YITP-18-99

Aspects of Massive Gauge Theories on Three Sphere

in Infinite Mass Limit

Kazuma Shimizu^{*}^{*}*
kazuma.shimizu(at)yukawa.kyoto-u.ac.jp

: Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

We study the partition function of three-dimensional supersymmetric U() SQCD with massive matter multiplets in the infinite mass limit with the so-called Coulomb branch localization. We show that in the infinite mass limit the specific point of the Coulomb branch is chosen and contributes to the partition function dominantly. Therefore we can argue whether each multiplet included in the theory is effectively massless or not in this limit even on and conclude that the partition function becomes that of the effective theory on the specific point of the Coulomb branch in the infinite mass limit. In order to investigate which point of the Coulomb branch is dominant, we use the saddle point approximation in the large limit because the solution of the saddle point equation can be regarded as a specific point of the Coulomb branch. Then we calculate the partition functions for small rank and actually confirm that their behaviors in the infinite mass limit are consistent with the conjecture from the results in the large limit. Our result suggests that the partition function in the mass infinite limit corresponds to that of an interacting superconformal field theory.

###### Contents

## 1 Introduction

In three dimension, the Yang-Mills coupling has positive mass dimension. This means that three-dimensional Yang-Mills theories are super-renormalizable. The Yang-Mills term is irrelevant and can not contribute to the infrared physics independently of the gauge group and the matter content. It might be expected that 3d gauge theories flow to the non-trivial infrared fixed point, which depends on its matter content. In fact, U() QCD with massless flavor, where is some critical value, might flow to an interacting IR fixed point while with massless flavors the theory is expected to flow to a gapped phase in the IR [1, 2, 3]. The number of the flavors plays an important role to determine the IR structure of the 3d gauge theories. However, it is generally difficult to know the non-perturbative properties of such a theory.

Three-dimensional supersymmetric gauge theories have several interesting features which four-dimensional supersymmetric gauge theories do not have. Especially, we are interested in the fact that there are real parameters: real mass and FI parameters. These are not given by the background chiral superfields. Thus the dynamics triggered by the real parameter deformation is not restricted by the holomorphy. This means that there can occur non-trivial phase transitions. When we give the matter fields infinite mass, the massive matter fields decouple from the theory. Decoupling of the flavors changes IR physics and an interesting phase transition would occur. Moreover, supersymmetric gauge theories are known to have the exactly calculable quantities such as the partition function on compact manifold by the localization methods in three dimension [20, 21, 22, 23]. In this paper, we focus on the round three-sphere partition function, which is given by the matrix type finite dimensional integral. The localization methods admit the real mass and FI terms deformation by weakly gauging a global symmetry and giving the background field which couple its current an expectation value. Then we can approach the non-trivial dynamics triggered by the real mass parameter with the localization methods. In [4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 13, 15] the phase structure of the mass deformed gauge theories on is argued.

In this paper, we focus on U() SQCD with pairs of chiral multiplets in the fundamental and anti-fundamental representation of U(). These theories are classified in [16] by their low energy properties. The authors define three types of the theories; “good”, “ugly” and “bad” theories.
A 3d gauge theory is a good theory if all the monopole operators obey the unitarity bound. In this case, the R-symmetry in the IR is the same as that in the UV. For an U() SQCD it is a good theory when . A gauge theory is called “ugly” if the monopole operators satisfy the unitarity bound, however several monopole operators saturate it. This type of theories are likely to flow to an interacting SCFT with R-symmetry visible in the UV with the decoupled free sector consisted of the monopole operators which saturate the unitarity bound. An U() SQCD is an ugly theory when . In a bad theory, there are the monopole operators with zero or negative R-charge of the R-symmetry manifest in the UV. Because the monopole operators violate the unitarity bound of the UV R-symmetry, a bad theory flow to an interacting SCFT whose R-symmetry is not manifest in the UV. An U() SQCD becomes an bad theory when ^{†1}^{†1}†1Recent progress of ”bad” theories in terms of the geometry of the moduli space of vacua is seen in [17, 18, 19].. It is known that whether the partition function diverge or not is related with the criterion of ”bad” theories. The partition function of a “bad” theory is divergent [21]. This might be because the localization methods use the R-symmetry manifest in the UV to put the gauge theory on a compact manifold. Thus we should carefully treat with the number of the flavors.

Our aim of this paper is to study the partition function of real mass deformed theories in the infinite mass limit^{†2}^{†2}†2
The infinite mass limit of the matrix model of 3d gauge theories is also considered in [30, 24, 27, 25, 26] in the context of finding new examples of Seiberg-like dualities [28, 29].
. For example, we consider that we give real mass to enough matter multiplets of ”good” theory for the theory to become a ”bad” theory after the massive matter fields are decoupled. It could be naively thought that the massive matter multiplets will be decoupled from the theory in that limit. However, a matrix model of a ”bad” theory is not well-defined^{†3}^{†3}†3
The magnetic theory of a ”bad” theory in terms of the Seiberg-like duality is considered as a good theory [27].. It is interesting to know what happens to this matrix model in the infinite mass limit. Hence our interest is to know which hypermultiplets become effectively massless or massive in the infinite mass limit on three-sphere. In order to argue decoupling of matter fields, we must choose a vacuum when a theory is put on the flat space. However, there are no choices of vacuum of the theories on three-sphere. Especially, we calculate the sphere partition function with the help of so-called Coulomb branch localization and it is given by the integral over the classical Coulomb branch parameters. Namely, the three-sphere partition function is represented by the integrals of a part of vacua in terms of the theory on flat space. Thus it is not simple to argue whether the massive multiplets will be decoupled when we take infinite mass or not.

For example, we consider that U(2) SQCD with pairs of hypermultiplets with real mass . The figure 1 shows the real parts of the two classical Coulomb branch parameter and there are some special points. When we fix a generic point of the Coulomb branch (blue dot), the effective theory is U(1) U(1) with massive matter fields and W-bosons while on a specific point like a green or red point, the effective theory has or massless hypermultiplets respectably. The origin (black dot) is also special in the sense that the gauge symmetry is enhanced to U(2). It is non-trivial which points dominantly contribute to the three-sphere partition function in the infinite mass limit because all the points of the Coulomb branch can contribute to it, including generic and above special ones.

To investigate this, we focus on the solution of the saddle point equation because the solution corresponds to a classical Coulomb branch point and in the large limit the solution gives a dominant contribution to the sphere partition function. Hence we can argue decoupling of the massive matter fields and also which theory will appear as an effective theory on a point of the Coulomb branch. We deduce the effective theory from the solution of saddle point equation in the infinite mass limit and confirm that the solution of the saddle point equation of the effective theory coincides with that of the original massive theory in the infinite mass limit.

Investigating the solution of the saddle point equation is just a way to know which point of Coulomb branch gives the dominant contribution to the partition function in the infinite mass limit. Even when we do not take the large limit, it is expected that there also exists a dominant point of the Coulomb branch and the matrix model becomes a specific effective theory in the infinite mass limit. This is because the mass infinite limit also corresponds to the decompactified limit ^{†4}^{†4}†4The mass must appear as the combination in the partition function. So we can not distinguish the infinite mass limit and the decompactified limit. In our convention we take to 1.
and thus the point of the Coulomb branch should be chosen in this limit. We check this in the matrix models for small and confirm the effective theory is the same as that which we deduced from the calculations in the large limit. We conclude that this vacuum selection does not need the large limit, just only the infinite mass limit.

The rest of this paper is organized as follows: In section 2 we review the localization methods and introduce the building blocks of matrix models. In section 3 we solve the saddle point equation of SQCD with massless or massive matter fields and investigate the theory which appears in the infinite mass limit. In section 4 we calculate the partition function of finite rank SQCDs and evaluate the leading part in the infinite mass limit. In section 5 we end with a conclusion and some discussion. In appendix A, we introduce the techniques of the resolvent methods which we use in this paper to solve the saddle point equation in the large limit. In appendix B, we introduce mixed Chern-Simons terms which must appear in the infinite mass limit as 1-loop effects. We try to interpret what happens in the infinite mass limit in terms of the mixed Chern-Simons terms. In appendix C, we discuss the convergent bound of the matrix model and reconsider the matrix model of the effective theory in the infinite mass limit from the viewpoint of the convergence bound of it. In appendix D, we introduce an example which becomes ABJM theory in the infinite mass limit while it is just a SQCD when .

## 2 Localization and matrix model

In this paper, we investigate the round three-sphere partition function of gauge theories. It is given by the finite dimensional integral instead of the path integral with the help of the localization technique [20, 21, 22, 23]. To use the localization technique we put a gauge theory on with preserving supersymmetry and deform the action on by a Q-exact term , where Q is a generator of the supersymmetry. The partition function of the deformed action is independent of the deformation parameter. Thus we take the parameter to infinite and the path integral reduces to the finite dimensional matrix integral since the path integral is determined by the finite dimensional saddle point configuration in the field configuration space. Since the saddle point approximation is one-loop exact, the action of the matrix model is written by classical and 1-loop parts as

(2.1) |

where is the usual Vandermonde determinant and and are 1-loop parts from the vector multiplets and matter multiplets respectably. The integral valuable corresponds to a eigenvalue of the scalar fields of a vector multiplet.

### 2.1 Vector multiplet

We consider here so-called Coulomb branch localization, where the integral valuables correspond to the eigenvalues of the scalar fields of the vector multiplet. We only consider U() gauge theories in this paper. The Yang-Mills term can not contribute to the partition function since the Yang-Mills term is Q-exact. On the other hand, the Chern-Simons term can contribute to it as a classical contribution, but, we do not consider it in this paper. The 1-loop part of the vector multiplets is given by

(2.2) |

where the denominator cancels against Vandermonde determinant, which appears when we choose the diagonal gauge of . When there is U(1) part of the gauge group, the FI term can be introduced and contributes to the partition function as a classical term

(2.3) |

### 2.2 Matter multiplet

Next we consider the contributions of chiral multiplets. The chiral multiplets can contribute to the partition function through only one-loop parts because its kinetic and superpotential terms are Q-exact. The 1-loop parts of chiral multiplets are determined by the representation both of the gauge group and the flavor symmetry. By weakly gauging a flavor symmetry we can couple its current with a background vector multiplets in a supersymmetric way. Thus we can give the corresponding scalar an expectation value and regard it as a real mass for the chiral multiplets. Moreover we can give the chiral multiplets R-charge [22, 23]. However, we do not consider such a deformation in this paper and we consider the chiral multiplets have canonical dimension ^{†5}^{†5}†5 In an vector multiplet there exists an adjoint chiral multiplets in terms of the language. Then it seems that we must consider the 1-loop part of it. However, because its canonical R-charge is 1, the adjoint chiral multiplet does not contribute to the partition function without axial mass parameter [21, 23]..

The 1-loop part of the chiral multiplets in the representation of U() is given by

(2.4) |

where is a weight vector of the representation . In [22] the function is defined as

(2.5) |

Its remarkable property we will often use is

(2.6) |

We focus on SQCDs, which are super Yang-Mills theories with pairs of chiral multiplets in the fundamental and anti-fundamental representation of U(). In this paper, we consider following two mass deformation: In case (i) we give real mass to flavors while we give real mass to the remaining flavors^{†6}^{†6}†6where we assume is integer. .
This breaks each SU() of the flavor symmetry SU()SU() to SU()SU(). Its total 1-loop part is given by

(2.7) |

In case (ii) we give flavors real mass while we give other flavors real mass . Then we keep the remaining flavors massless. This real mass assignment breaks each of SU() flavor symmetry of the matter fields to SU()SU()SU()^{†7}^{†7}†7We assume is integer.. Its total 1-loop part of the chiral multiplets is given by

(2.8) |

## 3 Large solution and Coulomb branch point

### 3.1 SQCD with massless hypermultiplets

In this subsection, we solve the saddle point equation of U() SQCD with massless hypermultiplets for latter use. The solution is given as the eigenvalue density function , which determine the large behavior of the theory. Its partition function is written by

(3.1) |

It is difficult to calculate the partition function exactly because there are dimensional integral. In principle, it’s leading part in the large limit can be evaluated by the saddle point approximation. The saddle point equation for this theory is given by

(3.2) |

We assume that the eigenvalues become dense in the large limit and we take the continuous limit as follwoing:

(3.3) |

The planer part of this saddle point equation is rewritten as a singular integral equation^{†8}^{†8}†8
We represent a principal value integral as

(3.4) |

where we also took infinite with finite and introduced the density function , which determines the large behavior of the theory, defined as

(3.5) |

and this means that we regard the values of the eigenvalues as the fundamental variable. The density function counts the number of the eigenvalues which exists between and and satisfy the normalization condition which depends on how to take the continuous limit.

(3.6) |

In order to solve the equation (3.4) and obtain the density function , we use the resolvent methods. We give a brief summary of the resolvent methods in appendix A. We take , and and define the resolvent and the potential as

(3.7) | ||||

(3.8) |

where and are intervals and respectably. The resolvent is determined from the analyticity and the one-cut solution of the resolvent is given by (A) as

(3.9) |

where because of the symmetry of the saddle point equation. We should carefully consider the branch of the square root. For latter convenience we introduced the following function:

(3.10) |

The density function defined on is given by (A.8) as

(3.11) |

The end of the cut is determined by the asymptotic behavior of the resolvent from (3.9). The asymptotic behavior in is decided by the following equation:

(3.12) |

The solution is given by

(3.13) |

where this solution exist only when since the right-hand-side of (3.12) is always more than as a function of .

Here we consider the relation between this large solution and a point of the classical Coulomb branch. The equation (3.13) means that when we take to infinite, and become 0 since the radius is recovered as and . Thus the saddle point solution becomes condensed to the origin. Taking the radius to infinite corresponds to considering the theory on the flat space. So this solution corresponds to a point of the Coulomb branch of the theory on the flat space and it is the origin of the classical Coulomb branch. The origin of the Coulomb branch is most singular in the sense that all the massive W- boson on the generic point of Coulomb branch become massless. On this point, the theory at the deep IR of the RG flow expected to be an interacting superconformal field theory. It is expected that the sphere partition function of SQCD with massless hypermultiplets always represents that of a non-trivial SCFT.

The solution exists when . This reflects the bound of the convergence of the matrix model. In this paper, we will add real mass to matter fields while keeping the special flavor symmetry. Even for that case this bound always appears in our analysis.

#### 3.1.1 Adding FI term

Here we consider U() gauge theories with massless hypermultiplets with FI term. Especially, we consider imaginary FI terms in this section for the latter part of this paper, where it appears when we take the infinite mass limit as one-loop effects, which are certain mixed Chern-Simons terms. The density function is almost the same as that in the previous section. However, an FI term breaks the symmetry of the saddle point equation under the simultaneous change of the sign of all eigenvalues . For this theory the matrix model is written by

(3.14) |

where is a imaginary FI parameter in the sense that ordinary FI terms are considered as . This FI term can be considered as the R-charge of monopole operator since the real part of the monopole operator is , where is R-charge of the monopole operator [37, 34]. Its saddle point equation in the continuous limit is

(3.15) |

where we also take and infinite while keeping

(3.16) |

finite to solve the saddle point equation. We can solve this saddle point equation in the large limit by resolvent methods. We define resolvent and potential for this theory as

(3.17) | ||||

(3.18) |

The resolvent is obtained by the same calculation of that which appear in the previous section since a FI term does not change the singular structure of the resolvent,

(3.19) |

The density function is given by the equation (A.8) as

(3.20) |

where and is determined by the equation from the asymptotic behavior of at

(3.21) | ||||

(3.22) |

Because a FI term breaks the symmetry under which in the saddle point equation, the and do not satisfy the condition . The solution of (3.21) and (3.22) is given by

(3.23) |

From (3.21) and (3.22) we recognize that the solution only exists when

(3.24) |

This condition is same as the condition that the matrix model converges in the large limit. In appendix C, we will discuss the convergence bound of the matrix model of SQCDs.

### 3.2 SQCD with massive hypermultiplets

In this subsection, we consider U() SQCD with pairs of chiral multiplets with real mass by weakly gauging its flavor symmetry and coupling its current to vector multiplets as background fields so that the matrix model is given by

(3.25) |

When , this matrix model becomes that of U() with massless fundamental hypermultiplets. When we take the infinite mass limit, if the massive matter multiplets decouple, the matrix model is not well-defined. Therefore we investigate what happens to this matrix model of the theory in the infinite mass limit.

The saddle point equation is written as

(3.26) |

and in the continuous limit it becomes

(3.27) |

Next we define the resolvent and potential as

(3.28) | ||||

(3.29) |

where . The resolvent is determined by its analytic properties (A) as

(3.30) |

The density function is given by (A.8) as

(3.31) |

The constant and are decided by the symmetry and the asymptotic behavior when ,

(3.32) |

This equation immediately means that exists when . We conclude that this type of mass deformation does not affect the bound of the existence of the solution. Here is given by

(3.33) |

In this theory, when we take the infinite mass limit, it is naively considered that the theory goes to a bad theory and its matrix model diverges. However, this argument is not correct in the following sense: the density function has the peak around and the eigenvalues gather around these peaks as becomes large. Thus in the large limit the partition function of this massive SQCD corresponds to that of the effective theory on the point of the Coulomb branch points where the half of the eigenvalues sit on and the others sit on like as

(3.34) |

In fact, this argument is confirmed following direction: we assume that the eigenvalues are separated like as

(3.35) |

where we assume that and do not depend on . The saddle point equations (3.26) for the first eigenvalues are written as

(3.36) |

where we took the infinite mass limit in the second line and we note that the first term can be interpreted as the gauge-R mixed Chern-Simons term [32, 31, 33] induced by integrating out massive gauginos and complex fermions of chiral multiplets. For the latter eigenvalues the saddle point equation in the large mass limit is almost same as (3.36),

(3.37) |

The equations (3.36) and (3.37) mean that in the infinite mass limit the matrix model (3.25) becomes ^{†9}^{†9}†9The overall factor of the matrix model can not be determined in this procedure.

(3.38) |

because the saddle point equation is equivalent to (3.36) and (3.37). The factor is a part of the decoupled free massive degrees of freedom. We can evaluate . This part can not be read from the saddle point equations. This matrix model is made by the two U() with fundamental hypermultiplets FI parameter ^{†10}^{†10}†10
Exactly speaking, the FI parameter is given by if we recover the radius of because in three-dimensional theory a FI parameter has mass dimension 1.
SQCD theories. As we note before, the FI term is induced by one-loop effects as the mixed Chern-Simons term from vector multiplets of gauge and R-symmetry by integrating out the effectively massive fermions. We argue this point in appendix B. This FI term can not appear when we consider the gauge theories on the flat space.

In fact, we check our claim by comparing the density function of the matrix model of the effective theory (3.2) with that of the matrix model (3.25) in the infinite mass limit. First, we consider the density function of SQCD with massive hypermultiplets (3.31) in the infinite mass limit. We rewrite as and assume is order . This procedure corresponds to the simultaneous shift of by and focusing on the peak of the density function around . We have to consider the expansion of (3.33) around . it is given by

(3.39) |

Thus the density function is expanded around as

(3.40) |

where in the infinite mass limit. Then we compare this with that of the part of the matrix model (3.2) since part corresponds to the part concentrated around of the eigenvalues of the massive SQCD. The solution of its saddle point equation (3.36) is just given by the calculation of section 3.1.1. In this case, and are

(3.41) |

and its density function is

(3.42) |

where the additional factor due to the fact that the effective theory has two U() as the gauge group and the normalization condition should be taken as

(3.43) |

The density function (3.40) and (3.42) are completely equivalent. Next we should consider the part concentrated around . In this time we must rewrite in (3.31) and the density function is in this limit

(3.44) |

where . Then we consider the part of the (3.2). The solution of its saddle point equation is just given by applying the result of 3.1.1 to (3.37) and we obtain

(3.45) |

and its density function is

(3.46) |

This is the same as (3.44). Therefore we conclude that SQCD with massive hypermultiplets we study here become two SQCDs in the infinite mass limit: U() SQCD with massless hypermultiplets and FI term . This result suggests that if we assume the massive matter fields will be decoupled, the sphere partition function of a massive theory which would become a bad theory always become that of a specific effective theory. This means that an interacting SCFTs on the specific singular point of the Coulomb branch appears in the infinite mass limit instead of appearing of a bad theory. This result may also suggest that the massive theory can not be used as the UV regularization of the bad theory. In section 4 we will check our claim from the exact calculation of the partition function of finite rank SQCD. It is expected that the partition function can be written as the sector of the decoupled free massive multiplets and that of the effective theory in the infinite mass limit.

### 3.3 SQCD with massive and massless hypermultiplets

In the previous subsection, all matter fields of the theory are massive. In this subsection, we consider the SQCD theory with both massive and massless matter fields. It is expected that the asymptotic behavior of the partition function in the infinite mass limit depends on the number of the massless matter fields since the sufficient number of matter fields make the matrix model to converge.

We consider U() SQCD with pairs of massive hypermultiplets with and massless hyper multiplets. We assume is integer. The matrix model is given by

(3.47) |

The saddle point equation is given by

(3.48) |

and we take the continuous limit of this. It is written as

(3.49) |

We define the resolvent and potential as

(3.50) | ||||

(3.51) |

where . The resolvent is obtained by (A) as

(3.52) |

The cut is determined by the following asymptotic equation;

(3.53) |

Unfortunately there are generally no explicit forms of the solution because this equation corresponds to the octic equation of . However, we can know the solution of it numerically or in the infinite mass limit. The density function of this case is given by (A.8) as

(3.54) |

Let us consider what happens to the matrix model when the number of the matter fields varies. When , the matrix model is still well-defined after we take the infinite mass limit and all massive matter fields decouple from the theory. In fact, in this case, the limit which takes mass to infinity and the integrals of the matrix model commute ^{†11}^{†11}†11
In this paper, we focus on just the leading part of the mass infinite limit. Namely, when there exist a finite constant and so that the relation
(3.55)
is satisfied for , which is a function of and , we say that the infinite integral commutes with the limit of .
. This immediately means that all the massive matter fields decoupled and the remaining theory is U() SQCD with massless matter fields. This situation is reflected into the equation (3.53). We assume that the solution does not depend on ^{†12}^{†12}†12This assumption means that the effectively massless degrees of freedom can not appear.
when we take mass to infinity. Then the equation (3.53) becomes the same equation as (3.12) for the case with the number of the flavor

(3.56) |

This means that the solution of (3.53) which does not depend on mass can exist when while the constant solution can not exist in the infinite mass limit when . The numerical analysis of (3.53) supports existence of such a solution. Indeed, the density function is the same as that of U() gauge theory with massless hypermultiplets and this means that all the massive hypermultiplets decouple from the theory since in the infinite mass limit the origin of the Coulomb branch is dominant.

On the other hand, when , is proportional to in the infinite mass limit and the density function has three peaks at the origin and two-point around . We show the behavior of the density function in figure 3. Then we study the effective theory which appears in this situation by analyzing the behavior of the density function when we take the infinite mass limit. First, we have to know how the gauge group U() is broken into. From the density function we know that U() breaks to three parts. Thus we assume

(3.57) |

The rank of each three gauge groups is determined by the ration of the number of the eigenvalues around each peak. The density function counts the number of the eigenvalues between and . Therefore we count the number of the eigenvalues which exist around each peak by integrating corresponding density function in the infinite mass limit and determine , and .

We assume that there exists a solution proportional to . The equation (3.53) becomes in the infinite mass limit

(3.58) |

where we assume . We immediately determine as

(3.59) |

In order to know the behavior of the density function around , we redefine by order variable as and take . The density function (3.3) becomes in this limit

(3.60) |

where . Next we look at the density function around the peak on by regarding as in (3.3). It becomes by the same calculation of (3.42).

(3.61) |

where . The final part is the density function around . To investigate this part of the density function, we assume is order . Then we take the infinite mass limit and the density function becomes

(3.62) |

where takes value .

To know , and we integrate (3.60),(3.61) and (3.62). We obtain

(3.63) | ||||

(3.64) | ||||

(3.65) |

This result means that the gauge group U() is broken to as following:

(3.66) |

where we assume that and are integer. This means that in the infinite mass limit the theory becomes the effective theory on a point of the Coulomb branch like as

(3.67) |

We assume that the eigenvalues are separated as

(3.68) |

With the similar calculation in the previous subsection the saddle point equation is rewritten as the following three parts:

(3.69) | ||||

(3.70) | ||||

(3.71) |

These equations mean that the matrix model (3.25) in the infinite mass limit becomes the following matrix model: