We consider domain walls that appear in supersymmetric SU(N) with one massive
flavour. In particular, for N > 3 we explicitly construct the elementary domain
wall that interpolates between two contiguous vacua. We show that these
solutions are BPS saturated for any value of the mass of the matter fields. We
also comment on their large N limit and their relevance for supersymmetric
gluodynamics.Comment: 4 pages, 1 figure, uses latex with hep99 class files. Presented at
  the International Europhysics Conference in High Energy Physics, Tampere
  (Finland) 15-21 July 199

de Carlos, B.

Moreno, J. M.

English

arXiv

We consider domain walls that appear in supersymmetric SU(N) with one massive flavour. In particular, for N > 3 we explicitly construct the elementary domain wall that interpolates between two contiguous vacua. We show that these solutions are BPS saturated for any value of the mass of the matter fields. We also comment on their large N limit and their relevance for supersymmetric gluodynamics.We consider domain walls that appear in supersymmetric SU(N) with one massive flavour. In particular, for N > 3 we explicitly construct the elementary domain wall that interpolates between two contiguous vacua. We show that these solutions are BPS saturated for any value of the mass of the matter fields. We also comment on their large N limit and their relevance for supersymmetric gluodynamics

Moreno, J.M.

CERN Document Server

Domain walls in supersymmetric QCDB. de CarloszTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandJ.M. MorenoxInstituto de Estructura de la Materia, CSICSerrano 123, 28006 Madrid, SpainWe consider domain walls that appear in supersymmetric SU(N) withone massive flavour. In particular, for N  3 we explicitly construct theelementary domain wall that interpolates between two contiguous vacua.We show that these solutions are BPS saturated for any value of the massof the matter elds. We also comment on their large N limit and theirrelevance for supersymmetric gluodynamics.Talk given at the EPS Conference on High Energy PhysicsTampere, Finland, July 1999(to be published in the proceedings)CERN-TH/99-322IEM-FT-197/99IFT-UAM/CSIC-99-41October 1999CERN-TH/99-322‡ e-mail address: Beatriz.de.Carlos@cern.ch§ e-mail address: jesus@makoki.iem.csic.esDomain walls in supersymmetric QCDB. de Carlos1 and J.M. Moreno21 Theory Division, CERN, CH-1211 Geneva 23, Switzerland2 Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, SpainE-mails: Beatriz.de.Carlos@cern.ch, jesus@makoki.iem.csic.esAbstractWe consider domain walls that appear in supersymmetric SU(N) with one massiveflavour. In particular, for N  3 we explicitly construct the elementary domainwall that interpolates between two contiguous vacua. We show that these solutionsare BPS saturated for any value of the mass of the matter elds. We also commenton their large N limit and their relevance for supersymmetric gluodynamics.The study of supersymmetric gauge theoriesthat are in the strong coupling regime has beenintensied recently by the realization that someof their constructions could admit exact solutions.In particular, the issue of domain walls in SU(N)supersymmetric gluodynamics, the theory of gluonsand gluinos, is one of the most exciting ones. Thesewalls arise because this theory has an axial U(1)symmetry broken by the anomaly to a discreteZ2N chiral symmetry. Due to non-perturbativeeects gluino condensates (hi) form, breaking thesymmetry further down to Z2. This leaves us witha set of N dierent vacua labelled byhTri = 3e2ik=N k = 0; 1; :::; N − 1 ; (1)where  is the condensation scale, and, as indicatedabove, a set of domain walls interpolating betweenthem. If we assume that they are BPS saturated,the energy density of these walls is exactlycalculable and given by [1, 2, 3] =N82jhTri1 − hTri−1j ; (2)In fact, it has been shown in Ref. [4] that, in thelarge N limit, these domain walls are BPS states.On the other hand, these solutions preserving halfof the supersymmetry would play an important rolein the D-brane description of N = 1 SQCD [5].In order to get to study pure gluodynamics, itis convenient to add matter elds to the theory andanalyze the limit where these extra elds (usuallytaken to be pairs of chiral superelds transformingas (N; N) under the colour group) become veryheavy. In the strong coupling regime, we shouldexpect the formation of squark condensates. Thesemodels were considered in Refs [6, 7], for the caseof (N − 1) flavours. Their analysis of the vaccuumstructure led to the conclusion that the existence ofBPS saturated domain walls was restricted to valuesfor the mass m of the squark elds below a certaincritical one. It therefore looked like it would beimpossible to recover pure gluodynamics by takingthe limit m ! 1, and this is precisely the mainissue we will address during this talk.In order to do that let us consider supersym-metric QCD with SU(N) gauge group and onecouple of chiral superelds (Q; Q) transforming asN; N . Non-perturbative eects become relevant atthe scale , where condensates form. The gauginoand squark colourless condensates are described bythe following composite eldsS =3322TrW 2 ;(3)M = Q Q ;where W 2 is the composite chiral supereld whoselowest component is . In this regime, therelevant degrees of freedom are described by a Wess-Zumino model, as shown in Ref. [8]. Its eectiveLangrangian is given byL = 14∫d4 K + 12[∫d2 W + h:c:]; (4)where K is the Ka¨hler potential and W is thesuperpotentialW = 23S lnSN−1M3N−1eN−1− 12mM ; (5)with m the mass for the matter superelds. Thissuperpotential has N extrema labeled by thedierent phases of the gaugino condensate. At theminimum we have the gaugino condensate xed toSN =34m3N−1 : (6)2The matter condensate is aligned with respect tothe former and given byM =1m43S : (7)We want to study domain wall congurations thatinterpolate between the dierent minima. Herea technical problem appears: the superpotentialhas several branches associated with its logarithmicpiece [9]. In the pure SUSY gluodynamicslimit described by Veneziano and Yankielowicz [8]this is a severe problem, since any congurationconnecting two vacua has to cross this branch. Thisis not necessarily the case when we include otherelds, given that the variation in the phase of thegaugino condensate can be partially compensatedby these new elds. In this case, this will be doneby matter elds.Let (S; M)a be a particular vacuum. Wecan continuously deform it into another vacuum,(S; M)b. For this path in the conguration space,we dene , w such thatSjb = eiSja ;(8)M jb = ei(+2w)M ja :Notice that Eq. (7) implies that w must be someinteger number. On the other hand, one necessarycondition to avoid crossing the logarithmic branchalong this path is(N − 1) + ( + 2w) = 0 : (9)Since we are interested in congurations interpo-lating between the vacua i and i − 1, we will x = − 2N and then w = 1.If we assume that there is a BPS domain wallconnecting these two vacua, it will be described bythe following dierential equationsKSS¯@z S = eiγ@W@S;(10)KMM¯@z M ii = eiγ@W@M ii;where K¯ is the induced metric from the Ka¨hlerpotential K, and γ is given byγ = −12( + ) =N− 2: (11)The conguration is described by four real functionsM(z) = jMj (z)ei(z) ;(12)S(z) = jSj R(z)ei(z) :Notice that we have dened (z); R(z) in such away that (1) = R(1) = 1. On the otherhand,  varies from 0 to 2(1 − 1=N) and  from0 to −2=N . A consistent ansatz under reflectionz ! −z is given by: (z) = (−z), R(z) = R(−z),(z) = −2=N − (−z) and (z) = 2(1− 1=N)−(−z). These relations x the boundary conditionsat z = 0. Eqs (10) imply the following BPSconstraintIm[eiγW(S; M)] = const : (13)In Refs [6, 7] a similar analysis was presentedfor Nf = N − 1 couples of matter elds. In thesepapers it was shown that these domain walls areBPS states only for squark masses lower than somecritical value, m, that depends on N and theKa¨hler potential. The existence of this bound isrelated to the presence of two dierent BPS domainwall solutions for small enough values of m, whichbecame identical at the critical value.Here we have worked in detail a dierent case:N = 3, Nf = 1, using the same Ka¨hler potential,i.e. K = (S S)1=3 + (M M)1=2. We have foundthat the equations can be solved for all values ofthe squark mass, and we have checked that thelogarithmic branch is never crossed. The proles forR are shown in Fig. 1 for several values of m (givenin units of ), focusing on their central region. Thespatial coordinate z is expressed in units of ~−1,where ~ = (3m4Λ )1=3N is the eective QCD scalethat arises in the large m limit.In our case there is only one BPS solution forevery value of m. This can be understood analyzingboth the large and small m limit, as explained indetail in [10]. These limits depend on the numberof flavours.Let us consider the limit m ! 1, that isexpected to describe pure gluodynamics. FromFig. 1 we see that there is a well dened gauginocondensate prole in that limit. In fact, thefollowing constraints apply in the asymptoticregions(z)ei(z) = R(z)ei(z) (z << −1=m) ;(14)(z)ei(z) = R(z)ei((z)−2) (z >> 1=m) :Therefore, using Eq. (14) in this large m limitwe can get rid of  and  in the BPS equations.Also the BPS constraint involves only the gauginocondensate and can be written asIm{ei[γ+(z)]R(z)[ln(R(z)ei˜(z))− 1]}= const ;(15)3−0.2 −0.1 0 0.1 0.200.20.40.60.81zRFigure 1.R(z) as defined in Eqs (12) versus z (in units of Λ˜−1), form = 2 (dotted line), 20 (dash-dotted), 100 (dashed), 200(solid). The thick solid line corresponds to the m → ∞solution given by Eq. (16).where ~(z) = (z) for z < 0 and ~(z) = (z) +2=N for z > 0. This constraint allows us toexpress  as a function of R, and we end up withthe following BPS equation for R(z)@zR(z) = 6N(R(z))4=3~ fcos(γ + ([R(z)])) ln R(z)− sin(γ + ([R(z)])) ~[R(z)]}; (16)Notice that, in the large m limit, the prole has twobranches which are smoothly connected for nitevalues of m. In this limit, the domain wall hastwo typical scales. One is associated to the gauginocondensate, while the other one is associated to theadditional eld that compensates the phase changesof the gaugino condensate, in our case the mattereldk.Let us now analyze the large N limit of thisasymptotic conguration [11]. In order to do thatwe expand the normalized modulusR(z) = 1− r(z)=N +O(1=N2) ; (17)and we rewrite (z) = b(z)2N . Then Eq. (15)impliesb(z) = −r(z) (18)for the left branch, i.e. for z < 0. If we want toanalyze the structure of the domain wall, we have‖ See [4] for other possibilities.to include properly the dependence of S on N . Oneexpects [4, 5]S  hTri  O(N) : (19)Then, we have to replace 3 by N3 in theabove equations. This implies that the energy ofthe previous elementary domain wall, as given byEq. (2), scales like   O(N). If we want to studythe width, i.e. how the energy is spread along thedomain wall, we need some information about theN dependence of kinetic terms of both gaugino andregulator -matter- elds.For example, if we assume that the gauginokinetic term term is O(Na), then r(z) is given by@jzjr(jzj) = −k2~N1−ar(jzj) ; (20)where the precise value of the positive constantk2 depends on the particular Ka¨hler potential.Under this assumption, the scale associated withthe gaugino eld variations is O(1=(~N1−a)).In summary, it is possible to build BPS domainwalls in SQCD with one flavour for any value ofthe mass of the matter elds. This allows us tostudy the limit where the theory approaches puresupersymmetric gluodynamics.The work of JMM was supported by CICYT ofSpain (contract AEN98-0816). We both acknowl-edge the British Council/Acciones Integradas pro-gram for the nancial support received through thegrant HB1997-0073.References[1] G. Dvali, M. Shifman, Phys. Lett. B396(1997) 64, Erratum Phys. Lett. B407 (1997)452.[2] A. Kovner, M. Shifman, A.V. Smilga,Phys. Rev. D56 (1997) 7978.[3] B. Chibisov, M. Shifman, Phys. Rev. D56(1997) 7990, Erratum Phys. Rev. D58 (1998)109901.[4] G. Dvali, G. Gabadadze, Z. Kakushadze,hep-th/9901032.[5] E. Witten, Nucl. Phys. B507 (1997) 658.[6] A.V. Smilga, A.I. Veselov, Phys. Rev. Lett. 79(1997) 4529, Nucl. Phys. B515 (1998) 163,Phys. Lett. B428 (1998) 303.[7] A.V. Smilga, Phys. Rev. D58 (1998) 065005.[8] G. Veneziano, S. Yankielowicz, Phys. Lett.B113 (1982) 231; T. Taylor, G. Veneziano,S. Yankielowicz, Phys. Lett. B218 (1983) 493.[9] I.I. Kogan, A. Kovner, M. Shifman,Phys. Rev. D57 (1998) 5195.[10] B. de Carlos, J.M. Moreno, Phys. Rev. Lett.83 (1999) 2120.[11] B. de Carlos, M. Hindmarsh, N. McNair,J.M. Moreno, work in progress.

Domain walls in supersymmetric QCD

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