Conformal Gauge Mediation

We propose a one-parameter theory for gauge mediation of supersymmetry (SUSY) breaking. The spectrum of SUSY particles such as squarks and sleptons in the SUSY standard-model and the dynamics of SUSY-breaking sector are, in principle, determined only by one parameter in the theory, that is, the mass of messengers. Above the messenger threshold all gauge coupling and Yukawa coupling constants in the SUSY-breaking sector are on the infrared fixed point. We find that the present theory may predict a split spectrum of the standard-model SUSY particles, m_{gaugino}<m_{sfermion}, where m_{gaugino} and m_{sfermion} are SUSY-breaking masses for gauginos and squarks/sleptons, respectively.


Introduction
Gauge mediation [1] is an attractive scenario to mediate supersymmetry (SUSY) breaking effects from the hidden to the SUSY standard-model (SSM) sector without inducing too large flavor-changing neutral currents. This is because the soft SUSY-breaking masses of squarks and sleptons are generated mainly by the standard-model gauge interactions and hence they are flavor independent. For the successful gauge mediation, we typically introduce messenger chiral superfields P andP which carry standard-model gauge indices with the following superpotential interaction to the hidden-sector field S: where the F -term of the chiral superfield S = m + θ 2 F S breaks the SUSY.
The above minimal gauge mediation model predicts a definite spectrum of the SSM SUSY particles. However, the prediction is at least based on two independent parameters m and F S , while they are required to conspire together to give a consistent spectrum.
Besides, in a large class of gauge mediation models, more free parameters are involved, and the spectrum of SUSY particles is not uniquely predicted. Furthermore, the effective SUSY-breaking parameter F S might not necessarily correspond to the dominant SUSYbreaking scale giving the gravitino mass.
Therefore, one can freely vary the gravitino mass while keeping, for instance, the mass of squarks or gauginos in the above gauge mediation models. Indeed, in the gauge mediated SUSY-breaking scenarios, an acceptable range of the gravitino mass is restricted only after taking into account the cosmological requirement. Then we face a problem from the particle-physics point of view even if the gauge mediation scenario is turned out correct: why is the gravitino mass just as we (will) observe in nature? Or if the gravitino mass can not be measured: at which scale is the SUSY broken?
In this paper, we propose "conformal gauge mediation", where all physical scales in the SUSY-breaking sector are fixed by only one parameter, the messenger mass scale.
Furthermore, since all the coupling constants in the SUSY-breaking sector are fixed by the conformal dynamics, there are no arbitrariness between the gravitino mass and gaug-

An example of the conformal gauge mediation
Dynamical SUSY-breaking model Now, let us consider an example of the conformal gauge mediation based on the IYIT SUSY-breaking model [2]. Concretely, we first introduce Sp(N) gauge group with 2(N +1) chiral superfields Q i (i = 1, . . . , 2(N +1)) transforming as a fundamental (2N-dimensional) representation of Sp(N). We add singlets S ij (= −S ji ) in (N + 1)(2N + 1) representation of the flavor SU(2N + 2) F symmetry of our theory. The superpotential is given by The effective low-energy superpotential is given by in terms of gauge invariant low-energy meson superfields M ij ∼ Q i Q j . Here Λ hol denotes the holomorphic dynamical scale of our theory [3]. When λ is small, we obtain where J ij = iσ 2 × 1 N ×N is the invariant tensor of Sp(N + 1) F ∈ SU(2N + 2) F . The effective low-energy dynamics is approximated by the Sp(N + 1) F singlet superfield S with the superpotential and the Kahler potential where Λ denotes the hadron mass scale of the low-energy effective theory. The η is a constant of order 1 and its signature is not calculable since perturbative calculations break down at the hadron mass scale Λ.
This model generates a dynamical SUSY breaking [2] with 1 where we take, in this paper, κ ≃ 0.1 as discussed below. We, furthermore, assume that η > 0, and hence, the model breaks also R-symmetry spontaneously as 2 which could be regarded as the definition of our hadron mass scale Λ.

Conformal extension of the SUSY-breaking model
To extend the IYIT model into the conformal regime, having the gauge mediation in mind, we add messenger chiral superfields P andP in the (anti-)fundamental representation of the standard-model (SM) gauge group (i.e. 5 and5 respectively in the grand unified SU(5) gauge group) [5] and in the fundamental representations 2N under the Sp(N) gauge group. We introduce the SUSY invariant mass term Well-above the physical mass scale of P andP , the theory is in the conformal regime as we will review momentarily [6]. The superpotential (9) breaks the conformal invariance and leads to dynamical SUSY breaking as in the original IYIT model below the decoupling scale of P andP .
So far, the rank of the hidden-sector gauge group has been arbitrary, but we now show N = 2 is the unique possibility in the conformal gauge mediation scenario. Perturbativity of the standard-model gauge interactions demands that the number of the messengers n mess (= 2(N + 1)) should satisfy, where we included the higher loop effects of the Sp(N) sector through the anomalous dimension γ P of P andP [7]. As we will see shortly, to obtain soft masses of order 100 GeV for the SSM SUSY particles, we take m phys ∼ 10 8 GeV, which yields n mess 8 as a bound for not so strong (i.e., |γ P | ≪ 1) hidden sector dynamics.
Thus, when N > 2, the standard-model gauge couplings will blow up well-below the Planck scale due to the large number of messenger chiral superfields. On the other hand, when N = 1, the hidden-sector gauge interaction becomes IR-free and we never achieve the IR conformal theory. We, therefore, set N = 2 in the rest of the paper.

Conformal fixed point
In the conformal gauge mediation scenario, we assume that our SUSY-breaking sector is The beta function for the hidden-sector gauge group (for a general Sp(N) gauge group with n F additional flavors; N = 2, n F = 5 in our case) is given by the NSVZ formula [7]: in terms of the anomalous dimensions of the chiral superfields γ P and γ Q , where α hid is defined in terms of the gauge coupling constant g hid of Sp(N) as α hid = g 2 hid /(4π) and µ denotes the renormalization scale. The beta function of the Yukawa coupling constant in Eq.(2) is also given in terms of the anomalous dimension factors of the hyperquarks, γ Q , and of the singlet chiral superfields, γ S , by where α λ is defined in terms of the coupling constant λ as α λ = λ 2 /(4π). Here and hereafter in the discussion of the conformal fixed point, we neglect the masses of the messenger quarks P andP .
All the beta functions are essentially governed by the wavefunction renormalization factor, or the anomalous dimensions of the chiral fields. The one-loop anomalous dimensions can be computed as 3 The requirement for the vanishing beta functions determines the coupling constants at the fixed point as The anomalous dimensions of the chiral superfields are computed accordingly as in the one-loop (or Banks-Zaks [8]) approximation.
Actually, one can obtain the exact anomalous dimensions by using the a-maximization technique [9] . We only present the final results here (see appendix for the details) We cannot compute the exact value of the coupling constant at the fixed point from the a-maximization technique, but the comparison between the one-loop results and the exact results for the anomalous dimensions suggests the accuracy of about 10%.

Decoupling and SUSY breaking
The anomalous dimensions of the chiral fields P andP determine the physical decoupling scale m phys with respect to the mass parameter m at the Planck scale in the superpotential (2). The wavefunction renormalization dictates the relation which leads to where M P l ≃ 2.4 × 10 18 GeV is the reduced Planck mass.
Let us now discuss relations among various scales. We note that the physical de- We should note that the hadron mass scale Λ does not necessarily coincide with the holomorphic dynamical scale. Later we will set a dynamical assumption Λ ∼ 0.3 × m phys .
Here we consider that the hadron mass scale Λ is the scale at which the holomorphic gauge coupling α hol becomes O(1). This means that the parameter κ ≃ 0.1 in Eq. (7).
Below the decoupling scale m phys for P andP , the dynamics of our conformally extended IYIT model reduces to the original SUSY-breaking IYIT model, albeit the Yukawa coupling λ is not so small. In this way, we obtain a SUSY-breaking and messenger sector where all the parameters are determined only by the messenger mass m phys , although it is difficult to determine them precisely since the interactions becomes strong below the messenger scale and perturbative calculations may break down.

Gauge mediation effects
In the present model, we have not introduced direct couplings between P ,P and S. 5 In such a case, the gauge mediation effects to the gaugino masses arise at not O(F S /m phys ) but at O(F 3 S /m 5 phys ) [10]. Since everything is strongly coupled at the SUSY-breaking scale, it is a challenging problem to yield precise values of the soft SUSY-breaking masses for the SSM SUSY particles although it should be possible in principle. In the following, we give a heuristic evaluation of the soft masses by combining knowledge from perturbative computations and its extrapolation by the naive dimensional analysis, instead of trying to solve this difficult problem in an exact way.
Above the threshold scale of the messengers P andP , they are in the conformal window. Since the anomalous dimensions of the messengers are not so large, we can integrate out the messengers at the scale m phys , perturbatively. Then, the effective interactions between the SUSY-breaking IYIT sector and the SSM sector are given by, at the leading order, up to O(1) constants. The SUSY-breaking effects will be mediated to the SSM sector from the above effective interactions. Notice that the gauge coupling α hid [µ] here is not the holomorphic coupling but the canonical one given at the renormalization scale µ. The dimension 8 interaction in (22) generates soft scalar masses but does not generate gaugino mass, 6 so we need to keep the next leading dimension 12 interactions.
After the decoupling of messengers P andP , the conformal invariance is broken and the gauge and Yukawa couplings of IYIT sector run quickly into a nonperturbative regime, leading to a dynamical SUSY-breaking at the scale In the following, we present our evaluation of soft masses for the SSM SUSY particles.
After integrating out Q andQ (together with the hidden-sector gauge interactions), the SUSY scalar mass is generated by the effective interaction from the effective three-loop amplitude, where Φ represents a standard-model superfield.
This interaction leads to a soft scalar mass where we have used S = Λ/λ(Λ) and F S = κλ(Λ)Λ 2 . Notice that the sign of the sfermion squared masses depends on the sign of the first operator in Eq. (22). 7 However, since the sfermion squared masses also receive higher-loop corrections after integrating out QQ and hidden sector gauge fields, we cannot track the sign of them perturbatively.
In the following, we simply assume that the sign of the squared mass of sfermions are positive.
On the other hand, we obtain the effective interaction for gauginos [ 7 This can be seen from the fact that the supertrace of the squared mass matrix of the messenger particles P andP is non-vanishing. In such cases, the sign of the sfermion masses is sensitive to the details of the hidden sector [11]. 8 Originally before integrating out messengers, they correspond to five-loop diagrams.
We note that the contribution is suppressed compared with the expression proposed in [5] in the context of the strongly coupled gauge mediation, by the factor of |F S | 2 /m 4 phys . 9 From (24) and (26), we obtain a relation between the gaugino and scalar masses as and κ. First of all, the prediction of our model depends sensitively on the ratio Λ/m phys .
As explained in the previous discussion, we take (Λ/m phys ) ∼ 0.3. 11 We suppose α λ [Λ] ≃ 1 since the Yukawa coupling tends to be smaller than the gauge coupling constant in the perturbative regime (but non-perturbatively large at the hadron mass scale Λ). We also recall that the above assumption on (Λ/m phys ) ∼ 0.3 corresponds to the parameter κ ≃ 0.1, since Λ hol ∼ 0.1 × m phys . Note that our results for gaugino and scalar masses depend only on a particular combination of κ and λ[Λ], namely κλ 2 [Λ].
Thus our dynamical assumptions above amount to Then, the ratio between the gaugino and the sfermion masses becomes m gaugino m sfermion ∼ 10 −3 .
(29) 9 As discussed in [10], the naive contribution to the gaugino mass m 1/2 ∼ α SM α 2 hid F S λS /m 2 phys should be suppressed either by |F S | 2 /m 4 phys or the standard-model loop factor α 2 SM from the gaugino screening mechanism. 10 The holomorphic gauge coupling α hol is of O(1) at the hadron mass scale Λ. However, the canonical coupling α hid may become much larger than α hol there, since it receives higher-order corrections. 11 The one-loop holomorphic dynamical scale Λ hol has a relation Λ hol ∼ 0.1×m phys as in (21). However, the higher loop corrections may render the hierarchy between Λ and m phys smaller.
Our model, therefore, predicts a split spectrum of the SSM SUSY particles. Under the same assumptions, we have gaugino masses of order 100 GeV for Λ ≃ 10 8 GeV.
It also determines soft SUSY-breaking masses m sfermion for squarks and sleptons and the gravitino mass as m sfermion ∼ 100 TeV and m 3/2 ∼ 1 MeV, where we have used

Comments on the model
Before we finish our discussion on the present model, several comments are in order.
Firstly, notice that the effective coupling of the messengers to the SUSY-breaking fields is governed by the Sp(2) gauge interaction, and hence it is independent of the charges of the messenger particles under the SSM gauge group. On the other hand, the mass of the colored messengers becomes heavier than the one of the SU(2) L -doublet messengers because of the difference of the wavefunction renormalization of the messenger particles due to the SSM gauge interactions. As a result, the gluino mass to the wino mass ratio is suppressed by a factor of, Once we accept some small deviations from the conformal theory, it may be natural to consider possible tiny Yukawa couplings of the S field to the massive quarks P and P . Interesting is that such a coupling λ ′ SPP induces a gaugino mass at the one-loop level and hence the gaugino mass is no longer suppressed. We find a mild hierarchy among soft SUSY-breaking masses as m gaugino ≃ O(0.1) × m sfermion for α λ ′ ≃ 10 −3 , for instance. 12 In this case, the gravitino mass can be as low as m 3/2 ≃ O(10) eV, provided m gaugino ≃ O(100) GeV and m sfermion ≃ O(1) TeV. Then, the model possesses no cosmological problems [13], but on the other hand, we lose a candidate for the dark matter. It should be also noted that the new Yukawa coupling λ ′ SPP generates a SUSY-preserving true vacuum at S = m phys /λ ′ . We find that the tunneling probability from our SUSYbreaking vacuum to the true vacuum is strongly suppressed for the above small Yukawa coupling λ ′ .

Other examples
The large disparity between m gaugino and m sfermion in the above model is originated from the smallness of the ratio, Λ/m phys ≃ 0.3. However, if the fixed-point value of the gauge coupling constant is of order 1, the ratio of Λ/m phys may become O(1) and we may have a much milder disparity of the SUSY spectrum without a direct coupling between messenger particles and the SUSY-breaking field. For example, let us consider the (uncalculable) dynamical SUSY-breaking model of SU(5) with 10 +5 [14]. We can add N f vector-like quarks 5 +5 to make it a conformal field theory (for 5 < N f < 13). 13 We assume that five pairs of additional N f quarks are charged under the standard-model gauge group and they serve as messengers. The anomalous dimensions of the chiral superfields at the 12 Here, we are treating λ ′ as a small perturbation to the conformal theory of α hid and λ. However, if there is another fixed point with λ ′ = 0, we can consider another candidate of the conformal gauge mediation around such a new fixed point [12]. For such a conformal model, the ratio between the m gaugino and m sfermion becomes the same as in the conventional gauge mediation models; m gaugino /m sfermion ≃ √ n mess . 13 A similar model has been studied in [15] (see also [16]). The detailed analysis will be given elsewhere [12].
conformal fixed point can be computed by using the a-maximization technique: The gauge coupling constant at the fixed point can be computed as

Conclusions
In this paper, we have proposed a concept of the conformal gauge mediation. The conformal invariance at the cut-off scale removes free parameters in the conventional gauge

A Anomalous dimensions from a-maximization
In this appendix we use the so-called a-maximization method [9] to determine the anomalous dimensions of the fields in the conformally extended IYIT model beyond the Banks-Zaks approximation presented in section 3.
The a-maximization method simply states that the conformal R current appearing in the superconformal algebra maximizes a particular 't Hooft anomaly a = Tr(3R 3 − R) , which is related to the conformal anomaly on a curved spacetime In our model of the Sp(N) gauge theory, the candidate of the conformal R current contains one free parameter x = γ Q , from which the corresponding R charges are determined as where ε = 2(N + 1) − n F .
The claim is that among these one-parameter R currents, the conformal one maximizes the anomaly a, which is obtained as follows: a = 2N(2N + 2) 3(R Q − 1) 3 − (R Q − 1) + 2(2N + 2 − ε)2N 3(R P − 1) 3 − (R P − 1) where we note that the R charges appearing in a are those of fermions (i.e. R ψ Q = R Q −1) because only fermions contribute to the anomaly. By maximizing a with respect to x, we can determine x * = γ Q | * . The unique local maximum is achieved by setting