aa r X i v : . [ h e p - ph ] S e p From Supercurrents to Soft Terms
Sibo Zheng and Jia-Hui Huang Department of Physics, Chongqing University, Chongqing 400030, P.R. China Center of Mathematical Science, Zhejiang University, Hangzhou 310027, P.R. China
Abstract
In this paper, hidden sectors of Ferrara-Zumino multiplets with contributions to softterms coming from quantum supergravity are investigated in framework of gravity media-tion. The two-point correlator of Ferrara-Zumino multiplets can be parameterized, whichimplies the wave function renormalizations of components fields in gravity supermultipletcan be evaluated in relatively simple form. Soft terms are calculated via supercurrent ap-proach. We find gaugino masses are independent of sfermion masses on general grounds.The unification of gaugino masses is not universal. In comparison with general gaugemediation, there are no sum rules for sfermion masses of each generation.August 2010
Introduction
Among mediation scenarios of supersymmetry breaking, gravity mediation [1] is a naturaloption. Although gravity mediation suffers from flavor problem that should be takencare of, it has virtues of achieving locally grand unification and electro weak breaking.The contributions to soft terms mainly come from anomaly mediation [2, 3]. In super-conformal theories, however, the contributions to soft terms arising from quantum effectsof supergravity are dominant. This paper is devoted to study soft terms that are inducedby quantum supergravity.The complicated Lagrangian of coupling supersymmetric models to supergravity is theorigin of hardly obtaining the structure of soft terms at next leading order. It is known thatthe leading order approximation of coupling supersymmetric models to supergravity canbe well described by linearized supergravity (see [4] and reference therein). According tothe symmetric properties, the supercurrents ( and corresponding linearized supergravity)can be classified into Ferrara-Zumion (FZ) multiplet [5], R multiplet [6] and S multiplet[7] and variant supercurrents [8, 9]. In this paper, we will discuss soft terms of FZmultiplet as hidden sector in gravity mediation. We impose two conditions so that we areable to evaluate the structure of soft terms in relatively simple form. The first conditionis perturbative validity of gravity coupling, which is always satisfied for soft terms nearhundred GeVs. The other condition is that the FZ multiplet has R symmetry, which isassumed to break either during embedding the FZ multiplet into supergravity or in thevisible sector. As we will show, even without the second assumption some importantresults can be still expected.Starting with FZ multiplet with R symmetry, we can parameterize the two pointcorrelator of FZ multiplet via a set of functionals. This procedure is similar to whatwe have experienced in general gauge mediation [11]. We find that gaugino masses areindependent on sfermons masses in general and the unification of gaugino masses is notuniversal. The sfermion masses is found to depend on both flavor and gauge quantumnumbers, which implies that there are no sum rule for sfermion masses of each generation.This property weakens the prediction of gravity mediation at LHC, however, also separategravity mediation from gauge mediation.This paper is organized as follows. In section 2, we discuss the parametrization ofFZ multiplet. Section 3 is devoted to the calculation of soft terms, with discussions onphenomenological implications. Finally, we make a few outlooks.1 Supercurrents in Hidden Sector
In this paper, we follow the conventions of Wess and Bagger [15]. We couple the hiddensector that is responsible for supersymmetry breaking to supergravity via approach ofsupercurrent. We consider the supercurrent of hidden sector belongs to the type that canbe described by FZ multiplet [5], which is viable for a lot of supersymmetric models weare familiar with. The constraint on supercurrent of FZ multiplet is,¯ D ˙ α J α ˙ α = D α S = 0 (2.1)It is understood that R symmetry is a necessary condition for supersymmetry breaking[14]. For hidden sector with this symmetry, S = 0. However, note that R symmetry has tobe spontaneously broken in order to permit Majorana gaugino mass of visible sector, weassume this happens when hidden sector embedded into supergravity or in visible sector.Introducing the two-point correlator of supercurrent, < J α ˙ α ( p, θ ) J ˙ αβ ( − p, θ ′ ) > ≡ I βα ( p, θ, θ ′ ) = − σ µα ˙ α (¯ σ ν ) ˙ αβ I µν ( p, θ, θ ′ ) (2.2)where I µν ( p, θ, θ ′ ) = < J µ ( p, θ ) J ν ( − p, θ ′ ) > (2.3)From constraint eq(2.1) we obtain a constraint on I βα , D α D β I αβ = D β D α I αβ = 0 (2.4)which implies a discrete symmetry α → β , p → − p and θ → θ ′ in the strucuture of I αβ . According to the definition of I βα in eq(2.2), this discrete symmetry requires I µν is asymmetric tensor. In this sense, the constraint equation eq(2.4) can be reformulated as, D I ( p, θ, θ ′ ) = 0 , I = η µν < J µ J ν > (2.5)The general solution to eq(2.5) has been considered in literature [11, 12], in which thegeneral form of I was given with four undetermined scalar functional F (1) , F (2) , F (3) and F (4) .Roughly there seems to have two options for transforming the scalar expression of thesolution to tensor expression. We can either modify the terms or the coefficients F ( i ) infront of these terms as required, or simply extend the coefficients F ( i ) to tensor functionals2 ( i ) µν . One can check the later choice is excluded by consistent considerations. And thesimplicity of structure for two point correlator is not kept in general. However, we canstill reduce the complexities to the form that can be handled for FZ multiplets. As wewill see, these parameters contain the information of supersymmetry breaking.According to the symmetric property, we can write the two point correlator as, < C µ ( p ) C ν ( − p ) > = F (3) µν ( p ) < χ µα ( p ) ¯ χ ν ˙ β ( − p ) > = p λ σ λα ˙ β F (2) µν ( p ) + ( p µ σ ν + p ν σ µ ) α ˙ β Z ( p ) < χ µα ( p ) χ νβ ( − p ) > = ǫ αβ M P F (1) µν ( p ) (2.6) < ˆ T µλ ( p ) ˆ T νκ ( − p ) > = − (cid:0) p η λκ − p λ p κ (cid:1) F (4) µν ( p )Here F µν ( i ) and Z are introduced to store the information of supersymmetry breaking.All the tensor functionals are symmetric. M P is the mediation scale of supersymmetrybreaking. The last formula in eq(2.6) is manifested by the conservation of energy-tensorof hidden sector ∂ µ ˆ T µν = 0. To derive this formula, we recall the embedding relation ofenergy-tensor of FZ multiplets, T µν = − h ˆ T µν + 2 g µν Re ( F S ) i = −
12 ˆ T µν (2.7)for FZ multiplets with S = 0. One can use some simple examples to check the validity ofeq(2.6) for FZ multiplets with R symmetry. Note that there are no two-point correlatorsof auxiliary fields M µ and N µ , whose contributions to Feynman diagram of soft terms aredenoted by their vacuum expectation values (VEV). Components λ µ and D µ are unrelatedto the calculations of soft terms in this paper, we do not discuss them.Coupling the supercurrent of hidden sector to supergravity via κ Z d θJ µ H µ = κ h C µ D µH − ( χ µ λ µH + c.c ) − ˆ T µν φ µνH + 12 (( M µ − iN µ )( M µH + iN µH ) + c.c ) (cid:21) (2.8)from which we obtain the wave function renormalizations of component fields of gravitymultiplets. κ = √ π G . According to the normalization taken in eq(2.8), all functions F (1) , F (2) , F (3) , and F (4) have mass dimension of two in momentum space.3 Soft Terms in Gravity Mediation
The supercurrent of supersymmetric standard model (SSM) [10] is given by , J visα ˙ α = 32 ¯ W ˙ α e V W α −
23 [ D α , ¯ D ˙ α ]( Q † e V Q ) + 2( D α Q ) e V ( ¯ D ˙ α Q † ) (3.1)where D α Q = D α Q +( e − V D α e V ) Q . Q i are quark superfields. Coupling the supercurrentof SSM to supergravity gives in components, κ Z d θJ µ H µ = κ (cid:16) C Jµ D µH − ( χ Jµ λ µH + c.c ) − ˆ T Jµν φ µνH + M Jµ M µH + N Jµ N µH (cid:17) (3.2)We divide the Lagrangian into L F and L B that are related to gaugino and sfermionmasses respectively, which are explicitly given by,1 κ L F = 16 D µH ( λσ µ ¯ λ ) − (cid:20) i ( λ µH σ ν ¯ λ ) (cid:16) F µν + ˜ F µν (cid:17) − QQ ∗ ( λ µH σ µ ¯ λ ) + c.c (cid:21) + 16 φ µνH (cid:0) i∂ ν λσ µ ¯ λ + c.c (cid:1) − φ µνH λσ µ ¯ λV ν (3.3) − h ( M µH − iN µH ) (cid:16) σ m ∇ m ¯ λ ) σ µ ¯ λ + √ Q ∗ ¯ λ ¯ σ µ ψ (cid:17) + c.c i and 1 κ L B = (cid:20) − D µH ( iQ ∗ ∂ µ Q ) + 2 √ M µH − iN µH ) (cid:0) Q ∗ ψσ µ ¯ λ (cid:1) + c.c (cid:21) + 43 h λ µH (cid:16) √ Q ∗ ∂ µ ψ + √ iQ ∗ V m σ µ ¯ σ m ψ + 2 QQ ∗ σ µ ¯ λ (cid:17) + c.c i − φ µνH (cid:2)(cid:0) η µν Q ∗ (cid:3) Q − η µν Q ∗ QV + 4 iη µν Q ∗ V m ∂ m Q (cid:1) + c.c (cid:3) (3.4) − φ µνH (cid:2)(cid:0) η µν ( D m Q ) + ( D m Q ) + 2( D µ Q ) + ( D ν Q ) (cid:1) + c.c (cid:3) where V m and F mn denote the gauge field and field strength respectively. ψ i refer tofermions of standard model and Q i their sfermions. The gauge covariant derivative isdefined by D m Q = ∂ m Q + iV m Q . F is the auxiliary field of quark superfield. The termswe neglect are conjugate terms.We would like to make a few comments about formulas eq(3.3) and eq(3.4). Someof operators in original Lagrangian do not contribute to generations of soft masses atleading order, for example for those terms that only carry derivative of Q scalar field,thus irrelevant to calculations of sfermion masses. Some of terms in L F and L B inducesoft mass terms at two loop while others at one loop, which are found to be same orderof O ( κ ). Finally, we use components λ H and φ µνH of gravity supermultiplet instead of4raviton and gravitino fields in evaluating the Feynman diagram. The propagators of λ H and φ µνH are obtained via the embedding relations for FZ supermultiplet.According to Lagrangian eq(3.3) and eq(3.4), the contributions to gaugino and sfermionmasses are composed of Fig.1 to Fig.6 and Fig.7 to Fig. 15 respectively. First, we com-puter the gaugino masses. We find the contributions to gaugino masses only come from5ig. 3 and Fig.4, which are explicitly given by, m / = − (2 κ ) Z d p (2 π ) M P p ( L ( p ) + L ∗ ( p )) (3.5)where L ( p ) = i c ( i ) (cid:2) F ab (1) p a p b (308 − i ) + F (1) p (418 − i ) (cid:3) + c ( i ) Z d q (2 π ) q ( p − q ) (cid:0) p a p b F ab (1) + 25 p F (1) (cid:1) (3.6) c ( f, i ) is the quadratic Casimir of the representation of f under the r gauge group.Here are a few comments about gaugino masses. First, it is observed that there areno contributions coming from those diagram associated with F µν (2) , F µν (3) and F µν (4) , all ofwhich are odd number of powers of momentum integrals. The gaugino mass is onlydependent on function F µν (1) via Feynman diagram Fig. 3 and Fig.4. In comparison withcalculations of sfermion mass as we will show, the sfermion masses are independent on F µν (1) . This structure of soft terms has been found to exist in general gauge mediation[11], but nerve also expected in gravity mediation. It can be verified function F µν (1) = 0 insupersymmetric limit on general grounds, which implies that gaugino masses vanish whensupersymmetry restored. Finally, the dependence of gaugino masses on gauge quantumnumbers is manifested by c ( i ). Thus, unification of gaugino masses is not universal ingravity mediation. This character is also shared by general gauge mediation [11].Now we compute the sfermion masses. The contributions to sfermion masses arecomposed of Fig.7 to Fig.15, m = − κ Z d p (2 π ) p "(cid:18) (cid:19) ˜ K ( p ) + (cid:18) c ( f, i )2 p (cid:19) ˜ U ( p ) + (cid:18) (cid:19) ˜ L ( p )+ p (cid:0) p η λκ − p λ p κ (cid:1) ˜ H λκ ( p ) i (3.7)where ˜ K ( p ) = p µ p ν F (3) µν ˜ U ( p ) = − p ( M + N ) − p a p b (cid:0) M a M b + N a N b (cid:1) (3.8)and ˜ L ( p ) contains the contributions from intermediate propagator of λ H , which is functionof F µν (2) and Z ( p ). Since the form of ˜ L ( p ) do not closely related to main conclusions below,6e do not explicitly evaluate it in this paper. Finally,˜ H λκ ( p ) = c ( f, i ) Z d k (2 π ) Z d k (2 π ) · (9) F λκ (4) p k k ( p + k + k ) + 9 F λκ (4) + F (4) η λκ p k k ( p + k + k ) ! + c ( f, i ) Z d q (2 π ) · (9) F λκ (4) p q ( p + q ) + (25 p η cλ η eκ − p κ p e η cλ − p c p λ η eκ + p c p e η λκ ) F (4) ce p q ( p + q ) ! − (cid:18) (cid:19) p F λκ (4) + c ( f, i ) Z d q (2 π ) Z d k (2 π ) · (9) F λκ (4) p q ( p + q + k ) ! (3.9)We want to mention that in Feynman diagrams associated with φ scalar fields one shoulduse the intermediate propagator < φφ µν > .As mentioned above, sfermion masses do not dependent on F µν (1) , which is manifestedby the Lagrangian eq(3.4). The vanishing of sfermion masses in supersymmetric limit isnot obvious. Given F (1) and other F ( i ) s of same order, the gaugino and sfermion massesare roughly comparable with each other. The dependence of sfermion masses on gaugequantum numbers is given by ˜ L ( p ) and ˜ H λκ ( p ). Unlike the universal dependence ofsfermon masses on gauge numbers in general gauge mediation [11, 13], the sfermonsmasses depend on their flavor and gauge symmetries. Thus, there are no sum rules ofsfermion masses of each generation in gravity mediation. This property weakens theprediction of gravity mediation at LHC, however, also separates it from gauge mediation.This helps identifying mediated mechanism of supersymmetry breaking when the primarycontributions to soft terms come from quantum supergravity.There are some interesting issues that should be studied in the future. First, thepositivity of soft terms, especially sfermion masses should be discussed at least in mod-els that are simple enough. There are some other supercurrent multiplets including R multiplets, S multiplets and variant supercurrents. It would be interesting to discuss softterms induced by quantum supergravity using the supercurrent approach proposed in thispaper. Acknowledgement
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