SU Compactification of spacetime in ( ) ∞ Yang-Mills theory 3 1 Kiyoshi Shiraishi 0 2 Institute for Nuclear Study, University of Tokyo, n Midori-cho, Tanashi, Tokyo 188, Japan a J Classical and Quantum Gravity 6 (1989) pp. 2029–2034 6 2 Abstract ] h t The compactification on a torus in SU(∞) Yang-Mills theory is con- - sidered. A special form of the configuration of a gauge field on a torus is p e examined. Thevacuumenergyandfreeenergyinthepresenceoffermions h coupledwiththisbackgroundinthetheoryarederivedandpossiblesym- [ metry breaking is investigated. 1 v Recently Floratos et al. offered SU( ) Yang-Mills (YM) theories [1] which 3 ∞ came from the study on membrane theories [2]. We consider, in this paper, 1 the compactification on torus in the SU( ) YM theory. A special form of the 2 ∞ 6 configurationofgaugefieldontorusisexamined. Thevacuumenergyandther- . modynamic potential in the presence of fermions coupled with the YM theory 1 0 in this situation are derived and possible symmetry breaking is investigated. 3 In order that our discussion should be self-contained, we start with a brief 1 review of SU( ) YM theory [1]. We denote the dimension of space-time as D. : ∞ v Thegaugefields aregivenbythe functionswhichdependonthe D-dimensional i coordinates xM as well as the coordinates of ‘sphere’, θ and φ; X r ∞ l a A (x,θ,φ)= Alm(x)Y (θ,φ) (1) M M lm l=1m=−l X X where Y are the spherical harmonics on S2. Note that the sum over l starts with l=1. The field strength is defined as F =∂ A ∂ A + A ,A (2) MN M N N M M N − { } where the bracket of two functions f and g is defined as ∂f ∂g ∂f ∂g f,g = . (3) { } ∂cosθ∂φ − ∂φ∂cosθ 1 The sequential operation of the bracket satisfies the Jacobi identity: f,g ,h + h,f ,g + g,h ,f =0 (4) {{ } } {{ } } {{ } } where f, g and h are functions of θ and φ. The gauge transformation of a gauge field is given by δA =∂ ω+ A ,ω . (5) M M M { } At the same time the transformation of the field strength follows δF = F ,ω . (6) MN MN { } The YM field equation is D FMN ∂ FMN + A ,FMN =0. (7) M M M ≡ { } For later use, we introduce the matter field ψ(x,θ,φ) in the ‘adjoint repre- sentation’. This field transforms as δψ = ψ,ω (8) { } and obeys the field equation D DMψ m2ψ =0 (9) M − where m is the mass of the ψ field. ψ is assumed to have coupling with the gauge field only. In the analysis here a set of spherical harmonics is chosen as a basis of generators. One can write the bracket relation as Ylm,Yl′m′ = flml′′m′′l′m′Yl′′m′′ (10) { } l′′m′′ X wheref isthe‘structureconstant’. Thebracketcorrespondstothecommutation for generators of the usual groups. We can find the Cartan subalgebrain this basis: if we pick up the spherical harmonics with m=0, then the following are trivially led Yl0,Yl′0 =0. (11) { } Next we consider spacetime compactification. We consider MD−1 S1 × ((D 1)-dimensional Minkowski spacetime circle) as the background space- − × time. The periodicity with respect to the coordinate on the circle gives rise to the ‘Kaluza-Klein’ excited states [3]. Furthermore, since S1 is a non-simply connected manifold, non-trivial Wilson loops can be defined on it [4]. In other words, there are vacuum expactation values of the YM field on a torus (S1) (modulogaugetransformation). Theycanbringaboutsymmetrybreakdownof 2 gauge groups in ordinary YM gauge theory [5, 6, 7, 8]. Thus the similar mech- anisms are extensively studied in the context of multidimensional unification theory [9]. In our model, we first write out the field equation. Setting the coordinates xM =(xm,y), m=0,1,2,...,D 2, the equation (7) decomposed to: − D FMn = D Fmn+D Fyn M m y = ∂ Fmn+ A ,Fmn +∂ Fyn+ A ,Fyn =0. (12) m m y y { } { } To obtain the equation of motion for A , we impose a gauge condition n ∂ AM =0. Further,ifweneglecttheself-couplingofYMfieldsA ,orconsider M n the coupling only to the ‘background gauge field’ A so as to get a free field y h i equation of motion, we obtain ∂2An+∂2An+2 A ,∂ An + A , A ,An =0. (13) m y {h yi y } {h yi {h yi }} We consider A = constantasausualcaseforargumentsforWilsonloops y h i [4], and then the background field strength F = 0 satisfies the equation of ym h i motion D FMN =0 automatically. M Now, we consider how many degrees of freedom A possesses. For an y h i ordinary gauge group such as SU(N) the degree of freedom is as many as the rank of the group, i.e. the dimension of Cartan subalgebra. This is true for an arbitrarydimensionaltorus. Inotherwords: supposeTa′ belongstotheCartan subalgebra. Then we can expand A as y h i A = Aa′ Ta′. (14) h yi h y i a′ X This form guarantees vanishing field strength automatically especially on a higher-dimensional torus. We assume A can be expanded in terms of the basis of the Cartan sub- y h i algebra even in the SU( ) YM theory. That is to say,by using components of ∞ the field, it follows ∞ A = Al0 Y (θ,φ). (15) h yi h y i l0 l=1 X Inagenericcase,the fieldequationforacomponentfieldAlm becomesaset n of simultaneous infinite number of equations: (∂2 +∂2)Alm+2f lm Al10 ∂ Al2m m y n l10 l2mh y i y n +fl10lml′mfl20l′ml3mhAly10ihAly20iAnl3m =0 (16) where the summations over l , l and l are implicit, while the sum over m is 1 2 3 unnecessary because of the ‘selection rule’ for the quantum number. To simplify the equations, we can take a new basis for A as y h i 1 1 A = A(1) Y + A(2) Y h yi 3h y i 10 15h y i 20 r r ∞ 1 1 1 + hA(yl)i√2l 1 √2l+1Yl0− √2l 3Yl−2,0 . (17) Xl=3 − (cid:18) − (cid:19) 3 In this basis, the bracket operation between A and A can be rewritten by y n h i {hAyi,An}= ∂∂hcAosyθi ∂∂Aφn = imhA(yl′+1)iAlnmYl′0Ylm. (18) l′ lm XX Thus, the use of the well-known formula for multiplication of Y [10] Y Y l1m1 l1m2 (2l +1)(2l +1) 1/2 1 2 = (l m l m l m )(l 0l 0l 0)Y (19) 4π(2l +1) 1 1 2 2| 3 3 1 2 | 3 l3m3 lX3m3(cid:26) 3 (cid:27) makesthecomponentequationssimpler. Intheaboveexpression(l m l m l m ) 1 1 2 2 3 3 | denotes the Clebsch-Gordon coefficient in the standard notation. (l) However, for a general set of A we also need a diagonalization of an y h i infinite-dimensional (mass) matrix. In this paper, rather than giving general discussions, we investigate the case for a specific form of A in detail. We y h i consider the following case: θ A(1) = √4π and A(2) = A(3) = = A(l) = =0 (20) h y i L h y i h y i ··· h y i ··· whereθ is aconstantandL is the length ofthe circumferenceofthe extraspace S1. This is the only case that the mass matrix is (already) diagonal. Since A can be expanded in a Fourier series with respect to the S1 coordi- n nate, i.e. ∞ Alm = Almkei2πky/L (0 y <L). (21) n n ≤ k=−∞ X We can make up the field equation for each excited mode: (2π)2k2 2π 1 ∂2 Almk 2 kmθAlmk m2θ2Almk =0. (22) m− L2 n − L n − L2 n (cid:18) (cid:19) Therefore the mass square of Almk in (D 1) dimensions is given by n − 1 (2πk+mθ)2 (23) L2 where k and m are integers. Based on this mass spectrum, we can evaluate the 1-loop vacuum energy. ThevacuumenergyintheSU( )YMtheoryisseeminglyanticipatedtodiverge ∞ becauseofaninfinitenumberof‘componentfields’. Asforourparticularmodel, we can first suppose that the component fields which have the label l N 1, ≤ − forafinite integerN. Inthis situation,the numberofcorrespondinggenerators are N−1 (2l+1)=N2 1 (24) − l=1 X 4 andthe number ofgeneratorswhichbelongsto the Cartansubalgebrais N 1. − These are precisely coincident with the case of the SU(N) group. According to the usual prescription [4, 5, 6, 8], the 1-loop vacuum energy is given formally as ∞ E = (D−2)VD−1 dtt−(D−1)/2−1 vac −2(4π)(D−1)/2 Z0 N−1 l ∞ 2 2 2π mθ exp t k+ (25) × l=1 m=−lk=−∞ (− (cid:18) L (cid:19) (cid:18) 2π(cid:19) ) X X X where VD−1 is the (D 1)-dimensional volume of the system. Using Jacobi’s − imaginary transformation [11] and regularising E by discarding an infinity, vac this reduces to ∞ (D 2)VD−1L 1 sin2(Nkθ/2) E = − Γ(D/2) 1 . (26) vac − πD/2LD kD sin2(kθ/2) − k=1 (cid:20) (cid:21) X Herefinitesummationshavebeenperformed. InthelimitN ,E diverges vac →∞ only at θ = 0 modulo 2π. This fact can be easily seen from taking a limit D . In the limit the only term with k = 1 in the sum remains. If we → ∞ assume a vacuum with minimum energy, the expectation value of θ is zero (mod 2π). (The periodicity of 2π in θ is explained with respect to a proper gauge transformation [4, 7].) Consequently,inthepureYMtheoryundertheassumptionofthisparticular A , gaugesymmetry isnotbrokenbecause θ =0 andthere appear(N2 1) y h i h i − masslessgaugebosons. Hereweshouldnotethatthereexistmanylocalminima in the potential, and the number of the local minima is N 2 in the range − 0<θ <2π. Nextweconsiderthematterfieldcoupledtothebackgroundgaugefield A . y h i Foratypicalexample,weexamineamasslessDiracfermionfieldinthe‘adjoint representation’ (recall (8)). For matter fields, we can take a ‘twisted boundary condition’ in the circle direction. Then we obtain the Fourier expansion of the field in the following form: ∞ ψlm = ψlmkei2πky/L+iδy/L (0 y <L) (27) n n ≤ k=−∞ X where δ is a constantwhich represents the ‘twist’. The mass spectrum is modi- fied as 1 (2πk+mθ+δ)2 (28) L2 where k and m are integers. The 1-loop vacuum energy is expressed as E (fermion) = NF2[D/2]VD−1 ∞dtt−(D−1)/2−1 vac 2(4π)(D−1)/2 Z0 5 N−1 l ∞ 2π 2 mθ+δ 2 exp t k+ (29) × l=1 m=−lk=−∞ (− (cid:18) L (cid:19) (cid:18) 2π (cid:19) ) X X X where N is the number of fermions, and after regularisationwe obtain F NF2[D/2]VD−1L E (fermion) = Γ(D/2) vac πD/2LD ∞ cos(kδ) sin2(Nkθ/2) 1 . (30) · kD sin2(kθ/2) − k=1 (cid:20) (cid:21) X Note the overall sign of E (fermion). vac In the case with δ = 0, provided that N is enough large to overcome the F contribution from YM fields, it is possible to get the non-vanishing vacuum gauge field expactation value at finite N, even after taking the limit N . → ∞ The minima of E (fermion) are located at θ =2πp/N, p=1,...,N 1. The vac − lowest energy of (degenerate) vacua is then NF2[D/2]VD−1L Γ(D/2),ζ(D) (31) − πD/2LD where ζ(z) is the zeta function. The vacuum energy, or the effective potential for θ, has an infinite number of degenerate minima in the limit N . →∞ Many massive fermions appear when θ is located at any minima according to the spectrum (28). On the other hand, symmetry-breaking pattern is rather complicatedinthecaseoffiniteN. Whenθ =2π/N,thereremainsonly(N 1) − masslessvectorbosonsassociatedwiththegeneratorsoftheCartansubalgebra. Thus a symmetry breakdown such as SU(N) [U(1)]N is expected. However, → forgeneralfinite N andforgeneralminimaofθ =2πp/N,weseemoremassless gauge bosons. For example, suppose N = 4 and the vacuum with p = 2. The statewithk =1andm=2(l=2or3)inthe spectrum(23)becomesmassless. Then the resulting symmetry can be larger than [U(1)]N. If N is a prime number, this ‘accidental’ symmetry does not emerge in any vacuum associated withθ =(2π/N) (integer). IfwetakeN ,wecansaythatthe minimaof × →∞ the vacuum energy as a function of θ are located at every point of 2πQ, where Q is a rational number, 0<Q<2π The free energy can be calculated in a similar way to obtain E [5]. The vac techniqueisthesameastheonein[12],whichtakestheimaginarytimedirection as a circle. One finds the following expression for the free energy F(fermion) with the fermion fields considered above: NF2[D/2]VD−1L F(fermion) = Γ(D/2) πD/2LD ∞ cos(kδ) sin2(Nkθ/2) 1 · (k=1 kD (cid:20) sin2(kθ/2) − (cid:21) X 6 N2 1 1 − 1 ζ(D) − βD − 2D−1 (cid:18) (cid:19) ∞ ∞ cos(kδ) sin2(Nkθ/2) + 2 1 (32) k=1n=1(L2k2+β2n2)D/2 (cid:20) sin2(kθ/2) − (cid:21)) XX where β is the inverse of temperature. For δ = 0, or δ at near 0, and sufficiently large D, no phase transition is expectedtooccuraslongastheformof A isconstrainedtoouransatz. That y h i is because what determines the shape of the ‘potential’ for θ is the term with k = 1 in the sum. For the case with δ takes the value near π/2 and in low dimensions, the k = 1 term does not necessarily dominate in the summation, and then the shape of the potential for θ is modified even at zero temperature; in addition, the phase transition can take place [8]. In conclusion, we see that gauge symmetry breaking in SU( ) YM theory ∞ is feasible under the assumption with a special form of the configuration of the gauge field on the extra S1 and in the presence of fermion fields. We did not persue the possibility of phase transition in the case of matter fields with a special twisted boundary condition on S1. We want to report the effect of general twist and the dimensionality of spacetime in an effective potential for a simpler group such as SU(3) elsewhere. For SU( ) YM theory, we must ∞ considerthe generalformof A by executing a diagonalizationof the infinite- y h i dimensional mass matrix from the beginning. Otherwise, we might miss the existenceofotherminimaorvacuawithlowerenergy,asintheproblemofHiggs potentials [13]. The construction of other ‘representation’ than the ‘adjoint representation’ is also an interesting task. We hope to investigate the above subjectsinrelationto the vacuumenergyandspontaneoussymmetrybreaking. Acknowledgements The author thanks S. Hirenzaki for useful comments. He also thanks A. Naka- mula for discussion and Y. Hirata for reading this manuscript. This work is supportedinpartbyaGrant-in-AidforEncouragementofYoungScientistfrom the Ministry of Education, Science and Culture (# 63790150). The author is grateful to the Japan Society for the Promotion of Science for the fellowship. He also thanks Iwanami Fu¯jukai for financial aid. References [1] E. G. Floratos,J. Iliopoulos and G. Tiktopoulos, Phys.Lett. B217 (1989) 285. [2] B. de Wit, J. Hoppe and H. Nicolai, Nucl. Phys. B305 [FS23] (1988) 545, and references therein. 7 [3] T.Appelquist,A.ChodosandP.G.O.Freund, Modern Kaluza-Kleln The- ories (Benjamin-Cummings, New York, 1987) [4] Y. Hosotani, Phys. Lett. B126 (1983) 309. D. J. Toms, Phys. Lett. B126 (1983) 445. N. Weiss, Phys. Rev. D24 (1981) 475; D25 (1982) 2667. [5] K. Shiraishi, Z. Phys C35 (1987) 37. [6] V. B. Svetovoˇi and N. G. Khariton, Sov. J. Nucl. Phys. 43 (1986) 280. A.T.DaviesandA.McLachlan,Phys.Lett.B200(1988)305;Nucl.Phys. B317 (1989) 237. A. Higuchi and L. Parker,Phys. Rev. D37 (1988) 2853. Y. Hosotani, Ann. Phys. (NY) 190 (1989) 233. [7] K. Shiraishi, Prog. Theor. Phys. 80 (1988) 601. [8] C.-L. Ho and Y. Hosotani, preprint IASSNS-HEP-88/48(October 1988). [9] M. Evans and B. A. Ovrut, Phys. Lett. B174 (1986) 63. K. Shiraishi, Prog. Theor. Phys. 78 (1986) 535; ibid. 81 (1989) 248 (E). A. Nakamula and K. Shiraishi, Phys. Lett. B215 (1988) 551; ibid. B218 (1989) 508 (E). J.S.DowkerandS.Jadhav,Phys.Rev.D39(1989)1196;ibid.D39(1989) 2368. K. Lee, R. Holman and E. Kolb, Phys. Rev. Lett. 59 (1987) 1069. B.H.Lee,S.H.Lee,E.J.WeinbergandK.Lee,Phys.Rev.Lett.60(1988) 2231. [10] I.E.McCarthy,Introduction to Nuclear Theory, (JohnWiley&Sons,New York, 1968). [11] A. Erdelyi at al., Higher Transcendental Functions (McGraw-Hill, New York, 1953). [12] L. Dolan and R. Jackiw, Phys. Rev. D9 (1974) 3320. [13] J. Breit, S. Gupta and A. Zaks, Phys. Rev. Lett. 51 (1983) 1007. 8