while in the magnetic theory is substituted by and are given in Eq. (33).
The arguments can be of course repeated for the anomaly. What remains to be discussed is the anomalous triangle. This case is harder to consider along the lines presented here. However, in the next section we will give some arguments of a more general nature indicating that this anomaly is also oneloop.
5 Cancellation of higher orders and holomorphy in the external field
Now we will show that the result derived above – cancellation of the higher order corrections in the external anomalies – has a very transparent interpretation in the language of the effective action in the external field. As an example of the external field one can keep in mind the field interacting with the baryonic charge, gravitational field and so on. The Wilson effective action is as follows:
(42) 
where is the inverse bare coupling constant and is the normalization point, the coefficients are defined through the generators of the gauge group in the representation , , ( is the invariant in the adjoint representation). The masses and are the regulator masses (one can keep in mind the supersymmetric PauliVillars regularization). In the covariant computation is the mass of the (chiral) ghost regulators, is the mass of the field regulator. Usually, it is assumed that all regulator masses are the same, . We keep them different for the purposes which will become clear shortly. Finally, is a superfield generalizing the stress tensor of the external gauge field in the same way as generalizes (in the previous section where the external baryonic current was considered as an example we dealt with ), the coefficients are defined similar to for the generators corresponding to the interaction with the external field. The superpotential may or may not be present in each particular model.
The property of holomorphy in the Wilson effective action means that the coefficients in front of and are given by one loop; higher order corrections in the coupling constants are absent. Higher orders enter only the factors; in taking the background field matrix elements of the last term in Eq. (42) higher orders in penetrate the answer.
Taking matrix elements of the operator action we proceed to the cnumber functional , the generator of 1PI vertices. Let us first discuss what happens at oneloop level. Then the last two terms in Eq. (42) are irrelevant for the issue of anomalies under discussion, and the oneloop description is given by
(43) 
From this expression one can easily read off anomalies by varying the regulator masses. For instance, the anomaly of current is obtained by applying the operator
(44) 
to the righthand side of Eq. (43). The anomaly in the Konishi current (see Eq. (27)) is generated by .
Moreover, the same expression (43) demonstrates the existence of the conserved current . Indeed, the first term in is invariant under the action of the operator
(45) 
where coefficients were defined and discussed in Sect. 3. The noninvariance of the second term in under the action of the operator (45) gives the external anomaly of the current.
Now what happens if we proceed to higher loops? The occurrence of the factors in manifests itself in in the following way [9]:
(46) 
We write down here only the part of containing and ; the part with classical superpotential is omitted. The inclusion of higher orders resulted in substituting the regulator mass by . The role of for the ghost regulator mass is played by (see Ref. [9]).
Invariance of the part of still persists. However, the transformation under which it is invariant corresponds now to the action of the operator
(47) 
where
(48) 
The application of operator (48) to the part of yields the external anomaly. Since in this part is also replaced by it is clear that the external anomaly remains oneloop.
A direct correspondence between the discussion of the external anomalies in Sect. 4 and the one given in this section is quite clear. However, the arguments of this section help demonstrate the general nature of the phenomenon. In particular it seems possible to deduce that the anomaly of the type is also oneloop.
6 Superpotentials
We have demonstrated that the conserved current contains higher order terms. The question arises about the selection rules for different terms in superpotentials which were obtained without these complications [8].
The change of the form of the current does not mean that these selection rules were incorrect. The same phenomenon of the cancellation of the higher order corrections as was described above takes place in the transformation laws of the chiral fields.
To elucidate this assertion it is instructive to analyze the form factor diagrams where the chiral matter scatters off the current . Taken at the vanishing momentum transfer these graphs yield the charge of the matter field. At a naive level one would start from the naive current , with all terms discarded, draw the tree graph plus two oneloop graphs (the diagram with the vertex correction and the diagram with the correction to the external line), and then one would conclude that the two oneloop graphs cancel each other in the same way similar diagrams cancel each other in the electric current in QED. This naive conclusion would be wrong! If the calculation is done supersymmetrically, in terms of the supergraphs, getting a nonvanishing result for the sum of the two oneloop graphs mentioned above is inevitable. The residual sum of these graphs is canceled, however, when one adds the term of the current as the vertex insertion in the tree graph. Effectively this means that the charge of the chiral matter is determined by the tree graph with at the vertex, i.e. coincides with Seiberg’s answer. In other words the very same assertion can be phrased as follows: the commutator of the charge with the (bare) matter fields contains an anomalous part that cancels the terms in the definition of the conserved charge. In particular, the commutator with the modular fields