We review some aspects of the correspondence between analytic gauge
invariants and supersymmetric flat directions for vanishing D-terms and propose
a criterion to include the F-term constraints.Comment: 8 pages, Late

Brax, Ph.

Savoy, C. A.

English

arXiv

We review some aspects of the correspondence between analytic gauge invariants and supersymmetric flat directions for vanishing D-terms and propose a criterion to include the F-term constraints.We review some aspects of the correspondence between analytic gauge invariants and supersymmetric flat directions for vanishing D-terms and propose a criterion to include the F-term constraints

Brax, P.

Savoy, Carlos A.

CERN Document Server

CERN-TH/2001-105T/01-040hep-th/0104077Supersymmetric Flat Directions and AnalyticGauge InvariantsPh. Brax1Theoretical Physics Division, CERNCH-1211 Geneva 232C. A. Savoy 3Service de Physique Theorique, CEA-Saclay F-91191 Gif/Yvette Cedex,FranceAbstractWe review some aspects of the correspondence between analytic gauge invari-ants and supersymmetric flat directions for vanishing D-terms and proposea criterion to include the F-term constraints.1email: philippe.brax@cern.ch2On leave of absence from Service de Physique The´orique, CEA-Saclay F-91191Gif/Yvette Cedex, France3email: savoy@spht.saclay.cea.fr1 Flat DirectionsFlat directions in supersymmetric theories are continuously connected de-generate supersymmetric vacua, modulo the group of gauge symmetry G,i.e., each vacuum corresponds to a G-orbit. The degeneracy of the classicalsolutions is described in terms of massless elds, the so-called moduli. Theflat directions, i.e., the moduli space V, dene the low energy regime of thesupersymmetric theory. Their analysis has led to the conjecture of dualities[1] between supersymmetric gauge theories as well as a new insight on con-nement in some of such models [1,2,3]. Even in the presence of low scalespontaneous supersymmetry breaking, the study of the flat directions in thesupersymmetric limit is still instrumental in many applications. The bestexample are the minimal supersymmetric extensions of the Standard Model,where the flatness of the scalar potential is lifted by the soft supersymme-try breaking operators. In general, the theory possesses other metastablevacua [4], which could invalidate it { or be important { for phenomenologicalpurposes.The relevance of the analytic gauge invariants in the study of supersym-metric theories was rst noticed a long time ago. Indeed, the F -terms inthe scalar potential are components of the gradient of the superpotentialwhich is an invariant analytic function. The D-terms are Hermitean func-tions of the scalar elds, but a beautiful result in algebraic geometry providesthe link with holomorphic invariants, at least for the analysis of the vacua[5,6,7,8]. The fundamental mathematical property of the (classical) modulispace (modulo the group of gauge symmetry) is its isomorphism with analgebraic variety of analytic gauge invariant polynomials in the primordialelds called the chiral ring. They can be classied in two categories, thosewhich have algebraic constraints among the invariants of the integrity basis,called syzygies, and those with a free basis. This classication is also relatedto the possible patterns of gauge and global symmetry breaking.Consider a set of chiral superelds φi whose scalar components zi belongto a Kahler manifold with Kahler potential K. The scalar potential of theglobally supersymmetric theory with gauge group G is the sum V = g2a2Da2 +Kij¯Fi Fj¯ where the F terms are the gradients of the gauge invariant analyticsuperpotential W (zi), i.e. Fi = ∂iW , and the D terms are Da = ∂iK(Taz)ifor the linear action δzi = a(Taz)i of the compact gauge group G on the2scalars zi. The matrices T a are the generators of G. The supersymmetricvacua V of the theory are dened by the conditions Fi = 0, Da = 0. Thegauge invariance of the Kahler potential and the superpotential implies thatthese solutions form G-orbits Ω = fgz0, g 2 G, z0 2 Vg. The solution z hasa little group Hz in G. The G invariance of the orbit through z implies thatthe little groups Hgz = g−1Hzg are conjugate. The little groups are orderedaccording to the relation: H  H 0 provided H is conjugate to a subgroupof H 0. Similarly G-orbits are ordered according to their little groups. Theorbits with the largest little groups are called critical.1.1 D=0The D-flatness condition is non-analytic, the F -flatness condition is analytic,i.e. involving only the scalars zi and not z i¯. The D-flatness conditions isnon-trivially related to analytic invariants. Analytic invariants I(z) satisfy∂iI(Taz)i = 0. A necessary and sucient condition for D = 0 is the existenceof an analytic invariant I and a non-zero constant c such that∂iI = c∂iK. (1)These solutions are extrema of I at xed K. This correspondence was pointedout in [6] and proved in [7], detailled discussions can be found in [8] and [9].Example 1 { Consider G = SU(3) with three quarks Qa and three anti-quarks Q¯a¯ , a = 1 . . . 3, α = 1 . . . 3. The most general invariant which is linearin the basic invariants, see section 2, is I = ρaQaQa¯ + µγabcQaQb Qγc +~µ¯¯γ¯a¯b¯c¯ Q¯a¯Q¯b¯Qγ¯c¯ where Maa¯ = QaQa¯, B = γabcQaQb Qγc andB =¯¯γ¯a¯b¯c¯ Q¯a¯Q¯b¯Qγ¯c¯ . General solutions can be found from (1) and correspondto orbits of the non-compact group U(3, 3)[6]. They correspond to combi-nations of the mesonic orbits: ∂aMaa¯ = c(Qa ) yields (Q11) = Q11, and thebaryonic orbits: ∂aB = c(Qa ) yields Q11 = Q22 = Q33, all others zero (andanalogously for antibaryons), up to a U(3, 3) transformation. The criticalorbit Q11 = Q12 = Q13, all others zero, breaks SU(3) ! SU(2) and gives I = 0and DU(1) 6= 0 for the broken U(1) in SU(2)U(1)  SU(3). It is not a flatdirection and its U(3, 3) orbit is not a D-flat direction too.Example 2 { Consider G = E6 with one representation 27. There is onlyone invariant I = dabczazbzc. Two critical orbits have little groups SO(10)3and F4 respectively. For z0 in the SO(10) critical orbit one nds I(z0) =∂iI(z0) = 0 and DU(1)(z0, z0) 6= 0 since z0 is charged under the U(1) suchthat SO(10)U(1) 2 E6. For z0 in the orbit with little group F4 one checksthat D = 0 with I(z0) 6= 0.It is important to note that critical orbits with a unique singlet corre-sponding to a broken U(1) such that Hz0  U(1)  G is maximal are notD−flat, as discussed below. These orbits will be identied with the openorbits of the complexied gauge group Gc.1.2 F=0The superpotential is G invariant and since it is an analytic function it isalso Gc invariant where Gc is the complexied gauge group Gc = fg =eiaTa, αa 2 Cg. For each solution of F i = 0 the whole complex orbit Ωc =fgz0, g 2 Gcg is a solution. Note that if z has a (complexied) little groupHcz , the number of non-trivial conditions ∂iW = 0 is equal to the number ofHcz singlets. Since ∂iW = 0, the superpotential cannot be chosen to be theinvariant I leading to solutions of D = 0. For a generic superpotential W (Ia)containing all the basic invariants Ia(z) with degree da, Fi = 0 implies that∑ada∂W∂IaIa(z0) = 0. (2)Now for D = 0 there is at least one invariant such that I(z0) 6= 0. This impliesthat directions corresponding to open critical orbits with all Ia(z0) = 0 areF -flat but not D-flat directions. They are characterized by a Hz0 singletin the coset G/Hz0 such that Hz0  U(1)  G. If z0 is critical, it is theonly Hz0 singlet, the action of the complexication of the U(1) is z0 ! λz0,∂z0Ia is the only potentially non-vanishing gradient direction, and z0∂z0Ia =zi∂iIa = daIa vanishes. Therefore ∂z0Ia = 0 for all invariants Ia. Now since(1) is necessary, D = 0 can only be satised for z0 = 0 which is not in theopen orbit but in its closure.This provides counter-examples to the statement that for each Gc-orbitwhich gives solutions to F i = 0 one can always nd one (G-orbit) solution ofD = 0. For a generic W and z0 such that D = 0, the Fi = 0 conditions liftat least one of the flat directions. If some of the Ia are excluded from W , e.g.with global symmetries, then they can be used to dene D-flat directionswhich are consistent with F = 0.4Example 3 { Let us consider a simplied supersymmetric version of theStandard Model, with gauge group SU(2)  U(1), and the following chiralsupereld content: two Higgs SU(2) doublets H1, H2, with U(1) chargesY = −1, +1, resp., two lepton doublets L1, L2, with Y = −1, and two leptonsinglets E1, E2, with Y = 2. We also introduce a lepton parity, with R = −1for leptons and R = 1 for Higgses. The most general invariant (at mostcubic) superpotential is of the form (up to some redenition), W = µH1H2 +λ1H1L1E1 +λ1H1L2E2. The following invariants have been excluded becauseof the R−parity: L1H2, L1H2, E1L1L2, and E2L1L2. By taking a linearcombinations of these invariants and those in W as the invariant I in equation(1) one nds D-flat solutions. The moduli space is its intersection with thesolutions of ∂iW = 0, and has components such that H2 and the lepton scalarsare non-vanishing. This breaks completely the gauge symmetry (in particularthe electromagnetic U(1)), but also the lepton parity. A similar situation isfound when families of quark multiplets are introduced with some kind ofbaryon or lepton symmetry, giving rise to potentially dangerous charge andcolour breaking directions of the moduli space [4]. The flat directions of theminimal supersymmetric extension of the Standard Model are listed in [5].2 Moduli and SyzygiesTo any solution of Da = 0 one can associate a holomorphic gauge invariantsatisfying (1). The proof of (1) is obtained by studying the closed orbits of thecomplexied Gc of the gauge group G and the ring of Gc{invariant analyticpolynomials. This ring is nitely generated: one can nd an integrity basis,i.e. a set of G-invariant holomorphic homogeneous polynomials fIa(z)ga=1dsuch that every Gc-invariant polynomial in z can be written as a polynomialin the Ia(z).Notice that orbits with all Ia(z) = 0, z 6= 0, are open, i.e. there existsa one parameter subgroup of Gc with generator T such that eT 2 Gc andeT z ! 0, ξ ! 1. The compact generator associated to T , i.e. iT , corre-sponds to a broken U(1) factor in the coset G/Hz with maximal subgroupHz  U(1)  G. In example 2, the SO(10) orbit is open, iT is the U(1)charge, while the F4 orbit is closed.The elements of an integrity basis are not always algebraically indepen-dent. In general, there exist algebraic relations (called syzygies) satised by5the fundamental invariants S (Ia(z)) = 0. In example 1, there is one suchrelation det M − B ~B = 0.To each closed Gc{orbit corresponds a vector in Cd made out of the valuestaken by the invariants fIa(z)g along this orbit and satisfying the syzygies.In that sense the algebraic manifold dened by the syzygies is identied withthe set of closed Gc{orbits. Notice that the origin fIa = 0g is associated withthe unique closed Gc{orbit of zi = 0.The existence of the syzygies can be related to the index of the mattereld representation, µ dened by tr(T aT b) = µδab. For low indices µ < µadj ,where µadj is the index of the adjoint representation, it has been shown thatthere are no syzygies. For µ > µadj the generic situation is that there aresyzygies, with a few exceptional cases with no syzygies.Equation (1) can be seen as a condition for the points of a closed Gc-orbit to extremize the Ka¨hler potential, i.e. ∂i(λaIa−K) = 0 with Lagrangemultipliers λa. A result of geometric invariant theory [7] states that the pointsextremizing the Ka¨hler potential on a Gc-orbit form a unique G-orbit andare solutions of{DA = 0}. Identifying the points on a same G-orbit, thereis a one-to-one correspondence between any two of the following sets: (i)the algebraic manifold MI defined by the syzygies; (ii) the closed Gc-orbits;(iii) the solutions of (1) modulo gauge transformations; (iv) the solutions ofDA = 0 modulo gauge transformations.Now imposing the F -flatness condition implies an extra relation∑a da@W@IaIa =0 to the syzygies. Geometrically one intersects the hypersurface deduced fromthe superpotential with the moduli space MI . It is a necessary condition.Three case are to be envisaged. If there is no intersection the only solu-tion is z = 0, if it exists. If z = 0 is not a solution then supersymmetry isspontaneously broken. If the intersection is reduced to one point the D-flatdirection is completely lifted. Finally in the generic situation the intersectionhas at most dimension dimMI − 1 corresponding to the lifting of at leastone flat direction.3 Dualities and ConfinementOne of the most striking result on supersymmetric gauge theories is the exis-tence of a new type of duality. This duality relates two apparently dierenttheories in the short distance regime that are described by the same eective6theory in the infrared limit. In the same vein the basic question concern-ing the issue of colour connement has been tackled and claried in thesenon-perturbative approaches for a large class of supersymmetric theories. Ef-fective theories have been written and argued to describe the IR behaviour ofsome gauge theories in terms of gauge invariant composite chiral superelds[1,2,3]. The non-perturbative quantum eects are xed by the holomorphyof the superpotential, by the global symmetries of the theory and by severaldescent links among series of such theories.A powerful and necessary criterion for the existence of an eective theorydescribing the IR regime of an asymptotically free gauge theory was statedby ’t Hooft [10]: there should be matching of the (formal) anomalies of theglobal symmetries calculated with either the UV or the IR massless fermions.The analysis of the isomorphism between the moduli space and the chiralring constrained by the syzygies leads to the following (partially proved)conjecture [11,12] The ’t Hooft conditions are satisfied for supersymmetricgauge theories if and only if the syzygies of the chiral ring of gauge invariantsderive from a superpotential W (Ia), through Sa = ∂W/∂Ia = 0.Now by comparing with the general structure of W required by the R-symmetries, one obtains the following condition: A necessary condition forthe matching of the anomalies is: µ = µadj +k, (k = 0, 1, 2). This is the con-nement condition for theories with µ  µadj [13]. There are some subtletiesand peculiarities that can be found in the literature.It is generally assumed that theories with 2 + µadj < µ < 3µadj havean infrared xed point [1] where they are described by a superconformaltheory. The syzygies are therefore exact quantum relations between thechiral primary elds at the superconformal xed point. Whereas in \electric"theories the gauge invariants are composite elds and so subject to syzygies,in the magnetic dual theories, some of them appear as elementary eldsand do not have a priori to satisfy the syzygies. These \magnetic syzygies"appear as the equations of motion from the superpotential generated nonperturbatively in the magnetic theories [14].These dualities amount to an identication between the moduli spaces,i.e., between the flat directions of the dual theories. It is worth noticing thatthe D-flatness in a theory corresponds to the F-flatness in the dual theory.7[1] N. Seiberg, Nucl. Phys. B435 (1995) 129.[2] T. R. Taylor, G. Veneziano and S. Yankelowiz, Nucl. Phys. B218 (1983)493.[3] N. Seiberg, Phys. Rev. D49 (1994) 6857.[4] J. A Casas, A. Leyda and C. Mu~noz, Nucl. Phys. B471 (1996) 3.[5] T. Gherghetta, C. Kolda, S.P. Martin, Nucl.Phys. B468 (1996) 37.[6] F. Buccella, J.P. Derendinger, S. Ferrara and C.A. Savoy, Phys. Lett.115B (1982) 375.[7] C. Procesi and G.W. Schwarz, Phys. Lett. 161B (1985) 117.[8] R. Gatto and G. Sartori, Commun. Math. Phys. 109 (1987) 327.[9] M.A. Luty and I. Washington Taylor, Phys. Rev. D53 (1996) 3399.[10] G. ’t Hooft et al., editors, Naturalness, chiral symmetry breaking andspontaneous chiral symmetry breaking, NATO advanced study, Cargese,France, Plenum, 1980.[11] G. Dotti and A.V. Manohar, Nucl. Phys. B518 (1998) 575.[12] Ph. Brax, C. Grojean and C. A. Savoy Nucl. Phys. B 561 (1999) 77.[13] C. Csaki, M. Schmaltz and W. Skiba, Phys. Rev. D55 (1997) 7840.[14] K. Intriligator and N. Seiberg, Nucl. Phys. Suppl. 45BC (1996) 1.8

Supersymmetric Flat Directions and Analytic Gauge Invariants

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