Compactification of spacetime in SU($\infty$) Yang-Mills theory

0.1 MB·English

by  Kiyoshi Shiraishi

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.
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