On quantization of massive non-Abelian gauge fields
aa r X i v : . [ h e p - ph ] O c t On quantization of massive non-Abelian gaugefields
Tsuguo MOGAMI ∗ RIKEN, Hirosawa 2-1, Wako-shi, Saitama, 351-0198 JapanJune 30, 2007
Abstract
A simpler method of quantization is given for massive gauge the-ories. This method gives the same results as those of the conven-tional massive gauge theory with ghost and Higgs fields under theHiggs mass. Besides, we point out physical importance of helicity zerostates in non-Abelian gauge theories even in massless case. Further-more, forms of mass terms that were impossible before, e.g. symmetricmass, are possible now. Applying our method to SU (2) × U (1) sym-metry has no particular difficulty, and gives a variant of the standardmodel without the Higgs boson. We consider a gauge field theory with the following Lagrangian density. L = −
14 ( F aµν ) − ξ ( ∂ µ A aµ ) + m A aµ ) , (1) F µν = ∂ µ A ν − ∂ ν A µ − ig [ A µ , A ν ] . Here A aµ is a SU ( N ) gauge field, g is a small coupling constant. We will takethe limit of ξ → ∞ after renormalization and obtain physical quantities[1].This is different from the unitary gauge, which takes the limit far beforerenormalization and becomes unrenormalizable. ∗ e-mail: [email protected] π µ = δL/δ ′ A µ and A µ . Their commutatorsare [ A aµ ( x , t ) , π bν ( y , t )] = iδ ab δ νµ δ ( x − y ) , (2)and the other commutators are all zero. After a tedious calculation to obtaineigenstates of the non-interacting part of the Hamiltonian, we have threeeigenstates with energy k = q k + m and one eigenstate with energy k ′ = q k + M for a 3-momenta k , where M = √ ξm . The commutation relationsfor their creation and annihilation operators are[ a aσ ( k ) , a b † σ ′ ( k ′ )] = δ ab δ σσ ′ δ ( k − k ′ ) , ( σ, σ ′ = L, + , − )[ a aS ′ ( k ) , a b † S ′ ( k ′ )] = − δ ab δ ( k − k ′ ) , (3)and zero for the others. The definition of the vacuum is a σ ( k ) | i = 0 ( σ = S ′ , L, + , − ). The state created by a † S ′ ( k ) is a negative norm state and will becalled “scalar polarization state” or “ S ′ state”. Please note that the energyof this state q k + M is positive, and this state have a polarization vector k ′ µ /M = ( k ′ , k ) /M . The other polarization states have polarization vectorsthat are orthogonal to k µ = ( k , k ). There is no reason to consider thosethree states to be unphysical since they have positive norm and a positiveenergy.The field A µ may be expressed in terms of these creation and annihilationoperators as A aµ ( x ) = Z d k (2 π ) / X σ = L, + , − a aσ ( k ) e − ikx √ k ε σµ ( k ) + a aS ′ ( k ) e − ik ′ x q k ′ q ξ ε S ′ µ ( k ′ ) + c . c . . (4)Here, ε σ ’s are polarization vectors and ε σ ( k ) · ε σ ′ ( k ) = − δ σσ ′ ,ε σ ( k ) · ε S ( k ) = 0 , ( σ, σ ′ = L, + , − ) ε S ( k ) · ε S ( k ) = 1 . (5)We chose the spatial three vector of ε Lµ to be proportional to k . If we write2own their explicit form when k is in 1 direction, ε Sµ = ( k , k, , /m,ε Lµ = ( k, k , , /m,ε + µ = (0 , , , + i ) / √ ,ε − µ = (0 , , , − i ) / √ . (6)The polarization represented by ε + µ will be called “longitudinal polarization”or “ L polarization” here . ε + and ε − are said to have “transverse” polariza-tion. Here we defined “ S state” to be a state having mass m and polarizationvector k µ /m for the convenience of later calculation, even though canonicalquantization of our theory (1) gives S ′ state and not S ′ state. Please notethat we have S ′ state and not S state in the expansion of A µ (4) too.The propagator of our gauge field is D µν ( k ) = − ik − m ( g µν − k µ k ν m ) + k µ k ν m − ik − M . (7)This propagator propagates S ′ state with mass M , but not S state with mass m , following the canonical quantization above.This theory satisfies the following basic physical requirements. First, thistheory is simply renormalizable since it has local operators of dimension lessthan or equal to four[2]. Since its S-matrix is matrix elements of e − iHT ,its unitarity is obvious. Because the particle with negative norm ( S ′ state)has large mass M ≡ √ ξm , negative norm state will not appear below thisenergy level. This theory will contain only positive norm particles aftertaking ξ → ∞ limit.On the other hand, the conventional massive gauge theory, which shouldbe compared to our theory, is a gauge theory that acquires mass with Higgsmechanism. Let us consider Higgs-Kibble model for instance. Its Lagrangiandensity is L HK = −
14 ( F aµν ) + | D µ Φ | − λ † Φ − | v | ) , (8)where its gauge group is SU (2), and D µ is covariant derivative. The fieldΦ is a two-dimensional complex field belonging to SU (2) doublet, and has Please be careful that some of the literatures use “longitudinal” to mean scalar polar-ization in this letter. v . It is convenient to parameterize Φ with realscalar fields: Φ( x ) = 1 √ χ ( x ) + iχ ( x ) , v + ψ ( x ) − iχ ( x )) T . (9)Here we add gauge-fixing terms: L gf = − (1 / ξ )( ∂ µ A aµ + ξmχ a ) + i ¯ c a [ ∂ µ D abµ + ξm δ ab + ( g/ ξm ( ψδ ab + f acb χ c )] c b , (10)where the numbers f abc are structure constants. This way of gauge-fixingis called “ R ξ gauge”[3]. The fields ¯ c and c are Grassmann fields, which arecalled Faddeev-Popov[4] ghosts. Using (9), we may write the free part of theLagrangian as L HK + L gf = −
14 ( F aµν ) − ξ ( ∂ µ A aµ ) + m A aµ ) + 12 ( ∂ µ χ )( D µ χ ) − M χ + 12 ( ∂ µ ψ ) − µ H ψ + g A aµ ( χ a ∂ µ ψ − ψ∂ µ χ a )+ i ¯ c ∂ µ ( D µ c ) + M ¯ cc, (11)where the interaction terms that is linear in A aµ are kept for the purpose oflater 1-loop calculation. We see that the gauge fields, Higgs χ , Higgs ψ andthe ghost fields acquire mass m (= gv/ , M (= √ ξm ) , µ H (= √ λv ) and M respectively.Our theory gives the same physical results as that of the conventionalHiggsed gauge theory when the energy is lower than the Higgs masses ( E ≪ µ H , E ≪ M ). That is because the difference between the two theories comesfrom Higgs and ghost fields and the effects of these heavy fields are small forlower energy phenomena.Here we introduce an easier method of calculating amplitudes, whichgives the same result as using propagator (7) and ordinary Feynman rules.Repeating small gauge transformation A µ ( x ) → A µ ( x ) + N − D µ φ ( x ) /m for N times and taking N → ∞ limit, A µ = T a A aµ is transformed into A µ → e igφ/m ( i∂ µ /g + A µ ) e − igφ/m = A µ + ∂ µ φ/m + ig [ φ/m, A µ ] + · · · ≡ A ′ µ , (12)4here T a ’s are generators of the symmetry group. The original Lagrangian(1) may be rewritten as L = −
14 ( F aµν ) + m A aµ ) − mA aµ ∂ µ φ a + 12 ξm φ by introducing a scalar auxiliary field φ a . Applying transformation (12), ourLagrangian is transformed into L → −
14 ( F aµν ) + m A aµ ) −
12 ( ∂ µ φ )( D µ φ ) + 12 M φ + (higher order terms in φ ) ≡ L φ . (13)To calculate n-point function Z D A e i R d x L ( x ) A µ ( x ) A ν ( x ) · · · ≡ h A µ ( x ) A ν ( x ) · · ·i A,ξ , (14)we use the fact that it is equivalent to= Z D φ D A e i R d x L φ ( x ) A ′ µ ( x ) A ′ ν ( x ) · · · ≡ h A ′ µ ( x ) A ′ ν ( x ) · · ·i φ,A,ξ (15)using transformation (12). In the latter form, A µ is now a Proca field andits propagator is D ′ µν ( k ) = − ik − m ( g µν − k µ k ν m ) , (16)which does not propagate S ′ state. The effects of S ′ states are replacedby those of the scalar field φ , since the two formula (14) and (15) shouldbe equivalent. This equivalence can also be proven by repeatedly applyingidentity (24) to the Feynman diagrams of the original theory (14).Now, let us check that S ′ states do not give any contribution to our theoryin the limit ξ → ∞ at 1-loop level. As an example, two point function h A µ ( x ) A ν ( x ) i A,ξ = h A ′ µ ( x ) A ′ ν ( x ) i φ,A,ξ (17) The higher order interaction terms of φ in L φ are unrenormalizable. The unrenor-malizable contributions to the Green functions from these terms will precisely be canceledwith the unrenormalizable contributions from the Proca field, because we know that theequivalent original theory is renormalizable from the beginning. h ∂ µ φ ( x ) A ′ ν (0) i /m will be discarded later even if wecalculate it. In general, because any physical quantity such as S-matrixis computed from the gauge invariant combinations of A µ , the differencebetween A ′ µ and A µ , i.e. ∂ µ φ/m + i [ φ/m, A µ ] + · · · , does not give any physicaleffect. Then, we only have to consider h A µ ( x ) A ν ( x ) i φ,A,ξ . (18)The lowest nontrivial order contributions to this two-point function are thegraphs shown in fig.1. (One can directly check that the scalar loop in fig.1is equivalent to the looping of a S ′ state in (14) by applying identity (24).)This scalar loop gives ig f abc f a ′ bc Z d k (2 π )
14 ( q − k ) µ ( q − k ) ν ik − M i ( q − k ) − M , (19)and the momentum integration gives= ig f abc f a ′ bc ( q g µν − q µ q ν ) ( − log Λ + (1 − M q ) / arccot s − M q ) (20)abbreviating the constant factors, and Λ is the cutoff. Only a small contri-bution of q O ( q /M ) is left after renormalization for q ≪ M , and it goesto zero in the limit of ξ → ∞ .Next, Let us check that our theory gives the same result as that of theconventional Higgsed theory. Our theory removes S ′ states by making itsmass M very large. On the other hand, the conventional theory removes S ′ states by using ghost and Higgs particles. The action of ghost fields takesthe form of + i ¯ c [ ∂ µ D µ + M ] c as is shown in (11). This is apparently same as the action of φ in L φ φ [ ∂ µ D µ + M ] φ, and the ghosts give just − q µ proportional part is ignored. Where, the6egative sign comes from anticommutation of the Grassmann field. Finally,the action for Higgs field is − χ [ ∂ µ D µ + M ] χ, and gives the same loop integration as (19). These contributions add up tocompletely cancel contributions from S ′ states. This cancellation is said tobe necessary to keep unitarity of S-matrix.This fact tells us the reason why it was believed that Higgs mechanismwas indispensable for massive gauge bosons. A loop of the S ′ state has asymmetry factor 1 /
2, because it is effectively a real scalar field φ . On theother hand, the ghost loop has factor −
1, which is twice as much as needed.This excessive correction is inevitable because anti-commuting field cannothave kinetic term of the form ( ∂ µ c )( ∂ µ c ), but only ( ∂ µ ¯ c )( ∂ µ c ) is possible. Toeliminate this overcorrection, another unphysical field of mass M is required.That is the Higgs field.Even in the limit of m →
0, the ghosts give twice excessive correctionover the effect from negative norm states (i.e. S ′ states). What does iteliminate? Originally, ghost fields are introduced as a trick to make onlypicking transverse polarizations up consistent[5]. Then the ghosts should becanceling not only S ′ states but also L states.Actually, the amplitude of scattering of particles giving rise to final statethat include L state does not vanish even in m → iM µν = ( ig ) ¯ v ( p + ) { γ µ T a p − 6 k − m f γ ν T b + γ ν T b k − 6 p + − m f γ µ T a } u ( p )+ ig ¯ v ( p + ) γ ρ ′ T c u ( p ) × D ρ ′ ρ ( k ) f abc V µνρ , (21)where m f is fermion mass, the antifermion has momentum p + and spinor¯ v ( p + ), the other fermion has p and u ( p ), momentum conservation requires p + + p = − k , k + k + k = 0, and V µνρ = g µν ( k − k ) ρ + g νρ ( k − k ) µ + g ρµ ( k − k ) ν . (22) The loop of the negative norm states has the same sign as that of positive norm scalarloop. Please do not confuse having negative norm with having negative sign for a loop. ε ∗ ν ( k ) = k ν /m . The amplitude gets iM µν ε ∗ µ ( k ) ε ∗ ν ( k ) = − ig ε ∗ µ ( k ) m − f abc ( k g µρ − k µ k ρ − m g µρ ) × D ρ ′ ρ ( k )¯ v ( p + ) T c γ ρ u ( p ) , (23)where we have used V µνρ k ν = − ( k g ρµ − k ρ k µ ) + ( k g ρµ − k ρ k µ ) , (24)and from Dirac equations for final state spinors u, ¯ v , k ρ ¯ vT c γ ρ u = − ¯ v ( p + ) T c [( p + + m f ) + ( p − m f )] u ( p ) = 0 . (25)When the other boson has L or transverse polarization ( ε · k = 0 , k = m ), iM µν ε ∗ µ ( k ) k ν = 0 . (26)When the other boson has ε ∗ µ ( k ) = k µ /m polarization we have, iM µν k µ k ν /m = 1 , (27)and finite in the m → S states will not appear in our theory,since our propagator (7) does not include S polarization of mass m , and S ′ states are too heavy to appear. Now let us calculate the amplitude for twolongitudinal final bosons. For calculation, we define ε Bµ ( k ) ≡ ε Lµ ( k ) − k µ m = m | k | + q k + m (1 , − k | k | ) . With this, we have ε Lµ ( k ) ε Lν ( k ) = − k µ k ν /m + ε Lµ ( k ) k ν /m + k µ ε Lν ( k ) /m + ε Bµ ( k ) ε Bν ( k ) . (28)We already calculated that the second and the third term is zero, and thefourth term is negligible in m → ε B is proportional to m. Thenwe get M µν ε L ∗ µ ( k ) ε L ∗ ν ( k ) = − M µν k µ k ν /m , (29) This vanishing occurs even in higher order in perturbation because this fact followsconservation of the current. And, this holds for every mass for k because neither k = m nor k = M was used. L -polarized bosons is finite andequal to (27) in the m → L stateparticles is finite as m →
0, since ( ε L ) keeps constant even though elementsof ε Lµ ( k ) get infinitely large in this limit.) Now we see that the ghosts areeliminating this appearance of L states too by that excessive factor 2.Only when m = 0, it is possible to remove not only S ′ states but also L states from the theory. This is the reason why it was possible only for masslesscases to quantize gauge theories using ghosts. Let us see it in an actual cal-culation. For the states with momentum ( q m + k , , , k ), the transversepolarization vectors are (0 , , ,
0) and (0 , , , k / q m + k into 1 direction, their momentum gets ( q m + k + k , k , , k ).Then the polarization vector (0 , , ,
0) is boosted into ε = ( k q m + k , q m + k q m + k , , , where k = ( k , , k ). This is not transversely polarized in this Lorenz framesince its spatial component is not orthogonal to k . Using the longitudinaland transverse polarization vectors of this frame ε L ( k ) = 1 m ( | k | , q m + k k | k | ) ,ε T ( k ) = 1 | k | (0 , k , , − k ) , vector ε is decomposed as ε = mk | k | q m + k ε L + q m + k q m + k ε T . (30)Thus, when the theory is massive, transverse polarization will be mixed with L polarization by boosting, and then we cannot make covariant theory with-out L states. Only when the theory is massless, it is possible to remove L states by hand, because the coefficient of ε L in (30) goes to zero. Thenthe conventional gauge theory with ghosts is consistent despite its excessivecorrection that eliminates L states.Now, we see that two different kinds of gauge theories are possible when m = 0. Both of the theories satisfy unitarity and have positive norms and9ositivity of energy. One kind is the conventional massless theories withghosts only. The other is m → L stateswith finite probability. Taking m → m = 0. The latter arepossible for general m including m → m → L polarization states may be restated as helicity= 0 states. We now knowthat these states have finite probability. Therefore, even in such theories asgravity and supersymmetry, we should consider adopting other helicity statesthan the highest and lowest helicity states.Furthermore, the forms of mass terms that were impossible before arepossible in our theory. For example, all the gauge bosons in a representationof SU ( N ) could not have the same mass except for N = 2 in the conventionaltheories. To restore the physical freedoms which were removed by the ghostfields, we need N − N real freedoms, it is impossible for all the gaugebosons to have mass except for N = 2 (Higgs-Kibble model). If the Higgsfield is in adjoint representation, the gauge boson related to the unbrokengenerator will remain massless. Our theory, however, does not have suchlimitation related to Higgs mechanism.If we apply our method to the electroweak theory[6], we obtain a renor-malizable theory, which gives the same physical results as the conventionalHiggsed theory under the Higgs mass. Our Lagrangian is L std = −
14 ( ∂ µ A ν − ∂ ν A µ − ig [ A µ , A ν ]) −
14 ( ∂ µ B ν − ∂ ν B µ ) (31) − ξ ( ∂ µ A aµ ) − ξ ( ∂ µ B µ ) + 12 M W (( A µ ) + ( A µ ) + ( A µ − tan θ W B µ ) ) , where A aµ and B µ are SU (2) and U (1) Y gauge fields respectively, M W is weakboson mass, and θ W is Weinberg angle. Unitarity and renormalizability are10bvious in this theory.We don’t have negative norm states in this theory (31). Let us call theunbroken U (1) gauge symmetry U (1) γ and the corresponding gauge boson“photon”. The S ′ polarization states of other gauge bosons will not have anyeffect, because they are simply heavy as is the case of Higgs-Kibble model.Finally, the photon S ′ states and L states will be neither emitted nor absorbedas they didn’t in QED, because the breaking of U (1) γ is proportional to 1 /ξ ,and negligible when ξ is very large.This theory (31) agrees with the conventional Higgsed electroweak theorywhen the energy level is much lower than the Higgs mass µ H and ξ is verylarge. The difference between these two theories in the R ξ gauge is theghost and Higgs fields. The Higgs fields and heavier ghosts will not appearbecause of their heavy mass √ ξM Z , √ ξM W and µ H . The massless ghostcorresponding to photon will be cancelled with the S ′ states and L statesowing to BRS symmetry[7] related to U (1) γ . Therefore, the two theoriesagree in that S ′ and L states will have no physical effect.Finally, let us consider the letponic part of the standard model. The trou-ble is that the fermion mass terms explicitly break SU (2) × U (1) symmetry.In the conventional understanding, BRS symmetry should be conserved toprevent appearance of negative norm states. Then, consistent theories areobtained[8] only when we introduced mass terms via the vacuum expectationvalue of the Higgs field. On the other hand, our theory don’t need such mech-anism and may simply introduce fermion mass terms, because the negativenorm states are ruled out by simply giving unphysically large mass. References [1] T. D. Lee, C.N. Yang, Phys. Rev. 128 (1962) 885.[2] N.N. Bogoliubov, O.S. Parasiuk, Acta Math. 97 (1957) 227.K. Hepp, Comm. Math. Phys. 2 (1966) 301.W. Zimmerman, Comm. Math. Phys. 15 (1969) 208.[3] K. Fujikawa, B.W. Lee and A.I. Sanda, Phys. Rev. D6 (1972) 2923.[4] L.D. Faddeev, V.N. Popov, Phys. Lett. 25B (1967) 29.[5] R.P. Feynman, Acta Phys. Polonica 24 (1963) 697.116] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264.A. Salam, in “Elementary Particle Theory” ed. by N.Svartholm(Almquist and Forlag, 1968) p367.S.L. Glashow, Nucl. Phys. 22 (1961) 579.[7] C. Becchi, A. Rouet and R. Stra, Comm. Math. Phys, 42 (1975) 127.I.V. Tyutin, Lebedev preprint FIAN No. 39 (in Russian) 1975.[8] G. ’tHooft and M. Veltman, Nucl. Phys. B50 (1972) 318.
Figure Legends
Figure1.
The 1-loop graphs contributing to the two-point function. Thewavy lines of the gauge bosons represent the propagator in (16) here.