The Possibility of Construction the Standard Model without Higgs Bosons in

the Fermion Sector

V.P. Neznamov RFNC-VNIIEF, 607190, Sarov, N.Novgorod region

e-mail: Neznamov@vniief.ru

Presentation at Conference “Physics of Fundamental Interactions”Nuclear Physics Section of RAS Department of PhysicsIHEP, Protvino

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 , gluons and photons.

Outline

1. Introduction

2. Foldy-Wouthuysen representation

3. Isotopic representation of Dirac equation

4. Isotopic Foldy-Wouthuysen transformation

5. Standard Model formulation without Higgs bosons in the fermion sector

6. Isotopic Foldy-Wouthuysen representation is a possible key to understanding the dark matter problem?

7. Conclusion

,W Z

2

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. V.P.Neznamov. The Necessary and Sufficient Conditions for Transformation from Dirac Representation to Foldy-Wouthuysen Representation. hep-th/0804.0333, (2008).

4. V.P.Neznamov. Voprosy Atomnoi Nauki I Tekhniki. Ser.: Theoretical and Applied Physics. 1988. Issue 2. P.21.

3

1. Introduction

As we know, to provide SU(2) – invariance of the theory, the Standard Model first considers massless fermions, which are provided with masses following the introduction of the mechanism of spontaneous symmetry violation, occurrence of Higgs bosons and postulation of their gauge invariant interaction with Yukawa-type fermions.

On the eve of the decisive LHC experiments one can question oneself, whether it is possible to formulate the Standard Model with initially massive fermions, while preserving the theory’s SU(2) – symmetry.

In this case, Higgs bosons are responsible only for the gauge invariance of the theory’s boson sector and interact only with gauge bosons , gluons and photons.

Within this statement of the theory, fermion masses are introduced from outside. The theory has no vertices of Yukawa interactions between fermions and Higgs bosons and, therefore, there are no processes of scalar boson decay to fermions , no quarkonium states including Higgs bosons, no interactions of Higgs bosons with gluons and photons via fermion loops, etc.

W , Z

H f f , , ggH

H

4

The answer to the question above has already been given in papers [1], [2], where the Standard Model is formulated in the modified Foldy-Wouthuysen representation. It has been shown that for its 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 this paper is to give a similar formulation of the Standard Model with initially massive fermions and spinors in the Dirac representation to meet the requirements of local SU(3) SU(2) U(1) symmetry.

5

2. Foldy-Wouthuysen transformation

As we know, Foldy-Wouthuysen transformation is performed using unitary operator . Dirac field operator and Dirac equation Hamiltonian are transformed as follows:

In the Foldy-Wouthyusen representation, Hamiltonian is block-diagonal with respect to the upper and lower components of field operator .

The second necessary condition for the transition to the FW representation for free motion and motion in static external fields is zeroing of either upper or lower components of .

Refs. [1], [4] have proposed a direct way for obtaining the Foldy-Wouthyusen transformation for interactions between fermions with arbitrary boson fields. The transformation matrix and relativistic Hamiltonian have been obtained as a series in powers of the coupling constant

Here, q is the coupling constant, is FW-transformation matrix for free Dirac particles, .

Quantum electrodynamics and the Standard Model have been considered with Hamiltonian in the FW-representation, a number of quantum-field effects have been calculated, an SU-2 – invariant formulation of the Standard Model with initially massive fermions without Yukawa interactions between Higgs bosons and fermions has been proposed [1], [2].

FW FW DU †

† FWFW FW D FW FW

dUH U H U iU

dt

0 2 31 2 31 ....FW FWU U q q q 2 3

1 2 3 ....FWH E qK q K q K 0FWU

2 2fE p m

FWU

FWH

FW

FW

6

3. Isotopic representation of Dirac equation

Let us consider the density of Hamiltonian of a Dirac particle with mass mf, which interacts with an arbitrary boson field

HD =

In Eq. (1), q is the coupling constant; are the left and right projection operators;

are the left and right components of the Dirac field operator

The density of Hamiltonian HD allows obtaining the motion equations for and

0 0

0 0 5

, 0,1,2,3; 1,2,3; ; ;

1, 0; ; , , ,

, 1, 2,3

k k

k

k

xy x y x y x y k p ix x

матрицыДиракаk

B

† †f L R f L Rp m q B P P p m q B P P

† † † †L L R R L f R R f Lp q B p q B m m

5 51 1,

2 2L RP P

,L L R RP P

L R

0

0

L L f R

R R f L

p p q B m

p p q B m

Below we use the following system of units:

are 4-vectors; the inner product is taken in the form1; , ,с х p B

(1)

(2)

7

8

Let us introduce an eight-component field operator

and isotopic matrices, ,

affecting the four upper and four lower components of operator . Now, Eqs. (2) can be written as

As commutes with the right-hand part of Eq. (3), field

is also solution to Eq. (3).

Finally, using the equality , Eqs. (3), (4) can be written in the form

It will be shown below that equivalent equations (3), (4), (5) in the Dirac representation in fact lead to different physical pictures of composition and interactions between elementary particles in the isotopic Foldy-Wouthyusen representation.

1R

L

Ф

3 1

0 0,

0 0

I I

I I

1Ф

0 1 1 1fp Ф p m q B Ф

0 2 1 2fp Ф p m q B Ф

2 1 1L

R

Ф Ф

0 1 1 1 2

1 1

2 2fp Ф p m q B Ф q B Ф

0 2 1 2 1

1 1

2 2fp Ф p m q B Ф q B Ф

2 1 1Ф Ф

1

(3)

(4)

(5)

4. Isotopic Foldy-Wouthyusen representation

9

Let us find the Foldy-Wouthuysen transformation in the introduced isotopic space for free motion Dirac equations (3), (4) without interaction with boson fields using the Eriksen transformation.

In expression (6) . Since we have

Expression (6) can be transformed to obtain the following expression:

Expression (7) is unitary transformation and

Thus, Eqs. (3), (4) in the Foldy-Wouthuysen representation have the form

1

20 3 3

3

1 11

2 2 4FW ErU U

1

1 2 2 2;fp mE p m

E

2 2

1 ,fp m E 2 1

12

0 3 3 1 3

33 1

3

1 11

2 2 2

11

2

FW Er

p m pU U

E E

E pm

E E p

†0 0 1FW FWU U

†0 01 3FW FW f FWH U p m U E

0 1,2 3 1,2FW FWp Ф E Ф

(6)

(7)

(8)

(9)

10

For the general case of interactions with arbitrary boson fields, one can use the author’s direct technique for transition from the isotopic representation of the Dirac equation to the isotopic Foldy-Wouthuysen representation. Application of the same isotopic Foldy-Wouthuysen transformation to Eqs. (3), (4), (5) yields: 2 3

0 1 3 1 2 3 1...FW FWp Ф E qK q K q K Ф

† 2 31 3 1 2 3 1...I

FW FW FWH Ф E qK q K q K Ф

2 30 2 3 1 2 3 2...FW FWp Ф E qK q K q K Ф

† 2 32 3 1 2 3 2...II

FW FW FWH Ф E qK q K q K Ф

2 3

0 1 3 1 2 3 1....2 2 2FW FW

q q qp Ф E K K K Ф

1 1 1

2 3

1 2 3 2...2 2 2 FW

q q qK K K Ф

2 3

0 2 3 1 2 3 2....2 2 2FW FW

q q qp Ф E K K K Ф

1 1 1

2 3

1 2 3 1...2 2 2 FW

q q qK K K Ф

1 1 1

2 3†1 1 2 3 2....

2 2 2FW FW

q q qФ K K K Ф

2 3†1 3 1 2 3 1....

2 2 2IVFW FW FW

q q qH Ф E K K K Ф

2 3

†2 3 1 2 3 2....

2 2 2FW FW

q q qФ E K K K Ф

1 1 1

2 3†2 1 2 3 1...

2 2 2FW FW

q q qФ K K K Ф

(10)

(11)

(12)

11

It is clear that one can construct density of Hamiltonian with

equations for fields from Eqs. (10), (11)

Expressions for the operators constituting the basis for writing the interaction Hamiltonian in the isotopic Foldy-Wouthuysen representation (i.e. terms ) have the following form:

(14)

(15)

IIIFWH

1 2,FW FWФ Ф

III I IIFW FW FWH H H

†0 0 0 0even

FW FWC U q B U qR B LB L R qR B L BL R

†0 0 0 0odd

FW FWN U q B U qR LB B L R qR L B BL R

1

†0 0 0 01 1 1 1 1

odd

FW FWC U q B U qR B L B L R qR B L BL R

1

†0 0 0 01 1 1 1 1

even

FW FWN U q B U qR L B B L R qR L B BL R

33 1

3

1;

2

E pR L m

E E p

(13)

1 1 11 2 3 1 2 3, , ..., , , ...K K K K K K

12

Basis orthonormal functions for free fermion motion are expressed through the left and right components of Dirac field and have the form:

0

1 1

0

, , 2iEt

FW FWL

Ф x t U Ф x t e Ex

E p

0

2 2

2, ,

0

LiEtFW FW

Ex

Ф x t U Ф x t e E p

0

2 2

2, ,

0

LiEtFW FW

Ex

Ф x t U Ф x t e E p

1 2,FW FWФ x Ф x

x

(16)

0

1 1

2, ,

0

RiEtFW FW

Ex

Ф x t U Ф x t e E p

5. Formulation of the Standard Model without Higgs bosons in the fermion sector.

13

For the free case, let us write the density of Hamiltonian, e.g. (13), taking into account basis functions (16)

HFW =

The density of Hamiltonian (17) bracketed between two-component spinors and has a form that is commonly used in the field theory

One can see that Hamiltonian (17) is SU(2)-invariant, regardless of whether the fermions are massive or massless.

† † ( ) † ( ) ( ) † ( )1 3 1 2 3 2 1 1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )FW FW FW FW FW FW FW FWФ E Ф Ф E Ф Ф E Ф Ф E Ф

† †( ) † ( ) ( ) † ( ) ( ) ( ) ( ) ( )2 2 2 2

2 2( ) ( ) ( ) ( )FW FW FW FW R R L L

E EФ E Ф Ф E Ф E E

E p E p

† †( ) ( ) ( ) ( )2 2L L R R

E EE E

E p E p

x

x

††2FWH E E

IIIFWH

(17)

14

In the presence of static external boson fields, basis functions in the Foldy-Wouthyusen representation in their isotopic structure are similar to those in Expr. (16). When solving applied problems in the quantum field theory using the perturbation theory, fermion fields are expanded in solutions of Dirac equations for free motion or for motion in static external fields. In our case, in the Foldy-Wouthuysen representation, we can also expand fermion fields over the basis of solutions of Eqs. (16) or over a similar basis of solutions of the Foldy-Wouthuysen equations in static external fields.

Hamiltonians in Eqs. (10)(12) are diagonal with respect to the upper and lower isotopic components by definition, and based on the above considerations, they are SU-2 –invariant regardless of whether the fermions are massive or massless.

15

The introduced isotopic space allows constructing the invariant Standard Model with massive fermions. In case of interaction with the gauge fields Lagrangian with covariant derivativeand fermion fields

can be written as

L .

The Lagrangian above allows obtaining motion equations for fermion fieldswith mass (see (3), (4), (5)).

Using the isotopic Foldy-Wouthuysen transformation (11), (19) we can obtain SU(2) – invariant densities of Hamiltonian and motion equations of fermion fields (see (10), (11), (12), (13)).

(2)SU Bm

D = iqB

1 2,R L

L R

Ф Ф

1 1 1 1 1 2 2 2 1 2f fФ D Ф Ф m Ф Ф D Ф Ф m Ф

1 2,Ф Фfm

6. Isotopic Foldy-Wouthuysen representation is a possible key to understanding the dark matter problem?

16

Consider densities of Hamiltonians and Eqs.(10)(12) from viewpoint of composition and possible interactions of elementary particles. Remember that Eqs. (10)(12) have been obtained from one and the same Dirac equation written in various forms in expressions (3)(5). Hence, implementation of the physical pictures corresponding to Eqs.(10)(12) is possible in our universe.

First, consider density of Hamiltonian and Eqs. (12). A symbolic physical picture is given in Fig. 1.

Fig. 1The left half-plane in Fig.1 represents the states of basis functions (12) with

isotopic spin and with positive value of energy E 0; the right half-plane

represents the states with and with negative value of energy E 0. Density of Hamiltonian contains the states of both the left and right fermions and

the left and right antifermions ; particles and antiparticles interact with each other both really (solid line with arrows in Fig.1) and virtually (dashed line with arrows in Fig.1). The physical picture in Fig.1 represents the real world around us.

IVFWH

IVFWH

3

1

2T

1 ,0

iEt RFW

A xФ x t e

2 ,0

iEt LFW

A xФ x t ie

3

1

2T

1

0, iEt

FW

L

Ф x t eA x

2

0

, iEtFW RФ x t ie A x

3

1

2T

3

1

2T

E 0E > 0

17

Now, consider density of Hamiltonian with equations from (10), (11). The symbolic representation of the structure and interactions of elementary particles is given in Fig.2.

Fig.2

The world in Fig.2 is poorer than that in Fig.1. The physical picture in Fig.2 has both left and right fermions and left and right antifermions. However, there are no interactions between real fermions and antifermions; there is only virtual interaction between them (a dashed line with arrows in Fig.2).

Fig.2 assumes there is a strong and electromagnetic interaction between particles and antiparticles without any real interaction between them. There are no processes of generation and absorption of real pairs “particle-antiparticle” and there are no coupled states of real particles and antiparticles, etc. Weak interactions are significantly poorer, as well, because there are no processes of simultaneous generation and absorption of real particles together with antiparticles.

IIIFWH

IIIFWH

3

1

2T

1 ,0

iEt RFW

A xФ x t e

2 ,0

iEt LFW

A xФ x t ie

3

1

2T

1

0, iEt

FW

L

Ф x t eA x

2

0

, iEtFW RФ x t ie A x

E 0E > 0

18

Let us consider densities of Hamiltonians with the corresponding equations from (10), (11). A symbolic picture is given in Fig.3 and Fig.4.

,I IIFW FWH H

3

1

2T

IFWH

1 ,0

iEt RFW

A xФ x t e

3

1

2T

1

0, iEt

FW

L

Ф x t eA x

IIFWH

3

1

2T

2 ,0

iEt LFW

A xФ x t ie

3

1

2T

2

0

, iEtFW RФ x t ie A x

Fig.

4

Fig.

3

E 0E > 0

E 0E > 0

19

It follows from Figs.3 an4 that densities of Hamiltonians , provide existence of either right fermions and left antifermions ( ), or left fermions and right antifermions ( ). There are no interactions between particles and antiparticles in the both cases.

The real world in Fig.3 or in Fig.4 has no electromagnetic and strong interactions, because for the sake of parity preservation such interactions require that both left and right fermions are present. The same requirement is valid for processes with weak neutral current interactions and because of this they are also absent in Figs.3 and 4.

The processes with weak charged currents and with participation of either left fermions (Fig.4), or left antifermions (Fig.3) also appear to be suppressed because of impossibility to emit, or absorb particles and antiparticles.

The vacuum state in Fig.3 and Fig.4 significantly differs from that in Fig.1 and Fig.2. The vacuum state in Fig.3 and Fig.4 demonstrates that there is no “bouillon” of virtual pairs “particle-antiparticle” and virtual carriers of interactions because of no interactions (except for the gravitational one).

IFWH II

FWHIFWH II

FWH

20

Here is the summary of physical pictures given in Figs. 1, 2, 3, and 4.

Owing to the prohibited strong and electromagnetic interactions and almost absolute absence of weak interactions the world shown in Figs.3, 4 has the following features:

• it does not emit/absorb light;

• it is electrically neutral;

• non-relativistic motion;

• it weakly interacts with the outer world.

The features above are the features of “dark matter” discovered during the last quarter of the past century, which constitutes 26% of the universe structure.

Hence, we may assume that “dark matter” is an implementation of the physical pictures shown in Fig.3 and Fig.4. It consists of either right fermions and left antifermions, or left fermions and right antifermions. A set of fermions and antifermions requires no changes in the composition of particles in the Standard Model.

21

As it has been mentioned earlier, the physical picture in Fig.1 is a real world of our part of the universe. The baryon matter (“light matter”) constitutes 4% of the universe structure. If we assume that in the past some part of the universe has undertaken transformation from the picture shown in Fig.1 to the physical picture shown in Figs.3 and 4, we assume generation of “dark matter” together with a significant reconstruction of vacuum resulted in disappearance of quark, gluon, and electro weak condensates. This leads us to a speculative idea that such reconstruction associates with the problem of “dark energy”, which currently constitutes ~70% of the universe according to observation data [7].

Transition from Fig.1 to Fig.2 requires no reconstructions of vacuum space. If we assume that in the past our part of the universe has undertaken transformation to the physical pictures shown in Fig.2 and returned to its previous state after some time, such speculative idea leads us to explanations of the current baryon asymmetry of the universe.

A principal question still requires an answer:

- How and why does the universe transform to various physical pictures of composition and interactions of elementary particles?

It should be noted once again that different physical pictures of composition and interactions of elementary particles have been obtained basing on one equation of the Dirac field interacting with boson fields described by equations (3)-(5) using the isotopic Foldy-Wouthuysen transformation that allows writing equations of fields with massive fermions and their Hamiltonians in the form invariant relative to SU-2 – transformations.

CONCLUSION

22

1.     Hamiltonians and the corresponding equations of fermion fields obtained in the isotopic Foldy-Wouthuysen representation are invariant with respect to SU(2) transformations, regardless of whether the fermions are massive or massless.

2.     This allows constructing the Standard Model without Higgs bosons in the fermion sector.

3.     Many processes of interactions between Higgs boson and fermions do not take place in this case. For example, there are no processes of scalar boson decay to fermions quarkonium states including Higgs boson are absent, there are no boson interactions with gluons and photons via fermion loops, etc.

4.     The four resultant Hamiltonians correspond to different physical pictures of composition and interactions of elementary particles in the isotopic Foldy-Wouthuysen representation. Two physical pictures are close in their features to the observed features of the “dark matter”.

In all physical pictures, compositions of elementary particles are not beyond the set of particles of the Standard Model.

,H f f , ,

ggH H