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The Without Higgs in the Sector V.P. Neznamov

RFNC-VNIIEF, 607190, Sarov, N.Novgorod region e-mail: [email protected]

Abstract The paper addresses the construction of the Standard Model with massive without introduction of the between Higgs bosons and fermions. With such approach, Higgs bosons are responsible only for the gauge invariance of the theory’s sector and interact only with gauge bosons W,Z± , and .

1 As we know, to provide SU(2) – invariance of the theory, the Standard Model first considers massless fermions that are given after the mechanism of spontaneous violation is introduced, Higgs bosons appear and their gauge invariant interaction with Yukawa-type fermions is postulated [1]. On the eve of the decisive LHC experiments one can question oneself whether it is possible to construct 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 WZ− , , gluons and photons. With the theory defined in this manner, 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 boson decay to fermions (Hff→ ) , no states ψ ,,ϒ θ including Higgs bosons, no interactions of

Higgs bosons with gluons ()ggH and photons (γγ H ) via fermion loops, etc. The answer to the question above has already been given in papers [2], [3], where the Standard Model is derived 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 construct, in a similar way, the Standard Model with initially massive fermions and in the Dirac representation to meet the requirements of local SU(3)× SU(2)× U(1) symmetry. ∂ The paper uses the system of units, where = = сх=1; ,pB , are 4-vectors; piµ = ; ∂xµ

2

the inner product is taken in the from

µ 00 kk xy==− yµ x y x y,µ = 0,1,2,3; k = 1,2,3;

1,µ = 0 µµµµ;;,,,005k αγγαβγαγγ===− k are Dirac matrices αµ,1,2,3==k

Consider the density of Hamiltonian of a Dirac with m f , which interacts with an arbitrary abelian boson Bµ

††GGµµ GG ℋD = ψ ()αβp++ mfLRfLR qB αµµ ψψ =( PP +)( αβ p ++ m qB α)() PP + ψ =

††††GGµµ GG =+ψ LLRRLfRRfL()()ααψψααψψβψψβψpqµµ B ++ pq B + m + m (1)

11−γ +γ In (1), q − is the ; PP= 55, =−are the left and right LR22 projection operators; ψ LL==⋅−PPψψ, RR ψ are the left and right components of the

Dirac field operator ψ .

The reason why the abelian case is considered for the field Bµ is simplicity. As will be shown below, using a general case of a Dirac particle interacting with non- abelian boson fields would not change the conclusions and implications of this paper.

Using the density of Hamiltonian ℋD the motion equations for ψ L and ψ R can be obtained:

GG µ ppqBm0ψ LLfR=+()ααψβψµ + (2) GG µ ppqBm0ψ RRfL=+()ααψβψµ +

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 having a mass. It follows from Eqs. (2) that

GG µ −1 ψ LfR=−−()ppqBm0 ααµ βψ (3) GG µ −1 ψ RfL=−−()ppqBm0 ααµ βψ

3 By substituting (3) to the right-hand side of Eqs. (2) proportional to β m f , we obtain

integro-differential equations for ψ R and ψ L

GG GGG GG G −1 ppqBB−−αααβ00 − − mppqBB −− αααβψ − m =0 ()00()ffL( 00())  (4) GG GGG GG G −1 ppqBB−−αααβ00 − − mppqBB −− αααβψ − m =0 ()00()ffR() 00()

One can see that equations for ψ R and ψ L have the same form, and, in contrast to Eqs.

(2), the presence of mass mf does not lead to mixing the right and left components of ψ . Eqs. (4) can be written as

GG GG −1 ppqBppqBm−−ααµµ −+− αα2 ψ =0 (5) ()00µµ()LR ,

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

1 α µ =  i . −α

G G µ If we multiply Eqs. (5) on the left side by term p0 +−ααpqµ B, we obtain second- order equations with respect to pµ GG GG p+−αα pqBµµ p −− αα pqB − m2 ψ =0 ()()00µµLR , (6)

For the case of (qeB==, µ Aµ ) , Eqs. (6) have the form

2 GGGG 2 G G ()peApeAmeHiE−−−−++2 σαψ =0 (7) 00() LR , G G G G ∂A In Eqs. (7) HrotA= is , and EA=− −∇ is electrical field, ∂t 0 G G σ ′ 0 i σ = G , σ ′ - matrices Pauli. 0 σ ′ Eqs. (7) coincide with the second-order equation obtained by Dirac in the 1920s [4]. However, in contrast to [4] (see also [5]), Eqs. (7) contain no “excess" solutions. The

2 operator γ 5 commutes with Eqs. (6). Consequently, γψ5 = δψ() δ==±1; δ 1 . The case

of δ =−1 corresponds to the solution of Eq. (7) for ψ L , and δ =+1 corresponds to the solution of Eq. (7) for ψ R . 4 Eqs. (5), (6) are SU (2) − invariant, but they are nonlinear with respect to the ∂ operator p = i . Linear forms of SU (2) − invariant equations for fermion fields 0 ∂t relative to p0 can be obtained using the Foldy-Wouthuysen transformation [6] in a specially introduced isotopic space.

ψ R We now introduce an eight-component field operator, Ф1 = , and isotopic ψ L

I 00I matrices, ττ31==, , acting on the four upper and four lower 00−II components of operator Ф1 . So, Eqs. (2) can be written as

GG µ p01Ф =+()ατβαpmqB 1f +µ Ф 1 (8)

ψ L As τ1 commutes with the right-hand side of Eq. (8), field ФФ211==τ  is also ψ R solution to Eq. (8). Further, consider Eq. (8) without boson field Bµ (free motion) GG p01,2Ф =+()ατβpm 1f Ф 1,2 (9)

Ф1,2 shows that Eqs. (9) are the same for fields Ф1 , Ф2 . Now we find the Foldy-Wouthuysen transformation in the isotopic space for free motion Eq. (9) using the Eriksen transformation [7].

− 1 2 0 11τλ33+ λτ UUFW==+ Er ()1 τλ3  + (10) 224 G G 1 2 α pm+τβ1 f G 222 GG 2 In expression (10), we have λ ==+; Epm(). Since ()ατβp +=1 mEf , E λ 2 =1. Expression (10) can be transformed to obtain the following expression:

− 1 GG GG 2 0 11τα3313pm+ ττβ τα p UUFW==+ Er 1 + = 222EE GG (11) Ep+τα3 1 =+1 GGττβ31 m 2EEp+τα3

† Expression (11) is a unitary transformation UU00=1 , and ( FW( FW ) )

5 00GG † HUFW=+ FW()α pτβ13 mU f() FW = τ E (12) Thus, Eqs. (9) in the Foldy-Wouthuysen representation have the form

p Ф =τ E Ф (13) 01,2()FW 3 () 1,2 FW When converting to the Foldy-Wouthuysen representation, in addition to the condition of the Hamiltonian being block-diagonal (13), one should necessarily meet the requirement that the upper or lower components of the field operators ФФ12, [8] should be zero. One can term this condition as reduction of fields ФФ12, . Let us check whether this condition is met in our case, or not. Given Eqs. (2), (3), normalized solutions to Eq. (9) for the field operators ФФ12, can be expressed as follows:

()+ G 1 − G ψ R ()x () G G − GGβψmxL () Ф()+ xt, = e−iEt Ф()− xt, = eiEt Ep+α 1 () 1 ()+ G ; 1 ()  GGβψmxR () ()− G Ep−α ψ L ()x (14)

()+ G 1 − G ψ L ()x () G G − GGβψmxR () Ф()+ xt, = e−iEt Ф()− xt, = eiEt Ep+α 2 () 1 ()+ G ; 2 ()  GGβψmxL () ()− G Ep−α ψ R ()x

(++−−) ( ) ( ) ( ) In (14), ФФФФ1212,;, are solutions with positive and negative energy, respectively.

6 G G σ p ()+ G 1+ ϕ ()x ()++GG11() Em+ Em+ ψγψRD()xx=+()1 5 () = GG 222E σ p ()+ G 1+ ϕ ()x Em+

G G σ p ()− G 1− χ ()x ()−−GG11() Em+ Em+ ψγψRD()xx=+()1 5 () = GG 222E σ p ()− G 1− χ ()x Em+ (15) G G σ p ()+ G 1− ϕ ()x ()++GG11() Em+ Em+ ψγψLD()xx=−()1 5 () = GG 222E σ p ()+ G −−1 ϕ ()x Em+

G G σ p ()− G −+1 χ ()x ()−−GG11() Em+ Em+ ψγψLD()xx=−()1 5 () = GG 222E σ p ()− G 1+ χ ()x Em+

G G In expressions (15), ϕχ(+−) ( x), ( ) ( x) are normalized two-component solutions of G the with positive and negative energy. In (14), (15), E and p are respective operators. According to (15),

GG(++) G GG( ) G GG(−−) G GG( ) G αψp RR()xpx= σψ (); αψp RR( xpx) = σψ ( )

GG(++) G GG( ) G GG(−−) G GG( ) G αψp LL()xpx=− σψ (); αψp LL( xpx) =− σψ ( )

7 Expr. (15) lead to the following normalizing conditions: GG GG GEp+σ G ψψ()++††()x ()()xx= ϕ() +() ϕ() +() x RR 2E GG GG GEp−σ G ψψ()−−††()x ()()xx= χ() −() χ() −() x RR 2E GG GG GEp−σ G ψψ()++††()x ()()xx= ϕ() +() ϕ() +() x (16) LL 2E GG GG GEp+σ G ψψ()−−††()x ()()xx= χ() −() χ() −() x LL 2E

0 By applying the transformation U FW (11) to ФФ12, (see (14)) we obtain

2E ()+ G ()++GG0 () −iEt GGψ R ()x Ф11FW()xt,,== U FW Ф ()xt e Ep+σ  0

0 ()−−GG0 () iEt  Ф ()xt,,== U Ф ()xt e 2E G 11FW FW ψ ()− x GG L () Ep+σ (17)

2E ()+ G ()++GG0 () −iEt GGψ L ()x Ф22FW()xt,,== U FW Ф ()xt e Ep−σ  0

0 ()−−GG0 () iEt  Ф ()xt,,== U Ф ()xt e 2E G 22FW FW ψ ()− x GG R () Ep−σ One can see from relations (17) that the reduction condition is fulfilled and the matrix

0 U FW is, indeed, the Foldy-Wouthuysen representation for the fields Ф1 ,Ф2 in the isotopic space we have introduced.

8 Eqs. (13) allow us to write the density of the free-motion Hamiltonian of fermions with mass m f as

††()†()()†()++ −− ℋFW = ()Ф131FWττE ()()ФФ FW+= 232 FWE ()()()ФФ FW 1 FWE ФФ 1 FW − ()() 1 FWE Ф 1 FW +

()†++ () ()† −− () () +††22EE () +− () () − +−()()Ф22FWE ФФ FW ()() 22 FWE Ф FW =()ψψψψ RGGEE R −() L GG L + Ep++σσ Ep

()++−−††22EE () () () +−()ψ LLRRGGEEψψ() GG ψ (18) Ep−−σσ Ep One can see that Hamiltonian (18) is SU(2)-invariant, regardless of whether the fermions are massive or massless. Expression (18) shows that two fermion field ФФ, operators, ()12FW () FW , need to be used to provide a complete description of the free motion of the right and left fermions. Given (15), the density of Hamiltonian (18) G G bracketed between two-component spinors ϕ (+) ( x) and χ (−) ()x has a form that is commonly used in the field theory,

=−2 ϕϕχ()++−† EE () ()† χ () − ℋ FW ().

In the presence of boson fields Bµ ()x interacting with fermion fields

Ф12(),x Ф (),x the Foldy-Wouthuysen transformation and Hamiltonian of Eq. (8) in the Foldy-Wouthuysen representation in the isotopic space can be obtained as a series in powers of the coupling constant using the algorithm described in Refs. [2], [9]. As a result, using denotations from Refs. [2], [9], we obtain

0 UUFW=++++ FW ()1δδδ123 ... (19)

23 p01,2()Ф FW==++++H FW ()Ф 1,2 FW()τ 3EqKqKqK 1 2 3 ...()Ф 1,2 FW (20)

The expressions for operators C and N constituting the basis for the interaction Hamiltonian in the Foldy-Wouthuysen representation obtained using the technique of Refs. [2], [9] can be written in the following form in our case:

† even G G G G CUqBU==−−−00αααµ qRBLBLRqRBLBLR 00 FWµ () FW ( ) () † odd GGGG NUqBU==−−−00αααµ qRLBBLRqRLBBLR 00 (21) FWµ () FW ( ) () GG Ep+τα3 1 RL==; GGττβ31 m 2EEp+τα3

9 The superscripts even, odd in (21) show the even and odd parts of the operators relative to the upper and lower isotopic components of Ф1 and Ф2 .

ФФ,, For Eqs. (20), the Hamiltonian density for fermion fields ()12FW( ) FW interacting with boson field Bµ ()x can be written as

†23 ℋFW = ()Ф13FW()τ EqKqKqK++ 1 2 + 3 + ...()Ф 1 FW +

†23 +++++(Ф23 )FW()τ EqKqKqK 1 2 3 ... (Ф 2 ) FW (22) The expression for the Foldy-Wouthuysen Hamiltonian in parentheses in equation (22) is, by definition, diagonal with respect to the upper and lower ФФ, components ()12FW () FW [6], [8], [9]. When solving applied problems in the 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 (17) or over a similar basis of solutions of the Foldy-Wouthuysen equations in static external fields. Then, Hamiltonian density (22) can be expressed through the functions (17), and it is obvious that this expression, similarly to (18), will be SU (2) -invariant due to the diagonality. Thus, expression (22) and Eqs. (20) are invariant relative to SU (2) - transformations regardless of fermions having or not having masses. Formula (22) demonstrates the necessity of using two fermion fields, ФхФх, ()()()12FW FW ( ) , in the formalism for constructing the Standard Model. If

Фх only()()1 FW is used in the theory, motion and interactions of the right fermions, as well as motion and interactions of the left anti-fermions remain. If, on the contrary, Фх only ()()2 FW is used in the theory, motion and interaction of the left fermions, as well as motion and interactions of the right antifermions remain.

More careful analysis shows that even with two fields, Фх1FW ( ) ,Фх2FW ( ) , Hamiltonian (22) contains no interactions between real and anti-particles. This happens due to the structure of expressions (17) in the introduced isotopic

10 space. The theory’s special feature in the Foldy-Wouthuysen representation is that the

Hamiltonian terms include interaction Kn (except K1 ) of an even number of odd operators N that couple states with positive and negative energy. Therefore, interactions between particles and can occur only between real and intermediate virtual states [2], [9]. In order to introduce interactions between real particles and antiparticles into the theory in Refs. [2], [9], the Foldy-Wouthuysen representation had to be modified. To solve the same problems in our case, let us use the following approach.

Write Eq. (8) for field Ф1 ( x) and the same equation for field Ф2 ( x) in the equivalent form

GG 11µµ p01Ф ()xpm=+()ατβ 1f Ф 1()xqB + αµµФ 1()xqB + ατ 12Ф ()x 22 (23)

GG 11µµ p02Ф ()xpm=+()ατβ 1f Ф 2()xqB + αµµФ 2()xqB + ατ 11Ф ()x 22

Eq. (23) uses the equality Ф211( x) =τ Ф ( x) . Further, performing the Foldy-Wouthuysen transformation (19) for Eqs. (23), we obtain equations for fields Ф12FW( x),Ф FW ( x) .

23 qq  q p Ф xE=+++++τ Ф xK K K.... Ф x 01FW() 3 1 FW ()  1 2  3 1 FW () 22  2

23 qq  q ++KKK + +... Ф ()x (24) 1232τττ111  FW 22  2

23 qq  q p Ф xE=+++++τ Ф xK K K... Ф x 02FW() 3 2 FW ()  1 2  3 2 FW () 22  2

23 qq  q ++KKK + +... Ф ()x 1231τττ111  FW 22  2

11 Denotations K in formulas (24) mean that the matrix τ is placed in the operators C, nτ1 1

N (21) next to the fields Bµ : GGG G CqRBLBLqRBLBLR=−−−τ00 τ τα τα τ1 ()11() 1 1 G G G G NqRLBBLqRLB=−−−τ00 τ τα τα BLR (25) τ1 ()11() 1 1 Equations (24) correspond to the Hamiltonian density

23 qq  q =+++++Ф† τ EK K K.... Ф ℋ FW13 FW 1 2  3 1 FW 22  2

23 qq  q +++++Ф† KKK.... Ф (26) 11FWτττ111 2  3 2 FW 22  2

23 qq  q ++++++Ф† τ EK K K.... Ф 23FW 1 2  3 2 FW 22  2

23 qq  q ++++Ф† KKK... Ф 21FWτττ111 2  3 1 FW 22  2 By analogy with [2], [9], Feynman rules for calculating specific physical processes in the quantum theory of interacting fields using perturbation theory methods can be derived using Hamiltonian density (26) and Eqs. (24). The isotopic space we have introduced allows constructing the SU (2) − invariant Standard Model with massive fermions. In case of interaction with

µ µµ gauge fields Bµ , the Lagrangian with covariant derivative D=∂− iqB and

ψ RL ψ fermion fields ФФ12==,  can be written as ψ L ψ R

µµ ℒ =−+−Ф1111122212γτγτµµD ФФm ffФФ D ФФm Ф (27)

Using this Lagrangian, the motion equations for fermion fields ФФ12, with mass m f (see (23)) can be obtained. Using the isotopic Foldy-Wouthuysen transformation (11), (19) one can obtain the SU (2) − invariant Hamiltonian density (26) and motion equations for fermion fields

(24) with appropriate definitions of operators CNK, , , K , K ... (see (25)). ττ11123 τ 1 τ 1 τ 1

12 To construct the Standard Model using Hamiltonian density (26), one must

µ replace, as it is done in [2], [3], the interaction vertex qα µ B in the operators

K , K ... , K , K ... with the interaction vertices of the Standard Model [1]. 1 2 1τ1 2τ1

µ µµµg2 −+32IIγγ55 2  qBα µ → eQfαθαθα Aµ +−() T ffW Qsin − Q fW sin  Zµ + cosθW  2  2

g2 −IIγγ55µ  −+ +=()f uααµ ()() f =+= d f ve () f =+ e Wµ Hermit.. conj + 2 22  g + 3 ()()f ==ud,,αλµ aa f ud G (28) 2 αβαβ µ

+ − a In (28), Aµ is the electromagnetic field; Zµµ,W are the fields; Gµ

3 are the fields; Qf is the fermion electric in the units of eT;1/2f = for

3 e f = vue ,; Tf =−1/2 for fed= ,;θW is the electroweak mixing angle; gg23= ; is sinθW the coupling constant; λ a are generators of SU (3). Expressions (28) is written out only for the first and family. For the second and the third families it is required to make appropriate substitutions

()veude ,,,→ () vµ ,µ ,, cs and (vtbτ ,,,τ ) and introduce the quark mixing. In (28) notations (f=u), (f=d), etc., imply that spinor FW-fields of associated fermions will be located at specified places in the Hamiltonian. The resulting Standard Model in the Fodly-Wouthuysen representation preserves the SU(3) × SU(2) × U(1)-invariance and all its theoretical and experimental implications with no interactions required between Higgs bosons and fermions. In this case, Higgs bosons are responsible only for the gauge invariance of the theory’s boson sector and interact only with gauge bosons WZ± ,, gluons and photons. The suggested version of the Standard Model is, most likely, renormalizable, because the theory’s boson sector remains massless till the Higgs spontaneous symmetry violation mechanism is introduced, as quantum electrodynamics with a massless and massive and is a renormalizable theory.

13 Nevertheless, the question of whether or not the suggested version of the Standard Model is renormalizable needs to be studied more profoundly. Of course, the results of forthcoming experiments on searching for scalar bosons using the CERN’s Large Collider would provide direct verification of the conclusions of this paper concerning the construction of the Standard Model without interactions between Higgs bosons and fermions.

14 References

1. S.Weinberg, The Quantum Theory of Fields (translated to Russian), V.1, 2 (Fizmatlit, Muscou, 2003). 2. Original Russian Text V.P.Neznamov, published in Fisika Elementarnykh Chastits I Atomnogo Yadra 37(1),152(2006); [Physics of Particles and Nuclei 37(1), 86; Pleiades Publishing, Inc. (2006)]. 3. V.P.Neznamov, hep-th/0412047, (2005). 4. P.Dirac, The Principles of (translated to Russian). (Nauka, Muscou, 1979). 5. S.S.Schweber, An Introduction to Relativistic Quantum Field Theory (translated to Russian) (Foreign Literature Publishing House, Muscou, 1963). 6. L.L.Foldy and 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.: Teoreticheskaya I Prikladnaya Fizika 2, 21 (1988).

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