JHEP10(2009)077 July 24, 2009 : October 2, 2009 October 27, 2009 y where super- : September 9, 2009 : : , Received a Accepted Revised Published the TeV-scale supersymme- natures at the LHC experi- e Hidden Local Symmetry in 10.1088/1126-6708/2009/10/077 ate a spin-two resonance into doi: sm, we propose a supersymmetric Published by IOP Publishing for SISSA and Masafumi Kurachi b [email protected] , Ryuichiro Kitano a 0907.2988 We present a simple formulation of non-linear supersymmetr Supersymmetry Phenomenology [email protected] SISSA 2009 Theoretical Division T-2, Los AlamosLos National Alamos, Laboratory, NM 87545, U.S.A. Department of Physics, Tohoku University, Sendai 980-8578, Japan E-mail: [email protected] b a c
Abstract: Michael L. Graesser, ArXiv ePrint: fields and partnerless fields can coexist.Standard Model Using this without formali the Higgsinotry as an breaking effective scenario. model for Wenon-linear also supersymmetry, consider where an we applicationthe can of theory naturally th incorpor in aments manifestly are discussed. supersymmetric way. Possible sig Keywords: Higgsinoless supersymmetry and hidden gravity JHEP10(2009)077 4 4 8 1 3 12 16 10 11 20 Λ, is ≪ tions to h . However, m Pl ∼ M W ]. We assume that m ≪ 8 , , is explained by dy- 7 d more fine-tuned, at ]. The big hierarchy, me very popular as a Pl 2 QCD , M ments, parameters in the hich we call the scale Λ), field to break electroweak ence of elementary Higgs 1 cs through direct couplings etry is dynamically broken ≪ arios with a weakly coupled brid of technicolor and SUSY io has several virtues: (1) the ties in this idea. One is that uilding [ owever, with the experimental – 1 – ]. 6 – 4 ]; (2) the hierarchy problem, Λ 11 ]; (3) one can hope that the little hierarchy, – 9 12 ]. Another is the difficulty in writing down the Yukawa interac 3 , is elegantly explained by the very same reason as Λ Pl M ≪ After the LEP-I experiments, supersymmetry (SUSY) has beco In this situation, it may be interesting to (re)consider a hy 4.1 Perturbative unitarity 4.2 Phenomenological signatures 2.1 Convention and2.2 superfields Non-linear SUSY 2.3 Gauge invariance W 1 Introduction Technicolor is an attractive idea in which electroweak symm 4 Hidden gravity 5 Summary 3 Higgsinoless SUSY by a strong dynamics operating around the TeV energy scale [ Contents 1 Introduction 2 Non-linear SUSY and invariant Lagrangian namical SUSY breaking [ m generate fermion masses in the Standard Model. along the similar spirit of the early attempts of SUSY model b natural scenario for the lightbound weakly coupled on Higgs the boson. lightest H minimal Higgs SUSY boson standard mass model fromleast (MSSM) the in are LEP-II the required experi conventional to scenarios be [ more an strong dynamics breaks SUSYwith at electroweak the symmetry multi-TeV energy breakingbetween scale triggered the (w by Higgs field the and dynami theYukawa interactions dynamical sector. can This be scenar fields written in down the by UV theory, assumingsymmetry which an mix at with exist low (or energy remain [ as) the Higgs it is well-known thatthe there are electroweak two precision phenomenological measurementslight difficul seem Higgs boson to [ prefer scen JHEP10(2009)077 2, / and 2 i 2 H h )Λ k π 4 = n. By com- her requires α/ 2 H oson mass from xperiments can m EP-II experiments can ting possibility, an ex- Or simply the Higgsino do not generically allow ector, so that the typical p quarks as usual. The ino field is absent in the adratic divergences from le Λ. owski space — one needs ologically motivated from TeV scale. rms among superfields and Higgs boson as the pseudo- the order of ( seudo-Goldstone boson. In ing as organizing principles: tic interaction although the ling to the dynamical sector an for the scenario without y. One therefore expects the sponsible for SUSY breaking this case from a loop diagram try breaking, m linearly under SUSY); and ere. In this paper, we take a he connection to string theory. wo kinds of fields are defined on depends on the UV completion, ming field because it is assumed ymmetry; (2) the quarks/leptons ]; (4) the cosmological gravitino prob- (100) GeV (in other words, the matter 13 O ). This divergence is not cancelled, simply 2 t TeV. Precision electroweak constraints, on (Λ). We may therefore need to assume that λ O × – 2 – ]. The dynamical scale may therefore have to be 16 (a few) i ∼ H h π 4 . being the coupling constant of the Higgs quartic interactio 10 TeV). To obtain a light Higgs boson at this larger scale eit k ]; (5) one can expect additional contributions to the Higgs b − 14 ]; and (6), there is an interesting possibility that the LHC e (6 with O 15 2 ≃ )Λ 2 π In constructing the effective Lagrangian, we take the follow Although the TeV-scale SUSY breaking scenario is an interes In this Higgsinoless model, the Higgs potential receives qu The stop potential also receives quadratic divergences, in The Lagrangian we construct needs to contain interaction te 16 k/ naturalness suggests Λ Nambu-Goldstone particle in the strong dynamics [ explained by either SUSY or some other mechanisms such as the the other hand, obtainedsuch from a the low LEP-II scale and without SLC fine-tuning experiments [ (1) the Lagrangian possessesand non-linearly gauge realized fields supers are onlymass weakly splitting coupled between to bosons the and SUSY breaking fermions s are plicit model realizing thisless scenario ambitious will approach not andspecifying be (while construct attempted hoping an h for effectiveand the Lagrangi its existence mediation. of) a UV theory re lem is absent [ probe the SUSY breaking dynamicsthe directly. point (1) SUSY (and is also phenomen (6)) in this framework in addition to t larger, Λ the SUSY breaking sector,be with evaded which [ the mass bound from the L ( paring with the quadratic term needed for electroweak symme fine-tuning, or some newappears weakly around coupled a few new TeV.) physicsgenerically It gives below is the also Λ. Higgs true boson that ( mass the to direct be coup and gauge fields are introduced(3) the as Higgs superfields boson which is transfor to introduced as be a directly non-linearly transfor coupledminimal to model. the SUSY breaking sector. The Higgs loop diagrams with thetop-quark gauge interactions loops and can therough Higgs be estimate quar cancelled of by the correction the to loops the of Higgs boson the mass scalar is to of involving the Higgs boson (and proportional to the Higgs boson isthis somewhat paper special we in simply ignore theand the dynamics, issue here e.g., because we a its only resolution p concerned with the effective theory below the also partnerless fields such as thedifferent Higgs spaces boson. — Since one these superspace t and the other the usual Mink because the Higgsinos are notstops present to in have the a low mass energy no theor smaller than a loop factor below the sca JHEP10(2009)077 ly ] where try (the 19 – 17 ersymmetric has ], Ivanov and Kapust- his technique to SUSY, ongly coupled field for a de possible by a warped invariant under the non- 17 per, we present a simple ity on a single space-time. w that there is a sensible r fields to constrained su- grangian, in particular the ] in the chiral Lagrangian. odel to describe the vector e down a supersymmetric uct a model “Hidden Grav- i space into the superspace is opposite to that and is, o field which is introduced o convert partnerless fields k [ the Higgs boson as a non- i space, on which superfields ation of the resonant single- free parameter in this model. 26 elated to the gauge coupling nd that the coupling constant (TeV) scale). ) and also matter and gauge – cause the unbroken symmetry up to Λ, another direction not io at the LHC. aviton is much lighter than the O rsa. One approach is to utilize e the graviton mass the effective 22 own formalisms. The essence is to of partnerless fields [ – 3 – ] by the same authors, where SUSY is broken on the ] have established a superfield formalism of non-linear 21 19 ]. There SUSY is broken explicitly at the Planck scale, and on 20 meson) as the gauge boson of the hidden vectorial SU(2) symme ρ ], and Samuel and Wess [ 18 As a related topic, a model in which the MSSM is only partly sup One can consistently incorporate the resonance as a non-str As a possible signature of the TeV-scale dynamics, we constr to convert the partnerless fields into superfields or vice-ve established formulations for constructing superfields out nikov [ SUSY by upgrading the Goldstino fermion and other non-linea the Goldstino field ismanifestly also supersymmetric promoted formulation to whereinto we a superfields, although do superfield. it not is totally try Inprepare equivalent two t to this kinds the of pa kn spaces:and the partnerless superspace fields and are the defined. Minkowsk by using By a embedding SUSY the invariant map, Minkowsk By one using can the define formalism, a one Lagrangian can dens Yukawa interactions, write only down with a a SUSY single invariant La of Higgs field. the Higgs We also quartic fi constant. interaction Therefore, can the be Higgs a boson mass free can parameter, be unr treated as a been proposed in ref. [ The Hidden Local Symmetry isresonance a (the manifestly chiral symmetric m the Higgs sector isextra-dimension remained (or to a be conformalrelatively supersymmetric speaking, dynamics). which closer is Our toIR ma ref. philosophy brane [ (or equivalently by some strong dynamics at the range of parameters andparameter small region range where of we energy.graviton can production perform Indeed, cross a section. we perturbative At sho theory calcul energies becomes not strongly far coupled abov andcut-off incalculable. scale, new If physics the is gr pursued required here. to complete We the discuss theory signatures of this graviton scenar 2 Non-linear SUSY andIn invariant Lagrangian this section welinearly present realized a method globallinearly to supersymmetry. transforming construct a field We Lagrangian (whichfields will we as introduce call superfields. aLagrangian non-linear We where field therefore both need kinds a of formulation fields to are writ interacting. Ro˘ce unbroken symmetry of thewe chiral obtain Lagrangian). a supersymmetric Whenas Lagrangian we a for apply graviton associated a t with massiveis a the spin-tw hidden Poincar´esymmetry. general covariance be ity,” which is an analogy of the Hidden Local Symmetry [ JHEP10(2009)077 2 (2.3) (2.4) (2.6) (2.5) (2.1) (2.7) (2.2) (2.8) tively. 2, unless licated, / space. , 1 ˙ α x − η ]. + ¯ ˙ 29 α ¯ θ = ) on the ˜ ˙ ′ α ¯ ] that establishes the θ 2.6 ′ ¯ θ 28 [ → , , ′ ˙ α )) bedding the theory into su- ¯ ,θ θ on superfields defined by the ′ x paper, we will use a simpler y transforming field. We will (˜ x 1 ). Note that a global SUSY ) with mass dimension , x − (˜ α g on in eq. ( 1 η, λ 2.1 λ η ( ] ¯ η. µ = =Ψ . + a , . a 30 λ , etry can be found in ref. [ ˙ α α α ¯ ¯ θ P Q ) is defined by Volkov and Akulov in η η θ ). The SUSY algebra is ˙ +∆ iξσ ¯ η β i a α µ = + − 2.2 c σ )+ ) + ¯ x,θ, ¯ ′ α ξ x x θ + a iηQ (˜ (˜ − −− ˙ = 2 + α α µ a )Ψ → ¯ λ x λ – 4 – iησ o 1 P (+ ˙ . a β α in eq. ( − = ˜ ≡ ¯ ic g Q ′ e ( g ) = ) = ) , ′ ′ µ r α , θ ) transforms as [ x x ˜ x = is a representation of ) ˙ (˜ (˜ α Q = η,ξ = diag ˙ g ¯ ′ α ′ α θ ( n 1 → 1 , ¯ a λ λ ab η,θ − α − µ ( η ∆ g g ˜ a x → → ,θ a ) ) ¯ θ r transforms as x x x (˜ (˜ +∆ ˙ α α ¯ θ a x,θ, ¯ λ λ c + Ψ are the Goldstino fermion and its complex conjugate, respec x,θ, a g x ¯ λ ] for a recent work.) Although the formalism is somewhat comp = 27 ′ factor is defined by a and x a λ → a ]. It is x 31 A superfield Ψ Under a group element, The generalization of this relationship to local supersymm Throughout this paper we will use the shorthand notation 1 2 second equality. 2.2 Non-linear SUSY The non-linear transformation under where the ∆ The transformation above satisfies the algebra in eq. ( 2.1 Convention and superfields We use the metric convention: perfields. (See [ under SUSY. The operation ref. [ correspondence between superfields and non-linear fields. transformation induces a general coordinate transformati using superfields is motivatedpergravity. there as As a we first areformalism step not where towards interested the em in Goldstinoalso field supergravity remains in use as this results a from non-linearl earlier work by Ivanov and Kapustnikov the superspace coordinate ( The fields indicated otherwise. JHEP10(2009)077 G ) = ′ ∈ ¯ θ (2.9) , space ) ′ (2.12) (2.13) (2.10) (2.11) (2.15) (2.14) x ( ,θ x ′ ). iπ x ¯ e θ ( ′ Ψ ≡ . The appro- ) x ) by using the x,θ, → x ( ) which we cannot invariant one by ¯ θ ξ 2.6 g on the ˜ x G ) can be constructed x,θ, ¯ θ and ˜ ment . x x,θ, . A Lagrangian which is , )) llan-Coleman-Wess-Zumino es still contains ˜ as the gauge connection. sformations are different. In H ′ ,θ , orentz indices. ) ˜ x ¯ λ, ation in eq. ( ) ] for a review.) There a global x a x (˜ (˜ φ 25 λ a λσ ν , µ ]: ) can be obtained simply by writing ) A ∆ ( )= x i∂ space. The SUSY transformation on µ ν 32 (˜ ,θ x ′ indices by defining the transformation ) and superfields, we need a “converter” − ) (˜ ˜ x ˜ φ x + x x x φ ∂ x ∂ transformation where the Maurer-Cartan (˜ : H (˜ ¯ λ ( g φ λ µ ) φ ¯ λ, λ. ( = H ), the transformation of the classical field (or expec- ∂ µ µ µ a and ∂ ∂ ′ 2.8 ]. (See also [ g, λ ∆ )= ˜ x ( µ µ ). For a classical superfield, Ψ( – 5 – a a 1 G x iλσ x 33 − ( (˜ − a 1 1 φ φ µ − ′ µ h − − x ) and ( a A µ A A r (Ξ) ) = = η 2.7 ′ r → x = ˜ µ (˜ . For example, a superfield Φ( space into a superfield in the superspace ( = = = ′ ) ˜ ≡ x 1 ¯ λ φ ). We can follow a similar prescription here by taking the x x ¯ a − ¯ Q transformation can be upgraded to a λ λ (˜ µ i ¯ g a a θ a → ) + µ A g,π ) ∇ ∇ H x ( A (˜ x ) by ] 1 (˜ iQλ x,θ, x φ − e (˜ gφ 28 h are invariant. φ ) Φ ) can be introduced on the ˜ x A supersymmetric action for ¯ λ ( x a (˜ 3 ). gξ ∇ φ ). 2.9 → 2.6 transforms as the vielbein under is spontaneously broken down to a subgroup ) and ) is the representation of the space-time translation actin x A λ ( 1 (Ξ) with Ξ = G a ξ − r h ∇ ( ). The transformation laws remain unchanged for fields with L r ¯ θ to be Matter fields Superfields and non-linear fields are living in different spac One can construct the Maurer-Cartan 1-forms [ The discussion is completely parallel to the formalism of Ca In the CCWZ formalism, a linear representation of a group ele In terms of the same notation in eqs. ( ξ 3 x,θ, operators is defined by [ an invariant action under the general coordinate transform defined in eq. ( whereas where ˜ tation values of the field) is: vielbein in eq. ( symmetry The matrix (CCWZ) for internal global symmetries [ making the Lagrangian invariant under a local 1-form (projected onto the unbroken generators) can be used identify as the same spaceorder to at write this down an stage interaction since term between their SUSY tran Ψ( which transforms a field in the ˜ invariant under the global plays a role of the converter between the converter from a non-linear field of priate identification is found to be At this stage, Φ is not defined yet because the last expression JHEP10(2009)077 ) x ( λ (2.22) (2.21) (2.20) (2.23) (2.19) (2.17) (2.16) (2.18) , x ) which ). With ¯ θ x (˜ ¯ λ x,θ, ] = pectively. The , ¯ θ 31 ] for constructing rmation about the ), = Ψ( in terms of 37 x (˜ – K x etric action using the λ 34 e ns will be terminated at , , = )) )) θ ) is a superfield, i.e., 19 ¯ λ, A, X, . . . ¯ θ ,θ a , ,θ ) at ) ∇ x 17 x (˜ ¯ θ, bit of complication because the (˜ A , x,θ, λ a λ, . ) λ ( a . ( x ′ µ λσ (˜ ∇ µ , ¯ µ θ ¯ λ ,θ ∆ A. ∆ ¯ ′ λ, is i∂ ′ − ˜ − x − ˜ x − ¯ θ + det µ as independent variables in constructing X x,θ, µ ¯ θ ˜ φ x ′ ˜ x ¯ λ ˜ x x 4 ]. (See also [ λ µ , λ, Ψ − = d ) ∂ − θ 28 a µ µ x 4 − ′ µ (˜ Z x – 6 – ¯ θ ∆ x ( λ , 4 xd ( ,θ , and ˜ iθσ ′ 2 4 , and is equal to 4 ¯ f − θ ) 4 − d A ,θ − x , µ ′ − ′ (˜ θ a x Xδ X δ x Z µ , = η , we obtain the Volkov-Akulov action [ , φ , θ x = = ) Φ . It is possible to iteratively solve ˜ S det det 2 = ¯ x x θ µ S ′ / ˜ ˜ (˜ x x a 4 ˜ x 4 4 φ µ f d X θd x,θ, − ) is a superfield. 4 Z · Ψ xd ) 4 2.15 λ d 1= − ×K Z θ ( = represents arbitrary superfields and non-linear fields, res 4 δ S φ = must be real and scalar under the general coordinate transfo K K defined by eq. ( , but the solution involves many terms although the iteratio x θ As an equivalent formulation, one can construct a supersymm coordinate. As an example of the invariant action, we can tak ˜ If we take gives the supersymmetric action, which consistently defines the superfield Φ [ x the delta function, one can treat superfields out of non-linearabove fields). equation There is is non-linear in still a ˜ little supersymmetric invariant where Ψ and which transforms in the same way as the Lagrangian. The invariant action can be written down as in the superspace integral. The Jacobian matrix function and finite steps. Nonetheless, one can explicitly check that Φ( In general, any function of because with ˜ JHEP10(2009)077 ] ) ]. 38 34 2.26 , ) and (2.24) (2.25) (2.28) (2.29) (2.30) (2.27) 3 32 [ ∂ ( ¯ λ ) and ( O , b λ ∇ a 2.24 -loop corrections ∇ n unt the number of ., c and/or ¯ λ . d invariant actions. are obtained by per- ) and ) and so on, with the is the decay constant µ d λ 4 x p a iance of f ( ∂ +h ( iθσ ∇ O or ˜ O ter ve dimensional analysis [ x + 2 g term begins at ¯ ¯ λ λ ave b . µ different numbers of derivatives, the external lines in each amplitude. For ) . ∇ ¯ λ, A, . . . x a iλσ (˜ λ, ) ˜ ∇ a L easily see that terms in eq. ( x − ] (2.26) ( ˜ x ∇ λ, µ ˜ ¯ 4 θ, ab G L a [ ˜ x d µ η b ∇ ν − Z )+ iθσ A λ, A F µ A R x external Goldstinos is a ( y µ − − )+ d L A – 7 – µ det 4 det x ( ,θ x ˜ x ˜ x ≡ x ) L 4 4 Y δ 4 = x d x d d (˜ µν 4 µ d Z G det y Z Z , φ 4 ˜ ) x Z 2 4 f = = − y,θ θd S 2 S = that forms a Lorentz invariant out of Ψ( yd 4 . 4 S F d πf 4 ×W Z √ = ∼ S ). When comparing with other terms in the action, we should co n +4 d p ( O One may generalize the Volkov-Akulov action. From the invar Lagrangian densities with a single space-time coordinate Superpotential-like terms can also be constructed as Although the Volkov-Akulov action involves many terms with 4 is SUSY invariant for any momentum expansion still makesexample, sense the once lowest we order fix (tree) the amplitudes number with of This contains the kinetic term for the Goldstino. The parame contain more derivatives than the Volkov-Akulov action. implies a cutoff Λ Another possibility is to consider the “metric” which represents the size of the SUSY breaking. Note that nai to that are which transforms asFor instance, a covariant tensor and can be used to buil is invariant under global SUSY transformations. The leadin derivatives with a fixed number of the Goldstino fields. We can involves two Goldstinos. Terms involving four Goldstinos h last term involving 8 Goldstinos and 6 derivatives. forming one of the space-time integrals, i.e., of the form: We now have a Lagrangian in a single space-time. with This action is of course identical to JHEP10(2009)077 ) y,θ ( (2.35) (2.34) (2.31) (2.37) (2.36) (2.33) (2.32) ) O x iant. (˜ φ O gV )) on the brane. 2 ., x c − . (˜ e λ 2 D )+h own a usual superspace as the Higgs boson, this chiral superfield ion should be invariant gV magine a set-up where a x ) defines the map from a ¯ 2 λ. (˜ φ x e rane fields can be written µ e it is not gauge invariant. (˜ φ ∂ 2 1 which is a translation in the λ a , pends on ) ) iθσ x ¯ λ, θ λ (˜ 2 µ . φ − , − Λ ) ¯ λ i θ iθσ , ( = ¯ λ . − 2 ) ¯ θ 4 ) µ e y,θ δ x λ, x ∂ ( (˜ − (˜ a = gV φ )) O ¯ φ 2 λ a ) a µ x,θ − ˜ T T iλσ e ) =˜ ) λ, θ † x x y,θ (˜
+ Λ iλσ ( y,θ a i – 8 – ( ∆( e ¯ λ a iα O + a e Λ O − i µ → = e ˜ x ˜ x λσ gV → µ 2 gV − W − φ 2 O i∂ e x − 2 O→ ( e + A 4 D a µ η gV det X δ 2 θ are charged under some gauge symmetry (as it is true in the e = 2 ): 1 2 a x φ det ˜ (˜ . Under this transformation, the action is clearly not invar xd µ 4 φ ji Y ˜ x d ) 4 A a and Z θd T ( 4 . The gauge transformation of the vector superfield is O det = a − ˜ xd x ˜ T S 4 4 a = as a bilinear of the quarks/leptons superfields and d d V ij O ) Z Z a = ˜ T = = V S There is an intuitive picture for this construction. We can i In order to maintain gauge invariance, we write the action as However, if where ( and The gauge transformation is defined by Standard Model), we need to modify the interaction term sinc 3-brane is embedded into a superspace. The Goldstino field where and gives a supersymmetric Yukawa interaction term. By taking The interaction terms betweendown by superfields using (bulk a delta fields) function. and b 2.3 Gauge invariance It is now possibleand to a non-linear write field down an interaction term between a from point on the braneLagrangian to a as point well inunder as the the superspace. a general brane coordinate Onesuperspace, can transformation localized write induces because action. d SUSY, a coordinate The transformation brane (which de act JHEP10(2009)077 ) x (˜ α ). We (2.40) (2.46) (2.41) (2.43) (2.44) (2.42) (2.45) (2.38) (2.39) x (˜ φ satisfying α . 2 . ) a x (˜ ∂ φ ∂x ˙ lex-valued α gV . ) nsformation (it is real in kinetic term for 2 a i − ) ), one can confirm that it ansformation at this stage. e θσ x λ , ) ( (˜ , i . x Λ φ , . − (˜ i Λ a . i † α gV + − θ 2 φ − ¯ λ e ( D A ˙ α − e gV 4 ) a ). ¯ 2 θ ∂ α δ α gV ∂ x 4 k − ∂ 2 λ De (˜ ∂x D e ˜ x, λ, − a a φ − Λ α Λ − e i ¯ i σ a † ∇ e ) a e D ( = D x 1 g i )) g gV g ˙ (˜ 2 α gV x iα φ 2 (˜ ∆) and ¯ + e + + D e ¯ φ ) with De ) = Λ a Λ Λ gV a − i x i 2 4 1 → – 9 – ≡ (˜ A , − − D ˜ x − ) φ e ( e a y, λ e ig ≡ ¯ x θ h ( a α ) ∂ − (˜ a ) x ∂x − A A this time. By using these superfields, the “covariant” φ ¯ θ (˜ λ is unrelated to the gauge coupling constant in con- x a Λ Λ a † ( α a i i x,θ, 4 φ − k e e D T ¯ δ 2 θ a θ ) = Λ a x,θ, α ( ≡∇ m ) is gauge invariant. Note, however, that the function x → → σ 4 (˜ A a a − δ a a g α i D A α A 2.36 = ) A ] g − . λ 39 α − kin ∂ and Λ = Λ ∂θ θ -functions, K ( δ a 4 = δ T a α = V D . = pot V K ). The quartic coupling 2.39 It is possible to define a covariant derivative to write down a Then the kinetic term can be written as define gauge superfields [ defined above can beWess-Zumino complex gauge). valued unlike the usual gauge tra eq. ( behaves as a covariant derivative: and Both are gauge invariant under the delta functions with comp The derivative operators are defined by The gauge transformations of these superfields are We defined derivative is constructed as This derivative operator isHowever, not under covariant the under the gauge tr trast to the prediction of the MSSM. The potential terms are By defining the gauge transformation of the interaction term in eq. ( JHEP10(2009)077 (3.3) (3.2) (3.7) (3.6) (3.5) (3.4) (3.1) . i L c j . ): i ), motivated by E λ x i (˜ . − c L h . 2 c (cov) j θ ( E D 4 , +h δ 1 2 ce. Soft SUSY breaking to construct interaction . cale Λ has something to king terms. The appear- ns and gauge superfields, gV ¯ λ ¯ quarks/leptons and gauge space, λ 2 − gV i x . We assume that such soft , he Yukawa interactions. The 2 e ) and † Q . The Yukawa interactions for stly supersymmetric. One can − ˜ x, λ, ˜ λ ) x, λ, c ) j e i x † U − (˜ ) α Q Ψ h c x θ † j . (˜ ( W ij e U 2 Ψ h 2 (cov) α ( A δ gV 2 Ψ ij e · 2 D W y ) + − α 2 1 Ψ 2 e x † / + (˜ 2 i W 1 A m 2 Ψ · h α Q D m 2 Ψ ) i c ij u j x W det gV Q A m D (˜ A 2 c 2 j ˜ · x / h – 10 – e ) 4 1 ) 2 D ij u d λ λ gV det y ≡ m 2 ˜ · − x Z − − 2 ) (cov) 4 ) e θ θ d λ † ( λ D ( 2 (cov) ) 2 4 2 1 − ∋− δ x Z δ − D (˜ θ θ S − h = ( ( 4 ij d 2 gV A δ ∋− δ = ⇒ 2 A h − S W − = e ) † soft λ ) = up ⇒ K x ) K − (˜ λ h θ soft ( − ij d 4 y W δ θ ( 4 = δ × A = K down K We introduce the Higgs field as a non-linear field on the ˜ -terms can also be written down by taking one can simply write downterms the can MSSM be Lagrangian written in down the by superspa using delta functions 3 Higgsinoless SUSY We are now ready to construct a Lagrangian. For quarks/lepto For down-type quarks and leptons, an assumption that thedo SUSY with breaking dynamics the at originterms the of has cut-off electroweak been s symmetry discussedup-type already breaking. quarks in the are The previous way subsection These are the same asance the of spurion the method Goldstino for interactionsalso the makes soft add these SUSY terms hard brea manife breakingand terms hard breaking by terms using arefields covariant are somewhat derivatives not suppressed participating because the the SUSY breaking dynamics. and It is not necessary toA introduce two kinds of Higgs fields for t Here we have used the covariant derivative: and JHEP10(2009)077 ], )= 40 ′ x (4.4) (4.2) (4.3) (4.5) (4.1) (˜ -space. ′ ab g x ) is → αβ ) g H x (˜ µν ab g H : g − space having a local x νβ h the enlarged gauge H he ˜ µα ) is a free parameter, the mation under structing a realistic model ]. The appearance of a massive meson in QCD. We further H , ( scription of a composite spin- ) 42 ansformation on the ]. ρ . x 2.46 e graviton can be found in ref. [ (˜ 43 αβ SUSY transformation, and one g s a scalar under ] is ρσ g µν 44 in ref. [ ν σ ′ , field is therefore a SUSY covariant gg ˜ ns are related by multiplying by the vielbein, x from the Minkowski metric describes ˜ in ref. [ x . global ∂ µν ∂ √ µν µν ab h 2 µν µ ρ η ′ G g H (TeV), unrelated to the four-dimensional P b ˜ x ˜ m x ν 2 8 ∂ O 2 P − ∂ m A ]. A more ambitious attempt to formulate Einstein a m µ + µν = 41 g A − – 11 – ′ µν -boson mass. It is not a very obvious result that ) ˜ = x η = Z g ( = µν µν ] it has been necessary to introduce an extra Higgs ′ µν gR H g G µν 35 space is realized as a local coordinate transformation in g √ → Note that this is a 2 P x 2 ) m 6 x (˜ ). − µν g 2.6 A is a mass parameter of det P 4 of Einstein gravity. With 2 ). We will not pursue this formulation any further. f m 2 x / (˜ − 1 µν − N g ˜ x G 4 b ν d A in the SUSY breaking dynamics analogous to the a is given in eq. ( µ Z 5 ′ A ). This local translation allows us to introduce a metric in t x = ≡ S ). This is an equivalent formulation, since the two definitio ) 2.6 The invariant action having the Fierz-Pauli form [ Specifically, we introduce a second “metric” whose transfor Since the quartic coupling of the Higgs boson in eq. ( An attempt to describe a spin-two resonance in QCD as a massiv We could have instead defined the spin-two field to transform a x x 5 6 (˜ (˜ ab ab Planck mass The space-time is always flat. The deviation of g g is a covariant tensor, defined previously. The where and the spin-two field. eq. ( we could writeinvariance. down a For Lagrangian example, with in a ref. single [ Higgs boson wit elaborate on this comparison towards the end of this section bound-state graviton in open string field theory can be found gravity as a composite of the Goldstino fermions can be found Higgs boson mass is not related to the boson, and that iswith claimed non-linear to SUSY. be a general requirement for con 4 Hidden gravity A SUSY transformation in the ˜ transformation law under the global SUSY.two This field provides a de whose supergravity extension is discussed in ref. [ where ˜ tensor. The scale should not be confused with the actual general coordinate tr JHEP10(2009)077 ]. 56 (4.7) oducing an wo Goldstinos constructed only precisely. Here the ] is that this vector r- and axial-type 1- o the Hidden Local oson, but no kinetic 26 duce a tadpole for the – n-two field is equivalent 22 mical. mass term is the unique d is not strongly coupled the global symmetry, it is , is made chirally invariant rgy scale. esonance can be consistently of [ e calculations make sense at V um and the absence of ghosts e form of the Fierz-Pauli mass − ffect is pushed to the cutoff [ + (four-fermion terms) (4.6) ]. Although loop corrections will V ) (4.8) j ) . 46 ν in QCD. By introducing a spin-two H , = ( to the spin-two field in a global SUSY etc V R ↔ 45 and up to two derivatives, such as V j m · j ˜ , gauge theory in the unitary gauge, which µ ) µν A g V ( h + ( R det ¯ λ · . , with µ , – 12 – µν vector meson). The action obtained in this way A V ) j ˜ h λσ ρ ν H V ( j det ˜ i∂ R · − g, 1) between the two terms appearing in the definition of g ¯ λ √ − ν √ ∂ µ are analogous to the , since the coordinates also transform. a µ iλσ A , the term Tr A µ + V µν transforms inhomogeneously under the chiral SU(2). By intr h P V 2 ). We first check this by looking at the elastic scattering of t j m m ( = O out of the pion fields. A chirally invariant Lagrangian can be µν , since vector boson H A Further assuming a Ricci scalar term in the action for the spi V,A j j V 7 is fixed by requiring that the Fierz-Pauli mass term not intro The analogy of the massive spin-two field as formulated here t Other interactions, such as There are other invariant terms involving In the chiral Lagrangian of QCD one can construct SU(2) vecto The other invariant is det 7 µν field having a localthen and possible to inhomogeneous introduce transformation theterm. under 1-forms into the action, in th and can be addedterm; to it can the be action. trivially integrated This out. gives The a key mass assumption to the vector b but these are forbidden byand the Lorentz tachyons. invariance of the Thattachyon vacu and is, ghost-free action the for Einstein a action spin-two field with [ Fierz-Pauli SU(2) out of one has obviously can have a sensible description up to some high ene Symmetry (HLS) of QCDMaurer-Cartan can 1-forms now be drawn more closely, though im to the physical assumption in HLS that the4.1 gauge boson is Perturbative dyna unitarity A question to be addressedintroduced here is in whether a this weakly newenergies spin-two coupled r of regime, in which perturbativ boson is dynamical (and describes the not preserve this form, the ghost pole is harmless since its e begin at higher than quadratic order in Note the relative coefficient (of coincides with the spontaneously broken SU(2) H graviton. The last term in the action gives a mass invariant way. with the same helicity.at threshold. Then we require that the spin-two fiel forms JHEP10(2009)077 ] )– 54 – (4.9) 4.9 (4.10) (4.11) (4.12) ze the 51 1), etc. . − o fermion 2 z ) is the same θ (3 2 1 4.2 cos ). By inspection, 2 3 )= z ( − ), and those from the 4.11 2 5 2 P comment. The second channel exchange of the ct four-point interaction , 4.10 z , 2 u ly, it has been known from f two gravitinos have been scattering in the Standard -dimensional theory [ this subsection that a spin- m 2 λλ nsate the integral over all of q. ( of eq. ( )= e up to some high energy scale e obtains m − exactly cancel. That is, at low z and cel the growth of the scattering M ( ) ) u 1 t WW θ raviton coupling in the Fierz-Pauli P + HH ( λλ is defined in the center-of-mass frame. (cos θ ℓ θ M P ) = 1, ) z cos θ ( 0 3 2 P 2. Substituting the expressions in eqs. ( ]. + / (cos appearing in the action in eq. ( ] that a massive vector boson can also partially d 55 – 13 – 2 5 1 1 f 49 − , , receives contributions from both the Volkov-Akulov Z 2 ) ]. The massive vector boson is indeed identified with 48 2 π m 1 λλ HH m 50 64 scattering amplitude as in the theory without a massive ( λλ − , t → = M 2 48 λλ ) is the contribution from s 2 + λλ 2 ℓ λλ s ) 2 8 ). The production angle m 8 A 4.11 M f m for 2 P f 2 P , m λλ (det 4.11 2 λλ 4 5 m s partner of the Goldstone boson, the Higgs boson, can unitari f 2 − − M ] and it has been shown that the scalar partners of the Goldsin M L 47 = = = ) ) A λλ HH ( λλ M λλ (det denotes the Legendre Polynomials: M M ℓ P scattering amplitude if it is light enough. But alternative The contributions from the spin-two action deserve further The partial wave amplitudes are defined by The amplitude In supergravity, the amplitudes of the elastic scattering o massive graviton, arising from the Goldstino-Goldstino-g mass term. The Fierz-Pauliinvolving mass the term Goldstinos, however giving also the has first a term conta in the r.h.s. term in the r.h.s. of eq. ( energies one obtains the same The contribution from the Volkov-Akulov action is given in e in the low-energy limit these two contributions to as the one in the original Volkov-Akulov action. the analysis of the Higgslesscancel the model amplitude, [ and theabove theory can the remain Kaluza-Klein perturbativ scale [ by using the technique of deconstruction [ spin-two action in eq. ( the one in HLS once the Higgsless model is formulated as a four calculated in ref. [ unitarize the amplitudes if theytwo field, are instead light of enough. theamplitudes. We scalar show fields, in This can is also partially analogous can to the discussion of the graviton. Therefore, the decay constant Model. The SU(2) WW Since the particles inphase the space final by state multiplying are by a identical, factor we of compe 1 where action and the action for the spin-two field. Specifically, on JHEP10(2009)077 )), . 4 f f . The 4 2 (4.13) . ) f 3 2 π . 2 ∼ . ((4 ∗ / ∼ 2 E ∗ s 2 ( E s these corrections O m f contributions. One ∼ 1+ ) interactions (i.e., the . Upper and lower curves m ν ν , one finds ln 1 H ∼ are chosen for illustration. s/f µ µ s ) terms, and the middle curve √ re 2 f kinds. First, the interactions ν ν H . The upper and lower curves scattering. 2 √ s m . , by the energy where the tree igh energies both the Goldstino H 3 the opposite sign (lower curve). litude, delaying the onset of the − are used as an example. ∗ 5. In the case of the example de- µ mplitude. The magnitude of the µ . λλ s/f E ive correction of f = 1 H µν 2 √ . ). At 4+ − H m πf =1 µν µν f 4 − , while the remaining terms come from H H m and √ 3 ) scattering at one-loop. Compared to the ( s A µν f O s 7 H . and 4 λλ and ( (det λλ 8 ) always have an opposite sign, and thus the f m ), respectively). The the middle curve represents 7 f – 14 – = 0 A 2 P . M → and ( P m 4.13 as a function of =0 4.13 A π m λλ 1 1 P 2) det 16 m / 4 − 2) det -wave amplitude for the f / -wave amplitude as a function of s 2 4 s − s f 2 scattering reaches the value 0 8 − m f 2 P λλ 4 2 0 1 2 3 4 s f m 0 ) terms in eq. ( 0.5 2 −0.5 π π 0.25
−0.25 1 s 5
( s−wave amplitude s−wave 16 32 O − = , the pure Goldstino amplitude (i.e., the upper curve) gives 1 ) and the expansion parameter is small. =0 2 ℓ λλ ) π M . The magnitude of the (4 / . There is no parameter to control the relative sign of the two ) -wave amplitude of (1 s ), we obtain the following Perturbativity imposes additional constraints, since at h We define the perturbative-unitarity-violation scale O HH ( λλ -wave amplitude is plotted in figure 4.11 level M can see that two tree-level amplitude, the one-loop amplitude gives a relat which suggests that the natural cut-off scale is The first term in the r.h.s. comes from represent contributions from ( and the graviton become stronglyin coupled. the These are Volkov-Akulov of action three modify Figure 1 graviton contribution partially cancels the growth of the a the total amplitude. The parameters s first and second row of the r.h.s. of eq. ( represent contributions from ( picted in figure represents the total amplitude. Here, contribution of the spin-two particleThis to contribution this partially amplitude cancels has thestrong pure coupling Goldstino amp regime. For the parameter values used in figu are ( JHEP10(2009)077 are . In /f m (4.14) can be 2 P . i π √ m (5) ∗ > Λ and , (5) ∗ ∗ ]. This estimate E h m/f 56 [ the theory becomes and Λ 5 / min P 1 . Factors of 4 3 in the parameter space led. As we have seen, ) m onsistently incorporated 4 m tional and stronger con- are satisfied. energy at which the low- >m m = m m< 2 ∗ e considerations. For illus- ≫ P m √ E 2 m ∼ m pin-two field to be suppressed ling is not far above the mass te becomes comparable to the E 2.5 esonance in the single-graviton r √ he coupling of the longitudinal = E ergy and become strong at ener- > (5) ∗ at E=m (5) ∗ . 2 5 / 1 and Λ f 4 and Λ non-perturbative / in the parameter space of
m
m 1.5 m m ]. For example, the one-loop contribution of P
m
this is of order Λ or larger. There is however 2 =
>m 2 >m
38 E √ ∗ π √ ∗ ∗ πm
4
– 15 –
strongly coupled strongly E 1
4 E > =