CC BY
32
UDC 539.12
Chiral 8-Spinor Model with Pseudo-Vector Interaction
Yu. P. Rybakov, N. Farraj, Yu. Umniyati
Department of Theoretical Physics Peoples' Friendship University of Russia 6, Miklukho-Maklaya str., Moscow, Russia, 117198
In the paper the structure of the chiral 8-spinor field model is discussed, the interaction with the electromagnetic, Yang-Mills and gravitational fields being included.
Key words and phrases: spinor, topological charge, solitons.
1. Introduction
The Skyrme's idea to describe baryons as topological solitons [1] proved to be fruitful in nuclear physics for modeling the internal structure of hadrons [2, 3] and light nuclei [4, 5]. In the Skyrme Model the topological charge Q = deg(53 ^ S3) is interpreted as the baryon number B and serves as the generator of the homotopy group k3(S3) = Z. The similar idea to describe leptons as topological solitons was announced by L.D. Faddeev [6]. In the Faddeev Model the Hopf invariant Qh is interpreted as the lepton number L and serves as the generator of the homotopy group k3(S2) = Z. The unification of these two approaches was suggested in [7], hadrons and leptons being considered as two possible phases of the effective 8-spinor field model.
The basic idea was to take into account the existence of the special 8-spinors identity discovered by the Italian geometer Brioschi [8]:
jWM - = s2 + P2 + v2 + a2, (1)
where the following quadratic spinor quantities are introduced:
s = p = #75^, v = ^ A^,
a = = fa = tf 7^75^,
with ^ = ^+7o and A standing for Pauli matrices in the flavor (isotopic) space. Here the diagonal (Weyl) representation for 75 = 7+ is used and ^ = 0, 1, 2, 3, designate the unitary Dirac matrices acting on Minkowsky spinor indices. If one defines 8-spinors as columns:
^ = col(^i,^2), = col(<£i,Xi), i = 1, 2,
with and Xi being 2-spinors, then one easily finds that the following identity holds
2jMj m = s2 + p2 + v2 + a2 + A2, (2)
showing the time-like character of the 4-vector where the denotation is introduced:
A2 = 8[(^Vi)(<^2) - |^2|2 + (X+X1)(X+X2) - IX+X2I2] > 0.
The structure of the identity (2) leads to the natural conclusion that Higgs potential
V in the effective spinor field model can be represented as the function of j^j^:
2
v = yOW" - k2)2, (3)
Received 18th May, 2012.
with a and k° being some constant parameters. If one searchs for localized soliton-like configurations in the model, one finds the natural boundary condition at space infinity:
lim = k2. (4)
As follows from the identity (2), the condition (4) determines the fixed (vacuum) point on the surface S8. Using (4) and the well-known property of homotopic groups of spheres: ^3(S") = 0 for n > 4, one concludes that the two phases with nontrivial topological charges may exist in the model (3). The first one corresponds to the choice ^3(S3) = Z (Skyrme Model) and the second one coresponds to the choice ^3(S2) = Z (Faddeev Model).
For example, if the vacuum state V° defines s(V°) = 0, then the configurations characterized by the chiral invariant s2 + a2 determining sphere S3 as the field manifold are possible, that corresponds to Skyrme Model phase. On the contrary, if only ^3(V°) = 0, then the SO(3) invariant v2 determines the S2 field manifold, that corresponds to Faddeev Model phase.
2. The effective nonlinear 8-spinor field model
In view of these topological arguments, using the analogy with Skyrme (or Faddeev) Model, we suggested in [7] the following Lagrangian density for the effective 8-spinor field model:
1 __e2
^spin = ^ W^tt + 4 W" - V (5)
where , stands for the antisymmetric tensor of Faddeev-Skyrme type:
Uu = 0*7^ (6)
with A and e being constant parameters of the model. It should be stressed that the first term in (5) generalizes the a-model term in Skyrme Model and includes the projector P = 7°7on the positive energy states. The second term in (5) gives the generalization of Skyrme (or Faddeev) term.
Now we intend to generalize the model by including the interaction of the spinor field with the electromagnetic, Yang-Mills and gravitational fields. To this end we consider the following structure for the extended covariant derivative:
= a** - ie°TeA^V + (Ajj + A* - r*)V, (7)
where A*, Aj, A* stand for the vector-potentials of the electromagnetic and left and right Yang-Mills fields respectively. r* in (7) denotes the spinor connection with the gravitational field, and re in (7) stands for the electric charge operator, e° being the corresponding coupling constant. The standard form of re, Aj and A* reads as follows:
re = 1(A3 - 1); ALjR = PL,Re^fA;L,RA- Pl,r = 1(1 ±75), (8)
with eiL, eir being the corresponding coupling constants.
However, the Yang-Mills interaction in (7) can be simplified if one accepts that for the leptonic sector it should vanish. Taking into account that in the leptonic sector the pseudo-vector part a = «V75AV = 0 one gets the following constraint:
eilA«L + eiRAf = 0. (9)
140 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 3, 2012. Pp. 138-141
In view of (9) one finds
\a .„t Xa
K + A? = T0eiL yzA;l = goTo YxAL = goToA;, (10)
where the new pseudo-vector field A; and the new coupling constant g0 were introduced. Now we take into account the invariance of the theory under the space reflection: xi ^ -xi, ^ ^ 70^, Ag ^-Ag, Ai ^ Ai, i = 1, 2, 3. (11)
The condition (11) stems the following structure of the Yang-Mills Lagrangian density:
¿ym = 32L Sp {FLLVF'-L + F*F;»R) =
32^g o
1
8n
= ^Sp^A, -d„A;)(dLA" -d»AL)+ g20[A;, Au ][AL, A^ ]}, (12)
where the strengths of the Yang-Mills fields were defined as
F^r = d,ALw'R - d„A^R + [A£>r, Alw'r\.
Thus the Lagrangian density of the model in question reads
£ = £spin + £YM + £em + £g, (13)
where dM in £spin is changed to DM and the gravitational part corresponds to the Einstein theory:
r 1 R 8nG
R being the scalar curvature and G — the Newton gravitational constant.
The electromagnetic part of the Lagrangian density was investigated in [7] and it corresponds to the Mie generalized electrodynamics:
£em = -[1 + GW] - ¿^V), (14)
where = d^Av - dvA^ and G(I), H(I) are some functions of the Mie invariant I = AMA^. As was shown in [7], the model (14) admits the existence of static solitons with fixed electric charge and positive energy.
The final generalization concerns the Higgs-like potential (3), where the scalar multiplier a2 is supposed to be taken in the form [9]:
M 2
a2 = «SKI ■ M = fifel^' (15>
with /i, I2 being Riemannian invariants:
h = ; h = -432rm^;trm^a't• (16)
The form (15) of the Higgs-like potential permits one to satisfy the quantum-mechanics correspondence principle, since the asymptotic behavior of the spinor field ^ at large distances from the soliton center is described by the Klein-Gordon equation with the Schwarzschild mass M of the soliton as the mass term.
6
3. Conclusion
Using Brioschi 8-spinor identity in the form (2) and introducing the Higgs-like potential (3), we considered, following some ideas of Mie [10], the effective nonlinear 8-spinor model that includes, via the mechanism of spontaneous symmetry breaking, Skyrme and Faddeev models as particular cases. Using the space-reflection invariance of the model we simplify the Yang-Mills part of the Lagrangian density by considering the single pseudo-vector field instead of the pair of vector fields: left and right ones. Finally we generalize the Higgs-like potential y, the latter being multiplied by the special Riemannian invariant to satisfy the correspondence with quantum mechanics.
References
1. Skyrme T. H. R. A Unified Field Theory of Mesons and Baryons // Nucl. Phys. — 1962. — Vol. 31, No 4. — Pp. 556-559.
2. Makhankov V. G., Rybakov Y. P., Sanyuk V. I. The Skyrme Model. Fundamentals, Methods, Applications. — Berlin: Springer-Verlag, 1993. — 260 p.
3. Zahed I., Brown G. E. The Skyrme Model // Phys. Reports. — 1986. — Vol. 142, No 1-2. — Pp. 1-102.
4. Braaten E., Townsend S., Carson L. Novel Structure of Static Multisoliton Solutions in the Skyrme Model // Phys. Lett. Ser. B. — 1990. — Vol. 235. — Pp. 147-152.
5. Light Nuclei of Even Mass Number in the Skyrme Model / E. Battye, N. S. Manton, P. M. Sutcliffe, S. W. Wood. —
6. Faddeev L. D. Gauge-Invariant Model of Electromagnetic and Weak Interactions of Leptons // Reports of Ac. of Sc. USSR. — 1973. — Vol. 210, No 4. — Pp. 807-810.
7. Rybakov Y. P. Soliton Configurations in Generalized Mie Electrodynamics // Phys. of Nuclei. — 2011. — Vol. 74, No 7. — Pp. 1102-1105.
8. Cartan E. Leçons sur la théorie des spineurs. — Paris: Actualités scientifiques et industrielles, 1938. — 223 c. [Cartan E. Leçons sur la theorie des spineurs. — Paris: Actualites scientifiques et industrielles, 1938. — 223 s. ]
9. Rybakov Y. P. Self-Gravitating Solitons and Nonlinear-Resonance Quantization Mechanism // Bulletin of Peoples' Friendship University of Russia. Ser. Physics. — 1995. — Vol. 3(1). — Pp. 130-137.
10. Mie G. Die Geometrie der Spinoren // Ann. der Physik. — 1933. — Vol. 17, No 5. — Pp. 465-500.
УДК 539.12
Киральная 8-спинорная модель с псевдовекторным
взаимодействием
Ю. П. Рыбаков, Н. Фарраж, Ю. Умнияти
Кафедра теоретической физики Российский университет дружбы народов ул. Миклухо-Маклая, 6, Москва, Россия, 117198
В работе обсуждается структура 8-спинорной полевой модели, включающей взаимодействие с электромагнитным полем, полем Янга—Миллса и гравитацией.
Ключевые слова: спинор, топологический заряд, солитоны.
