ЯДЕРНАЯ ФИЗИКА, 2013, том 76, № 9, с. 1203-1207
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
INTERPRETATION OF NEUTRINO-MATTER INTERACTIONS AT LOW ENERGIES AS CONTRACTION OF GAUGE GROUP
OF ELECTROWEAK MODEL
©2013 N. A. Gromov*
Department of Mahematics, Komi Science Center UrD RAS, Syktyvkar
Received May 29, 2012
The very weak neutrino—matter interactions are explained with the help of the gauge group contraction of the standard Electroweak Model. The mathematical contraction procedure is connected with the energy dependence of the interaction cross section for neutrinos and corresponds to the limiting case of the Electroweak Model at low energies. Contraction parameter is connected with the universal Fermi constant of weak interactions and neutrino energy as j2(s) = y/Gps.
DOI: 10.7868/S0044002713080151
1. INTRODUCTION
The standard Electroweak Model is a gauge theory based on the group SU(2) x U(1) and gives a good description of electroweak processes. Mathematically this theory is very complicated with nonlinear dynamics of the involved fields. Therefore special significance takes investigations of different limit cases of the model which correspond to limit values of physical parameters. Similar investigations promote better understanding of the model. In physics it is the well known operation of group contraction [1], which transforms a starting group to a new one noniso-morphic to the initial group. For symmetric system contraction of its symmetry group corresponds to limit case of the system under investigation.
In this paper we discuss the modified Electroweak Model with the contracted gauge group SU(2; j) x x U(1). We explain with the help of contraction of gauge group the vanishingly small interactions of neutrinos with matter especially for low energies and the decrease of the neutrino—matter cross section when energy tends to zero already at the level of classical fields. We connect dimensionless contraction parameter j ^ 0 with neutrino energy and demonstrate that contraction of the gauge group corresponds to the low energy limit of the Elec-troweak Model. The limit case of the boson sector of the standard Electroweak Model was discussed in [2]. (See [3] for contraction of the Electroweak Model with 3-dimensional spherical geometry in the matter field space.) The modified Electroweak Model, where boson and quark sectors are the same as in the
standard model, but the lepton sector is described by the contracted gauge group, was considered in [4, 5].
2. STANDARD ELECTROWEAK MODEL
We shall follow the books [6—8] in description of standard Electroweak Model. Its Lagrangian is the sum of boson, lepton, and quark Lagrangians
L = Lb + Ll + Lq. (1)
Boson sector LB = LA + L^ involves two parts: the gauge field Lagrangian
1 - Л2 1
= - = 11
(2)
= -^[(it)2 + (F%)2 + ~ 4
and the matter field Lagrangian
where 0
J4>
= (Ф1> \ф2>
(3)
e C2 are the matter fields.
The fermion sector is represented by the lepton Ll and quark Lq Lagrangians. The lepton Lagrangian is taken in the form
Ll = b\ifßDßLi + e\irßDßer -
(4)
- he [e¡(ф%) + $ф)ег],
E-mail: gromov@dm.komisc.ru
where Li = (v) is the SU(2) doublet, er is the SU(2) singlet, he is constant, t0 = T0 = 1, fk = -rk, r^ are Pauli matrices and er, el, vl are two-component Lorentzian spinors.
1203
2
6
*
The quark Lagrangian is given by
Lq = Q\ifpD^Qi + ul ir^D^Ur + (5)
+ dl ir^ Dn d hd[dlr&lQi) + (Qk)dr] -
- hu[ulr((¡>lQi) + (Ql4>)ur],
where left quark fields form the SU(2) doublet Ql = = idl), right quark fields ur,dr are the SU(2) singlets, (i = €ik(j)k, eoo = 1,€ii = -1 is the conjugate representation of SU(2) group and hu, hd are constants. All fields ul,dl,ur,dr are two-component Lorentzian spinors.
las:
D„
The covariant derivatives are given by the formu-
u = dßu - (1W+T+ + W-TJ) u -
— %
cos 0
W
-Z, (T3 - Q sin2 0w) Li - %eAßQLh
wî = -m (4 =F O
V2
=
A, =
1
i
2
{gAl - g'Bß),
(<g'Al + gBi
are expressed through the fields
Ai(x) = -%g^Tk Al(x),
k=l
Bi(x) = -%g'Bß(x),
3. ELECTROWEAK MODEL WITH CONTRACTED GAUGE GROUP
We consider a model where the contracted gauge group SU(2; j) x U(1) acts in the boson, lepton, and quark sectors. The contracted group SU(2; j) is obtained [2, 3, 9] by the consistent rescaling of the fundamental representation of SU(2) and the space C2
z'(j) =
jz'i
(6)
a jß -jß a = u(j)z(j),
det u(j) = \a\2 + j2\ß\2 = 1, in such a way that the Hermitian form z]z(j) = j2\zi\2 + |z2\2
= (9)
u(j)uj (j ) = 1
(10)
D^er = d^er - ig'QAe cos 6W + ig'QZ^e
where Tk = k = 1,2,3, are the generators of SU(2), T± = Ti ± iT2, Q = Y + T3 is the electrical charge, Y is the hypercharge, e = gg'(g2 + g'2)-i/2 is the electron charge and sin dW = eg-1. The gauge fields
(7)
remains invariant, when contraction parameter tends to zero j — 0 or is equal to the nilpotent unit j = 1, 12 = 0. The actions of the unitary group U(1) and the electromagnetic subgroup U(1)em in the fibered space C2(i) with the base {z2} and the fiber {zi} are given by the same matrices as on the space C2.
The space C2(j) of the fundamental representation of SU(2; j) group can be obtained from C2 by substituting zi by jzi . Substitution zi - jzi induces other ones for Lie algebra generators Ti — jTi, T2 — jT2, T3 — T3. As far as the gauge fields take their values in Lie algebra, we can substitute the gauge fields instead of transforming the generators, namely:
jAl,
A, -
A3 A3 Al — A,,
A2 Al
Bi
jAl
Bi
(11)
(8)
which take their values in the Lie algebras su(2) and u(1), respectively.
From the viewpoint of electroweak interactions all known leptons and quarks are divided on three generations. Next two lepton, and quark generations are introduced in a similar way to (4) and (5). Full lepton and quark Lagrangians are obtained by the summation over all generations. In what follows we shall regard only first generations of leptons and quarks.
->1 — i-
Indeed, due to commutativity and associativity of multiplication by j
SU(2; j) 3 g(j)= (12)
= exp {Aj) + Aj) + A3T } =
= exp {(jA^)Ti + (jAl)T2 + AIT3} .
For the gauge fields (7) these substitutions are as follows:
W± - jW±, The left lepton Li
Zi — Zi, Ai — Al. (13)
- g) and quark Qi = Q) fields are SU(2) doublets, so their components are transformed in the similar way as components of the vector z, namely:
vi — jvi, ei — ei, ui — jui, di — di. (14)
The right lepton and quark fields are SU(2) singlets and therefore are not changed.
HŒPHAfl OH3HKA TOM 76 № 9 2013
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INTERPRETATION OF NEUTRINO-MATTER INTERACTIONS
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After transformations (13), (14) and spontaneous
symmetry breaking with ф?
V2
the boson La-
grangian (2), (3) can be represented in the form [2, 3]
LbU) = 4fU) + Lf(j) = - {d,xf - (15) 1 2 2 1 1 2 1
+ .f + m^W+W-} + L%\j),
where as usual second-order terms describe the boson particles content of the model and higherorder terms Lgij ) are regarded as their interactions. So Lagrangian (15) includes charded W bosons with identical mass mw = \gv, massless photon A^, neutral Z boson with the mass mz = § \/g2 + g'2
and Higgs boson x,mx = \/2Xv. The lepton Lagrangian (4) in terms of electron and neutrino fields takes the form [5]
1
eei faA a ei
- g cos UWerTßAßßr + g' sin 6>w4TßZßer + j2^ vjifßdßщ +
+
9 vjfßZßvi +
2 cos Uw
+
л/2
VfaW+ ei + 4fßW- vi
= Ll,ö + j2 Llj .
Q ~ \sin2 9w) ~
cos Uw
- ^g' cos 9wdlTßAßdr + ^g' sin 9wdlrßZßdr + + j2^ulifdßui + ulirßdßUr - mu(u[щ + u\ur) +
+
g
cos Uw \2 3
i - \ sin2 6>w ) u\fßZßui +
2e
+—u\TßAßui +-j=
g
ЩfßW+di + dlTßW - ui
+
22
+ —g' cos ewulrßAßUr - -g' sin ewulrßZßur j>
= LQ,b + j2LQ,f,
where me = hev/V2 and mu = huv/y/2, ma = = hdv/л/2 represent electron and quark masses. The full Lagrangian of the modified model is the
LlU) = e]lifßdßei + e\ir^e - (16)
, + +4 g cos2UW + ^
- me{elrei + e er) + —-—e!TßZßei
i 2cosUW i и p
- eelfaAßei - g cos Uwerт^А^ +
The quark Lagrangian (5) in terms of u- and d-quark fields can be written as
lQ(j) = + 4ir^d^dr - (17)
- md(d\di + d\dr) - ^dlf^A^di -
sum
L(j) = Lb (j) + Lq(j) + LL(j ) = Lb + j2Lf.
(18)
The boson Lagrangian LB(j) was discussed in [2, 3] where it was shown that masses of all particles involved in the Electroweak Model remain the same under contraction j2 — 0 in both formulations: standard one [2] and without Higgs boson [3]. In this limit the contribution j2Lf of neutrino, W-boson and u-quark fields as well as their interactions with other fields to the Lagrangian (18) will be vanishingly small in comparison with contribution Lb of electron, d-quark and remaining boson fields. So Lagrangian ( 18) describes very rare interaction of neutrino fields with the matter. On the other hand, contribution of the neutrino part j2Lf to the full Lagrangian is risen when the parameter j2 is increased, that again corresponds to the experimental facts. So contraction parameter is connected with neutrino energy and this dependence can be obtained from the experimental data.
4. DECOMPOSITIONS OF PHYSICAL SYSTEMS
The standard way of describing a physical system in field theory is its decomposition on independent more or less simple subsystems and then introduction of interactions between them. In Lagrangian formalism this implies that some terms describe independent subsystems (free fields) and the rest terms correspond to interactions between fields. When subsystems are not interacting with each other the composed system is a formal unification of subsystems and symmetry group of the whole system is direct product G = G1 x G2, where G1 and G2 are symmetry groups of the subsystems. The Elec-troweak Model gives an example of such approach. Indeed, there are free boson, lepton, quark fields in Lagrangian, and terms which describe interactions between these fields.
The operation of group contraction transforms a simple or semisimple group G to a nonsemisimple one with the structure of a semidirect product G = A x x Gi, where A is Abelian and Gi С G is untouch
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