On the class of chiral symmetry representations with scalar and pseudoscalar fields
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19. April, 2008
On the class of chiral symmetry representationswith scalar and pseudoscalar fields
Peter MinkowskiInstitute for Theoretical PhysicsUniversity of BernCH - 3012 Bern, SwitzerlandE-mail: [email protected]
Abstract
In the following few pages an account is given of a theme , which Ibegan in 1966 and continued to the present.
Σ = √ ( σ − i π ) scalar - pseudoscalar fieldsand the class of their chiral symmetry represen-tations Lets denote by t , s , n · · · quark flavor indices with t , s , n · · · = 1 , · · · , N ≡ N fl (1)and by λ a the N hermitian U N matrices with the normalization λ a = (cid:16) λ a (cid:17) ts ; tr λ a λ b = δ ab a = 0 , , · · · , N − λ = √ N − / ( ¶ ) N × N tr λ a = 0 for a > λ a | conv. = √ λ a (2)In order to maintain clear quark field association we choose the convention and restriction projecting out color and spin degrees of freedom from thecomplete set of q q bilinearsΣ s ˙ t ∼ q ˙ c ˙ t ( 1 + γ R ) q cs γ R = i γ γ γ γ ; c, ˙ c = 1 , , logical structure of Σ - variables is different, when used to derive thedynamics of quarks, i.e. QCD, or before this, when used in their own rightas by M. Gell-Mann and M. L´evy [1] , or else associating chiral symmetrywith superconductivity as by Y. Nambu and G. Jona-Lasinio [2].Here the chiral U N fl R × U N fl L transformations correspond to
U N fl R : ( 1 + γ R ) q cs → V ss ′ ( 1 + γ R ) q cs ′ U N fl L : ( 1 − γ R ) q cs → W ss ′ ( 1 − γ R ) q cs ′ l Σ → V Σ W − (4)The construction in eq. 4 can be interpreted as group-complexification ,discussed below. The Σ-variables arise as classical field configurations ,Legendre transforms of the QCD generating functional driven by generalx-dependent complex color neutral mass terms.1he latter represent external sources with U N fl R × U N fl L substitu-tions aligned with the Σ - variables − L m = m ˙ ts ( x ) (cid:8) q ˙ cs ( 1 − γ R ) q ct (cid:9) + h.c. ∝ tr (cid:0) m Σ † + Σ m † (cid:1) m → V m W − ←→ Σ → V Σ W − (5)The so defined (classical) target space variables form– upon the exclusion of values for which Det
Σ = 0 –the group GL ( N , C ) = { Σ | Det Σ = 0 } (6)the general linear group over the complex numbers in N dimensional target-space .We proceed to define the hermitian chiral currents generating U N fl R × U N fl L ( global ) pertaining to Σ j aµ R = tr Σ † (cid:18) λ a i ⇋ ∂ µ (cid:19) Σ ∼ q γ µ λ a P R qj aµ L = tr Σ † i ⇋ ∂ µ Σ (cid:0) − λ a (cid:1) ∼ q γ µ λ a P L qA ⇋ ∂ µ B = A ∂ µ B − ( ∂ µ A ) B ; P R ( L ) = ( 1 ± γ R )(7)We avoid here to couple external sources to all other q q bilinears except thescalar - pseudoscalar ones as specified in eq. 5 for two reasons1) – to retain a minimum set of external sources capable to reproducespontaneous chiral symmetry breaking alone as a restricted but fullydynamical spontaneous phenomenon.2) – in order to avoid a nonabelian anomaly structure . The latter wouldforce either the consideration of leptons in addition to quarks , or theinclusion of nonabelian Wess-Zumino terms obtained from connectionsformed from the Σ fields [3] . The notion of target-space is used as defined in modern context of string theories . q q (cid:2) j a R ( t , ~x ) , j b R ( t , ~y ) (cid:3) = i f abn j n R ( t , ~x ) δ ( ~x − ~y ) (cid:2) j a L ( t , ~x ) , j b L ( t , ~y ) (cid:3) = i f abn j n L ( t , ~x ) δ ( ~x − ~y ) (cid:2) j a R ( t , ~x ) , j b L ( t , ~y ) (cid:3) = 0 (cid:2) λ a , λ b (cid:3) = i f abn λ n (8)The GL ( N , C ) group structure defined in eq. 6 enables bilateral multi-plication of the Σ , Det Σ = 0 elements , of which the left- and right-chiralcurrents defined in eq. 7 are naturally associated with the Lie-algebra of U N fl R × U N fl L through the exponential mapping with subgroups of GL ( N , C ) R × GL ( N , C ) L . These (sub)groups act by multi-plication of the base-group-manifold by respective multiplication from theleft ↔ G R and from the right ↔ G L . The reverse association– here – is accidental GL ( N , C ) R ( L ) → G R ( L ) = G Σ ∈ G ; g ∈ G R ; h ∈ G L : G R • G ↔ Σ → g Σ G L • G ↔ Σ → Σ h − G R ⊗ G L • G ↔ Σ → g Σ h − Σ = Σ ( x ) ; g , h : x-independent or ’rigid’ (9) The exponential mapping and compactification(s) of G ( Σ )The condition Det Σ = 0 in the restriction to GL ( N , C ) ( eq. 6 ) is veryspecial and surprising in conjunction with the field variable definition.In fact such a condition is completely untenable and shall be discussed below.This was a stumbling block for a while .3his condition is equivalent to the relation with the Lie algebraof GL ( N , C ) through the exponential mapping and its inverse ( log )Σ = exp b ; b = b a λ a ; λ = ( 2 N ) − / ( ¶ ) N × N Det
Σ = exp ( tr b ) = exp β ; β = q N b Det
Σ = 0 ↔ ℜ β = − ∞ ; β ∼ β + 2 π i ν ; ν ∈ Z (10)Of course eliminating – from general dynamical Σ-variables – the subset with
Det
Σ = 0 affects only the non-solvable ( and non-semi-simple ) part ofthe associated group, whence the former are interpreted as a manifold, whichsimply is not a group . It may thus appear that the restriction in order toenforce a group structure is characterized by the notion of ’group-Plague’,infecting the general structure at hand .This said we continue to treat Σ-variables as if they were identifiable with GL ( N , C ) .The next reductive step is to consider the solvable ( simple ) subgroup SL ( N , C ) ⊂ GL ( N , C ) ⊂ { Σ } SL ( N , C ) = n b Σ (cid:12)(cid:12)(cid:12) Det b Σ = 1 ob Σ ∼ Σ / ( Det
Σ ) /N ; allowing all N roots (11)The advantage of the above reduction to SL ( N , C ) is that it allows theexponential mapping to an irreducible ( simple ) Lie-algebra ,refining eq. 10 b Σ = exp b b ; b b = b b a λ a ; a = 1 , , · · · , N − b b = 0 ; tr λ a = 0 (12)i.e. eliminating the unit matrix ∝ λ from the latter . The words testify to the fight for definite mathematical notions . .1 Relaxing the condition Det Σ = 0 and the unique association Σ −→ Det Σ = 0 GL ( N , C )We transform Σ s ˙ t as defined or better associated in eq. 3 by means of the N hermitian matrices λ a in eq. 2 .Σ s ˙ t = Σ a (cid:16) λ a (cid:17) s ˙ t Σ a = tr λ a Σ ; a = 0 , , · · · , N − a are components of a complex N -dimensional space C N and in one to one correspondence with thematrix elements Σ s ˙ t C N = n (cid:16) Σ , Σ , · · · Σ N − (cid:17) o (14)This serves to become aware of the second algebraic relation ( ⊕ ) , beyond( ⊗ ) , i.e. to add matrices and not to just multiply them . The ⊕ operation is also encountered upon ’shifting’ general (pseudo)scalarfields relative to a spontaneous vacuum expected value . This is relevant here for spontaneous breaking of chiral symmetry .It arises independently for the SU L -doublet scalar (Higgs) fields .Hence the idea that the combination of ⊕ and ⊗ – which form the fullmotion group ( of matrices ) – are related to ’fields’ ( ’K¨orper’ in german ).Thus we are led to consider quaternion- and octonion-algebras in the nextsections . Let q = q i + q a i a ; a = 1 , , (cid:0) q , ~q (cid:1) ∈ R i = ¶ ; i a i b = − δ ab i + ε abn i n | for a , b , n = 1 , , q = q i − q a i a (15)denote a quaternion over the real numbers .Then a single octonion is represented ( modulo external automorphisms ) Elements of a N × N -matrix can equivalently be arranged along a line . These automorphisms form the exceptional group G .
5y a pair of quaternions ( p , q ) with the nonassociative multiplication rule o = ( p , q ) = p j + p a j a + q j + q a j a o α = ( p α , q α ) ; α = 1 , , · · · o ⊙ o = (cid:0) p p − q q , q p + q p (cid:1) o = ( p , − q ) → for o = o ; o = (cid:0) p , − q (cid:1) o ⊙ (cid:0) o = o (cid:1) = (cid:16) p p + q q , − q p + q p (cid:17) = n (cid:12)(cid:12) p (cid:12)(cid:12) + (cid:12)(cid:12) q (cid:12)(cid:12) o j + 0 j = ¶ , j , j ; j , , ≃ i , , (16)In eq. 16 we used the involutory properties q = q ; o = o (17)It follows that unitary quaternions ( q q = q q = ¶ ) are equivalent to S ≃ SU ⊂ R , whereas unitary octonions ( o o = o o = ¶ ) areequivalent to S ⊂ R .This leads together with the complex numbers to the algebraic association of N = 1 and N = 2 − Σ variables to the three inequivalent ’field’-algebras1 N = 1 ↔ C ≃ R ⊃ S N = 2 ↔ Q ≃ R ⊃ S N = 2 ↔ O ≃ R ⊃ S (18)The group structures of cases 1 - 3 in eq. 18 correspond to1 : S ≃ U ↔ U R ⊗ U L S ≃ SU ↔ SU R ⊗ SU L S ↔ U L ⊗ U R (19)While the model introduced by M. Gell-Mann and M. L´evy [1] correspondsto case 2 ( eq. 18 , 19 ) , it is case 3 ( also for N = 2 ) which is different and the only one extendable to N > N = 3 and from there back to case 3 with N = 2 in the next section. 6 .3 Σ = √ ( σ − i π ) for N = N fl = 3 ( m u ∼ m d ∼ m s ) For N = 3 the Σ − variables describe a U fl − nonet of scalars andpseudoscalars (one each) . I shall use the notation Σ → π , K , η , η ′ labelledby the names of pseudoscalars , yet denoting associated pairs scalars ↔ pseudoscalars Σ = Σ Σ π − Σ K − Σ π + Σ Σ K Σ K + Σ K Σ Σ = √ Σ η ′ + √ Σ π + √ Σ η Σ = √ Σ η ′ − √ Σ π + √ Σ η Σ = √ Σ η ′ − √ Σ η (20)In the chiral limit m u,d,s → π , (3) ; K , K , (4) ; η , (1) , whereas η ′ and all 9 scalarsremain massive. π ↔ η ↔ η ′ − mixing – eventually different for scalars relative topseudoscalars – is not discussed here [4] .Projecting back on case 3 and N = 2 in the limit m s → ∞ an SU fl − singlet pair – denoted Σ η (2) – forms as ( singlet ) combinationsof Σ η , Σ η ′ and a corresponding isotriplet pair Σ π → ~ Σ π .Instead of the 2 × N = 2 wecan equivalently display the double quaternion basis from the octonion -structure ( eq. 16 ) p ↔ (cid:0) σ η (2) , ~π (cid:1) → [1] q ↔ (cid:0) η (2) , ~σ π (cid:1) (21)7 From h Σ i as spontaneous real parameter to f π As shown in section 1 , the Σ − variables are chosen such , that the spon-taneous breaking of just chiral symmetry can be explicitely realized .For N equal ( positive ) quark masses it folows h Σ i = S ¶ N × N S = √ N (cid:10) σ (cid:11) ; Σ = √ ( σ − i π ) N × N j aµ R = i S tr λ a ∂ µ (cid:0) Σ − Σ † (cid:1) + · · · = S ∂ µ π a + · · · Σ − Σ † = − i π b λ b h Ω | j aµ R (cid:12)(cid:12) π b , p (cid:11) = i f π p µ δ ab for a, b > − S = f π ↔ − (cid:10) σ (cid:11) = (cid:0) N (cid:1) / f π ; f π ∼ . ~π (22) Acknowledgement
The present account was the subject of a lunch-seminar, 16. April 2008in Bern. The discussions especially with Uwe-Jens Wiese, ´Emilie Passemarand Heinrich Leutwyler are gratefully acknowledged .8 eferenceseferences