Gauge bosons in a noncommutative geometry

Volume 217, number 4

PHYSICS LETTERS B

2 February 1989

GAUGE BOSONS IN A NONCOMMUTATIVE GEOMETRY M. D U B O I S - V I O L E T T E , J. M A D O R E Laboratoire de Physique Thborique et Hautes Energies ~, Universitb de Paris-Sud, B~ztirnent 210, F-91405 Orsay, France

and R. K E R N E R Laboratoire de Physique ThOorique des Particules Elementaires, UniversitOPierre et Marie Curie, 4 Place Jussieu, F-75252 Paris, France

Received 28 October 1988

A noncommutative extension of abelian gauge theory is proposed and is compared with standard nonabelian gauge theory.

1. Introduction The basic idea of noncommutative geometry is to replace the algebra of smooth functions defined on a manifold by an abstract associative algebra ~/which is not necessarily commutative. We refer to ref. [ 1 ] for a general introduction to this subject and for references to the previous literature. One can then attempt to reformulate the objects o f interest in differential geometry such as vector fields, differential forms, metrics and connections, in terms of the corresponding objects involving .~¢. We refer to ref. [2 ] for a simple example of this and for references to the previous literature. We propose here to develop classical gauge theory using as algebra the tensor product of an algebra W of functions on space-time and the algebra M,, o f n × n complex matrices: . ~ = c~® M , . We refer to refs. [ 2 - 4 ] for a description o f basic noncommutative geometry using this algebra and for more details o f the application to n o n c o m m u t a t i v e gauge theory. As in refs. [ 2 - 4 ] we use the differential calculus defined in ref. [ 5 ]. Laboratoire associ6 au CNRS. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

2. Mathematical preliminaries We shall choose Wto be the algebra of smooth complex-valued functions on R4. The elements of the algebra ~/are what replace the functions on N4. The Lie algebra D (W) o f vector fields on the manifold N4 can be identified with the algebra of derivations of ~¢, that is, with the algebra of linear maps of W into itself which satisfy the Leibnitz rule. In the noncommutarive case we must consider the derivations D ( d ) of the algebra .J. This Lie algebra was shown in ref. [ 3 ] to be the direct sum of the ordinary derivations of W and the W-module generated by the inner derivations o f M,. Let 2a, for 1 <~a<~n 2 - 1, be a basis of the Lie algebra o f the special unitary group in n dimensions, chosen so that the structure constants Ca~,c are real. The Killing metric is given by d,,b= -Tr(2~2~,). We shall raise and lower indices with this metric. The set 2a is a set of generators of the matrix algebra Mn. It is not a minimal set but it is convenient because of the fact that the derivations e,, = ~c- ~ad (2 ~) form a basis over the complex numbers for the derivations of M,. They satisfy the commutation relations [e~, eb] =lc-IC~,~t, ec.

(2.1)

A length scale K has been introduced to keep the di485

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mensions correct. The mass scale m will be given as the inverse of x. Let x ~ be coordinates of R4, Then the set (x a, 2 ") is a set o f generators o f the algebra M. We define the exterior derivative of an element o f M as usual. For example, if f is an element of M~, then d f i s defined [ 5 ] by the formula

e~= (e,~, ea) as a basis o f D(.~/) over ~. We can write

4f( ea) = e . ( f ).

d=dH +dr.

(2.2)

This means in particular that

dA"(et,)=m[2t,,2 a ] =mCc~ ~2q

(2.3)

We define the set o f 1-forms 12 ~(sO) to be the set o f all elements of the form f d g or dgf, with f a n d g in M subject to the relations d (fg) = ( d f ) g + f d g . We similarly define 12t (M~) and f2~(~g), the definition of the latter being of course the usual one. The algebra M c a n be viewed as an algebra o f matrix-valued functions and it is convenient to consider it as such at times. However then one would not use the operator d which has just been defined. The kernel of d, the elementsfwith df = 0, is the set o f f which are not only constant when considered as matrix-valued functions on N4, but which are also multiples of the identity. The set of d2 ~ forms a system of generators of IZ ( M , ) as a left or right module, but it is not a convenient one. For example 2~d2b ¢ d262". However because of the particular structure of M , [ 2 ] there is another system of generators completely characterized by the equation 0~ (e~,) =3],.

(2.4)

t2 j (M) as a direct sum gZ ( J ) =£2~ Gf2~.

(2.7)

The horizontal part ~2~ has basis 0 ~ and the vertical part ~26 has basis 0 a. We shall decompose the exterior derivative into horizontal and vertical parts also:

We shall introduce the quadratic form, of signature d - 2, given by ds 2=

~ijO'@0 j= rlc~O°~@O'e+3.t, Oa@O h.

( 2.8 )

The ~/-B is the Minkowski metric. We shall refer to this quadratic form as a metric although it contains two terms of a slightly different nature. To within a rescaling the 3~t, are the components of the unique metric gv for M, with respect to which all the derivations ea are Killing derivations. From the basis elements 0" we can construct a lform 0 in ~ ,

O= - m 2 a O ~,

(2.9)

which from (2.5) and (2.6) satisfies the zero-curvature condition

dO+02=O.

(2.10)

We shall see below in section 3 that 0 is gauge invariant. It satisfies with respect to the algebraic exterior derivative dv similar conditions to those which the Maurer-Cartan form satisfies with respect to ordinary exterior derivation on the group SU..

It is related to d2 ~ by the equations

d2~=mC"bc2~'O ~, O~=x262~d2 b,

(2.5)

and it satisfies the same structure equations as the components of the Maurer-Cartan form on the special unitary group SU.:

dO" = - ~mC~b~Ob ^ Oq

(2.6)

The 0" c o m m u t e with the elements of M,, and in fact 12~( M , ) can be identified with the tensor product of M~ with the dual o f the vector space o f derivations. Choose a basis 0~dx x o f l 2 ~(~g) over Cgand let e, be the pfaffian derivations dual to 0 ~. Set i = (ce, a), 1 <~i<4+n 2 - 1, and introduce 0~= (0% 0 ~) as generators of 12~(M) as a left or right M-module and 486

3. Gauge fields In the commutative case a connection o) on the trivial principal Ul-bundle equipped with the associated canonical flat connection is an anti-hermitian 1-form which can be split as the sum of a horizontal part, a 1-form on the base manifold, and a vertical part, the Maurer-Cartan form d a on UI, co=A + dc~.

(3.1)

The gauge potential A is an element of O~ (~g) and using it we can construct a covariant derivative on an associated vector bundle. The notion o f a vector bundle can be generalized to the noncommutative case

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as an ~¢-module which in its simplest form, a free module of rank 1, can be identified with ~,' itself. This is in fact the natural generalization to the algebra we are considering of a trivial U~-bundle since M , has replaced ~'in our models. So the U~ gauge symmetry we shall use below comes not from the rank of the vector bundle, which we shall always choose to be equal to 1, but rather from the factor M, in our algebra J . The noncommutative generalization of A is an anti-hermitian element of £2~( J ) , which we saw in the previous section in turn can be split as the sum of two parts, called also horizontal and vertical. We shall here designate by ~o such an element of £2~(~¢) since we wish to reserve the letter A and the name gauge potential for the horizontal part in this latter sense. We write then

oJ=A+O+O,

(3.2)

where A is an element off2~ and we have introduced the Higgs field ~, an element o f £2~v. We have noted that 0 is in many respects like a M a u r e r - C a r t a n form. Formula (3.2) with 0 = 0 and formula (3.1) are formally similar but the meaning of the words horizontal and vertical in the two cases is not the same. We have then a bundle over a space which itself resembles a bundle. This double-bundle structure, which is what gives rise to a quartic Higgs potential as we shall see below, has been investigated in previous articles [6,7]. Let N be the group o f invertible elements o f st, considered as functions on R 4 with values in GL,, and g an element o f N. Let o~/,,be the subgroup o f f¢ o f elements which satisfy gg* = 1. We shall choose it to be the group of local gauge transformations. A gauge transformation defines a mapping o f g2~(.z¢) into itself of the form

o9' =g-~o)g+g-~dg.

(3.3)

We define

O' =g-IOg+g-ldvg, A' =g-lAg+g-ldHg,

(3.4)

and so 0 transforms under the adjoint action o f ~',:

(~' =g-lOg. It can be readily seen that in fact 0 is invariant under f¢,

2 February 1989

o' =0,

(3.5)

and so the transformed potential o9' is again of the form (3.2). We define the curvature 2-form £2 and the field strength F as usual f2=d(z)+o) 2, F=dHA+A 2. In terms o f components, with 0 = 0,0" and A =A,O ~ and with

g2=ly2,jO' AtT, F=½F,/jO" AO tj,

(3.6)

we find

~

=F.~,

£2,t, = [0,, 0~,1-- mC%t, Oc.

(3.7)

The gauge-invariant lagrangian for the gauge potentials A and the Higgs fields ~ is given by ~ = ¼Tr (£2~/s~/).

( 3.8 )

It contains all the terms, including the Higgs potential, a quartic polynomial in 0 which is fixed and has no free parameters apart from the mass scale m. Written out explicitly using (3.7), it becomes

5°= ITr(F,~F ~ ) + ½Tr(D,0~D~0 ~) - V(O), (3.9) where the Higgs potential V(O) is given by

V(O) = - t T r (£2,h~"').

(3.10)

From (3.7) we see that V(0) vanishes for the values 0~=0,

O,=m2~.

(3.11)

The orbit o f the second value under the action o f the gauge group can be identified with SUn. The Higgs potential has therefore two absolute minima, a point at the origin and a submanifold of dimension n 2 1, separated by a potential barrier. There are therefore two stable phases. We shall consider separately the two phases, first the symmetric phase, ~ = 0 and then the broken phase, ~ = rn2~. In the symmetric phase the masses of all the Higgs modes are equal and they are real since the corresponding value of the potential is a stable minimum. We find that the mass is given by

m ~ = n m 2.

(3.12) 487

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The gauge bosons of course have vanishing mass in the s y m m e t r i c phase. In the b r o k e n phase, to calculate the Higgs masses we use the vertical part a)a0" o f the connection as a new Higgs field and we e x p a n d it in terms o f the basic field modes: ~oa =2,,0)2 + ( i / x / n ) to °. The analysis here is rather messy and we give it only in the case n = 2 . We d e c o m p o s e o)~a,=rb,,o)~ into its irreducible parts, a trace, a trace-free s y m m e t r i c and an a n t i s y m m e t r i c part:

o9,,~= ('r / , ~ )gah + a,b + aab. The masses are respectively given by

m ~ = 2 m 2, m 2 = 8 m 2, m 2 = 0 ,

rn~=2m 2, (3.13)

where mo is the mass o f the m o d e s co°. The three degrees o f f r e e d o m in the a are the three gauge degrees o f freedom. They correspond to the modes which have been absorbed as the longitudinal part o f the now massive gauge bosons. In the b r o k e n phase the mass o f the A° r e m a i n s equal to zero. The mass term for the r e m a i n i n g vector bosons is ~m" T r ( [ A , , 2,1 [A", 2~1 ).

(3.14)

The mass o f these bosons is given therefore by the equation

m 2 =2nm 2.

(3.15)

If we c o m p a r e this with (3.13) in the case n = 2 , we obtain a numerical prediction. Some o f the masses o f the Higgs bosons are larger than that o f the vector bosons by a factor x/2 a n d some are smaller by the same factor.

4. Nonabelian gauge models The lagrangian (3.9) is a generalization of the Y a n g - M i l l s - H i g g s - K i b b l e lagrangian, with a m o r e elaborate Higgs sector. Since we have replaced complex-valued functions by functions with values in the matrix algebra M,, it is to be expected that each ~a takes its values in Mn. The most original part is the potential term V(~) which comes from the curvature of the vertical part of the connection. It is not the most 488

2 February 1989

general gauge-invariant p o l y n o m i a l in the Higgs field which would be allowed and we have i n t r o d u c e d an e x t e n d e d BRS o p e r a t o r [4] to insure that the form is m a i n t a i n e d during renormalization. C o n s i d e r the case n = 2 a n d c o m p a r e the above m o d e l with that of the s t a n d a r d model for the electroweak interactions. We have considered U,, gauge bosons. In anticipation o f a parity-breaking mechanism we could equally well have considered the chiral Un × Un gauge bosons. Everything we have done could be repeated by replacing the matrix algebra Mn by the product algebra Mn × M,, and the connection ~o by the sum ~o~+ ) + a) ~- ) o f the right a n d left helicity components. The Higgs potential has then four stable m i n i m a . We could suppose that nature has chosen that m i n i m u m which corresponds to the ~ ~+ ) in the s y m m e t r i c phase and the ~ t - ) in the b r o k e n phase. The massless non-abelian bosons would be confined and not a p p e a r as asymptotic states. We would then have as physical states two massless U ~gauge bosons, one o f which could be identified with the photon and n 2 - 1 massive SUn gauge bosons which could be coupled only to left-handed fermions. There is of course no analogue o f the Weinberg angle. The massive gauge bosons are neutral and they have all equal masses. In the case n = 3 we would have a model for gluons plus a spurious U~ abelian gauge potential p r o v i d e d we suppose that we are in the s y m m e t r i c phase. The m a j o r difference with the s t a n d a r d model for the strong interactions lies in the fact that here the symmetric phase is a stable phase. The Higgs bosons are coloured and would not appear as asymptotic states.

References [ 1] A. Connes, Publ. I,H.E.S. 62 (1986) 257. [2] M. Dubois-Violene, R. Kerner and J. Madore, Noncommutative differential geometry of matrix algebras, Orsay preprint (October 1988 ), to be published. [3]M. Dubois-Violette, R. Kerner and J. Madore, Noncommutative differential geometry and new models of gauge theory, Orsay preprint (November 1988), to be published. [4] M. Dubois-Violette, R. Kerner and J. Madore, Classical bosons in a noncommutative geometry, Orsay preprint (October 1988), to be published. [ 5 ] M. Dubois-Violette, C.R. Acad. Sci. Paris, 307, Sdrie 1 ( 1988 ) 403. [6] N.S. Manton, Nucl. Phys. B 158 (1979) 141. [7] R. Kerner, L. Nikolova and V. Rizov, Lett. Math. Phys. 14 (1987) 333.