CC BY
33
UDC 539.12
Topological Soliton Configurations in 8-Spinor Nonlinear
Model
Yu. P. Rybakov, N. Farraj, Yu. Umniyati
Department of Theoretical Physics Peoples' Friendship University of Russia 6, Miklukho-Maklaya str., Moscow, 117198, Russia
We study the structure of the charged topological solitons in the lepton sector of the nonlinear 8-spinor model, at small distances the closed-string approximation being used. The mass, the spin and the magnetic moment of the soliton configuration with the unit leptonic number are estimated. The model is based on the well-known 8-spinor identity suggested by the Italian geometer Brioschi. Due to the identity the Dirac current appears to be time-like 4-vector that permits one to introduce the special form of the Higgs potential depending on the current squared. Within the framework of this model the natural classification of leptons and baryons can be realized via the Higgs mechanism. Concentrating on the lepton sector we study the simplest soliton configuration endowed with the unit Hopf index playing the role of the lepton number. Investigating the behavior of solutions at large and small distances we obtain the numerical estimate of physical characteristics of the topological soliton. The special symmetry group is used in our calculation, the combined rotations in ordinary and isotopic spaces being considered. The corresponding equivariant spinor fields involve phase functions linear with respect to azimuthal and toroidal angles. This property permits one to find explicit value of the topological invariant for the axially-symmetric configuration and to investigate the dependence of the physical characteristics on topology.
Key words and phrases: 8-spinor, topological charge, solitons.
1. Introduction
In our previous paper [1] the nonlinear 8-spinor model was suggested for the unified description of leptons and baryons as topological solitons in Faddeev [2] and Skyrme [3] models respectively. This unification is based on the special 8-spinor identity discovered by the Italian geometer Brioschi [4]:
jWM - = s2 + P2 + v2 + a2, (1)
where the following quadratic spinor quantities are introduced:
s = p = 75^, v = ' A',
a = 75 A', = '7m'> ^ = '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 7^, ^ = 0,1, 2, 3, designate the unitary Dirac matrices acting on Minkowski spinor indices.
Taking into account the time-like character of the 4-vector the topological distinction between leptons and baryons can be realized via the Higgs mechanism, the
special form of the Higgs potential being used:
2
^ = yOWM - k2)2, (2)
with a and k0 being some constant parameters. If one searches for localiRyb8zed soliton-like configurations in the model, one finds the natural boundary condition at space infinity:
lim = K2. (3)
Received 6th June, 2013.
As follows from the identity (1), the condition (3) determines the fixed (vacuum) point on the surface S8. Using (3) and the well-known property of homotopic groups of spheres: k3(Sn) = 0 for n > 4, one concludes that the two phases with nontrivial topological charges may exist in the model in question. The first one corresponds to the choice n3(S3) = Z (Skyrme Model) and the second one to the choice k3(S2) = Z (Faddeev Model).
For example, if the vacuum state defines s(^0) = 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(^0) = 0, then the SO(3) invariant v2 determines the S2 field manifold, that corresponds to Faddeev Model phase.
In view of these topological arguments, using the analogy with Skyrme (or Faddeev) Model, we suggested in [1] the following Lagrangian density for the effective 8-spinor field model:
1 _ e2
L = 2Â2 D^ + J f "" - V + £em, (4)
where stands for the antisymmetric tensor of Faddeev-Skyrme type:
U„ = (5)
with A and e being constant parameters of the model. It should be stressed that the first term in (4) generalizes the a-model term in Skyrme Model and includes the projector P = on the positive energy states. The second term in (4) gives the
generalization of Skyrme (or Faddeev) term. Ryb8 Here the interaction of the spinor field with the electromagnetic one is introduced via the extension of the derivative:
= d^ — ieoTeA^, (6)
where A^ stands for the vector-potential of the electromagnetic field and re = | (A3 —1) in (6) stands for the electric charge operator, e0 being the corresponding coupling constant.
The electromagnetic part of the Lagrangian density was investigated in [5] and corresponds to the Mie generalized electrodynamics:
¿em = — F,JlW [1 + G(I)] — ^H (I), (7)
where FI1U = — dvA^ and G(I), H(I) are some functions of the Mie invariant
I = A^A^. As was shown in [5], the model (7), for the power series representation of these functions:
G(I ) = H (I ) = ^ [jnIn
n=2
admits the existence of static soliton-like configurations with fixed electric charge and positive energy.
2. Structure of Lagrangian in Leptonic Sector
In the lepton sector the 8-spinor ^ is invariant under the space reflection [1] ^ ^ 70^ and therefore it reduces to 4-spinor = col(^i,^2), with <p1,<p2 being 2-spinors. As was shown by Faddeev [2], the configurations endowed with the nontrivial Hopf index Qh are similar to closed twisted strings. The appropriate description of these configurations, which are also typical for the Kerr solution in General Relativity [6], can be obtained by using the toroidal coordinates x G [0, to), y G [—$ G [0,2^]
with the metric
d s2 = di2 - a2e2a(dx2 + dy2 + sinh2 xd02), (8)
where e-a = cosh x - cos y and a is the length parameter (the closure radius of the string).
The model admits the following infinitesimal spinor transformations defining the correspondent groups:
G1 : 5' = i5a'; G2 : 0 ^ 0 + 5$, S' = i50J3'; G3 :y ^ y + 5y, 5' = i5ypy
where the following operators are introduced:
J3 = -id^ + I«*, Py = —dy,
the latter group being effective in the asymptotic region (x ^ 1). Let us construct the following group of combined transformations:
G = diag(Gi ®G2) <8> diag(Gi »G3), (9)
with the corresponding equivariant field having the form:
=col(u1, v1 exp [i(0 - n1 y)]), = col(v2 exp [-%(0 - n2y)] ,u2), (10)
where Ui, Vi do not depend on the azimuthal angle 0, and n1, n2 are some integers.
Now we investigate the structure of topological solitons in the model in question using the perturbation method with respect to electromagnetic coupling constant e0. In the first approximation we neglect the electromagnetic field and find the following action functional using the substitution (10):
CO TV
' 2 e7,
A = -2na3 J dx J dy |^|2 [(Ôi^)2 + (^)2 + n2|<ui|2 + n2|^+
0 -n ^
+ 2n2lm(u*Ö2u2) + 2mIm(v*02vi) + e2(a-7) (|vi|2 + |^|2)] +
2
+ ^ e7-2° a4
2
(2Im[(^+ôi^)(Ô2^+^)j -Ôi|^|2(m|vi|2 -n2|^|2)2)' +
+ e2^) [(d^2)2 + (fc^2)2] (|^ - |^212)2 +
+ 2a2e2«+7 (|^|2 - K2)2}, (11)
where e7 = sinhx ea and ^ = col(u1, v1, v2, u2).
Now it is worth-while to study the structure of the Hopf invariant Qh playing the role of the lepton charge in our model.
3. Structure of Hopf Invariant
Hopf invariant Qh as the generator of the homotopic group k3(S2) classifies the mappings n : R3 ^ S2, where the sphere S2 is given by the unit vector n = v/|v|.
Qh can be represented by the Whitehead integral [7], [8]:
QH = (¿2/ d3^ (c rot c), (12)
where the vector c is defined by the relation
dzck — dk Ci = eabcdtnadk nbnc. (13)
Hopf suggested an elegant method of calculating the integral (12) via the inverse Hopf mapping S3 ^ S2. To this end let us introduce the auxiliary 2-spinor
% = col (cos A + 1 sin A cos B, sin A sin B exp(iC)), (14)
with A, B, C being angular coordinates on S3. Then the following relations hold:
rot c = —2«[Vx+Vx], c = Im(x+Vx), n = (x+Ax). (15)
Inserting (15) into (12), one finds that
Qh = f d3^ sin2 A sin B([VAVB]VC) = deg(S3 ^ S3), (16)
(2^)2 J
the latter formula expressing the identity of the homotopic groups k3(S2) = k3(S3) = Z. If one introduces the new variables ft and p by putting
sin A sin B = sin(ft/2), tan A cos B = tan p,
one can get the more compact expression for the Hopf invariant:
Qh = — 8^2 f d3x sinP([VPVp]VC). (17)
Taking into account that n\ + xn2 = sin ft exp(^), one finds the relation
7 = C — p (18)
and the final expression for the Hopf invariant:
Qh = j d3x sin ]VC). (19)
In our case the angular coordinates /3, j can be found from (10) and the definition of the vector v = ^A^ stemming the relation
7 = p(x,y) — $. (20)
In view of (18) and (20) one can choose in (19) for the axially-symmetric configurations C = —0, that is
CO -K
Qh = — J d3® sin ]V0) = — ^ J dx J dy sin ¡3(pxpy — px). (21)
0 —-k
Taking into account (10) and (20), one finds the following value of the leptonic charge L for our configuration (10):
Qh = L = rn + «2, with the boundary condition for fi being imposed:
cos^ (œ = 0) = 1, cos^(œ = to) = -1.
(22)
4. Configuration with Unit Leptonic Charge
Let us consider the simplest state with the unit leptonic charge L = n1 = 1 when = 0, i>2 = 0. In this case the following approximation is appropriate: A1 = A2 = 0, Ao = 0, A3 = 0, that permits using the substitution:
u1 = VM sinBsin$1, u1 = VR sinBcos$1, u2 = VR cosB. (23)
Inserting (23) into the Lagrangian (4), one gets the following action functional:
A = 2^a / dœ
7 (
J dy e7 <
— tr ^
2 Â2
,fi2e2aa2e2/cos2 0 - 4(d#)2-
+
+ 8 e 2^2
- 0)2 - sin2 0 ((ô$i)2 + cos2 coth2 œ) (ô#)2 cos4 0 - ^J7(sin2 0 cos2 $i - e0^s cos2 0)2)
„-2a
— sin4 0 cos4 $i(ôi^)2
2
- 2a2a2e
2„2„2a / d2
- f) -
p 2aa2 1
(/) + 8^ M))
= "27
(dAo)2 - (ôAs)2
where the denotations are used:
e7 = sinhœ ea, / = A0 -
= -27
0 -"^3, )2 = (ôiK)2 + (Ô2K)2
Let us first study the behavior of functions 0, A0, A3 as solutions to the equations of motion at œ ^ to:
R = Ro + R12(1 - tanhx), B = Bo + B12(1 - tanhx),
^ _
= 2 - 2arctane x, Ao = Aoo - A012(1 - tanhx), A3 = A30 - A312(1 - tanhx), with the following constraints being imposed:
R1 = 8R0 sin2 Bo, B1 = sin 2Bo, A01 = S0A00, A31 = S0A30, where we use the denotations:
^0 =
1 + G( /0)
H'(/0) - ^0 cos2 00
/0 = A00 - 4a
a
2
2 A30
(24)
2
2
a
and introduce the small parameter
c = aaXxo « 1, (26)
neglecting the corresponding terms in the first approximation.
The behavior of fields at large distances (x ^ 0, y ^ 0) can be derived from linear equations of motion valid in the vicinity of the vacuum state for which
ui = y/ Ko/2:
R = (1 -ze-a), 0 = 2 -uo tanhxe-a/2, $i = 2 — k tanhxe-a/2,
2 n (27)
A0 = e-a/2, A3 = A tanh2 xea/2, ay 2
with q being electric charge of the particle-soliton and the parameter A proportional to its magnetic moment.
Now we intend to estimate the mass of the soliton through smooth matching the functions (24) and (27) at some point with the coordinates x = x0 and y = n/2. From this condition one derives the numerical values x0 = log(\/2 + \/3), S0 = 2 and all other parameters: 00 ~ 0.347826; k « 0.57277; z « 0.05694; R0 ~ 0.25991^0; Ri ~ 0.241575k0. To estimate the energy £, we divide the interval [0, to) into two parts: [0, x0] and [x0, to) and calculate the corresponding integrals in (11) using the trial functions (27) and (24) respectively. The resulting energy £ = —A, electromagnetic contribution being omitted, reads:
£ = 2n2a3K0 ^+ Aok2 + joa2K^j ,
(28)
where «0 = 0.209223; ^0 = 8.06583 ■ 10-3; 70 = 1.205782.
Minimization of the energy (28) with respect to the radius a gives
2 1
a2 =
6a2K2J0
/ 2 \ 1/2
-12 + ( S + 12a2Kle2AoJo)
Thus, the mass of the particle-soliton is given by
m = ^ (2«0 + M02a270) , M0 = 2a\x0.
5. Spin and Magnetic Moment of Soliton
Now we calculate the spin of the particle-soliton using the well-known expression for the z-projection of the angular momentum:
S = J d3x 2Re( oj^ji^ = —2 ^ J d3x M2M2. (29)
Introducing the density of the electric charge
dC
dAo1
(30)
one finds from (29) and (30) that S = -— f d3x pe = -—. Therefore for the standard
V 7 V 7 2 e 0J 2 e0
choice = 0 one gets S = 1/2.
P
e
To calculate the magnetic moment m of the particle-soliton, we use the classical electrodynamics formula for the vector-potential of the point-like magnetic moment:
A = 1 [mr].
Comparing this expression with the azimuthal component of the vector-potential
„ e-7 , |m| . a aA^2 . a
As =-A3 = sin § = —— sin §,
a r2 r2
one gets the relation
|m| = a^2A. (31)
On the other hand, in the first approximation with respect to e2 one finds
/0 « (ft a2 - ax)-1 = A2o - ^
a2
where for q = e0 one obtains that A00 = 33/4e0/(2a), hence
1/2
A30 =
33/2 f - (ft -ai/a2)-1
However, the smooth matching of A3 gives the relation
A = i/cosh^0 coth:r0A30 = 31/4^/ 3/2A30.
Inserting this value of the constant A into (31), one can calculate the magnetic moment of the particle-soliton:
|m| = 33/4^ 33/2e0 - 4(^2 -ai/a2)- 1/2
m, _
2
6. Conclusion
Using some ideas of Mie [9], the effective 8-spinor model unifying the models by Skyrme and Faddeev was suggested [5] permitting, via the mechanism of spontaneous symmetry breaking, to describe the particles as topological solitons. We consider the leptonic sector of the model in question and study the structure of axially-symmetric configuration with the unit lepton charge. Using the behavior of fields at large and small distances, one can estimate the mass, the spin and the magnetic moment of the particle-soliton. For the natural choice of the electric charge of the particle = 0 the spin proves to be 1/2.
References
1. Rybakov Y. P., Farraj N., Umniyati Y. Chiral 8-Spinor Model with Pseudo-Vector Interaction // Bull. of Peoples' Friendship University of Russia, Series "Mathematics. Information Sciences. Physics". — 2012. — No 3. — Pp. 138-141.
2. 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.
3. Skyrme T. H. R. A Unified Field Theory of Mesons and Baryons // Nucl. Phys. — 1962. — Vol. 31, No 4. — Pp. 556-559.
4. Cartan E. Lecons sur la theorie des spineurs. — Paris: Actualites scientifiques et industrielles, 1938. — 223 p.
5. Rybakov Y. P. Soliton Configurations in Generalized Mie Electrodynamics // Phys. of Nuclei. — 2011. — Vol. 74, No 7. — Pp. 1102-1105.
6. Burinskii A. Some Properties of the Kerr Solution to Low Energy String Theory // Phys. Rev. D. — 1995. — Vol. 52. — Pp. 5826-5831.
7. Whitney H. Geometric Integration Theory. — Princeton, New Jersey: Princeton University Press, 1957. — P. 534.
8. Whitehead J. H. C. An Expression of Hopf's Invariant as an Integral // Proc. Roy. Irish. Acad. Sci. — 1947. — Vol. 33. — Pp. 117-123.
9. Mie G. Die Geometrie der Spinoren // Ann. der Physik. — 1933. — Vol. 17, No 5. — Pp. 465-500.
УДК 539.12
Топологические солитонные конфигурации в 8-спинорной
нелинейной модели
Ю. П. Рыбаков, Н. Фарраж, Ю. Умнияти
Кафедра теоретической физики Российский университет дружбы народов Россия, 117198, Москва, ул. Миклухо-Маклая, 6
Изучается структура заряженных топологических солитонов в лептонном секторе нелинейной 8-спинорной модели, когда на малых расстояниях используется приближение замкнутых струн. Оцениваются масса, спин и магнитный момент солитонной конфигурации с единичным лептонным числом. Модель основана на хорошо известном 8-спинорном тождестве, предложенном итальянским геометром Бриоски. В силу этого тождества дираковский ток оказывается временно-подобным 4-вектором, что позволяет ввести специальную форму потенциала Хиггса, зависящего от квадрата тока. В рамках этой модели может быть реализована естественная классификация лептонов и барионов благодаря механизму Хиггса. Ограничившись лептонным сектором, мы изучаем простейшую солитонную конфигурацию, наделённую единичным индексом Хопфа, который играет роль лептонного числа. Исследуя поведение решений на больших и малых расстояниях, мы получаем численную оценку физических характеристик топологического солитона. В наших расчётах используется специальная группа симметрий, включающая комбинированные вращения в обычном и изотопическом пространствах. Соответствующие эквивариантные спинорные поля включают фазовые функции, линейно зависящие от азимутального и тороидального углов. Это свойство позволяет найти явное значение топологического инварианта для аксиально-симметрической конфигурации и исследовать зависимость физических характеристик от топологии.
Ключевые слова: 8-спинор, топологический заряд, солитоны.
