Avenues of cognition of nongravitational local gauge field theories

This controbution is devoted to present basic fearures of a unifying local gauge field theory, prevailing up to a mass scale of approximately 10 16 GeV , allowing the neglect of gravitational curvature effects – indicated by the attribute : ’nongravitational’ in the title above .


Introduction
The two main mass generation mechanisms within a general gauge field theory in 3 + 1 uncurved space-time dimensions -henceforth called gravitationless gauge field theory --minimally the neutrino mass extended standard model based on the gauge group S U3 c × S U2 L × U1 Y and one scalar doublet with respect to S U2 Lform the basis of the present outline, supposed to be an integral part of an enveloping unifying gauge group , minimally SO10 ≡ spin10.
1) the Bose condensation of some components of elementary scalar fields scalar stands here for scalar and/or pseudoscalar Yukawa couplings 2) the Bose condensation of the gauge field-strength bilinear -gauge-and renormilzation group invariant with respect to the unbroken gauge group S U3 c The key features of point 1) in table 1, are outlined in section 2.
The main topic here, point 2) in table 1, is elaborated on in section E. Recent assessments can be found in the talks of François Englert -[1-2013] and, more historically oriented, Peter Higgs [2-2013] at the Nobel Prize 2013 awards ceremony.We base the general properties of spontaneous gauge breaking of an enveloping gravitationless gauge field theory , based on a gauge group G env ⊃ G min in the sense of gauge group unification beyond the minimal case G min = S U3 c × S U2 L × U1 Y , presented in the introduction.
Within larger simple enveloping groups the exceptional chain is singled out ,  , most importantly because it offers the possibility of canceling all gauge-and gravitational anomalies in the product gauge group  , ] Following the hypothesis of an underlying unifying gauge group the top down of gauge breaking is initiated by a primary breaking followed last by the electroweak gauge breaking .

2-1 Primary gauge breaking
Primary gauge beaking is linked to the unifying gauge group scale M env assumed and also restricted by limits on direct observation of baryon decays and lepton flavor violation to be much larger than the electroweak scale In eq. 4 v denotes the v.e.v. of the unique doublet scalar field using the quaternion associated basis for the local scalar fields In eq. 5 the symbol * denotes hermitian conjugation of individual complex and/or real field components.Thus the quantity z , defined in eq. 5 , shows its quaternionic representation The four 2 × 2 matrices , displayed in eq. 5 , form a 1 to 1 true representation of the base quaternions q m q n = − δ mn q 0 + ε mnr q r for m, n, r = 1, 2, 3 As we will see , the final stage of the ( nu-mass extended -) standard model gauge breaking involving just 1 doublet of scalars with respect to the elecroweak part S U2 w × Y w represents a case for perfectly semiclassical , driven Bose condensation , eventually contrasting with intrinsic properties of primary breakdown.This is so, because there is precisely 1 invariant with respect to S U2 w × Y w of which a general invariant is a function .This would not remain true for more than one scalar doublet .
As a consequence of eqs. 4 -8 , the electroweak gauge breaking is driven and semiclassical We shall discuss 2 types of primary gauge breaking denoted a) and b) below .The v.e.v.v in eq. 9 corresponds to the classical minimum of the quartic potential , uniquely restricted to depend on two parameters The minimum conditions become The second derivatives with respect to Z at the minimum of the potential become Expanding the deviation of the potential up to quadratic terms around the minimum thus yields It is customary to denote the shifted hermitian field Δ Z 0 ( x ) From eq. 14 we read off the mass of the field H ( x ) as well as the vanishing of the mass of the other three fields However in the case at hand the existence of would-be long range forces represented by the S U2 L × U1 Y gauge field interactions does not permit the existence of goldstone-modes .
The three would-be Goldstone fields, defined in eq.15, through their space-time gradient, mix with the gauge bosons to become massive, vis.W ± and Z ICNFP 2014 in such a way as to obtain masses of the 3 massive gauge bosons m W , m Z and physically form the longitudinal spin components of the resulting massive states (resonances) .In eq.16 the neutral massive gauge boson is denoted Z not to confuse it with the sclar field components Z 0 , Z .
In tree approximation the mass-square of the H-field is twice the value of the Z 0 field in the unbroken case, i.e. for μ The detailed description of the mixing as stated in eq.16 is not given here.A complete derivation can be found in the textbook ref.We introduce notation for the ensemble of scalar fields adapted to primary gauge breaking, generalizing S U2 L doublets as defined in eqs. 5 -7 .
Lets fix for definiteness in the following.A general irreducible representation of SO10 shall be denoted [D] , where D is equivalenced to its dimension.As entry point we take the [16] representation for one fermion family ( the lightest in mass ) in the left chiral basis [16] : It is instructive to illustrate the use of the basis defined in eq.19 considering the Majorana logic characterized by N e,μ,τ derived from the subgroup decomposition spin10 → SU5 × U1 J 5 (20) Among the 3 generators of spin10 commuting with S U3 c , I 3 L , I 3 R , B − L and forming part of the Cartan subalgebra of spin10 there is one combination, denoted J 5 in eq.20, commuting with its largest unitary subgroup SU5 .The charges Q ( J 5 ) form the pattern as in eq.21 [16] : with D ( S U5 ) equivalenced with the dimension of the representation.The suffix Q is added in eq.22, since in the present context SU5 multiplets necessarily occur embedded in SO10 representations.This is derived in ref. 4 -[4-2008] .
It is instructive to decompose the Q-value pattern in eq.21 into an S U2 L+R invariant part and the remainder proportional to From eq. 25 we obtain the identification of the spin10 Cartan subalgebra hermitian components ( charges ) In eq. 26 B and L denote baryon and lepton number respectively .Two remarks shall follow , concerning the recognizable key features inherent to spontaneous gauge breaking of an enveloping gravitationless gauge field theory , based on a gauge group G env = spin10 1) primary gauge breakdown must be much different than on the lowest -i.e.electroweak -scale level pertaining to This is so because empirically well established candidate symmetries , like baryon and lepton number conservation are broken on the primary level and imply very large scale of unification M env = O 10 16 GeV .
As examples let me quote the upper limits of the μ + → e + γ and μ + → e + + e + e − branching fractions 2) the power of the set of scalar fields involved in primary gauge breaking does not follow any principle of minimal selection of spin10 representations pertaining to scalar fields .

specific notation for scalar field variables
We proceed defining notation for scalar field variables suitable for primary gauge breaking | broken down to real coordinates (28) In eq.28 D (ν) ; ν = 1, 2, • • • denote a complete set of unitary irreducible representations of spin10 , finite dimensional; constructively defined through the method of Peter and Weyl [14-1927] .n ν stands for the multiplicity of a given representation D (ν) .
For D (ν) , D (ν) beeing a pair of inequivalent , relative complex conjugate representations the coor- broken down to real and imaginary parts count as 2 (complex) dim ( D ) components over real numbers.This is the meaning of the attribute 'broken down to real coordinates' on the right hand side of eq.28 .
Thus choosing real values for the components of z ∈ R M as defined in eq.28 it follows we find to get an idea of the power of the set of scalars We illustrate the order of M , in eqs.29 , 30 by the representations of the fermion bilinears from left and right chiral bases, adapting the scalar variables to their definition in eq.29 , allowing for complex linear combinations applied to complex representations ( from section E and ref. 4 -[4-2008] ) .
H f ermion mass ←− (32) The real and complex representations in the multiplication table in eq.32 are denoted D R , D C respectively Choosing minimally multiplicities 2 for real and 1 for complex representations in eqs.32 and 33 yields We conclude from the example multiplicities leading to eq. 34

2-1-a Primary gauge breaking -type a)
The gauge breaking in type a) gravitationless gauge field theory is 1) driven by pre-established quadratic, cubic and quartic scaler field self interactions 2) not reducible to semiclassical approximation for vacuum expected values for scalar variables and their composite operators, necessarily include gauge variant ones in order to qualify for gauge breaking For clarity let me remark that condition 2) above is necessary, since in the case of exclusively gauge invariant vev's for composite scalar operators they are part of an alternative case -in conjunction with other gauge invariant composite field variables -of spontaneous mass generation without gauge breaking.This will be discussed within QCD with 3 light flavors -without scalars -elsewhere .

2-1-b -Primary gauge breaking -type b)
The gauge breaking in type b) gravitationless gauge field theory is 1) driven by pre-established quadratic, cubic and quartic scaler field self interactions like in type a) 2) reducible to semiclassical approximation for vacuum expected values for scalar variables and their composite operators.This is the usual case discussed in the literature, as e.g. in refs.10 = [10-1013] -and 15 = [15-2013] , while in the latter reference the main topic is electroweak gauge breaking .

ICNFP 2014
03025-p.7 The semiclassical approximation -with respect to vev's of scalar fields and their composite local operators as well as composite operators involving other fields -means E The spin10 product representations 16 ⊕ 16 ⊗ 16 ⊕ 16 We follow the spin10 decomposition discussed in section 2-1 ( eq. 37 repeated below ) spin10 → SU5 × U1 J 5 (37) Further let us denote representations of spin10 as opposed to those pertaining to SU5 and associated J 5 quantum number by spin10 : [dim] ; SU5 × U1 J 5 : {dim} J 5 (38) Thus we obtain In turn SU5 representations shall be decomposed along the standard model gauge group SU3 c ⊗ SU2 L ⊗ U1 Y , where Y denotes the electroweak hypercharge with a factor 1 2 included The brackets on the right hand side of eq.41 are reversed in order not to confuse spin10 -and standard model representations.
Then the base 16 16 decompose to The product representations 16 ⊕ 16 ⊗ 16 ⊕ 16 generate all SO10 antysymmetric tensor ones, of which we encountered the fivefold antisymmetric in section 2-1 .
To elaborate we specify the n-fold antisymmetric tensors obtained from the 10-representation of SO10 The quantities [ t n ] defined in eq.43 form irreducible real representations of SO10 , except for n = 5 , which is composed of the relatively complex irreducible representations 126 and 126 .
The correspondence of product representations of the 16 + 16 = 32 associative Clifford algebra with the sum of antisymmetric tensor ones follows from the completeness of all products of γ matrices forming the spin10 algebra i.e. are of dimension

EPJ Web of Conferences
a e-mail: mink@itp.unibe.chDOI: 10.1051/ C Owned by the authors, published by EDP Sciences, 9 = [9-1995]  .The simplicity of the presumably lowest in scale gauge breaking relative to the SM gauge group S U2 L × U1 Y prompted most discussions of primary gauge breaking to be of the same type b) presented below, i.e. driven -semiclassical: for an example of primary gauge breakdown patterns see e.g.ref.10 = [10-1013]  .This brings us to the two types a) and b) -of an enveloping gravitationless gauge field theory.For both types the scalar field variables turn out to involve a complex ensemble of irreducible representations of the enveloping gauge group -as e.g.SO10 , in particular for the generation of masses for neutrino flavors, both light and heavy, as discussed e.g.inrefs.4 = [4-2008]  and 10 = [10-1013] through Yukawa couplings to basic fermion bilinears .Two recent papers deserve mention here byGuido Altarelli ref.11 = [11-2014]  andFerruccio Feruglio,  Ketan M. Patel and Denise Vicino ref. 12 = [12-2014] with charges as given in eq.21 represents a hermitian generator of the Cartan subalgebra of spin10 , unique up to a (real) multiplicative factor, which commutes with the the SU5 subgroup of EPJ Web of Conferences spin10 .The flavors within one family sharing the same Q-charges in fact form irreducible representations of SU5 , which shall be labeledD ( S U5 ) → {D ( S U5 )} Q (22)with D ( S U5 ) equivalenced with the dimension of the representation.The suffix Q is added in eq.22, since in the present context SU5 multiplets necessarily occur embedded in SO10 representations.Thus the Q values of the[16]  representation on the right han side of eq.21 translate to[16] = {1} 5 + {10} 1 + 5 −3 (23)The sequence of Q values : Q = 5 , 1 , −3 as arranged in the sequence on the right hand side of eq.23 -decreasing in steps of 4 -is related to the properties of binomial coefficients 10 n for n = 0 , 4 , 8 and n = 10 , 6 , 2 10 −12 Br ( μ + → e + + e + e − ) < 1.0 .10 −12 in ref.11 -[13-2013] -) scalar fields in the 126 representation of SO10 F , G = I, II, III : fermion family indices Clebsch-Gordan coefficients projecting the product of two ( fermionic ) 16 representations on irreducible spin10 representations .We reproduce the 4 product representations of 16 and 16 representations from ref. 4 ( Appendix E ) reduce the [16] ⊗ [16] product with respect to J 5 , SU5 and SU3 c × SU2 L × U1 Y .The individual products are s (a) : (athe (anti)symmetric products ( [16] ⊗ [16] ) s = [10] ⊕ [126] and ( [16] ⊗ [16] ) a = [120] with respect to SU5 ⊗ U1 J 5 using eq.47 ( [16] ⊗ [16] ) s = [10] ⊕ [126] roman indices I,II in eq.50 indicate that appropriate linear combinations of the two 5 2 representations form parts of [10] and [126] respectively .
The tenfold antisymmetric invariant corresponds to [ t n=10 ] .The product of two full Clifford algebras pertaining to spin10 contains all [ t n ] ; n = 0 • • • 10 representations exactly once .Treating the n = 5 tensor as one representation -it is reducible only over C -the dimensions of the [ t n ] representations follow Pascal's triangle .