Massive Connection Bosons

Transcription

1 arxive: v SB/F/ Massive Connection Bosons Gustavo R. González-Martín a Dep. de Física, Universidad Simón Bolívar, pdo , Caracas 1080-, Venezuela. It is shown that eometric connection field excitations acquire mass terms from a eometric backround substratum related to the structure of space-time. Commutation relations in the electromanetic su() Q sector of the connection limit the number of possible masses. Calculated results, within corrections of order, are very close to the masses of the intermediate W, Z bosons h, 14.70Fm, 14.70Hp, 0.0.Qs a Webpae

2 Massive Connection Bosons Introduction In the standard model of particle theory it is accepted that interaction fields may acquire mass by the Hi s mechanism. 1 On the other hand, many properties of particle are related to the space-time structure of special relativity. Therefore it should not be a reat surprise that there is a curved eometry, fundamentally associated to the spacetime structure of eneral relativity, whose field equations determine a particular substratum solution that provides a constant mass parameter for the linear Yukawa 3 equations obeyed by its excitations. The connection excitations may represent eometric bosonic particles. The eometry is related to a connection Γ in a principal fiber bundle (,M,G). The structure roup G is SL(4,R) and the even subroup G is SL 1 (,C). The subroup L (Lorentz) is the subroup of G with real determinant, in other words, SL(,C). lectromanetism is associated to an SU() Q subroup. Matter is represented by a current J expresed in terms of a principal fiber bundle section e, which is related by charts (coordinates) to matrices of the roup SL(4,R). This structure defines a frame of SL(4,R) spinors over space-time. The connection is an sl(4,r) 1-form that acts naturally on the frame e (sections). It has been shown 4 that fermionic particle mass ratios may be obtained usin a substratum solution and related symmetric subspaces of the coset K = G G. We have chosen that the substratum matter be referred to itself. In this manner, the local matter frame e b, referred to the reference frame e r becomes the roup identity I. ctually this eneralizes comovin coordinates (coordinates adapted to dust matter eodesics). 5 We adopt coordinates adapted to local substratum matter frames (the only non arbitrary frame is matter itself, as are the comovin coordinates). If the frame e becomes the identity I, the substratum current density becomes a constant. t the small distance λ, characteristic of excitations, the elements of the substratum, both connection and frame, appear symmetric, independent of space time. These are the necessary conditions for the substratum to locally admit a maximal set of Killin vectors 6 of the space-time symmetry of the connection (and curvature). This means that there are space-time Killin coordinates such that the connection is constant non zero in the small reion of particle interest. flat connection does not satisfy the field equation. The excitations may always be taken around a symmetric non zero connection. If we look at the left side of the field equation we notice that for a constant connection form ω, the expression reduces to triple wede products of ω with itself, which may be put in the form of a polynomial in the components of ω. This cubic polynomial represents a self interaction of the connection field since it may also be considered as a source for the differential operator. Substratum solutions of this field equation are discussed in the appendix. They provide constant parameters for the fluctuation equations of the eometric field equation. Particular xcitation Solution In particular, we assume a connection excitation with alebraic components only in the complex Minkowski plane K k. These excitations exist around a constant odd complex solution which was constructed in the appendix by extendin the real substratum solution to complex functions. The connection form, in terms of the eometrical fundamental unit of lenth is ( i ) ± ˆ ± ω ω δω mdx κ ˆ κ ˆ δω = = ±, (1) ( ) ˆ dd ± ± δω ˆ ω δω ˆ ω δω ˆ J δ = π δ ˆ δ xx δ δ δ δ 4. () Given a representation, a solution to these coupled linear equations may always be found in terms of the Green s function of the differential operator and the current excitation δj, which may be an extended source. In order to decouple the equations, it is necessary to assume that δω vanishes. If this is the case then all equations are essentially the same and the solution is simplified. We shall restrict ourselves to consider the equation for a unit point excitation of the odd current and its solution which is the Green s function. If we assume a time independent excitation with spherical symmetry, the only relevant equation would be the radial equation. Since ω ± is constant, it represents an essential sinularity of the differential equation, an irreular 3 sinular point at infinity. The correspondin solutions have the exponential Yukawa behavior. Let us interpret that the absolute value ω ± of the curved substratum ives an effective rane ω 1 to the linear excitations. quation () time independent for a point source is, desinatin the fluctuation as a weak field W, ( ) ( ) e W x ωw x π =4 δ( xx ) π δ( xx ) e 4, (3)

3 Massive Connection Bosons 3 where we explicitly reconize that the current fluctuation is of order or equivalently of order chare squared. Furthermore, we realize that the assumed excitation is the odd part of an su() Q representation and should explicitly depend on the odd part of the chare. The eometric chare that enters in this first order perturbation equation is the su() Q chare defined by the oriinal nonlinear unperturbed equation, which is the electric chare quantum e=1. We should define a factor that expresses this eometric chare quantum in terms of its odd component, as a unit of a weak chare. The factor () is not part of the δ function which represents a unit odd chare. This equation may be divided by (), obtainin the equation for the odd excitations produced by the unit point odd chare or equation for the Green s function, 1 ( ) ω W x W x = xx ( ) ( ) πδ( ) 4. (4) We may define a new radial coordinate r=x 1, rationalized to a new rane 1 for odd excitations, which incorporates the constant. Takin in consideration the covariant transformations of W and δ, the radial equation is then 1 ( rw ) W =4 πδ( r r ). (5) r r Since is constant, the Green s function for this differential operator is x x r 1 e 1 e G = = 4π xx 4π r (6) whose total space interation introduces, in eneral, a rane factor 4π r d Ω = 1 1 dr r e 4π 0 0 The rane may be evaluated usin eq. (), ( ) CC. (7) ω = ω ω = ω ω = 8 m, (8) ivin m ω = =. (9) Massive SU() xcitations. The su() Q connection excitations may be considered functions over the sphere SU()/U(1). In similarity with the quantization of the anular momentum, we concluded in a previous paper 7 that the SU() Q electromanetic excitations have indefinite azimuthal directions but quantized polar directions determined by the possible translation values on the sphere. The internal direction of the potential must be alon the possible directions of the electromanetic enerator in su() Q as indicated in the fiure. The components must be proportional to the possible even and odd translations. In consequence, the total vector must lie in a cone defined by a quantized polar anle θ relative to an axis in the even direction and an arbitrary azimuthal anle. The fundamental state that represents an electromanetic quantum is the SU() Q state with chare 1, correspondin to the ½, ½ eienvalues of the electromanetic rotation. The correspondin θ anle is ( ) ( π 1 ) 1 1 = = 3 = tan tan θ 3. (10) The su() connection excitations over the bidimensional sphere SU()/U(1) are enerated by the odd enerators. The complex chared raise and lower enerators ± are defined in terms de the real enerators,

4 Massive Connection Bosons 4 ± = 1 ± i, (11) which raise or lower the chare q of an eien state as follows c q = N c q±1. (1) ±,,( ) Therefore, these excitations, defined as representations of the SU() Q roup, require a substratum with a preferred direction alon the even quaternion q 3 in the su() Q sector. Only the complex solution discussed in the appendix, eq. (46), ives an adequate substratum. We shall call this substratum the odd complex substratum. The odd complex substratum solutions in the su() sector reduce to ( i ) ω= ω κ ± κκ κ = ω 0 ± 0 ± ± q and should correspond to the two odd sphere enerators. This odd su() subspace enerated by the complex external solution is physically interpreted as an SU() electromanetic substratum, a vacuum, with potential ω which determines ranes which may be interpreted as masses for the excitations of the electromanetic connection in its surroundins. The rane 1 is determined by ω, the odd fundamental bidimensional component in the substratum equatorial plane. The even and odd components of the su() Q excitation in the direction have to obey the relations, =. (14) Due to the quantization of the su() connection, there only are two possible definite absolute values, associated to a fundamental representation, which are related by (13) 1 = = sin θ1. (15) n odd su() Q connection excitation should be the spin 1 boson (SU() S representation) which also is a fundamental SU() Q representation where the three component enerators keep the quantized relations correspondin to electromanetic rotation eienvalues ½, ½, similar to the frame excitation representation (proton). Of course, they differ reardin the spin SU() S because the former is a spin 1 representation and the latter is a spin ½ representation. We shall call this excitation by the name fundamental complete excitation. fluctuation of the odd substratum 0 ( ) = ( ) q δ ω δ ω ± ± does not provide a complete su() Q excitation. The excitation ranes are proportional to the absolute values of a connection. The only possible ranes associated with an su() Q excitation should be proportional to the only possible values of the quantized connection (16) =. (17) The su() Q currents are similarly quantized and are related by j e 1 = = sin θ1 =. (18) j e The factor that expresses the eometric chare quantum in terms of its odd unit component is = cscθ 1 1 and we obtain, omittin the polar anle indices, ω m = = and sin θ (19) (0)

5 Massive Connection Bosons 5 m =. (1) On the other hand, the excitation ranes should be provided by the excitation field equation () as the absolute value of some substratum connection. The odd complex substratum admits a related substratum with an additional even component ω connection that should provide the value of the smaller rane 1. This even part ω increases the modulus of the total connection. The possible value of ω should correspond to the allowed value of the total substratum connection absolute value. We shall call this substratum the secondary complex substratum. The value represents the bound enery of a complete (with its three components) su() Q fundamental excitation around the secondary complex substratum with equation ( ) πδ ( ) x = xx 4. () Since the orientations of the potential and the backround connection ω are quantized, their decompositions into their even and odd parts are fixed by the representation of the connection. Therefore this enery term may be split usin the characteristic polar anle θ, ( cos sin ) ( cos ) ( sin ) = θ θ = θ θ. (3) The corresponds to the even component ω associated with the U(1) roup enerated by the κ 5 even electromanetic enerator. This enery may not be decomposed without dissociatin the complete su() excitation due to the orientation quantization of its components. If the complete excitation is disinterated into its partial components, the correspondin equation for the even component separates from the odd sector in K, as indicated in eq. (), and has an abelian connection. Therefore, no mass term appears in the even component equation, physically consistent with the zero mass of the photon. The enery associated to ω in the κ 5 direction is available as free enery. For a short duration the splittin takes enery from the substratum. On the other hand, corresponds to the odd complex substratum and when the complete excitation is disinterated, the enery appears as the mass term in the coupled equations () associated to the pair of odd enerators ±. ( ) ( ) ( δω) πδ( ) x L1 =4 xx. (4) We may nelect the couplin term L 1, as was done previously in eq. ()), so that the equation may approximately be written ( ) πδ( ) x = xx 4. (5) The fundamental real fermionic solution that represents the proton has a mass (enery), iven by eq (34) as indicated in the appendix, m p = 4 m. (6) The odd complex substratum solution has a similar relation in terms of m, eq. (48), 1 ω= tr ω ω = m =. (7) 4 Considerin these relations, the θ polar anle also determines the ratio of the two eneries or masses associated to the fundamental excitation, m m = sin θ. (8) These relations determine that the SU() connection excitation eneries or masses are proportional to the proton mass m p, m = m mp =. Gev mz. Gev = = = , (9)

6 mp sin θ m = msin θ = = Gev mw = Gev Massive Connection Bosons 6. (30) These values indicate a relation with the weak intermediate bosons. They also indicate a possible relation of this polar anle with Weinber s anle 8 : Weinber s anle would be the complement of the polar anle. They admit corrections of order. For example if we use the corrected value for θ, obtained from its relation to the experimental value of Weinber s anle, mp sin θ m = = Gev mw = Gev. (31) Let us define a monoexcitation as an excitation that is not a complete SU() Q representation and is associated to a sinle enerator. collision may excite a resonance at the fundamental connection excitation enery m determined by the secondary complex substratum. The fundamental connection excitation which is an SU() representation may also be dissociated or disinterated in its three components (, ± ), when the enery is sufficient, as two ± free monoexcitations, each one with mass (enery) determined by the odd complex substratum, and a third free U(1) monoexcitation, with null mass determined by the even complex substratum component ω which is abelian. This simply determines the existence of four spín-1 excitations or boson particles associated to the SU() connection and their theoretical mass values: 1. The resonance at enery of the fundamental potential, which has to decay neutrally in its components (, ± ) and may identified with the Z particle;. The pair of free monoexcitations ±, chared ±1 with equal masses, which may identified with the W ± particles; 3. The free excitation, neutral with null mass which may identified with the photon. In this manner we may ive a eometric interpretation to the W, Z intermediate bosons and to Weinber s anle which is then equivalent to the complement of θ 1/ polar anle. The complex substratum ppendix It has been shown 9 that there is a real backround (substratum) connection solution, ˆ ( ˆ ) ω= e m dx κ e e de= mj e de, (3) which allows the calculation of mass quotients. The solution corresponds to the local movin frame rest system, where all derivatives are assumed zero in the non linear differential operator. The value of m is determined by the only remainin connection contribution to the field equation left side, which is the connection triple product in the covariant derivative, c a bν ω ( ω ω) ( ω ω) ω = δν ωω ω [ c, ab] c a b = ωω ω c, [ a, b] = πj 4. (33) The real substratum is obtained by restrictin the enerators to the orthonormal set κ. The definition of mass in terms of the Cartan-Killin metric is m = 1 tr( J 4 Γ ), (34) and ives a relation between the fundamental inverse lenth of the eometry m and the mass of a fundamental particle, which we take as the proton, 9 m p = 4 m. (35) The symmetric space K has a complex structure 10. The center of G, which is not discrete, contains a eneratin element κ 5 whose square is -1. We shall desinate by J the restriction of ad(κ 5 ) to the tanent space TK k. This space, that has for base the 8 matrices κ, κ β κ 5, is the proper subspace correspondin to the eienvalue -1 of the operator J, or, ( ),, λ λ 1 λ λ λ λ J x κλ y κλκ κ κ x κλ y κλκ 5 = 4 =x κλy κλκ (36)

7 Massive Connection Bosons 7 The endomorphism J defines a complex structure over K. The real substratum connection may be extended to the complex field usin this complex structure ω= ω ωκ5, (37) a a where we indicate the eiht K space enerators by a. If we restrict the latin enerator indices from 1 to 9 we may write the left side of the field equations with the sinle contribution of the connection triple product, ( ) ( ) c a b ν [, ] c a b ω ω ω ω ω ω = δν ωω ω c ab = ωω ω c, [ a, b], (38) here indicated by X, in the followin form,,[, ],[, 5 ] c, [, ] a,[, ] 5 c 5 5,[ b, a],[ b, ] 5 5 5,[, ],[, ] X = ωω ω ωω ω κ c b a c b c b a c b c a c ω ω ω κ ω ω ω κ κ b a b a 5 5 a 5 5 ωω ω κ ωω ω κ κ ω ω ω κ κ ω ω ω κ κ κ5, (39) The total sl(4,r) odd vector subspace, spanned by κ and κ 5 κ β, is isomorphic to K k, the tanent space of the symmetric coset K at a point k. The complex structure allows us to define a complex metric in its tanent space usin the Cartan-Killin metric. Thus, the total sl(4,r) odd vector subspace may be considered as a complex Minkowski space with metric η, CC ( X, Y) C ( X, Y) =η ( X, Y). (40) Since the enerator κ 5 commutes with the even sector and enerates the complex structure, eq. (39) becomes [ ] 5 { } {, } c b a c b X = ωω ω c, b, a ωω ωκ c, b c a b a ω ω ω κ ω ω ω ω ω ω. (41) 5 c a b a We may use the K complex structure to take ω as a complex functions and the enerators as κ, [ ] c b a c b X c, b, = ωω ω κ κ κa 4ωω ωκη 5 cb c a b a 4ω ω ω κ η 4 ω ω ω κ 4 ω ω ω κ. (4) 5 ca b a Finally, we evaluate the K components in the term X, c b a b a ( [ ] ) 1tr dx 1 κ tr κd ω ω ω κc, κb, κ 4 = 4 a ω ω ω κb ω ω ω κa, (43) c c ( ω ω ω ω ω ω ) ( ω ω ω ω ω ω ) X d = c d d c d d 4 4 (44) and, usin this expression for the left side X write the field equation as (( c d d c ) ( d d )) c c ( ) ω ω ω ω ω ω ω ω ω ω ω ω κ c c d ω ω ω ω ω ω κ5 =πδ. (45) c c J If we assume that the even connection component ω is zero we obtain the odd complex substratum solution which has the form, ( ) ( ) ω= κ ± κ κ ± κ ˆ± ˆ± mdx ˆ ˆ m i ˆdx 5 1, (46) with the same value for the constant m iven by eq. (3). We interpret this complex connection solution as a complex substratum with a complex matter current.

8 (( ) ( ) ) ( ) Massive Connection Bosons 8 J λ = 1 i κ λ 1 i κ λ = 1 i κ λ. (47) We may represent the odd complex substratum connections ω ± iven by this solution as points on the complex Minkowski space K k, with equal moduli, ( ) C ( ) C ( ) ω = η ω ω = ω ω = ω ω ± ± ± ± ± The points define complex vectors q ±, ( ( ˆ ˆ m κκ ˆ κκκκ 5 ˆ 5 )) 1 = tr = m. (48) 4 q = κ iκ κ i 5 = (49) x y z and their complex conjuates, which form a base on the K k sector, related to the associated complex coordinates z. This pair of vectors toether with the vector determined by the static connection ω, 3 q = κ5, (50) provide a quaternion base on a compact subspace for the subalebra su() Q. If we assume that the connection even component ω is non zero we should extend the source current accordinly. The field equations (45) determine that only any one component of the 1-form ω may be a non zero constant. This ω is either time-like or space-like and static and accordinly may be called electrostatic or manetostatic. Thus, we obtain a series of related complex substratum solutions with different possible constant values of ω ± and ω. The modulus of the total connection solution ω is not iven by eq. (48). Furthermore, it may be shown that the modulus of any of these related complex solutions increases due to ω and is larer than the value iven by this equation. Connection xcitations To obtain field excitations we perform perturbations of the eometric objects in the field equations. Then the linear differential equation for any perturbation of the connection takes the eneral form ( [ ] ) δω 4ω ω δω L δω L δ = 4 π δj. (51) ν c 1 c ν ν d c d d c d ν d In particular, we assume that we can obtain a connection excitation solution with alebraic components only in the complex Minkowski plane K k. Then the complex connection and its excitation have the form ˆ! ± ( ) ω = ω δω ˆ, (5) ( i ) ± ˆ ± ω mdx κ ˆ κ ˆ = ±. (53) The excitation equation becomes, usin the complex metric of the complex Minkowski plane K k, that involves lowerin the indices and takin the complex conjuate, ( ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ) ( ˆˆ χ ˆˆ ˆˆ ˆ ˆ) = ddδω 4 δω ω ω ω δω ω ω ω δω ν χ βν χ βν χ βν δˆ ν 4 π δj. (54) δβ δ χβ δ

9 Massive Connection Bosons 9 (References) 1 P. W. His, Phys. Lett. 1, 13 (1964); P. W. His, Phys. Rev. Lett. 13, 508 (1964). Gustavo R. González-Martín, Phys. Rev. D 35, 15 (1987); Gustavo R. González-Martín, Gen. Rel. and Grav., 481 (1990). 3 Hideki Yukawa, Proc. Phys. Math. Soc. (Japan) 17, 48 (1935). 4 Gustavo R. González-Martín, Rev. Mexicana de Física 49 Sup. 3, 118 (003). 5 W. Misner, K. Thorne, and J. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco,1973), Vol. 1, p W. Killin, J. Reine new. Math. 109, 11 (189). 7 Gustavo R. González-Martín, arxive physics/ Robert. Marshak, Conceptual Foundations of Modern Particle Physics (World Scientific, Sinapore, 1993), p Gustavo R. González-Martín, arxive physics/ ; arxive physics/ S. Helason, Differential Geometry and Symmetric Spaces (cademic Press, New York 196), p Gustavo R. González-Martín Physical Geometry (U. Simón Bolívar, Caracas, 006).

10 Massive Connection Bosons 10 q 3 Θ q 1 q Polar anle Θ in the su() Q alebra determined by the even and odd enerators