RFNC-VNIIEF, , Sarov, N.Novgorod region

Transcription

1 The Standard Model Without Higgs Bosons in the Fermion Sector V.P. Neznamov FNC-VNIIEF, 679, Sarov, N.Novgorod region Abstract The paper addresses the construction o the Standard Model with massive ermions without introduction o the Yukawa interaction between Higgs bosons and ermions. With such approach, Higgs bosons are responsible only or the gauge ± invariance o the theory s boson sector and interact only with gauge bosons W,Z, gluons and photons.

2 As we know, to provide SU() invariance o the theory, the Standard Model irst considers massless ermions that are given masses ater the mechanism o spontaneous symmetry violation is introduced, Higgs bosons appear and their gauge invariant interaction with Yukawa-type ermions is postulated []. On the eve o the decisive HC eperiments one can question onesel whether it is possible to construct the Standard Model with initially massive ermions, while preserving the theory s SU() symmetry. In this case, Higgs bosons are responsible only or the gauge invariance o the theory s boson sector and interact only with gauge bosons W, Z, gluons and photons. With the theory deined in this manner, ermion masses are introduced rom outside. The theory has no vertices o Yukawa interactions between ermions and Higgs bosons and, thereore, there are no processes o scalar boson decay to ermions ( H ), no quarkonium states, ϒ, θ including Higgs bosons, no interactions o Higgs bosons with gluons ( ggh ) and photons ( γγ H ) via ermion loops, etc. The answer to the question above has already been given in papers [], [], where the Standard Model is derived in the modiied Foldy-Wouthuysen representation. It has been shown that or its being SU()-invariant, the theory ormulated in the Foldy-Wouthuysen representation does not necessarily require Higgs bosons to interact with ermions, while all theoretical and eperimental implications o the Standard Model obtained in the Dirac representation are preserved. The goal o this paper is to construct, in a similar way, the Standard Model with initially massive ermions and spinors in the Dirac representation to meet the requirements o local SU() SU() U() symmetry. The paper uses the system o units, where = с= ; х, pb, are 4-vectors; p = i ;

3 the inner product is taken in the rom k k y y y y k = =, =,,,; =,,;, = k 5 α = ; γ = γ α ; β = γ, α, γ, γ k α, = k =,, are Dirac matrices Consider the density o Hamiltonian o a Dirac particle with mass interacts with an arbitrary abelian boson ield B α β α = α β α = H D = p m q B ( P P )( p m q B )( P P ) m, which = αp qα B αp qα B βm βm ( ) ( ) () γ 5 γ 5 In (), q is the coupling constant; P =, P = are the let and right projection operators; = P, = P are the let and right components o the Dirac ield operator. The reason why the abelian case is considered or the ield B is simplicity. As will be shown below, using a general case o a Dirac particle interacting with nonabelian boson ields would not change the conclusions and implications o this paper. be obtained: Using the density o Hamiltonian H D the motion equations or and can p = αp qα B βm () p = αp qα B βm ( ) ( ) One can see that both the density o Hamiltonian H D and motion equations have a orm, which is not SU () invariant because o the Dirac ermion having a mass. It ollows rom Eqs. () that = p αp qα B βm ( ) ( ) = p αp qα B βm ()

4 By substituting () to the right-hand side o Eqs. () proportional to β m, we obtain integro-dierential equations or and p αp q α B αb βm p αp q α B αb βm = p αp q α B αb βm p αp q α B αb βm = ( ( )) ( ( )) ( ( )) ( ( )) One can see that equations or and have the same orm, and, in contrast to Eqs. (), the presence o mass m does not lead to miing the right and let components o. Eqs. (4) can be written as p p q B p p q B m α α α α, = In epression (5),, shows that equations or and α = i α. (4) (5) have the same orm; I we multiply Eqs. (5) on the let side by term p αp qα B, we obtain secondorder equations with respect to p = ( p p q B )( p p q B ) m α α α α, (6) For the case o quantum electrodynamics ( q e, B A ) = =, Eqs. (6) have the orm p ea p ea m eσh iαe, = (7) A In Eqs. (7) H = rot A is magnetic ield, and E = A is electrical ield, t σ, i σ = σ - matrices Pauli. σ Eqs. (7) coincide with the second-order equation obtained by Dirac in the 9s [4]. However, in contrast to [4] (see also [5]), Eqs. (7) contain no ecess" solutions. The operator 5 = = =±. The case γ commutes with Eqs. (6). Consequently, γ ( 5 δ δ ; δ ) o δ = corresponds to the solution o Eq. (7) or, and δ = corresponds to the solution o Eq. (7) or. 4

5 Eqs. (5), (6) are SU () invariant, but they are nonlinear with respect to the operator p = i. inear orms o SU () invariant equations or ermion ields t relative to p can be obtained using the Foldy-Wouthuysen transormation [6] in a specially introduced isotopic space. = We now introduce an eight-component ield operator, Ф ( α τ β α ), and isotopic I I matrices, τ =, τ =, acting on the our upper and our lower I I components o operator Ф. So, Eqs. () can be written as p Ф = p m q B Ф (8) As τ commutes with the right-hand side o Eq. (8), ield Ф τ Ф solution to Eq. (8). Further, consider Eq. (8) without boson ield B (ree motion) p Ф = αp τ βm Ф ( ),, Ф, shows that Eqs. (9) are the same or ields Ф, Ф. = = (9) is also Now we ind the Foldy-Wouthuysen transormation in the isotopic space or ree motion Eq. (9) using the Eriksen transormation [7]. U Er ( ) τλ λτ = U = τλ 4 p α τβ m λ = ; E = p m E In epression (), we have λ =. Epression () can be transormed to obtain the ollowing epression: U ταp ττβm ταp = U = = E E E ταp ττβ m E E τα p Er = Epression () is a unitary transormation ( ) U U =, and (). Since αp τ βm = E, () 5

6 H = U α p τβm U = τ E Thus, Eqs. (9) in the Foldy-Wouthuysen representation have the orm τ,, () p Ф = E Ф () When converting to the Foldy-Wouthuysen representation, in addition to the condition o the Hamiltonian being block-diagonal (), one should necessarily meet the requirement that the upper or lower components o the ield operators Ф, Ф [8] should be zero. One can term this condition as reduction o ields Ф, Ф. et us check whether this condition is met in our case, or not. Given Eqs. (), (), normalized solutions to Eq. (9) or the ield operators Ф, Ф can be epressed as ollows: ( ) ( ) iet Ф ( t, ) = e ( βm ) E α p ( ) ( ) iet Ф ( t, ) = e ( βm ) E α p ( ) iet βm Ф E p t e α = ( ) ( ; ) (, ) ( ) iet βm Ф E p t e α = ( ) ( ; ) (, ) (4) In (4), Ф, Ф ; Ф, Ф are solutions with positive and negative energy, respectively. 6

7 σ p ϕ ( ) ( ) E m E m = γ5 D = E σ p ϕ E m ( ) ( ) σ p χ ( ) ( ) E m E m = γ5 D = E σ p χ E m σ p ϕ ( ) ( ) E m E m = γ5 D = E σ p ϕ E m ( ) ( ) ( ) ( ) (5) σ p χ ( ) ( ) E m E m = γ5 D = E σ p χ E m ( ) ( ) ( In epressions (5), ϕ ), χ are normalized two-component solutions o the Dirac equation with positive and negative energy. In (4), (5), E and p are respective operators. According to (5), ( ) ( p = σ p ) ; ( α ) α p = σ p ( ) ( p = σ p ) ; ( α ) α ( ) p = σ p ( ) 7

8 Epr. (5) lead to the ollowing normalizing conditions: ( ) ( ) ( ) E σ p ( ) = ϕ ϕ E ( ) ( ) ( ) E σ p ( ) = χ χ E ( ) ( ) ( ) E σ p ( ) = ϕ ϕ (6) E ( ) ( ) ( ) E σ p ( ) = χ χ E By applying the transormation matri U () to Ф, Ф (see (4)) we obtain iet Ф t, U Ф t, e E p E ( = ) = σ ( ) ( ) ( ) iet Ф t, = UФ ( t, ) = e E E σ p (7) iet Ф t, U Ф t, e E p E ( = ) = σ ( ) ( ) ( ) iet Ф t, = UФ ( t, ) = e E E σ p One can see rom relations (7) that the reduction condition is ulilled and the matri U is, indeed, the Foldy-Wouthuysen representation or the ields Ф, Ф in the isotopic space we have introduced. 8

9 Eqs. () allow us to write the density o the ree-motion Hamiltonian o ermions with mass m as ( ) ( ) ( ) ( ) H = ( Ф ) τ E( Ф ) ( Ф ) τ E( Ф ) = ( Ф ) E( Ф ) ( Ф ) E( Ф ) ( ) ( ) ( ) ( ) ( ) E ( ) ( ) E ( ) ( Ф ) E( Ф ) ( Ф ) E( Ф ) = ( ) E ( ) E E σ p E σ p ( ) E ( ) ( ) E ( ) ( ) E ( ) E (8) E σ p E σ p One can see that Hamiltonian (8) is SU()-invariant, regardless o whether the ermions are massive or massless. Epression (8) shows that two ermion ield operators, ( Ф ),( Ф ), need to be used to provide a complete description o the ree motion o the right and let ermions. Given (5), the density o Hamiltonian (8) bracketed between two-component spinors ϕ ( ) and χ ( ) has a orm that is commonly used in the ield theory, ( ) ( ) ( ) ( ) H ϕ ϕ χ χ = E E. In the presence o boson ields B interacting with ermion ields Ф, Ф( ), the Foldy-Wouthuysen transormation and Hamiltonian o Eq. (8) in the Foldy-Wouthuysen representation in the isotopic space can be obtained as a series in powers o the coupling constant using the algorithm described in es. [], [9]. As a result, using denotations rom es. [], [9], we obtain (...) U = U δ δ δ (9) ( τ ) p ( Ф ) = H ( Ф ) = E qk q K q K... ( Ф ) (),,, The epressions or operators C and N constituting the basis or the interaction Hamiltonian in the Foldy-Wouthuysen representation obtained using the technique o es. [], [9] can be written in the ollowing orm in our case: even C U qα B ( U ) = = q( B B ) q( αb αb) odd N U qα B ( U ) = = q( B B ) q( αb αb) E ταp = ; = ττβ m E E τα p () 9

10 The superscripts even, odd in () show the even and odd parts o the operators relative to the upper and lower isotopic components o Ф and Ф. For Eqs. (), the Hamiltonian density or ermion ields ( Ф ) ( Ф ) interacting with boson ield B can be written as H = ( Ф ) τ E qk q K q K... ( Ф ) ( τ ),, ( Ф ) E qk q K q K... ( Ф ) () The epression or the Foldy-Wouthuysen Hamiltonian in parentheses in equation () is, by deinition, diagonal with respect to the upper and lower components ( Ф ), ( Ф ) [6], [8], [9]. When solving applied problems in the quantum ield theory using the perturbation theory, ermion ields are epanded in solutions o Dirac equations or ree motion or or motion in static eternal ields. In our case, in the Foldy- Wouthuysen representation, we can also epand ermion ields over the basis (7) or over a similar basis o solutions o the Foldy-Wouthuysen equations in static eternal ields. Then, Hamiltonian density () can be epressed through the unctions (7), and it is obvious that this epression, similarly to (8), will be SU () -invariant due to the diagonality. Thus, epression () and Eqs. () are invariant relative to SU () - transormations regardless o ermions having or not having masses. Formula () demonstrates the necessity o using two ermion ields, ( Ф ) ( х ), ( Ф ) ( х ), in the ormalism or constructing the Standard Model. I only( Ф ) ( х) is used in the theory, motion and interactions o the right ermions, as well as motion and interactions o the let anti-ermions remain. I, on the contrary, only ( Ф ) ( х ) is used in the theory, motion and interaction o the let ermions, as well as motion and interactions o the right antiermions remain. More careul analysis shows that even with two ields, Ф ( х ), Ф ( х ), Hamiltonian () contains no interactions between real particles and anti-particles. This happens due to the spinor structure o epressions (7) in the introduced isotopic

11 space. The theory s special eature in the Foldy-Wouthuysen representation is that the Hamiltonian terms include interaction K n (ecept K ) o an even number o odd operators N that couple states with positive and negative energy. Thereore, interactions between particles and antiparticles can occur only between real and intermediate virtual states [], [9]. In order to introduce interactions between real particles and antiparticles into the theory in es. [], [9], the Foldy-Wouthuysen representation had to be modiied. To solve the same problems in our case, let us use the ollowing approach. Write Eq. (8) or ield Ф ( ) and the same equation or ield Ф ( ) in the equivalent orm pф p m Ф q B Ф q B Ф = ( α τ β ) α α τ pф p m Ф q B Ф q B Ф = ( α τ β ) α α τ () Eq. () uses the equality Ф τ Ф =. Further, perorming the Foldy-Wouthuysen transormation (9) or Eqs. (), we obtain equations or ields, Ф Ф. q q q pф = τ EФ K K K... Ф q q q K τ K τ K τ... Ф (4) q q q pф = τ EФ K K K... Ф q q q K τ K τ K τ... Ф

12 Denotations Kn τ in ormulas (4) mean that the matri τ is placed in the operators C, N () net to the ields B : Cτ = q τ B τ B q ταb ταb ( ) ( ) Nτ = q τ B τb q ταb ταb ( ) ( ) (5) Equations (4) correspond to the Hamiltonian density H = q q q Ф τ E K K K... Ф q q q Ф K τ K τ K τ... Ф (6) q q q Ф τ E K K K... Ф q q q Ф K τ K τ K τ... Ф By analogy with [], [9], Feynman rules or calculating speciic physical processes in the quantum theory o interacting ields using perturbation theory methods can be derived using Hamiltonian density (6) and Eqs. (4). The isotopic space we have introduced allows constructing the SU () invariant Standard Model with massive ermions. In case o interaction with gauge ields B, the agrangian with covariant derivative D = iqb and ermion ields Ф =, Ф = can be written as Фγ D Ф Фτm Ф Фγ D Ф Фτm Ф = (7) Using this agrangian, the motion equations or ermion ields Ф, Ф with mass (see ()) can be obtained. Using the isotopic Foldy-Wouthuysen transormation (), (9) one can obtain the SU () invariant Hamiltonian density (6) and motion equations or ermion ields (4) with appropriate deinitions o operators Cτ, N τ, K τ, K τ, K τ... (see (5)). m

13 To construct the Standard Model using Hamiltonian density (6), one must replace, as it is done in [], [], the interaction verte q α B in the operators K K..., K τ, K... with the interaction vertices o the Standard Model []., τ α q B g I γ5 I γ5 eqα A T Q sin θw α Q sin θwα Z cosθ W g I γ5 I γ5 ( = u) α ( = d ) ( = ve ) α ( = e) W Hermit. conj. g a ( =, ) α λ ( =, ) ud ud G α a αβ β (8) In (8), A is the electromagnetic ield; Z, W are the gauge boson ields; a G are the gluon ields; Q is the ermion electric charge in the units o e; T = / or v, ; e u = e T = / or = e, d; θw is the electroweak miing angle; g = ; gis sinθ W a the quantum chromodynamics coupling constant; λ are generators o group SU (). Epressions (8) is written out only or the irst lepton and quark amily. For the second and the third amilies it is required to make appropriate substitutions ( v, e e, u, d) ( v,,, c s) and ( vτ, τ, t, b) and introduce the quark miing. In (8) notations (=u), (=d), etc., imply that spinor -ields o associated ermions will be located at speciied places in the Hamiltonian. The resulting Standard Model in the Fodly-Wouthuysen representation preserves the SU() SU() U()-invariance and all its theoretical and eperimental implications with no interactions required between Higgs bosons and ermions. In this case, Higgs bosons are responsible only or the gauge invariance o ± the theory s boson sector and interact only with gauge bosons W, Z, gluons and photons. The suggested version o 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 photon and massive electron and positron is a renormalizable theory.

14 Nevertheless, the question o whether or not the suggested version o the Standard Model is renormalizable needs to be studied more prooundly. O course, the results o orthcoming eperiments on searching or scalar bosons using the CEN s arge Hadron Collider would provide direct veriication o the conclusions o this paper concerning the construction o the Standard Model without interactions between Higgs bosons and ermions. 4

15 eerences. S.Weinberg, The Quantum Theory o Fields (translated to ussian), V., (Fizmatlit, Muscou, ).. Original ussian Tet V.P.Neznamov, published in Fisika Elementarnykh Chastits I Atomnogo Yadra 7(),5(6); [Physics o Particles and Nuclei 7(), 86; Pleiades Publishing, Inc. (6)].. V.P.Neznamov, hep-th/447, (5). 4. P.Dirac, The Principles o Quantum Mechanics (translated to ussian). (Nauka, Muscou, 979). 5. S.S.Schweber, An Introduction to elativistic Quantum Field Theory (translated to ussian) (Foreign iterature Publishing House, Muscou, 96). 6...Foldy and S.A.Wouthuysen, Phys.ev 78, 9 (95). 7. E.Eriksen, Phys. ev., (958). 8. V.P.Neznamov, The Necessary and Suicient Conditions or Transormation rom Dirac epresentation to Foldy-Wouthuysen epresentation. hep-th/84., (8). 9. V.P.Neznamov, Voprosy Atomnoi Nauki I Tekhniki. Ser.: Teoreticheskaya I Prikladnaya Fizika, (988). 5