THE STANDARD MODEL AND FERMION MASSES

THE STANDARD MODEL AND FERMION MASSES V.P.Neznamov RFNC-VNIIEF, 607190 Sarov, Nizhniy Novgorod region e-mail: [email protected] Abstract The paper formulates the Standard Model with massive fermions without introduction of the Yukawa interaction of Higgs bosons with fermions. With such approach, Higgs bosons are responsible only for the gauge invariance of the theory’s boson sector and interact only with gauge bosons Wµ± , Z µ , gluons, and photons.

2

The papers [1], [2] offer the Standard Model in the modified FoldyWouthuysen representation. It has been shown that for the being SU(2)-invariant, the theory formulated in the Foldy-Wouthuysen representation does not necessarily require Higgs bosons to interact with fermions, while all theoretical and experimental implications of the Standard Model obtained in the Dirac representation are preserved. The goal of the present paper is to give a similar formulation of the Standard Model in the Dirac representation with massive fermions meeting the requirements of local SU(3) × SU(2) × U(1) symmetry. The paper uses the system of units with = = с = 1; х, p, B are 4-vectors; the inner product is taken in the form xy = x µ yµ = x 0 y 0 − x k y k , µ = 0,1, 2,3; k = 1, 2,3; p µ = i

∂ ∂ ;∂µ = ; ∂xµ ∂xµ

1, µ = 0 αµ =  k ; γ µ = γ 0α µ ; β = γ 0 ,α k , γ µ , γ 5 − Dirac matrices α , µ = k = 1, 2,3

Consider the density of Hamiltonian of a Dirac particle with mass m f interacting with an arbitrary boson field B µ GG

GG

ℋD = ψ † (α p + β m f + qα µ B µ )ψ = ψ † ( PL + PR ) (α p + β m f + qα µ B µ ) ( PL + PR )ψ = GG GG = ψ L† (α p + qα µ B µ )ψ L + ψ R† (α p + qα µ B µ )ψ R + ψ L† β m f ψ R + ψ R† β m f ψ L

In (1) q − is the coupling constant; PL =

.

(1)

1− γ 5 1+ γ 5 , PR = − are the left and 2 2

right projective operators; ψ L = PLψ , ψ R = PR ⋅ψ − are the left and right components of the operator of the Dirac field ψ . The density of Hamiltonian ℋD allows obtaining the motion equations for ψ L and ψR : GG p0ψ L = (α p + qα µ B µ )ψ L + β m f ψ R . GG p0ψ R = (α p + qα µ B µ )ψ R + β m f ψ L

(2)

One can see that both the density of Hamiltonian ℋD and motion equations have a form, which is not SU (2) − invariant because of the Dirac fermion having a mass. For this reason we first consider massless fermions in the Standard Model to

3

provide SU (2) − invariance and then Dirac fermions are given masses after the mechanism of spontaneous violation of the symmetry has been introduced and Higgs bosons have appeared. [3] There is a question: Is it possible to find the form of Hamiltonian and the forms of motion equations for massive fermions with their SU (2) − symmetry preserved? If follows from equations (2) that GG

−1

GG

−1

ψ L = ( p0 − α p − qα µ B µ ) β m f ψ R

.

ψ R = ( p0 − α p − qα µ B µ ) β m f ψ R

(3)

Substitute (3) into the right-hand parts of equations (2) proportional to β m f and obtain the integro-differential equations for ψ R and ψ L :  p − αG pG − q α B 0 − αG BG − β m p − αG pG − q α B 0 − αG BG f 0 0 0  0  p − αG pG − q α B 0 − αG BG − β m p − αG pG − q α B 0 − αG BG f 0 0 0 0 

( (

(

))

(

))

( (

(

))

(

))

−1

β m f ψ L = 0

 −1 β m f ψ R = 0 

(4)

It is clearly seen that equations for ψ R and ψ L have the same form and, in contrast to equations (2), the presence of mass m f does not lead to mixing the left and right components of ψ . Equations (4) can be written as ( p − αG pG − qα B µ ) − ( p + αG pG − qα B µ )−1 m 2 ψ = 0 . µ µ 0  0  L , R

(5)

In expression (5), ψ L, R shows that equations for ψ L and ψ R have the same form; 1

αµ = 

−α

i

. GG

If we multiply (on the left side) equations (5) by term p0 + α p − qα µ B µ , we obtain the second-order equations with respect to p µ : ( p0 + αG pG − qα µ B µ )( p0 − αG pG − qα µ B µ ) − m 2 ψ L , R = 0 .  

For quantum electrodynamics

( q = e, B

µ

= Aµ )

( p − eA )2 − pG − eAG 2 − m 2 + eσG HG + iαG EG ψ = 0 . 0  0  L , R

(

)

(6) equations (6) have the form (7)

4

G G G G ∂A In equation (7), H = rot A − is magnetic field, E = − − ∇A0 − is electrical field. ∂t

Equations (7) coincide with the second-order equation obtained by Dirac in the 1920s [4]. However, in contrast to [4] (see also [5]), equations (7) contain no «excess» solutions. The operator γ 5 commutes with equations (6). Consequently, γ 5ψ = δψ (δ 2 = 1; δ = ±1) . The case of γ 5 = −1 corresponds to the solution of equation

(7) for ψ L and γ 5 = +1 corresponds to the solution of equation (7) for ψ R . Equations (5), (6) are invariant relative to SU (2) − transformations, however, they are nonlinear relative to operator p0 = i

∂ . Linear forms of SU (2) − invariant ∂t

equations for fermion fields relative to p0 can be obtained using the FoldyWouthuysen transformation [6] in the specially introduced isotopic space. ψ L   and isotopic matrices ψ R 

Introduce the eight-component field operator Ф1 = 

I 0  0 I   , τ1 =   affecting the four upper and four lower components of  0 −I   I 0

τ3 = 

operator Ф1 . Now, equations (2) can be written as GG p0Ф1 = (α p + τ 1 β m f + qα µ B µ ) Ф1 .

(8)

Owing to commutation between τ 1 and the right-hand part of equation (8), field ψ  Ф2 = τ 1Ф1 =  R  is also solution to equation (8). ψ L 

Further, consider equation (8) without boson field B µ (free motion): GG p0Ф1,2 = (α p + τ 1 β m f ) Ф1,2 .

(9)

Ф1,2 is written to show that equations (9) are the same for operators Ф1 , Ф2 .

Find the Foldy-Wouthuysen transformation in isotopic space for the free motion equation (9) using the Eriksen transformation [7]. U

0 FW

= U Er

1  1 τ λ + λτ 3  = (1 + τ 3λ )  + 3  2 4 2 

−1

2

(10)

5

GG

In expression (10) we have λ =

(αG pG + τ β m ) f

1

2

α p + τ1β m f E

1 G ; E = ( p 2 + m 2 ) 2 . Since we have

= E 2 , then λ 2 = 1 .

Expression (10) can be transformed to obtain the following expression: GG G G −1 1  τ 3α p + τ 3τ 1 β m   1 τ 3α p  2 U = U Er =  1 +  +  = 2 E   2 2E  GG  E + τ 3α p  1 = G G τ 3τ 1 β m  1 + 2E  E + τ 3α p  0 FW

(11)

(

)

0 Transformation (11) is a unitary transformation U FW (U FW0 ) = 1 and 0 U FW (αG pG + τ1β m f

) (U ) 0 FW

†

†

= τ3 E

(12)

Thus, equations (9) in the Foldy-Wouthuysen representation have the form p0 (Ф1,2 ) FW = τ 3 E (Ф1,2 ) FW .

(13)

Now, let us check whether the sufficiency condition for transformation to the Foldy-Wouthuysen representation [8] is fulfilled, or not. With regard to relations (2), (3) the normalized solutions to equation (9) for field Ф1 can be written, as follows: Positive-energy solution

Ф1( + ) = e − iEt

GG   E +α p ψR ψR    GG  2E E + τ 3α p   = e− iEt   1     1 m β ψ 2E R  E − αG pG  G G β mψ R    E E p) α 2 − (  

(14)

Negative-energy solution

Ф1( − ) = eiEt

1   G G β mψ L  − G G  − 1 β mψ  2E ( E + α p ) GG E + τ 3α p  L  E +α p = eiEt  G G     2E   E −α p ψL     ψL 2E  

(15)

0 to Ф1( + ) , Ф1( − ) we obtain By applying matrix U FW

(Ф1(+ ) ) (Ф )

FW

(−) 1 FW

ψ ( + )  0 = U FW Ф1( + ) = e− iEt  R   0  . 0   0 = U FW Ф1( − ) = eiEt  ( − )  ψ L 

(16)

6 0 One can see from (16) that the sufficiency condition is fulfilled and matrix U FW is,

indeed, the Foldy-Wouthuysen transformation for spinor Ф1 in the isotopic space we have introduced. ψ L   we can obtain ψ R 

Similarly, for field Ф2 = 

(Ф2(+ ) )

FW

(Ф )

(−) 2 FW

ψ ( + )  0 = U FW Ф2( + ) = e− iEt  L   0   0  0 = U FW Ф2( − ) = eiEt  ( − )  ψ R 

(17)

Equations (13) allow writing the density of the free motion Hamiltonian for fermions of mass m f in the form ℋFW = (Ф1 )†FW τ 3 E (Ф1 ) FW + (Ф2 )†FW τ 3 E (Ф2 ) FW = (Ф1( + ) )†FW E (Ф1( + ) ) FW − (Ф1( − ) )†FW E (Ф1( − ) ) FW + + (Ф2( + ) )†FW E (Ф2( + ) ) FW − (Ф2( − ) )†FW E (Ф2( − ) ) FW = (ψ R( + ) ) Eψ R( + ) − (ψ L( − ) ) Eψ L( − ) + †

†

(18)

+ (ψ L( + ) ) Eψ L( + ) − (ψ R( − ) ) Eψ R( − ) . †

†

In the presence of boson fields B µ ( x) interacting with fermion fields Ф1 ( x), Ф2 ( x), the Foldy-Wouthuysen transformation and Hamiltonian of equation (8) in the Foldy-Wouthuysen representation in isotopic space could be obtained as a series in powers of the coupling constant using the algorithm described in [1], [9]. As a result, using denotations from [1], [9] we obtain 0 U FW = U FW (1 + δ1 + δ 2 + δ 3 + ...)

(19)

p0 (Ф1,2 ) FW = H FW (Ф1,2 ) FW = (τ 3 E + qK1 + q 2 K 2 + q 3 K 3 + ...) (Ф1,2 ) FW .

(20)

Expressions for operators C and N constituting the basis for the interaction Hamiltonian in the Foldy-Wouthuysen representation can be written in the following form in our case: † 0 0  C = U FW qα µ B µ (U FW )  

even

GG GG = qR ( B 0 − LB 0 L ) R − qR α B − Lα BL R

(

)

G G † odd 0 0  = qR ( LB 0 − B 0 L ) R − qR LαG B − αG BL R N = U FW qα µ B µ (U FW )   GG E + τ 3α p 1 ;L= R= G G τ 3τ 1 β m 2E E + τ 3α p

(

)

(21)

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The superscripts even and

odd

in expressions (21) show the even and odd parts of

operators relative to the upper and lower components of Ф1 and Ф2 . For equations (20) we can write the Hamiltonian density for fermion fields

(Ф1 ) FW , (Ф2 ) FW , interacting with boson field

B µ ( x) .

ℋFW = (Ф1 )†FW (τ 3 E + qK1 + q 2 K 2 + q3 K3 + ...) (Ф1 ) FW + + (Ф2 )†FW (τ 3 E + qK1 + q 2 K 2 + q 3 K 3 + ...) (Ф2 ) FW

(22)

Considering the fact that the expression for the Foldy-Wouthuysen Hamiltonian in parentheses in equation (22) has, by definition, the diagonal form relative to the upper and lower components (Ф1 ) FW , (Ф2 ) FW [6], [8], [9], it is clear that the Hamiltonian density (22) is SU (2) − invariant. Thus, expression (22) and equations (20) are invariant relative to SU (2) transformations independently of the presence or absence of fermion masses. Expression (22) demonstrates the necessity of using two fermion fields -

(Ф1 ) FW ( х ) , (Ф2 ) FW ( х ) - in the formalism. If only (Ф1 ) FW ( х ) is used in the theory, motion and interactions of the right fermions, as well as motion and interactions of left antifermions remain; and vice versa, if only (Ф2 ) FW ( х ) is used in the theory, motion and interaction of the left fermions, as well as motion and interactions of the right antifermions remain. We can arrive at using the two fermion fields - (Ф1 ) FW ( х ) , (Ф2 ) FW ( х ) - to describe the motion of a fermion with mass m f , if we agree with the necessity to include the states of equations (2) with the sign changed before the mass term [1]. Changing the sign before mass m f in Dirac equation, in itself, leads to no physical consequences, however, when changing to the Foldy-Wouthuysen representation, a complete description of the motion and interactions of both the left and right fermions can be obtained using two fermion fields. In this case, Dirac equations for fields Ф1 , Ф2 have the form GG p0Ф1 ( x) = (α p + τ 1 β m f + qα µ B µ ) Ф1 ( x) GG p0Ф2 ( x) = (α p − τ 1 β m f + qα µ B µ ) Ф2 ( x)

(23)

8

It is still possible to write the Foldy-Wouthuysen transformation of the form 0 U FW = U FW (1 + δ1 + δ 2 + δ 3 + ...)

U

0 FW

=

(24)

GG  E + τ 3α p  1 G G τ 3τ 1 β m  1 ± 2E  E + τ 3α p 

(25)

The signs in (25) and (24), as well as in operators C and N , are chosen depending on whether the transformation is applied to field Ф1 ( x) (sign + ), or it is applied to field Ф2 ( x) (sign − ). Upon transformation (25), the form of the transformed functions (16), (17) and equations (13) remains unchanged. The free motion Hamiltonian density and the Hamiltonian density in the presence of interacting fields can be written in the form similar to (18), (22). The SU (2) − invariant Standard Model with massive fermions can be derived with the isotopic space introduced. In case of interaction with gauge fields Bµ , the Lagrangian

with

covariant

derivative

D µ = ∂ µ − iqB µ

and

fermion

fields

ψ  ψ  Ф1 =  R  , Ф2 =  L  can be written as ψ L  ψ R 

ℒ = Ф1 γ µ D µ Ф1 − Ф1 τ 1m f Ф1 + Ф2 γ µ D µ Ф2 + Ф2τ 1m f Ф2 .

(26)

The Lagrangian allows obtaining the motion equations for fermion fields Ф1 , Ф2 with fermion mass m f (see (23)). GG p0Ф1 ( x) = (α p + τ 1 β m f + qα µ B µ ) Ф1 ( x) GG p0Ф2 ( x) = (α p − τ 1 β m f + qα µ B µ ) Ф2 ( x)

Using the isotopic Foldy-Wouthuysen transformation (24), (25) one can obtain the SU (2) − invariant Hamiltonian density and motion equations for fermion fields with the sign in operators U FW , C , N , K1 , K 2 , K3 ... chosen properly (see (25)). ℋFW = (Ф1 )†FW (τ 3 E + qK1 ( + m f ) + q 2 K 2 ( + m f ) + q3 K3 ( + m f ) + ...) (Ф1 ) FW +

(

)

+ (Ф2 )†FW τ 3 E + qK1 ( − m f ) + q 2 K 2 ( − m f ) + q 3 K 3 ( − m f ) + ... (Ф2 ) FW

(27)

9

(

)

(

)

p0 (Ф1 ) FW = τ 3 E + qK1 ( + m f ) + q 2 K 2 ( + m f ) + q 3 K 3 ( + m f ) + ... (Ф1 ) FW p0 (Ф2 ) FW = τ 3 E + qK1 ( − m f ) + q 2 K 2 ( − m f ) + q 3 K 3 ( − m f ) + ... (Ф2 ) FW

(28)

In (27), (28) the notation K i ( ± m f ) indicates the sign of expressions containing fermion mass m f . Thus, with SU(3) × SU(2) × U(1)-invariance and each of its theoretical and experimental applications and consequences preserved, the Standard Model can be formulated without the requirement of interactions between Higgs bosons and fermions. Here, boson fields are responsible for the gauge invariance of the theory’s boson sector only and interact only with gauge bosons Wµ± , Z µ , gluons and photons. With such formulation of the theory, fermion masses are introduced from the outside. The theory has no vertices of Yukawa interactions between fermions and Higgs bosons and, therefore, there are no processes of scalar boson decay and generation of fermions ( H → f f ) , no quarkonium states ψ , ϒ,θ including Higgs bosons, no interactions of Higgs bosons with gluons ( qqH ) and photons (γγ H ) via fermion loops, etc. The suggested version of the Standard Model is, most likely, renormalizable, because the boson sector of the theory remains massless till the introduction of the Higgs mechanism of spontaneous violation of the symmetry, and quantum electrodynamics with a massless quantum and massive electron and positron is the re-normalizable theory. Nevertheless, issues of re-normability of the suggested version of the Standard Model are to be studied profoundly. Of course, the results of forthcoming experiments on searching for scalar bosons using the CERN’s Large Hadron Collider would provide direct verification of the conclusions made in this paper concerning the formulation of the Standard Model without Higgs bosons’ interactions with fermions.

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References 1. V.P.Neznamov. Physics of Elementary Particles and Atomic Nuclei (EPAN),Vol.37, N 1, (2006). 2. V.P.Neznamov. hep-th/0412047, (2005). 3. S.Weinberg. The Quantum Theory of Fields, V.I, II. Cambridge University Press, 2001. 4. P.Dirac. The Principles of Quantum Mechanics. 4-th edition. Oxford at the Clarendon Press, 1958. 5. S.S.Schweber. An Introduction to Relativistic Quantum Field Theory. ROW, Peterson and Co. Evanston, Ill.,Elmsford, N.Y., 1961. 6. L.L.Foldy, S.A.Wouthuysen, Phys.Rev 78, 29 (1950). 7. E.Eriksen//Phys.Rev. 111.1011.1958. 8. V.P.Neznamov. The Necessary and Sufficient Conditions for Transformation from Dirac Representation to Foldy-Wouthuysen Representation. hep-th/0804.0333, (2008). 9. V.P.Neznamov.Voprosy Atomnoi Nauki I Tekhniki. Ser.: Theoretical and Applied Physics. 1988, Issue2, P.21.