Current Correlators for General Gauge Mediation

In the gauge mediation mechanism, the effects of the hidden sector are characterized by a set of correlation functions of the global symmetry current of the hidden sector. We present methods to compute these correlators in cases with strongly coupled hidden sectors. Several examples are presented to demonstrate the technique explicitly.


Introduction
Gauge mediation has been an attractive mechanism for mediating the supersymmetry breaking effects. Recently, Meade, Seiberg and Shih [1] presented a very general definition of the gauge mediation mechanism that includes most models with the supersymmetry breaking sector and the messengers, and models with the direct mediation. They also computed several physical quantities such as gaugino and sfermion masses on a general ground. Many of the physically observable effects mediated to the MSSM can be encoded by the current-current correlators of the global symmetry of the hidden sector. This allows us to extract the soft supersymmetry breaking terms even in the strongly coupled hidden sector 1 .
In this paper, we provide methods to compute the zero-momentum current correlators in some classes of strongly coupled gauge theories. The way we calculate the current correlators is to weakly couple the system to some 'spectator' gauge theory. Sometimes the coupled system is solvable, in which case, by taking the decoupling limit, we are able to obtain useful information about the current correlators. We will review the idea of general gauge mediation of [1] briefly and then understand that the correlators of the global currents essentially characterize a wide range of gauge mediation models. Then we go on to the basic scheme of our computation of current correlators. To illustrate the technique, we provide a number of computable examples.

Review of General Gauge Mediation
In this section, we briefly review the definition of the gauge mediation and the determination of the effect of the hidden sector via current correlations in [1]. Readers who are familiar with the subject may skip this section safely. In [1], a careful definition of the gauge mediation is given. Following their definition, a model has the gauge mediation mechanism if the theory decouples into the MSSM and a separate hidden sector that breaks SUSY in the limit the MSSM gauge couplings α i all vanish. In this setup, we may be able to compute various quantities in perturbation theory in the gauge coupling α i , but the hidden sector may be strongly coupled. The information of the hidden sector, however, can be parameterized by the current correlation functions in the hidden sector. In the supersymmetric gauge theory, a global current superfield J A has the component form and satisfies the current conservation conditions with j µ satisfying ∂ µ j A µ = 0. Here A is an index for the adjoint representation of the global symmetry group. We follow the notation of [3]. The current-current correlators have the following general forms: where M is the characteristic mass scale of the theory. B 1/2 may be complex but all C a are real. There could also be nonzero one-point function J(x) , but it vanishes for nonabelian currents. When supersymmetry is not broken spontaneously, we have the relations Since supersymmetry is restored in UV, whether SUSY is spontaneously broken or not, We now gauge the global current by coupling it to a gauge field. The part of the original Lagrangian for the gauge field is given by where Imτ = 4π g 2 and we use normalization such that TrT A T B = δ AB . After integrating out the hidden sector, its effect is determined by the current correlation functions. Here we will only consider one gauge group and not a product of gauge groups, such as in the MSSM, for simplicity. Ignoring higher derivative terms, the change of the effective Lagrangian is HereC a andB are Fourier transforms of C a and B, respectively: (2.8) The gaugino and sfermion masses are also determined by the current correlation functions. The gaugino mass at tree level can be read off from the change of the Lagrangian (2.7): When the one point function J is zero, the sfermion mass occurs at one loop. In this case, we have to know the current correlation functions at the momentum of order M, the typical scale of the hidden sector. The sfermion mass is then where c 2f is the quadratic Casimir of the representation of f under the gauge group and

Basic Idea
In this section, we present the basic idea to compute the current correlators in non-supersymmetric vacua, which encode the gaugino and sfermion masses. Schematically, the full theory can be written in the following form: where L hid is the supersymmetry breaking hidden sector, L M SSM is the visible MSSM sector, and L int is the interaction between the two sectors, which transmits the supersymmetry breaking effects to the MSSM. When we integrate out L hid + L int , we produce the soft terms that break supersymmetry in the MSSM Lagrangian.
This can be done explicitly if the hidden sector is weakly coupled, but it is hard to do so for the strongly coupled case in general. However, we can circumvent the difficulties in certain cases. As we have reviewed in the previous section, many of the parameters of the soft terms can be determined by calculating the current correlation functions. To achieve this, we will replace the MSSM with another theory such that we have control over the combined theory. That is, we are going to consider the following Lagrangian: Here, if the gauge coupling constant g ′ for the spectator Lagrangian goes to zero, the spectator fields decouple. Supposing this new theory is solvable, we integrate out the hidden sector and Now, by taking g ′ small and extracting nontrivial terms, we get the desired soft terms for the original theory.
In the following sections, we will be more specific and gauge the flavor symmetry of the hidden sector. Suppose the hidden sector is a gauge theory with gauge group G 1 and global symmetry group G 2 , and each field lives in the representation (A i , B i ) under G 1 × G 2 . Now by weakly gauging G 2 , we get the theory with the product gauge group G 1 × G 2 : (3.5) Let the dynamical scales associated with gauge groups G 1 and G 2 be Λ 1 and Λ 2 , respectively, and g ′ gauge coupling for the gauge group G 2 . We set the scale Λ 1 ≫ Λ 2 , so that we can treat the gauge interaction for G 2 to be very weak at the scale we probe. If we can integrate out the whole Lagrangian (3.5) in this limit, we are able to extract information about the current correlation functions of the flavor symmetry of the hidden sector. Note that this probe Lagrangian should not be confused with the Lagrangian for the real visible sector, although the structure is very similar. Here, this is just a probe to extract the information of the hidden sector. In the next section, we provide certain classes of examples in which we can follow this procedure. Before we go on, let us briefly mention that we do not really have to gauge the full global symmetry. It is actually enough to gauge any subgroup H of the global symmetry group G, because the global currents are only sensitive to group theory factors. To see this, suppose the global current for G is in a representation R. Let the generators of G be {T a } and H {t α }. The global current correlators can be written as where we have omitted Lorentz indices. This is the only possible form by symmetry. The current correlators come with the group theory factor where D G (R) is the Dynkin index of the group G in the representation R and we have omitted the Lorentz indices. Now, let's decompose the representation in terms of the representations of the subgroup H by R = ⊕ i r i where r i is a representation of H. Then, the correlators we get by gauging the subgroup can be written as Therefore, we can multiply the currents obtained by gauging the subgroup by the group theory factors to get the desired current correlators.
Here we will consider N = 2 SU(N c ) Seiberg-Witten theory with N f hypermultiplets [4,5] with an appropriate superpotential as the hidden sector. This theory has U(N f ) global symmetry.
We will weakly gauge the SU(N f ) part of the global symmetry. So the hidden sector is given by the Lagrangian where a = 1, · · · , N f and W (Φ) is a superpotential for the adjoint chiral superfield Φ. Note that M in the mass term MQ a Q a is proportional to the N f × N f identity matrix, which is of the most general form to preserve SU(N f ) global symmetry. We will consider the Coulomb branch of the hidden sector, whose special coordinates are denoted by a i where i = 1, · · · , N c with a i = 0. With a suitable choice of the superpotential W (Φ), the hidden sector can be in a metastable SUSY breaking state. For example, we can use a Kähler normal coordinate truncated at some finite order as a superpotential to get a metastable SUSY breaking state at a generic point in the Coulomb branch [6,7]. Note that a i may have nonzero F -component, which we denote by F i . Our purpose is to compute the current-current correlators of the SU(N f ) global current below the typical scale of a i .
To achieve this, we gauge the flavor symmetry of the hypermultiplets. Since our purpose is to calculate the current-current correlators, we need not gauge the flavor symmetry using the MSSM. To facilitate computations, it is better to gauge the flavor symmetry by another N = 2 SU(N f ) SW gauge theory. So the total Lagrangian can be written as where primes denote similar multiplets in the SU(N f ) gauge theory and the trace in the last line is over the flavor SU(N f ) indices. Also, in the second line, the first term may be explicitly expressed as where i, j = 1, · · · , N c and a, b = 1, · · · , N f . The gauge coupling of the spectator gauge theory is assumed to be very weak, so τ ′ = 4πi/g ′2 with g ′ very small.
Treating the theory as N = 2 SU(N c )×SU(N f ) SW gauge theory, the low energy effective theory in the Coulomb branch is given by where a i and m a are eigenvalues of Φ and Φ ′ , respectively, and subscripts under F denote differentiations. Note that the prepotential F depends both on a i and m a . Usually, it is hard to compute F for product gauge groups. However, using the fact that the spectator gauge theory is weakly coupled, we may obtain necessary information out of F 0 of the original N = 2 SU(N c ) SW theory. To see this, note that the low energy effective theory of (4.1) is where F 0 (a) is the prepotential for N = 2 SU(N c ) SW theory. Let us consider the case where g ′ goes to zero. In such a limit, m a is treated as constant. Note that the dynamics of a i in (4.4) and (4.5) agree when F i = F 0i and m a ∼ 0. m a ∼ 0 is necessary since we set all nonabelian mass parameters to vanish in (4.5). The condition m a ∼ 0 really means that the typical scale of m a is much smaller than that of a i . That is, of the moduli space of the Coulomb branch of the product gauge group, in the region where |a| ≫ |m|, we may interchange F i and F 0i freely. Therefore, if our interest is to consider the low energy effective theory whose typical scale of m is much lower than that of a, we may use the prepotential of the original N = 2 SU(N c ) SW theory to compute quantities.
In the Seiberg-Witten curve language, one can obtain the prepotential in this limit by taking appropriate mass deformation of the curve, and regarding it as the moduli of the gauge theory. To see this, let's compare (4.1) and (4.2). In the limit where the second gauge ab acts as a mass term for the hypermultiplets. Therefore, by identifying the massive deformation parameterm a to be √ 2m a , and computing the period integrals, we can obtain the prepotential F and its derivatives in the limit of |m| ≪ |a|.

Calculation ofB
To show what is going on explicitly, let us write down the Lagrangian in terms of the component fields ignoring terms involving derivatives: Here I = (i, a) and the prepotential F has the superfields A I = (a i , m a ) as its arguments.
Note that W a = 0. g IJ = 1 4π ImF IJ is the metric for the product group gauge theory. After integrating out the auxiliary fields F I , we can read off the coefficients of λ I λ J and these are given by If the hidden sector is very weakly interacting with the spectator gauge theory, g KLW L should nearly be the same as (g ij 0W j , 0) where g ij 0 is the inverse metric of the original N = 2 SU(N c ) gauge theory. To see this, note that among the components of the metric g IJ , g ab is very large compared with g ij and g ia in the limit g ′ → 0. From the expression below δ K J = g IJ g JK = g ij g ib g aj g ab g jk g jc g bk g bc , we see that g ij is much larger than g ib and g ab and this is more so as g ′ → 0. Also g ij is the inverse of g ij , which is the same as g 0ij , all in the limit g ′ → 0. Therefore, g aLW L can be ignored compared to g ijW j and Next, we need to integrate out λ i fields in (4.7). When we do this, we get where (F ijm F m ) −1 is the inverse of the matrix X ij = (F ijm F m ) and we use the relation (4.9) in our limit. There are two terms in the coefficient of λ a λ b . The first term is the usual one, but the second is from integrating out the gaugino λ i in the hidden sector. However, when we gauge the SU(N f ) flavor current, the second term can be neglected. The point is that, in the second term, F is differentiated only once by m a . Note that F 0 in the original N = 2 SU(N c ) SW theory depends smoothly on the symmetric polynomials of m a 's, such as a≤b m a m b , a≤b≤c m a m b m c , · · · . Hence, when differentiated once, F 0 necessarily contains at least one factor of m a and so goes to zero when |m|/|a| → 0. On the other hand, F 0 itself or its second derivatives by m a need not vanish when m a is small. Since F 0i = F i in our limit, in the region of the moduli space where |a| ≫ |m|, the first term dominates and we can safely ignore the second term. Therefore, when the hidden sector fields are integrated out, we may say that the relevant gaugino mass term is Note that, if we gauged the U(N f ) flavor current, the prepotential would depend on a m a and we would not be able to ignore the second term in (4.10). Next, we will look for a theory at the scale of |a| which gives the low energy effective theory described above. That is, we hide all signals of the hidden sector into the gauge coupling of the spectator gauge theory and consider N = 2 SU(N f ) SW theory where τ 2 = 4πi/g ′2 + θ 2 F τ is the gauge coupling of the probe Lagrangian at the scale of |a|. This is a theory at the scale of |a|. We will go to the low energy effective theory in the Coulomb branch where the superfield Φ ′ has eigenvalues m a , a = 1, · · · , N f (so m a = 0).
Although |a| ≫ |m|, the coupling g ′ is so small that one loop correction is enough when we consider the dynamics at the scale of |m|. The prepotential at one loop is given by 13) and the low energy effective theory is (4.14) Note that the θ 2 component of F is not corrected by the one-loop effect. So the gaugino mass term is given by Note that we use coordinates for the second gauge group such that N f a=1 λ a = 0. Comparing this with (4.11), we obtain at m a = 0. Note that, since F k depends on the masses m a by the combination m a − b m b /Nc, taking derivatives with respect to m a and m b , F kab in the right-hand side has the same index structure as that in the left-hand side. From (2.7), It is instructive to actually calculate the gaugino mass using this formula in the semiclassical regime. That is, we check the expressions in the case where the expectation value of the chiral superfield Φ of the hidden sector is much larger than the scale of the hidden sector gauge theory. Additionally, we assume that the hypermultiplets Q andQ are massless: i.e. M = 0 in (4.2). The chiral superfield Φ ij has nonzero F -term F Φij where i and j are the gauge group indices for the hidden sector. Note that we are in the Coulomb branch of the hidden sector.
We use gauge transformation such that Φ ij is diagonal with diagonal elements a i ( a i = 0). Let F i be the corresponding F -term of a i . We will calculate the gaugino mass in this setup. We start with the Lagrangian (4.2) and go to the low energy effective theory at the scale of |m|, where Φ ′ has eigenvalues of m a with the constraint m a = 0. The a i dependent part of the prepotential F , given by [8] with the constraint m a = 0 built in, is where τ = 4πi/g 2 , τ ′ = 4πi/g ′2 . Λ is the scale for the first gauge group and Λ ′′ is some scale which does not change the answer that follows. Then after imposing tracelessness conditions. Since the scale of the spectator gauge theory does not enter into the expression, we may set F 0kab = F kab . In the limit m b → 0, we have, from (4.17), which gives the usual one-loop gaugino mass through (2.9)

Calculation ofC a (0) and sfermion masses
Using the same technique, we may computeC a (p 2 /M 2 ) at zero momentum. Note that the effect of integrating out the hidden sector fields is to change the gauge coupling τ ′ to τ 2 as shown in (4.12). Also, in this case, allC a (0) are the same. Using (2.7), When we go to the low energy effective theory, τ 2 (a) gets renormalized. But since g ′ is very small, it is enough to consider only the one-loop effect. So τ 2 (a) gets only additive renormalization of the form log m, which is independent of a. Hence we have Integrating this equation, we are able to obtain τ 2 (a), which then may be fed into (4.22) to getC a (0). The additive constant is determined by noting thatC a (0) goes to zero in the limit where a → ∞.
Since we are required to calculateC a (p 2 /M 2 ) when p 2 /M 2 is of order 1, the information we have just obtained is not enough to calculate the sfermion masses of the MSSM. Alternatively, we can introduce a matter multiplet charged with respect to SU(N f ) and evaluate its low energy effective action terms. A work in this direction is in progress.

Generalization to other gauge groups
Let us briefly comment on how the expressions (4.17) and (4.23) change for other non-abelian groups. We start from (4.12). Let T A be a basis of the adjoint representation of the flavor symmetry group and H a be a basis of the Cartan subalgebra. The prepotential as a function of the N = 2 adjoint chiral superfield Φ ′ at classical level is given by where c adj is the Dynkin index for the adjoint representation: In the Coulomb branch, all massive modes are integrated out and Φ ′ is diagonalizable due to D-term constraint. Hence Φ ′ = a m a H a and the prepotential above becomes where the summation is over all positive roots of the flavor group. Therefore λλ term in (4.14) becomes We now compare this with the λλ term in the low energy theory of the product gauge group (4.11). Therefore Since the θ 2 component of τ does not receive corrections at one loop, this F τ can be used to calculate MB 1/2 (0). That is, the part of λλ consisting of the Cartan subalgebra part in (2.7) is (4.29) Therefore the relation corresponding to (4.17) is Similarly, corresponding to (4.23), we have (4.31)

Hypermultiplet condensation
It may be of interest to check whether the hypermultiplet bilinearQ a Q b develops a nonzero expectation value. The Lagrangian for the product SU(N c ) × SU(N f ) gauge theory (4.2) has some terms containing the F component of Φ ′ : (4.32) Therefore we can calculate the expectation values of the bilinearQ a Q b by differentiating the partition function with respect to F Φ ′ . More precisely, we will calculate the traceless part of Q a Q b since the source F Φ ′ is traceless. To get the effective Lagrangian for the spectator gauge theory, we start with (4.6) and set all vev's for the fermions to zero. Then we integrate out the F and D terms for the hidden sector. The equation of motion for D i sets D i = 0. The equation of motion for F i isFj Plugging the result into the Lagrangian, we get Now we can read off the linear term in F a : where the second line follows in the limit g ′ → 0. But as we argued below (4.11), g iā vanishes as m a goes to 0 since we differentiate the prepotential F only once with respect to the second gauge indices to get g iā . Therefore, the linear term vanishes and the hypermultiplet bilinear can have at best an expectation value of the form Having derived the formula for the quark condensate (4.37), let us verify it in the semiclassical regime. The leading contribution of Figure 1, is (4.38) Note that the interaction is insensitive to the cutoff Λ 0 . Let's compare this with (4.37). The relevant part of the prepotential is similar to (4.18): Note we do not impose the constraint m a = 0 since (4.37) is valid when gauging U(N f ) symmetry. The metric component g ia at weak coupling is given by Let's go to the limit m a → 0. Then g jī 0 W j = −F¯i and using (4.37), The result agrees with (4.38).

Geometrically Realized Models
As another controllable model, we study geometrically induced supersymmetry breaking configuration in Type IIB string theory on A 2 -fibered geometry. This has been studied in [9] somewhat in a different context. Consider A 2 fibred geometry [10] defined by There are three singular points, t = a 1,2 and a 3 = (m 1 a 1 + m 2 a 2 )/(m 1 + m 2 ). Wrapping N c anti-D5 and N f D5 branes on two S 2 s that resolve the singularities at t = a 1 and a 2 respectively, we can construct a supersymmetry breaking configuration. We do not wrap any brane at t = a 3 , because this can decay into a lower energy configuration. Our present setup does not have an unstable mode as has been discussed in [11]. Therefore we can take field theory limit. Here we claim that as long as the size of S 2 at t = a 2 is much bigger than that at t = a 1 , there is a field theory description for this brane/anti-brane system. According to the conjecture proposed in [9,[12][13][14], there is a glueball description with respect to the hidden sector gauge group corresponding to a partial geometric transition. Thus it is reasonable to claim that the low energy field theory description is an interacting product U(1) × U(N f ) gau ge theory. The kinetic term is where · · · includes higher derivative terms and U(1) gauge kinetic terms. F 0,0 (S 1 ) is the prepotential for the geometry after the transition, where Λ 0 is a cutoff scale of this description. The superpotential terms are In application to phenomenology we will identify a subgroup of SU(N f ) as the standard model gauge group. So the adjoint field Φ 2 for SU(N f ) gauge group should be integrated out by taking m 2 → ∞. In the limit, the superpotentialW 2 becomes a relatively simple function, Our goal in this section is to compute the function τ (S 1 ) and extract C i s and B 1/2 from it. In the open string description we can say that this S 1 dependence is generated by the bifundamental matter. On the other hand, after the transition in closed string point of view, it is generated by closed string modes. To extract the interacting part, we use the glueball description for U(N f ) gauge group as well and assume that the glueball fields and U(1) ⊂ U(N f ) gauge supermultiplet w α are background fields. Following [15,16], we use glueball description for evaluating the interacting part even though the SU(N f ) theory is weakly coupled and is not confined. Turning on these backgrounds modifies the geometry slightly. At the leading order of the modification, we read off the kinetic term for the gauge group. The low energy description is given by where W (S i ) is Gukov-Vafa-Witten superpotential [17] generated by the flux. Solving the equation of motion for F 1 , we obtain the potential where we ignored U(1) gauge fields. The metric is defined by Im ∂ i ∂ j F 0 . Since we are interested in coefficients of correlation functions B 1/2 and C i , which are related to linear terms in S 2 and F 2 , we can put these to be zero when we evaluate the minimum of the potential, To find the minimum it is useful to expand the prepotential F 0 for A 2 geometry as where we ignored S 1 independent part, which can be combined with the classical action of SU(N f ) and construct the one-loop running coupling constant. In the matrix model computation, F 0,b are contributions of diagrams with b boundaries, which are perturbatively calculable order by order. With this expansion, the superpotential and metric become In our setup, the disk and annulus amplitudes are exactly known, [18][19][20] 2πiF 0,1 = S 1 log ∆ + ∆ 2 − 4S 1 where ∆ = a 1 − a 2 . With these expressions, we see that W (S 2 = 0) reproduces the superpotential in (4.43) in the limit m 2 → ∞ and S 2 → 0. At the leading order, the minimum of the potential is given by Note that N f > 0 > N c . Since there is an exponential suppression factor, vev of S 1 exists in physical region, which we regard as a dynamical scale of the theory on the anti-D5 branes. Expanding the potential (4.45) around the minimum we can read off coefficients of linear terms in S 2 which yield the gaugino mass term for SU(N f ) part, where we supplied a dimensionful parameter Λ. In the field theory limit, we take the string scale to be infinity, keeping the scale Λ finite which should be identified with the scale of S 1 in (4.46) in our model. The α 1 , which is the size of P 1 , also has to scale appropriately [21] 2 .
The vev of F 1 is cut-off independent and a finite quantity in the limit, where the β is defined 3 by 2i Imᾱ 1 ∼ β log Λ 0 , which encodes geometric data of the P 1 . On the other hand, another correlation functions can be read off from the linear term in F 2 .
2πiC i (0) = −2πi 16π 2 Re Im (4.48) Finally let us comment on the diagrammatical computation of the gaugino mass. Although our present geometric configuration does not include an unstable mode, we do not know explicitly the UV Lagrangian for the brane/anti-brane system. Thus it is not easy to compute the correlation functions studied above from Matrix model computations directly. However, the flop of the S 2 wrapping the anti-brane is a smooth process because its physical volume can never be zero [9,13]. The new geometry yields the brane/brane configuration. The world volume theory on the branes is quiver gauge theory with a superpotential, W SU SY = m 1 2 tr(Φ 1 − a 1 ) 2 + m 2 2 tr(Φ 2 − a 2 ) 2 + Q 12 Φ 2 Q 21 + Q 21 Φ 1 Q 12 .
Using this explicit Lagrangian and technology developed in [15,16,22,23], we can compute the non-perturbative effect from perturbative Feynman diagram computations. In fact, explicit formulae for F 0,0 , F 0,1 and F 0,2 have been perturbatively computed by this method.