We present a (2+1)-dimensional Skyrme-like model with a symmetry-breaking potential, which in R_3 has charge -n instanton solutions, and in the static limit in R_2 a sphaleron solution

Müller-Kirsten, H. J. W.

Tchrakian, D. H.

English

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A (2+1 )-dimensional model withinstanton and sphaleron solutions1) Department of Mathematical Physics, St. Patrick’s College, Maynooth, Ireland2) School of Theoretical Physics, Dublin Institute for Advanced Studies,10 Burlington Road, Dublin 4, IrelandS) Department of Physics, University of Kaiserslnutern, PostIach 3049, D—6750Kaiserslautern, GermanyAbstract: We present a (2+1)—dimensional Shyrme—like model with a symmetry—breakingpotential, which In ha., charge —n in.sta.nton solutions, and in the static limit in asphaleron solution.—•1—While the quantum tunnelling between lopologically distinct vacua of the WelnbergSalam (gauge—Iliggs) field theory is known to be negligible”, it is pouible that at sufficiently high temperatures transitions may occur essentially danically via subeleron fieldconfIgurations, leading to an appreciable violation of baryon—number conservation. Thismechanism was first suggested by Manton’1 and was further developed in ref. [31•Using the sphaleron field configuration of the Weinberg—Salam model, which waspreviously known as the DIIN slujonS, the eirnationS of the baryon—number violationof electroweak theory can be a task of considerable complexity in the quantum theory. Forthis reason, much attention has been devoted to carrying out this programme employingsimplified toy models in lower (than physics]) dimensions” Notable among thesemodels are those in 1+1 dimensions, where the sphaleron in question is a constant staticsolution on S’, of the+4—model and the sine—Gordon model respectively7’5). In the latterexample”, an extended version of the 0(3) sigma model in 2 dimensions has been proposedas the corresponding dynamical system in 1+1 dimensions. In both these models,7” ualso in the original DuN solution on 13, the sphaleron is an unstable field configurationwith finite energy. The energy is the d—dimensionsi integral of the static field configuration, namely d=3 for the DHN case, and d=1 for the toy models of refs. [7J and [8].The sphaleron field’s energy is then regarded a.s the energy—barrier between the topologically distinct vacua of the non—static theory. In all these models, the topological chargescharacterizing the distinct vacua are defined by the usual topological Invariant. In theWemberg—Salam theory, this is taken to be the integral of the Ckeru—Pontryagm densityon 14, while in the 0(3) model of ref. [8], the topological charge is the winding number ofthe order—parameter field defined on 12. In both cues, the dynamical models on d+idimensions, supporting stable instanton field configurations, differ from the dynamicalmodels on d—dimensions, which supports unstable sphaleron field configurations.The purpose of the present note is to propose a new model in 2+1 dImensions, whichsupports stable instanton field configurations on 13, and in the static limit supports unstable sphaleron field configurations on 1). As such, it is an mt mwIiii* “K[—’I’lI—91/9DI,-srt’-H- 2Dii. Tchrakianl)2) and H.J.W. M(iller—KirstenPACS: 03.65.Sq, 1l.10.Lm, ll.15.Kc—3--d=2, between the WIN cue4’ with d=3 md the soliton cases71 with d=I. This toy—model aspect though is not the main reason for proposing it. It. most important propertyIs, that unlike the d=1 and dz3 examples discussed above, the Instanton jgd sphaleronfield configurations are supported as solutions by one and the same model.To help us arrive at our model, we shall first note a common feature of both theDUN and the extended—O(3)—model sphalerons. In each case, respectively in d=3 andd=l, the scaling properties of the models are consistent with there being finite energysolution.. Such solutions could be topologically stable if there were topological Inequalitiessnpplylng lower bounds to the energy Integrals. In turn, such topological Inequalities can befound only for specific fleld—multiplets defining the dynamical coordinates. Specifically, forthe 513(2) Yang—Mill—Tliggs model on 13, such a topological charge (the monopole charge)can be defined lithe Riggs field I. in the adjoint representation of SU(2), and, for thesoilton model in one dimension, such a topological charge (the kinh—number) can bedefined If the field variable consists of one real scalax quantity. The (unstable) .phaleronsolutions on the other hind do not occur In the two models just described. Instead in theDUN case’1,the Higga field I. an isospinor and consist, of four real components as opposedto the three of an adjoint represeatatlon Higgs, and In the extended—O(3) model case51,the order—parameter has two real component. as opposed to the single component of thescalar field of the soliton model. In each case (d=3 and 1), the additional omponent of thedynamical field variable serves to parametrize the noncontractible orbif through theinstability point.In the light of these observations, we proceed to consider the model of ref (9] on-sq2IxI—4cx)the volume integral of (1) ii minimized by topologically stable field configurations, bvirtue of the topological inequalityd2x 2ic J ‘,IV 8,ç8 d2x. (3)Following our above descriptions of the d=3 and dm1 sphalevons, we modify the model (I)by augmenting the dynamical coordinate ‘ with an additional component 4. flns, Isplace of = + our new field variable lit 4- In terms of the Psull spin matrirm. This yieldsCo =—tr + f(q2-4,I) + V(,j2—+) (4)where we use the notation t: = and Iii: m [G+1. AgaIn JI. a symbolic functionrepresenting the, now non Abellan, quadratic kinetic term t. One should note that thescaling properties of the integral of (4) over 2’ are still consistent with the existence offinite energy solutions, but now we have lost the topological lieqinality (3). ThIs is sobecause, the corresponding topological charge density sjtr JV t8t can be neen azl tobe a total divergence, In contrast with the density on the right hand side of (3) defined Interm. of the complex field . As a conseqUence, we would expect any finite energy iclutlonsto the equations of motion that may be found, to be unstable. But this Is predsdy whatwould be expected of a spbaleroa fldd, expedally If we zemernber that the source of thisnew instability Is the additional component of the niultiplet I, over and above the numberof degrees of freedom of the old field In (1). We adopt (4) therefore u the static versionpotential, and us a symbolic (unction representing the quadratic kinetic term Lis regarded as the static limit of a Ligranglan £ In 2+1 dimensions.It was shown in ref. [9] that subjoct to the asymptotic condition(2)I,j = 1,2,= (l J8j7’)2+q2—jsoI, I8,I2) + V(q2—Ijpj), (I)where Is a complex scaiat field and q2 is the (absolute) scale. V is a symmetry breaking—4—of a candidate for a (2+l)—dimensionaj model with instanton anti sphaleron solutions, andproceed to verify these properties. For technical reasons, we consider the instantonproperties first.Insta.ntons; It is useful to specialize the Lagrangian (4), considered on 13, to analyzethe stability of the instanton solutions. This problem w&.s considered in some detail, andanalyzed in ref. [10]. To avoid the ubiquity of models afforded by the symbolic functions fand V in (4), we specialize to some specific choices of these functions.To start with, according to the viriai theorem or scaling argument, it is necessary tokeep only the lint and second, or, the first and third terms in£ = — tr 4’, + f(q2—+,4’) + V(i12—4’), (5)to enable finite action solutions on 13. Here js = 1,2,3, labels the coordinate x of 13.However, as explained in detail in refs. [10,11], in the absence of the second term J,topological stability would dictate the indusion of an additional aextic kinetic termwhich we wish to avoid here. We therefore must retain the second term fin (5). Topological stability does not demand the presence of third term V. Nevertheless, we shall retainV, In anticipation of a similar scaling argument, for the static Lagrangia.n to of (4), ifl 1.2.Retaining both f and V in (5), we opt to specialize (5) to the simplest sub—model arisingfrom the direct descent from the 8—dimensional conformally invariant generalized Yang—Mills system”. The distinguishing feature of this model, other than its relative simplicity,Is that it Involves no dimensional constants apart from the constant q setting the scale ofthe field +. Our choice is£ = — ir + tr (S,+)2 + tr S4tr [i+..,,_ —1 c,{S,+)]2 0. (7)Adding the positive—definite term 2 tr S4 to the left—hand--side 01(7) without disturbingthe iLequality, and expanding (7), we have£ 2 i trthe right—hand--aide of which can be shown to be a total divergence””, whose 1ntral,subject to the asymptotic conditiontr4’2— ,,,2 (9)Ixt—ooguarantees a non—zero lower bound for the action which is proportional to a windingnumber n. Thus the model (6) ii endowed with a stable instanton field configuration In 13.Since the instanton field configurations of this (and other) models(s) on 13 werediscussed in some detail in ref. [10], we suffice here by recalling that these Instinton,correspond to topologically distinct vacua characterized by a winding number a, which Inthis case is the topological charge given (up to normalization) by the integral of the right—hand—side of (8). The n—dependence of these field configurations is given’5’ by—5—where S: = q2—i’ and { , ) means anticommutation. We stress that our choices for f and Vin (6) are not unique.The topological stability of the instanton is then a consequence of the inequality(8)(6)+1 =+(R)sinScos n,2 =+(R)sinOsin n,$3=+(R)cosB, (10)where R= ‘1i, S and , are the polar and azymuthal angles in 3 dimensions, anddefines +=I and jp parametrize both the field and the space S2 C R3.The italic version of (6), defined on£0=—itr+’1+tr {S,+1)2+trS4 (11)will now be shown to have a sphaieron solution. First we recall that according to thescaling argument, the equations of motion for (11) can have finite energy solutions irrespective of the absence/presence of the second term quadratic in +. We also note that now, wehave no topological inequality analogous to (8), so that the finite energy configurations areriontopological.We consider the following Ansats for the (unstable) sphaieron field configurations+ =a1iif(r) sinp conS + o2tjf(r) imp sinS +a3,g(r) coep, (12)where r2= x.x (I = 1,2), and 0 is the azymulhal angle, while p is a constant which weexpect will parametrize the noncontractible path between topologically distinct vacua, andso the instability of the energy functional. Before proceeding to demonstrate thisinstability, we must check the consistency of this Aunts. This involves the verificationthat the Euler—Lagrange equation of the system (11) on81 [++] + 81 {{S,+),S) = 2{I,S3)— (13)for the field configuration (12), are solved by the Euler—Lagrange equations for the one—dimensional subsystem with Lagrangian Lf,g], defined by S = JL rdrdO a 2rSLdr, orL [I{r),f’(r);g(r),g’(r)] a 2n £[f,f’;g,g’I (14)in terms of the coordinates f,g and their “vdodtleeM f’ s df/dr and g’. This Is a verystraightforward if tedious task, and we limit ourselves to stating that indeed the uIer—Lagrange equations arising from the variations of f(r) nd g(r), respectively, for (14), solvethe equations (13) for the field configuration (12) These equations axe rather 1esgthyexpressions, and are not recorded here, but we make a pertinent comment: that If we setf(r) = g(r) in the Ansats (12), the consistency of this Ansats Is lost. We shall return to thedetailed discussion of this inconsistency elsewhere.The existence” of the sphaleron field configuration (12) then follows from thepositive definiteness of the energy integralE[f,g,p] = 4r {4q sin2ji!(g’2coep+f’sin2p)+ 2q6r [1—(g2coep+(2s1ng4] [(g’2oo.p+f’2slap) + ! sla2p]+ qr [1—(g2coep+ 1sIn2p)J4)dx. (16)The all—Important property of instability Is manifest, parametrised by thep-dependeuce of the Integrand In (15). The actual ephaleron is the (unstable) field configuration atthe top of the barrier separating the two distinct vacua, for which the value of (15) is amaximum. This occurs for p = , and by varying p between I) and a-, In the two directionsaway from the sphaleron value of , the value of E can be lowered.Tovologlcal charges. We have shown above that the (2+1)-dimensional model givenby the Lagranglan (6) Is endowed with charge—n Instanton solutions In 13, and Its staticversion (10) with a sphaleron solution In 12. As the latter Is expected to be the energybarrier given by the static fields, between the topologlcally distinct vacua of (he samemodel in 13 it remains for us to demonstrate this property by verifying that the(topological) charge integral—8——9—q Idtd2xi, = 2/ it,,1, tt S di d2x, (16)h0 = (gf2)2= [gf2( +2 )]. (20b)(cf. eq. (8)) for a (2+1)—dimensionai field configuration including the sphaleron field (12),can be evaluated as a surface integral whose value ii controlled by the topological proper- The integrals (19a,b) are then immediately evaluated using the asymptotic conditionsties of the field I, in 12. To this end, we follow the procedure first suggested in refs. [2,3],and employed in ref. [13]. This involves adopting a field configuration +(,t) given by g(oo) f(oo) 1 (21)(12), where the functions f and g depend on the radial variable r of 12, but where thewhich is consistent with the finite—energy conditioncoordinate p ii taken to be a function oft, p = p(t). Writing dp/dt p, the integral (16)tr .2r-_-, oo q2 (22)q i Jdid i ii f I di d2x S(+tl1I+ +i+14j+4’ijt)’ (17)for the fIeld (12), analogous to the finite—action condition (9), for the field (10). The bonn-can then be expressed udary condition at the origin of r isf(0) = 0, (2.3)q=q+q1 (18)which is also the necessary condition for the singlevaiuedness of the field (12). This defines,q 2v frdi fdi i amp [gf’+(fg’—gf’) cos2p] (18a) up to normalization, the topological charge of the (2+1)—dimensional model (6), which hascharge —n insta.nton solutions (10) in 13, and in the static limit a sphaleron solutIon (12) In= 2a jrdr jdt siup ((2f2) cos2p+fj [(fg’—g!’) coa2p+gf’] (18b) 1. Thus one can associate a finite value of the instanto’i (topological) charge with the(nontopological) aphaleron.Now allowing p(t) to vary between 0 and a- as I varies from —oo to +oo, we can performthe integrals (18a,b) as integrals with respect to cosp, between the limits cosp(t & oo) = This work was supported in part by GKSS (Geesthacht/Germany) and EOLAS (Dublin/& 1. The result isIreland).q0=4a-J b0(r)dr (19a)00References= 4a- 1oh1(r) dr, (lob)1) 0. t’Ilooft, Phys. Rev. LeIt. 7, 8 (1976).where both integrals can be evaluated simply by using the topologically meaningful bonn-2) N.S. Manton, Phys. Rev. 2019 (1983).dary values of f and g, by virtue of the fact that the functions h0 and b, are given as the3) F. Klinkhainer and N.S. Mutton, Phys. Rev. j), 2212 (1984).derivatives-5) p. Arnold and L. Mcterran, Phys. Rev. J, 581 (1987).(1989).7)8)9)10)11)12)13)DYn. Grlgoriev, V.A. Rubakov and ME. Shapoehrikov, Phys. LetI. 172N.S. Manton and T.M. Sarnols, Phy.. Lelt. fJfl, 179 (1088).E. Mottola and A. Wipi, Phys. Rev. 588 (1989).l1J.W. Moller—Kirsten and D.H. Tchrakian,J. Phys. A, (1990) L363; Phys. Rev. D, July (1991).DII. Tchrakian, Phys. Leti. Z44.D (1990) 458.DII. Tchrakian, J. Phys. A 4 (1901) 1959, and references therein.J. Buralaff, Nuci. Phy.. 282 (1984).K. Funakubo, S. Otsuki and F. Toyoda, Progr. Theor. Phys. , 118 (1990).,resp4wtively.Fb equation” •“.,t WI’ Cn’”

A (2+1)-dimensional model with

instanton and sphaleron solutions

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A (2+1)-dimensional model with instanton and sphaleron solutions

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