Local SU(3) gauge invariance in Weyl 2-spinor language and quark-gluon plasma equations of motion
aa r X i v : . [ h e p - t h ] J a n Local SU(3) gauge invariance in Weyl 2-spinorlanguage and quark-gluon plasma equations ofmotion
January 5, 2021
J. BuitragoDepartment of Astrophysics of the University of La Laguna, Avenida Fran-cisco Sanchez, s/n, 38205, La Laguna, Tenerife, Spain and Instituto de As-trofisica de Canarias (IAC), E-38200 La Laguna, Tenerife, [email protected] 5, 2021
Abstract
In a new Weyl 2-spinor approach to Non abelian Gauge Theories,starting with the local U(1) Gauge group of a previous work, we studynow the SU(3) case corresponding to quarks (antiquarks) interacting withcolor fields. The principal difference with the conventional approach isthat particle-field interactions are not described by means of potentialsbut by the field strength magnitudes. Some analytical expressions showingsimilarities with electrodynamics are obtained. Classical equations thatdescribe the behavior of quarks under gluon fields might be in principleapplied to the quark-gluon plasma phase existing during the first instantsof the Universe.
In a previous paper [1] (hereafter Paper I) a classical U (1) local gauge invariantlagrangian in terms of the electric and magnetic field strengths was introduceddiffering to the usual lagrangian quantum approach leading to the Dirac Equa-tion interacting with an external electromagnetic field described by the fourpotential A µ . Now, in this paper, the mentioned formalism will be extendedto non abelian gauge theories. Given that the Yang-Mills fields, in the SU(2)1ase, should be massless and are only partially incorporated in the electroweaktheory (devoid of hardly any classical interpretation in the weak sector), it isclear that any classical equations of motion would have only an academic inter-est. On the other hand, the SU(3) case, involving 8 massless color fields, mightbe of interest, not only by itself but also in the study of the initial evolution ofthe Universe (for energies above 100 GeV) when the running coupling constantwould be small and thus accesible to a non perturbative study.The similitudes between the gluon fields and the ordinary electric and mag-netic fields have been noted by many whose names (given their number) I do notwant to remember. In electromagnetism we had, to start with, the Coulomb law(known centuries ago), while for the forces between quarks, at least analytically,there is not something of such a simplicity to start with. It is the main purposeof this work to try to fill this gap and give the quark-color field interaction aclassical support. In section 3 we shall see that for colorless (not singlet) colorfields it is possible to find, in the special case of a constant electric-like field, thesame analytical solution of electrodynamics. For the other color gluon fields, wehave not been able to see (or I have not been able to show) whether any similarresult holds, consequently, we are not able to assert that color fields are justreplicas of the electromagnetic field. In fact, the non abelian group and the nonlinear aspects of the interaction seems to point out in the oposite direction.As a possible application, at least in principle, the classical equations ofmotion might be applied to a primordial see of quasi-free quarks and gluonsduring the first instant of the universe.The classical SU(3) local gauge invariant picture that will be presented in thefollowing sections will be symmetrical in both quarks and antiquarks, assumingthat the total color charge, and by extension total electric charge, in the earlyuniverse was zero (The reader should be warned about the possibility that in ahigh energy scenery in which both electromagnetic and strong running couplingconstant could eventually be comparable the quark-gluon dynamics descriptionmight be in practice incomplete). As in some parts of what next follows we shall implicitly deal with an scenarioin which asymptotically free quarks and antiquarks coexist, it will be convenientto change the notation used in Paper I where complex conjugate spinors, such as¯ η A ′ , are denoted with a bar over Greek and Roman symbols and primed capitalletters as superscript and subscripts. The bar-over is conventionally used forantiparticles and we shall obey this rule naming complex conjugate spinors, offirst and higher rank, simply η A ′ , ϕ A ′ B ′ .... and so on (a discussion about thevarious conventions used in the 2-Spinor formalism can be found in the book ofPenrose and Rindler opus cit. [2]).Assuming, for simplicity, a single flavour of quark, which can exist in three2olor-charge states, the four momentum can be written in the same way thatwas done in Paper I, namely p AA ′ c = 1 √ h π A π A ′ + η A η A ′ i c , (1)while for an antiquark p AA ′ c = 1 √ h ¯ π A ¯ π A ′ + ¯ η A ¯ η A ′ i c , (2)where c can be any of 1 = red, green, blue . Since p AA ′ c is to representthe four-momentum of a massive particle, must be time-like and, for any of thethree possible values of c , fulfill the condition: p AA ′ p AA ′ = m . (3)For a free quark (or antiquark putting bars over the symbols) in one ofthe three colors (anticolors), the classical lagrangian density (with dimensionsenergy per unit length) would be L f = π Ac ˙ η Ac , (4)(dot denoting derivative respect to proper time τ ). As η Ac and π Ac are triplets,the notation should again be changed relative to Paper I to assure that everyterm in the lagrangian is a scalar. The η and π spinors will be represented by η A η A η A , (5)and ( π A π A π A ) . (6)The Euler-Lagrange equations associated to η Ac are ddτ ∂ L ∂ ˙ η Ac − ∂ L ∂η Ac = 0 , (7)leading to ˙ π Ac = 0 = ⇒ π Ac = const. (8)The free lagrangian is not invariant under SU (3) local phase transformations η Ac → η ′ Ac = exp (cid:16) i g s λ q .α ( τ ) q (cid:17) η Ac , (9)3nd π Ac → π ′ Ac = exp (cid:16) − i g s λ q .α ( τ ) q (cid:17) π Ac , (10)where λ q ( q = 1 , , ...,
8) are the Gell-Mann SU (3) matrices.To restore invariance, the free lagrangian need to acquire an additional term.The procedure similar to the conventional in non abelian gauge theories is toreplace the ordinary derivative by the covariant one and consider an infinitesimaltransformation. In our classical approach the covariant derivative is definedtrough the condition of transforming in the same way as the spinors: Ddτ η Ac → exp (cid:16) i g s λ q .α ( τ ) q (cid:17) Ddτ η Ac . After some lengthy algebra, it is found that the interacting gauge invariantlagrangian is L f = π Ac ˙ η Ac − g s m π Ba (cid:2) λ q W ABq (cid:3) ab η Ab . (11)The eight field strengths W ABq are related to the usual symmetric W iAB following the rule W iAB = ǫ BC W CiA while the gauge fields W iAB transform as( i, j, k = 1 , , ..., W iAB → W iAB − im ˙ α i ǫ AB − f ijk α j ( τ ) W kAB W iA ′ B ′ → W iA ′ B ′ + im ˙ α i ǫ A ′ B ′ − f ijk α j ( τ ) W kA ′ B ′ , being f ijk the structure constants of SU (3). The corresponding eight fourth-rank antisymmetric spinor fields are represented, in the conventional form, as G iAA ′ BB ′ = W iAB ǫ A ′ B ′ + W iA ′ B ′ ǫ AB (12)and, as consequence of the non abelian field, G iAA ′ BB ′ is not invariant but trans-forms as G iAA ′ BB ′ → G iAA ′ BB ′ − g s (cid:2) f ijk α j ( τ ) G kAA ′ BB ′ (cid:3) . with the structure constant of the group: f = 1 , f = f = f = f = f = 1 / , f = f = √ / . H = ∂ L ∂ ˙ η Aa ˙ η Aa − L , (13)being just the interaction term: H = gm π Ba (cid:2) λ q W ABq (cid:3) ab η Ab . (14)Using Hamilton Equations ˙ π Aa = − ∂ H ∂η Aa (15)˙ η Aa = ∂ H ∂π Aa . (16)The Equations of motion are finally˙ η aA = gm h λ q W qAB i ab η bB (17)˙ π Aa = − gm π Ba h λ q W qAB i ab . (18)The goal is then to solve the previous equations in a given environment orbackground field given by the matrix on the right side and obtain the evolutionof the four-momentum (1) as a function of proper time. Note that as a classicalequation there is no distinction between quarks and antiquarks. In the standard version of QCD, the interaction is described by eight colorpotentials B lµ related to the field strengths by G lµν = ∂ ν B lµ − ∂ µ B lν + ig s f jkl B jµ B kν (19). Since spacetime derivatives do not change color, in the high energy regimecorresponding to small running coupling constant g s , the color fields seems tobe just a copy of the electromagnetic field and, therefore, in the early almosthomogeneous universe, we can expect that three-gluon vertex (proportional to g s ) as well as four-gluon vertex (proportional to g s ) are almost absent .5ach of the massless (spin one) color field potentials B lµ has two helicitystates and they dress themselves in eight different color combinations corre-sponding to the color octet [3]: B → ( r ¯ b + b ¯ r ) / √ B → − i ( r ¯ b − b ¯ r ) / √ B → ( r ¯ r − b ¯ b ) / √ B → ( r ¯ g + g ¯ r ) / √ B → − i ( r ¯ g − g ¯ r ) / √ B → ( b ¯ g + g ¯ b ) / √ B → − i ( b ¯ g − g ¯ b ) / √ B → ( r ¯ r + g ¯ g − b ¯ b ) / √ . (20)Two of them, B and B , are colorless while the rest mix color charges. In ourdescription, this will be reflected in the diagonal and non diagonal terms in theequations below. In a homogeneous early universe the background gluon fieldshould have been homogeneous and isotropic enough, otherwise the universecould possibly not have evolved to the state revealed by the 3K radiation field.Once expanded, the classical Equations of Motion looks like ˙ η A ˙ η A ˙ η A = g s m W BA + W BA √ W BA + iW BA W BA + iW BA W BA − iW BA − W BA + W BA √ W BA + iW BA W BA − iW BA W BA − iW BA − W BA √ η B η B η B (21)( ˙ π A ˙ π A ˙ π A ) = − g s m ( π B π B π B ) W BA + W BA √ W BA + iW BA W BA + iW BA W BA − iW BA − W BA + W BA √ W BA + iW BA W BA − iW BA W BA − iW BA − W BA √ . (22)In the precedent equations, the diagonal terms do not change the quark colorwhile the non diagonal connect different colors. However, since color mustbe conserved in any of the resulting differential equations, it is convenient tocheck further this point and also see if there is agreement between the fieldstrengths distribution in (21) and the color assignment to the different quarkand antiquark wave functions that describe interactions at the quantum level.In our approach, embedded in the eight field combinations are color quantawhich should be classified according to the assignment (20). First let us see thediagonal terms: W BA + W BA √ → ( r ¯ r − b ¯ b ) √ r ¯ r + b ¯ b − g ¯ g ) √ √ √ (cid:18) r ¯ r − b ¯ b − g ¯ g (cid:19) . (23)6n the same way − W BA + W BA √ → √ (cid:18) b ¯ b − r ¯ r − g ¯ g (cid:19) (24) − W BA √ → √ (cid:18) g ¯ g − r ¯ r − b ¯ b (cid:19) . (25)As should be expected, all diagonal terms have a similar status thus respecting SU (3) symmetry.For the non diagonal terms we have the following association: W BA + iW BA → r ¯ b + ¯ rb √ r ¯ b − b ¯ r √ √ r ¯ b (26) W BA − iW BA → √ b ¯ r (27) W BA + iW BA → √ r ¯ g, (28) W BA − iW BA → √ g ¯ r, (29) W BA + iW BA → √ b ¯ g, (30) W BA − iW BA → √ g ¯ b. (31)We cannot forget that our equations are classical. A quark of a certain colorcan interact with a gluon and change its color but this kind of interaction, de-scribed in QCD, is difficult to describe in classical lenguaje. If we can obtainsome information about the nature of color forces, we have to get some informa-tion about how does a quark move under a certain color field. To this end weshall first assume that there is a small lapse of time in which the particle movesunder an specific color field and to simplify even more, a colorless constant gluonfield. The color gluon field case will be discussed afterwards.Obviously, in the high energy regime, and in the conventional approach, oncethe nonlinear terms in the potentials B iµ are removed, the similarities with theelectromagnetic field are complete. In what follows and with the double purposeof showing in some detail how to obtain ~p ( t ) from the spinor equations and andto see whether the solution coincides with the electromagnetic one, they will besolved in the simple case of a constant electric-like field in some direction. Inthe electromagnetic case we know the solution: ~p ( t ) = ~Et. Assume that we have an isolated quark of color 3 in a background of electric-like E and magnetic-like B color fields (see equation (50) in Paper I). In such a case,the expression for the field would be 7 AB = 12 (cid:20) E E + i E E − i E −E (cid:21) + i (cid:20) B B + i B B − i B −B (cid:21) . (32)If for simplicity the field is pure electric-like, constant and in the z direction: W AB = (cid:18) E −E (cid:19) . (33)The equations to solve are˙ η A = g s m (cid:18) − √ W AB (cid:19) η B (34)and ˙ π A = − g s m π B (cid:18) − √ W BA (cid:19) (35)The solution in terms of the physical contravariant spinor components is givenby η = √ m exp (cid:18) − g s m √ E τ (cid:19) (36) η = √ m exp (cid:18) g s m √ E τ (cid:19) , (37)with equal solution for π A (it has been assumed that at proper time τ = 0 theparticle is at rest thereby the factor √ m ). Now, from the spinor transcriptionof the four momentum given by equation (1) it is relatively easy to find out theexpressions for the energy and spatial components as function of proper time: p = E = 12 √ m cosh (cid:18) g s m √ E τ (cid:19) (38) p = 12 √ m sinh (cid:18) g s m √ E τ (cid:19) . (39) p = p = 0Finally from the relationship between laboratory time and proper time whenthe driving force is constant t = 1 C sinh( Cτ ) , (40) p = 1 √ (cid:18) g s √ E (cid:19) t. (41)As in electrodynamics, this is an exact solution for a test particle (quark) ignor-ing radiation effects and under the special environment considered. We wouldlike to point out that equations (21) and the next, being the result of local8auge invariance, are exact. To clarify further this point note that the spinorfield strength G iAA ′ BB ′ is just the translation to the 2-spinor lenguaje of thefield tensor G iµν , in the particular case of G µν , the non linear term is G µν nl = ig s f ij B jµ B kν , (42)with f = f = √ /
2. To show explicitly that this term is included in G AA ′ BB ′ , one only needs to rewrite Eq (12) in terms of the potentials insteadof the field strengths.From the similitude found in equations (23) to (25) for the three colors, itis apparent that the result found for a quark in color 3 would also be the samefor the other two (as required from the SU (3) symmetry).Before entering in a few calculations related to the non diagonal elements in(26) that incessantly change the quarks colors, it seems in place a few commentson the difference between the color fields associated to diagonal entries and therest of them. While the diagonal fields, that do not change the quark colorcharge, fit in the classical picture of something laying in the background thatsilently act on the particles. The non diagonal fields do not fit into this kind ofpicture, instead, its representation or whatever one might call, is more adequateinside the QCD theory describing fields as some kind of quanta interactingwith the quarks in the particle sense. In plain words, a sea of color quanta ininteraction with another sea of quarks and antiquarks. As we shall see, in suchcircumstances, our classical equations enter in some trouble.In contrast to the diagonal terms, for the non diagonal entries in (21) wehave the following set of three differential equations system, connecting twodifferent color charges: ˙ η A = g s m ( W AB + iW AB ) η B ˙ η A = g s m ( W AB − iW AB ) η B (43)˙ η A = g s m ( W AB + iW AB ) η B ˙ η A = g s m ( W AB − iW AB ) η B (44)˙ η A = g s m ( W AB + iW AB ) η B ˙ η A = g s m ( W AB − iW AB ) η B (45)Comparing with the gluon-color content of the six relations (26) to (31), wesee that the quark color changes at a quark-gluon vertex in a way consistentwith the quantum description (see [3] Chapter 9). Let us now choose one of thesystem, for instance the second one. By derivation respect to proper time andconsidering both fields as constant: d dτ η A = g s m ( W AB + iW AB )( W AB − iW AB ) η B . (46)9o avoid the complications of the non commutative matrix algebra, both fieldswill be chosen as pure electric-like and in the z-direction. In such a case, theequation to solve is d dτ η A = g s m W AB η B , (47)with W AB = (cid:18) E + E E + E (cid:19) . (48)In the rest frame of the red quark, the solution is the same for both componentsof the spinor, omitting the color index for clarity η , = + √ m cosh( kτ ) , (49)where k = g s m q E + E . The solution for π A is similar only with a minus sign for the π component π , = ±√ m cosh( kτ ) . (50)Now from Eq.(1) for p AA ′ we find the interesting but unphysical result for a m = 0 particle: p = E = p = m √ ( kτ ) , (51)and p = p = 0.We know that conservation of energy and momentum forbids an electronto absorb, or emit, one photon (in its rest system or in any other) and thesame should apply to quarks and gluons. Looking at the initial system of equa-tions and the colors associated to the intervening fields in the process, from thequantum perspective, the process could be described as: ”a red quark absorbsa g ¯ r gluon becoming a green quark and shortly afterwards (to allow permis-sion from the Heisenberg principle) emits a r ¯ g gluon becoming again a redquark”. Notwithstanding that the sequence of color changes (see (29) and (28))is coherent in the equations involved, it is clear that the classical (continuous)description is in those cases, at least, inadequate, thereby my prior commentsabout the somewhat failure of the field classical description for the colored (nondiagonal) gluon fields. As mentioned at the Introduction, It was my purpose to explore the classicalaspects of QCD and obtain some results that might be of relevance. In fact thereis a similitude between the Lorentz Force of electrodynamics and its equivalentin chromodynamics . In Eq. (1), we find the translation to spinor form ( p AA ′ )10f the four momentum p µ which follows from the isomorphism between realfour-vectors and hermitian spinors of second rank. From (12) we could buildthe chromodynamic ”Lorentz Force” analogue as dp AA ′ dτ = G AA ′ BB ′ i p BB ′ . (52)However, in this form many (sometimes non classical) aspects revealed in thespinor equations (21)-(22) would be hidden (see Paper I). In the diagonal termsof the mentioned spinor equations of motion, we have found a certain paralelismwith electrodynamics while in the equations related to non diagonal terms thereare aspects either paradoxical or, perhaps, unveiling properties of color dynamicsthat should be worth of further study. References [1] Buitrago J. Results in Physics 6 (2016) 346-351[2] R. Penrose and W. Rindler
Spinors and Space-Time , Cambridge Mono-graphs in Mathematical Physics, Vol. 1, Cambridge Universtiy Press, Cam-bridge, England (1984/1986).[3] D. Griffiths