Suppression of Supergravity Anomalies in Conformal Sequestering
aa r X i v : . [ h e p - ph ] S e p DESY 07-129
Suppression of Supergravity Anomalies in Conformal Sequestering
Motoi Endo
Deutsches Elektronen Synchrotron DESY,Notkestrasse 85, 22607 Hamburg, Germany
Abstract
We show that the anomaly-mediated supersymmetry breaking via the K¨ahler and sigma-modelanomalies is suppressed by conformal dynamics in the supersymmetry breaking sector. . INTRODUCTION Low-energy supersymmetry (SUSY) is one of the most plausible extensions of the stan-dard model (SM). So far, low-energy experiments such as measurements of flavor-changingneutral currents (FCNCs) have imposed constraints on its breaking mechanism and media-tion. We often assume to put our world be secluded from the SUSY breaking sector. Then,the SUSY breaking is mediated only via the gravitational effects [1, 2, 3], and the dangerousFCNCs are suppressed naturally.It was proposed that the separation is achieved by geometrical configuration in higherdimensions [1]. This mechanism is simple and easy to imagine. However, it has been notedthat moduli fields in the bulk may induce the dangerous couplings. The contributions dependon the background, and the warped one, namely the AdS space, is successful, because theyare warped away [4].On the other hand, the separation is realized in the four dimensional setup by assuminga conformal dynamics in the SUSY breaking sector. This scenario is called as the conformalsequestering [5]. The renormalization group (RG) evolution of the conformal dynamicssuppresses the contact couplings between the SM and SUSY breaking sectors.These two mechanisms are suggested to be dual to each other according to theAdS/conformal field theory (CFT) correspondence [6]. This implies an equivalence of themass spectrum of the superparticles. It has been studied that the tree-level mediation ofthe SUSY breaking is suppressed in both cases [1, 5]. Then the soft parameters arise atthe quantum level. There are three anomalies in supergravity (SUGRA), which are knownto mediate the SUSY breaking [1, 2, 3]. In the AdS setup, the mediation is given by theSuper-Weyl (SW) anomaly, while the other two anomalies in SUGRA, called the K¨ahler andsigma-model anomalies, are known to cancel to each other [3]. In contrast, any cancellationor suppression has not been discussed in CFT. In this letter, we will show that the conformaldynamics suppresses the K¨ahler and sigma-model anomalies are suppressed.
II. ANOMALY MEDIATION
The anomaly-mediated SUSY breaking (AMSB) with respect to the SW, K¨ahler andsigma-model transformations is represented by the non-local operators in SUGRA [3]. How-2ver, the result is not easy to discuss the conformal dynamics. They are easily obtainedfrom the superconformal formula of SUGRA [7]. Only the leading terms with respect to1 /M P are phenomenologically significant. Then the Lagrangian is expanded as L = [ φ † φ Q † Q ] D + [∆ K ] D −
16 [ K ] D + [ φ W ] F + · · · , (1)where K and W denotes the K¨ahler and superpotential in the Einstein frame. The chiralsuperfield field Q denote the visible and hidden mattes. It is noted that φ is the chiralcompensator field to fix the gauge degrees of freedom of the superconformal symmetry.Namely, the frame is not fixed before giving a VEV for φ . The notation [ · · · ] D,F means totake D - and F -components in the global SUSY, respectively. Further, we simply assume acanonical normalization for the matters. The second term in the right-handed side representsthe higher dimensional terms, potentially including direct couplings between the visible andhidden sectors. The third one is obtained after expanding − e − K/ . The neglected termsare phenomenologically irrelevant, since they correspond to higher order terms of 1 /M nP inthe Einstein frame.The chiral compensator field, φ is a source to mediate the SUSY breaking via the SWanomaly. It is easy to introduce the Pauli-Villas (PV) fields Q ′ to see AMSB. Essentially,the superpotential involves the mass term, W = M ′ Q ′ ¯ Q ′ (2)with the regularization scale M ′ . After canonically rescaling Q ′ , the SUSY breaking B termis evaluated as B = M ′ F φ in addition to the mass term M = M ′ φ . Thus similarly to theevaluation of the gaugino mass in the gauge-mediated SUSY breaking, the loop diagrammediating Q ′ gives M λ = α π F φ φ . (3)This has a sign opposite to that of the gauge-mediation because Q ′ is the PV field. Wenotice that the result is independent of M ′ and finite even for M ′ → ∞ . The Einstein frameis realized by taking a φ = e K/ (cid:20) θ (cid:18) e K/ W ∗ + 13 K i F i (cid:19)(cid:21) . (4) a See [3] for the terms involving spinors. Z , δK = cZQ ′ Q ′† + h . c . gives δF Q ′ = − cF Z Q ′ , leading to B = − M ′ cF Z by combining to the mass term (2) (e.g. see below). Note that φ does not contribute to thesigma-model anomaly. Thus the gaugino mass becomes M λ = − α π cF Z . (5)This result is generalized to the result in [3] straight-forwardly. Then the anomaly is onlyfrom the U (1) subgroup of the connection, Γ jij ≡ K jℓ ∗ K iℓ ∗ j . It is also commented that thisresult depends on the higher dimensional operator in K and can appear in global SUSYmodels [8].Let us discuss the K¨ahler anomaly. The third term of the right-handed side plays a roleto mediate the SUSY breaking in (1). It looks like a higher dimensional operator in the D -term, [ · · · ] D , and the B term becomes B = 2 / M ′ K Z F Z for both Q and ¯ Q , similarly tothe sigma-model anomaly. So the gaugino mass is M λ = α π K Z F Z . (6)It is stressed that although the result depends on the linear term of K , it substantially comesfrom the higher dimensional operator in (1).From (3), (5) and (6), we obtain the complete AMSB for the gaugino mass which iscoincide with the result in [3]. In the literature, the operator is denoted by the superfields,involving the gravity superfield, R . We can see that the superfield representation of thenon-local terms is derived from the second and third terms in (1) for the K¨ahler and sigma-model anomalies. However, only a part is obtained for that of the SW anomaly, because wefocus on a source of AMSB and introduced only φ in this letter.The B terms are essential to derive AMSB in the above. For the K¨ahler and sigma-model anomalies, they come from the higher dimensional operators. The K¨ahler potentialis generally written as (here and in the following, we omit a prime of fields for simplicity) K = | Z | + | Q | + | ¯ Q | + h d Z + c Q Z | Q | + c ¯ Q Z | ¯ Q | + h . c . i + · · · , (7)and the mass term is W = M Q ¯ Q . Here the coefficients c Q, ¯ Q , d may depend on the (hidden)4atters as a background. Expanding e K/ , we obtain the higher dimensional operators; − e − K/ ⊃ ( c Q − d/ Z | Q | + ( c ¯ Q − d/ Z | ¯ Q | + h . c .. (8)These terms are a source of mediating the SUSY breaking in the K¨ahler and sigma-modelAMSB. The B term is easily obtained by solving the equation of motion of F Q and F ¯ Q .Another approach is to erase them by rescaling, Q → Q [1 − ( c Q − d/ Z ]. Then the massterm is modified as M Q ¯ Q −→ M " − c Q − c ¯ Q − d ! Z Q ¯ Q. (9)This involves the B term, and provides the gaugino masses. It is noted that the tadpoleterms of Z are irrelevant after the expansion.The contributions from the K¨ahler and sigma-model anomalies, (5) and (6), exactly cancelto each other, if the K¨ahler potential is the sequestered form [3], K = − (cid:20) −
13 ( | Q | + | Z | ) (cid:21) . (10)This cancellation is easily seen in (1). The second and third terms in the right-handedside are a source of the SUSY breaking for the sigma-model and K¨ahler anomalies. If wesubstitute (10) for the K¨ahler potential in (1), they cancel to each other. From anotherpoint of view, they correspond to the higher dimensional operators of − e K/ . Namely, thehigher dimensional operators in the Einstein frame are practically equivalent to those in theconformal frame [9]. In the conformal frame, since (10) does not have the contact termsbetween the visible and the SUSY breaking sectors, the K¨ahler and sigma-model anomaliesare absent, and only the SW anomaly remains. III. CONFORMAL SEQUESTERING
Let us discuss the K¨ahler and sigma-model anomalies under the conformal dynamics. Inthe previous section, we saw that they are related to the higher dimensional operators in(1). Thus we focus on the evolution of them in the conformal dynamics.At the cutoff scale, the Lagrangian is assumed to be general, involving the (flavor-violating) higher dimensional operators. Let us first discuss the case when the operatorsin the D -term linearly depend on the matters in the SUSY breaking sector, S . This means5hat c Q, ¯ Q and d in (7) are independent of the SUSY breaking fields. To see a suppression ofthem, we rescale the visible matters as Q → Q [1 − ( c Q − d/ S ]. Then the AM contributionsis derived from a coupling of S in front of the mass term in the superpotential, giving the Bterm. Its evolution is represented by the anomalous dimension of S . Near the fixed point,the B term behaves as (see e.g. [10, 11]) W ∼ (cid:18) µM ∗ (cid:19) γ ∗ S M SQ ¯ Q, (11)where γ ∗ S is the anomalous dimension at the fixed point. Since S should be gauge-singlet, γ ∗ S is positive. Thus the B term becomes suppressed in the infrared limit.The bilinear terms with respect to the SUSY breaking fields in the D -term can alsobe a source of mediating the SUSY breaking if the field has a finite vacuum expectationvalue. Regarding the visible fields as a background, their evolutions are represented by theanomalous dimensions [5]; (∆ ln Z ) = e Lt (∆ ln Z ) . (12)Here the scale is t = ln( µ/M ∗ ) and (∆ ln Z ) is defined as (∆ ln Z ) ≡ ln Z + γ ∗ t . Sincethe SUSY breaking sector usually consists of multiple fields, L forms a matrix. If it ispositive, i.e. all eigenvalues are positive, (∆ ln Z ) approaches to zero for the infrared limit t → −∞ . Then the contact terms are absent from the low-energy effective Lagrangian,because they arise as (∆ ln Z ) ⊃ cQQ † . Therefore the conformal sequestering is realized for L > c and d in (9) may depends on the hidden matters more complexly, they canbe treated similarly, or are practically irrelevant for phenomenology.Consequently, the B terms relevant for the K¨ahler and sigma-model anomalies are sup-pressed, and so they are absent in the conformal sequestering. In contrast, the SW anomalystill remains after the dynamics, since φ arises as an overall factor in front of the D -term b .Let us comment on a choice of the regularization scheme. So far, we used the PVregularization. If we apply the other scheme (see e.g. [3, 8, 13]), the discussions in the b The conformal dynamics may affect K i F i / M (dec . ) λ . The UV insensitivity tells us that it exactly cancels with thatfrom the regularization, that is, the AMSB mass, M (AM) λ . Thus if we evaluate the gauginomass from the matter threshold by postulating a hypothetical mass term, we obtain theAMSB mass as M (AM) λ = − M (dec . ) λ . Repeating the same discussions in this letter, we obtainthe same result.So far, we focused on the gaugino mass. The soft SUSY breaking effects also containscalar masses, scalar trilinear couplings, and holomorphic scalar mass terms. The SUGRAanomalies mediate the SUSY breaking to the parameters. Nevertheless, the complete resulthas not been known for the K¨ahler and sigma-model anomalies (see also [14]). On the otherhand, the SUSY breaking is mediated by the higher dimensional operators in (1). The softparameters other than the gaugino mass are also considered to originate in the terms. Wesaw that they are suppressed in the geometrical and conformal sequestering. Thus, if thesequestering is realized in nature, the K¨ahler and sigma-model anomalies do not contributeto the soft parameters. IV. DISCUSSION AND CONCLUSIONS
In this letter, we discuss the suppression of the K¨ahler and sigma-model anomalies inthe conformal sequestering. The contributions are obtained from the higher dimensionaloperators in the D -term, namely after expanding − e − K/ . Since the conformal dynamicssuppresses them, the anomalies are found to vanish.A dynamics of the gauge term R d θZW W is treated by using the anomalous dimen-sions [11]. However, the operators we focus on now are represented by the non-local oper-ators at the Planck scale [3], so its evolution is non-trivial. Instead, the counter term mayexist at the cutoff, and can affect the gaugino mass [3]. If it has a form of R d θf ( Z ) W W ,where f ( Z ) = αZ + · · · is a function of Z , its contribution tends to be suppressed by theconformal dynamics.The method in this letter can also be applied to discuss the anomaly-induced inflatondecay [9, 15]. The decay into the SUSY breaking sector is obtained by the higher dimensionaloperators of Z in the D -term for the K¨ahler and sigma-model anomalies. Thus they are7aturally suppressed by the conformal dynamics, even when the SUSY breaking fields donot always appear explicitly in the operators [16]. Acknowledgment
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