Gauge dependence of the gauge boson projector
aa r X i v : . [ h e p - ph ] J u l Gauge dependence of the gauge boson pro jector
E. M. Priidik Gallagher, Stefan Groote and Maria Naeem
Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia
Abstract
The propagator of a gauge boson, like the massless photon or the massive vec-tor bosons W ± and Z of the electroweak theory, can be derived in two differentways, namely via Green’s functions (semi-classical approach) or via the vacuum ex-pectation value of the time-ordered product of the field operators (field theoreticalapproach). Comparing the semi-classical with the field theoretical approach, the cen-tral tensorial object can be defined as the gauge boson projector, directly related tothe completeness relation for the complete set of polarisation four-vectors. In thispaper we explain the relation for this projector to different cases of the R ξ gauge andexplain why the unitary gauge is the default gauge for massive gauge bosons. Introduction
As it is familiar for the scalar and Dirac propagators, the propagator of the vector boson V between two space-time locations x and y can be considered as a two-point correlator,i.e. as the vacuum expectation value of the time ordered product of the vector potential atthese two locations, D µνV ( x − y ) = h |T { V µ ( x ) V ν ( y ) }| i . (1)However, in order to get to the momentum space representation of this propagator, oneneeds to use the completeness relation for the polarisation four-vectors. This is not aneasy task, as this completeness relation is not given uniquely for a complete set of fourpolarisation states. As it is well known, a massless vector boson like the photon hastwo polarisation states. For a massive vector boson ( W ± or Z ), in addition there is alongitudinal polarisation state. However, the addition of a time-like polarisation state isnot unique and depends on the gauge we use, as we will show in this paper. In order toget to this point, we construct the propagator of the vector boson in a semi-classical wayas Green’s function obeying the canonical equation of motion, derived as Euler–Lagrangeequation from the Lagrange density containing a gauge fixing term, L = − ∂ µ V ν ( ∂ µ V ν − ∂ ν V µ ) + 12 m V V µ V µ − ξ V ( ∂ µ V µ ) , (2)a result which will be derived in Sec. 5. ξ V is the gauge parameter in general R ξ gauge.The solution of the Euler–Lagrange equation leads to a propagator D µνV ( x − y ) = Z d k (2 π ) − iP µν ( k ) e − ik ( x − y ) k − m V + iǫ , P µνV ( k ) := η µν − (1 − ξ V ) k µ k ν k − ξ V m V (3)with a definite second rank tensor structure P µνV which we call the gauge boson projector.( η µν ) = diag(1; − , − , −
1) is the Minkowski metric.The paper is organised as follows. In Sec. 2 we introduce the gauge boson projector.As a naive extension of the completeness relation for the polarisation vectors fails, we offera pragmatic solution which will be explained in the following. In Sec. 3 we start with theLagrange density of the photon and explain why the solution of the corresponding Euler–Lagrange equation needs a gauge fixing term. For a general R ξ gauge we solve the equationfor the Green’s function. A recourse to historical approaches is needed to understand theoccurence of primary and secondary constraints. In Sec. 4 the quantisation of the photonfield is continued in a covariant manner. In Sec. 5 we explain the appearance of a mass term2ia the Higgs mechanism and the restriction of the gauge degrees of freedom in this case,leading to the unitary gauge as the default setting for massive vector bosons. In Sec. 6we explain and give an example for the gauge independence of physical processes. Ourconclusions and outlook are found in Section 7. For the basics we refer to Refs. [1, 2, 3, 4, 5]. The gauge boson projector as central tensorial object P µνV ( k ) in Eq. (3) takes the simplestform P µνV ( k ) = η µν for the Feynman gauge ( ξ V = 1). For Landau gauge ξ V = 0 one obtainsa purely transverse projector P µνV ( k ) = η µν − k µ k ν /k , and for the unitary gauge ξ V → ∞ one has P µνV ( k ) = η µν − k µ k ν /m V which is transverse only on the mass shell k = m V .But why do we talk about a projector at all? A comparison with the construction of thefermion propagator can help to explain the conceptual approach employed in this paper. As for the gauge boson propagator, there are in principle two ways to construct the fermionpropagator. As a Green’s function the fermion propagator has to solve the equation( iγ µ ∂ µ − m ) S ( x − y ) = iδ (4) ( x − y ) (4)equivalent to the Dirac equation ( iγ µ ∂ µ − m ) ψ ( x ) = 0 as the corresponding Euler–Lagrangeequation. In momentum space this equation reads ( p/ − m ) ˜ S ( p ) = i (with p/ := γ µ p µ ) whichcan be solved by ˜ S ( p ) = i/ ( p/ − m ). Note that the inverse of the matrix ( p/ − m ) is welldefined, since ( p/ − m )( p/ + m ) = p − m . Back to configuration space one has S ( x − y ) = Z d p (2 π ) ie − ip ( x − y ) p − m + iǫ ( p/ + m ) , (5)where we have added an infinite imaginary shift + iǫ to obtain a Feynman propagator. Onthe other hand, the fermion propagator is defined again as two-point correlator, i.e. as thevacuum expectation value of the time-ordered product of the spinor and the adjoint spinor, S ab ( x − y ) = h |T { ψ a ( x ) ¯ ψ b ( y ) }| i = X i =1 Z d p (2 π ) E ( ~p ) h u i ( ~p )¯ u i ( ~p ) e − ip ( x − y ) + v i ( ~p )¯ v i ( ~p ) e ip ( x − y ) i ab Z d p (2 π ) E ( ~p ) h ( γ µ p µ + m ) e − ip ( x − y ) + ( γ µ p µ − m ) e ip ( x − y ) i ab = ( iγ µ ∂ µ + m ) ab Z d p (2 π ) E ( ~p ) h e − ip ( x − y ) − e ip ( x − y ) i = ( iγ µ ∂ µ + m ) ab Z d p (2 π ) Z dp πi − e − ip ( x − y ) p − m + iǫ = ( iγ µ ∂ µ + m ) ab Z d p (2 π ) ie − ip ( x − y ) p − m + iǫ = Z d p (2 π ) ( p/ + m ) ab ie − ip ( x − y ) p − m + iǫ , (6)where we have started with the field operators ψ ( x ) = X i =1 Z d p (2 π ) q E ( ~p ) (cid:16) b i ( ~p ) u i ( ~p ) e − ipx + ˜ b † i ( ~p ) v i ( ~p ) e ipx (cid:17) (7)and ¯ ψ ( x ) = ψ † ( x ) γ with the only non-vanishing antimutators { b i ( ~p ) , b † j ( ~p ′ ) } = (2 π ) δ ij δ (3) ( ~p − ~p ′ ) , { ˜ b i ( ~p ) , ˜ b † j ( ~p ′ ) } = (2 π ) δ ij δ (3) ( ~p − ~p ′ ) , (8)where we have used the completeness relations X i =1 u i ( ~p )¯ u i ( ~p ) = γ µ p µ + m, X i =1 v i ( ~p )¯ v i ( ~p ) = γ µ p µ − m, (9)and, finally, where we have used Cauchy’s theorem to write the integral in a compactfour-dimensional form. The result is quite obviously the same as the one obtained via theGreen’s function. Still, one might become aware of the central link, given by the complete-ness relations. A similar construction should work also for the gauge boson propagator. As for the quantisation of the fermion field operator we summed over the spin polarisationstates i = 1 , λ .Still, the (silent) assumption that the summation runs over all possible (four) polarisationstates will have to be looked over again, as it will turn out. Up to that point, we usethe summation sign indexed by λ without specifying the set of polarisations it runs over.Therefore, starting with V µ ( x ) = X λ Z d k (2 π ) q ω ( ~k ) h ε µ ( ~k, λ ) a ( ~k, λ ) e − ikx + ε µ ∗ ( ~k, λ ) a † ( ~k, λ ) e ikx i (10)4ith [ a ( ~k, λ ) , a † ( ~k ′ , λ ′ )] = (2 π ) δ λλ ′ δ (3) ( ~k − ~k ′ ) and ω ( ~k ) = ~k + m V , the calculation of thetwo-point correlator leads to D µνV ( x − y ) = h |T { V µ ( x ) V ν ( y ) }| i = X λ Z d k (2 π ) ω ( ~k ) h ε µ ( ~k, λ ) ε ν ∗ ( ~k, λ ) e − ik ( x − y ) − ε ν ( ~k, λ ) ε µ ∗ ( ~k, λ ) e ik ( x − y ) i . (11)However, what kind of completeness relation we can use in this case? We know that thereare at least two physical polarisation directions which are orthogonal to each other and atthe same time orthogonal to the wave vector ~k , ~k · ~ε ( ~k, λ ) = 0 , ~ε ( ~k, λ ) · ~ε ( ~k, λ ′ ) = δ λλ ′ (12)( λ, λ ′ = 1 , ~ε ( ~k, ~ε ( ~k,
2) and ~k/ | ~k | span an orthonormal frame. Therefore, in particularthe usual three-dimensional basis ~e i can be expressed in this frame, ~e i = X λ =1 (cid:16) ~e i · ~ε ( ~k, λ ) (cid:17) ~ε ( ~k, λ ) + ( ~e i · ~k ) ~k~k = X λ =1 ε i ( ~k, λ ) ~ε ( ~k, λ ) + k i ~k~k . (13)As the usual basis is orthonormal, we conclude that δ ij = ~e i · ~e j = X λ =1 ε i ( ~k, λ ) ε j ( ~k, λ ) + k i k j ~k , (14)which can be rewritten as a first (three-dimensional) completeness relation, P ijV ( ~k ) = X λ =1 ε i ( ~k, λ ) ε j ∗ ( ~k, λ ) = δ ij − k i k j ~k . (15)Finally, considering ~ε ( ~k,
3) := ~k/ | ~k | as a third orthonomal polarisation vector, one obtains P ijV ( ~k ) = X λ =1 ε i ( ~k, λ ) ε j ∗ ( ~k, λ ) = δ ij , (16)where the complex conjugate has no effect on a real-valued basis but allows for the gener-alisation for instance to a chiral basis. A generalisation of this completeness relation tofour-vectors (with time component set to zero) is straightforward and leads to P µνV ( k ) = X λ =1 ε µ ( k, λ ) ε ν ∗ ( k, λ ) = η µ η ν − η µν (17) We will not make the chiral basis explicit though as we reserve λ = ± for something else. η µ = η µ . As before, an attempt can be done to switch the non-covariant part of theright hand side to the left hand side by defining a fourth (time-like) polarisation. However,in this simple form this attempt fails. ε ( k,
0) = ( η µ ) = (1; 0 , ,
0) does not give the correctsign, and the more involved trial ε ( k,
0) = ( i ; 0 , ,
0) is of no help here as the product withthe conjugate will remove the effect of the imaginary unit.
The canonical field quantisation in Eq. (10) is based on plane waves. This issue becomesproblematic if we consider a vector field complemented by a gauge fixing term, leading toa nontrivial dispersion of the solution in the case of a massive vector boson [6]. As wewill see in Sec. 5, the Proca equation can no longer be considered as a vector extension ofthe Klein–Gordon equation. Instead, the mass of the vector boson depends on the gaugeparameter ξ V . Accordingly, the canonical quantisation based on a particle with fixed masscannot be applied. However, in our approach we are able to circumvent the problem relatedto the canonical quantisation by using Green’s functions. Note that Green’s functions areclassical and, therefore, independent of the quantisation scheme. At this point we offer a pragmatic solution. As we know the explicit form of the gaugeboson propagator from the Green’s function approach employed before, we conclude that P µνV ( k ) = X λ ε µ ( k, λ ) ε ν ∗ ( k, λ ) = η µν − (1 − ξ V ) k µ k ν k − ξ V m V = P µνV ( k ) . (18)Therefore, the completeness relation depends on the gauge. The pragmatic solution tellsus that for Feynman gauge ξ V = 1 for instance one obtains P λ ε µ ( k, λ ) ε ν ∗ ( k, λ ) = η µν ,independent of whether we know which polarisations are summed over and how the explicitpolarisation vectors look like. However, we can speculate about how these two are relatedto each other. We can assure ourselves that a gauge boson on the mass shell has onlyvector components. In this case we obtain the Landau projector ( ξ V = 0) [7, 8, 9] − X λ =1 ε µ ( k, λ ) ε ν ∗ ( k, λ ) = η µν − k µ k ν k = P µν ( k ) (19)6ontaining only the vector component of the polarisation. Eq. (19) can be explicitly seenin the rest frame of the massive vector boson. For k = ( m V ; ~
0) one obtains η µν − k µ k ν m V = − = − X λ =1 ε µ ( ~k, λ ) ε ν ∗ ( ~k, λ ) (20)with ε ( ~k,
1) = (0; 1 , , ε ( ~k,
2) = (0; 0 , ,
0) and ε ( ~k,
3) = (0; 0 , , ξ V → ∞ ), containing also a scalarcomponent [8], X λ,λ ′ =0 η λ,λ ′ ε µ ( k, λ ) ε ν ∗ ( k, λ ′ ) = η µν − k µ k ν m V = P µν + k µ k ν k F S ( k ) = P µν ⊕ (21)with F S ( k ) = 1 − k /m V as the offshellness dominating the scalar component. Theappearance of the components η λλ ′ of the metric tensor η in polarisation space seems tosuggest that the summation over λ can be understood as the contraction of covariant withcontravariant components in polarisation spacetime, reserving for the polarisation vectorsthe role of a tetrad between ordinary spacetime and polarisation spacetime. This will beworked out in more detail in Sec. 4 in case of the photon (cf. Eq. (42)). In order to investigate the relation between completeness relation and propagator in detail,we start with the Lagrange density of the photon, L A = 12 ( ~E − ~B ) = − F µν F µν , F µν = ∂ µ A ν − ∂ ν A µ (22)Containing only the self energy of the photon, the Euler–Lagrange equations can be ob-tained by variation of the action integral S A = R L A d x . One obtains δS A = − Z δF µν F µν d x = − Z ( ∂ µ δA ν − ∂ ν δA µ )( ∂ µ A ν − ∂ ν A µ ) d x = − Z ∂ µ δA ν ( ∂ µ A ν − ∂ ν A µ ) d x = Z δA ν ∂ µ ( ∂ µ A ν − ∂ ν A µ ) d x, (23)where for the last step we have used integration by parts. In order to vanish for an arbitraryvariation δA ν of the gauge field, one has to claim that ∂ µ ( ∂ µ A ν − ∂ ν A µ ) = ∂ A ν − ∂ µ ∂ ν A µ = ( ∂ η µν − ∂ µ ∂ ν ) A µ = 0 . (24)7owever, the corresponding equation (a factor i for later convenience)( ∂ η µν − ∂ µ ∂ ν ) D µρA ( x ) = iη νρ δ (4) ( x ) (25)for the Green’s function D µρA ( x ) cannot be solved, as the operator ( ∂ η µν − ∂ µ ∂ ν ) is notinvertible. As found by Faddeev and Popov in 1967, this problem turns out to be deeplyrelated to the gauge degree of freedom [10]. The solution for this problem is given byamending the Lagrange density by a gauge fixing term, L A + = − F µν F µν − ξ A ( ∂ µ A µ ) , (26)the introduction of which can be understood on elementary level also as the addition ofa Lagrange multiplier times the square of ∂ µ A µ , restricting the solutions to those whichsatisfy the Lorenz gauge condition ∂ µ A µ = 0 proposed exactly a century earlier [11]. Thiscondition does not fix completely the gauge but eliminates the redundant spin-0 componentin the representation (1 / , /
2) of the Lorentz group, leaving a gauge degree of freedom A µ → A µ + ∂ µ f with ∂ f = 0. However, as the gauge field is not constrained a priori but via a Lagrange multiplier, instead of a single gauge condition one obtains a wholeclass of gauge conditions subsumed under the name of R ξ gauges. For ξ A → ξ A = 1 one obtains theFeynman gauge, and for ξ A → ∞ one ends up with the unitary gauge, to name a few. Varying the amended action functional with respect to the gauge field, in this case oneobtains ( ∂ η µν − (1 − ξ − A ) ∂ µ ∂ ν ) A µ = 0 and, therefore, ∂ η µν − − ξ A ! ∂ µ ∂ ν ! D µρA ( x ) = iη ρν δ (4) ( x ) (27)for the Green’s function. This equation can be solved. In momentum space the equationreads − k η µν − − ξ A ! k µ k ν ! ˜ D µρA ( k ) = iη ρν , (28)and by using the ansatz ˜ D µνA ( k ) = ˜ D g η µν + ˜ D k k µ k ν one obtains ( ξ A −
1) ˜ D g − k ˜ D k = 0and − k ˜ D g = i , i.e. D µνA ( x ) = Z d k (2 π ) e − ikx ( ˜ D g η µν + ˜ D k k µ k ν ) = Z d k (2 π ) − ie − ikx k η µν − (1 − ξ A ) k µ k ν k ! . (29)8epending on how the convention for the poles at k = 0 (i.e. at k = ± ω ( ~k ) = ±| ~k | ) isset, one obtains a retarded, advanced, or Feynman propagator (the latter not to be mixedup with the Feynman gauge). In the following we restrict our attention to the Feynmanpropagator, adding an infinitesimal imaginary shift + iǫ to the denominator. Even though the solution of Faddeev and Popov allows to deal with the calculation in aquite straightforward manner, in order to understand the situation more deeply it is worthto have a look at older approaches. A very valuable reference for this is the handbook ofKleinert [12] which will be used for the following argumentation.Starting again with the free Lagrange density (22), for a canonical field quantisation wehave to obtain the Hamilton density by performing a Legendre transformation. However,while the spatial components of the canonical momentum are given by the components ofthe electric field, the time component vanishes, π i ( x ) = ∂ L A ( x ) ∂ ˙ A i ( x ) = − F i ( x ) = E i ( x ) , π ( x ) = ∂ L A ( x ) ∂ ˙ A ( x ) = 0 . (30)According to Dirac’s classification [13], the property π ( x ) = 0 is a primary constraint onthe canonical momentum. Using the Euler–Lagrange equations, we get to the secondaryconstraint ∇ ~E ( ~x, t ) = 0 which is Coulomb’s law for free fields. The secondary constraintleads to an incompatibility for the canonical same-time commutator[ π i ( ~x, t ) , A j ( ~x ′ , t )] = iδ ij δ (3) ( ~x − ~x ′ ) . (31)This problem can be solved by introducing a transverse modification of the delta dis-tribution [12]. For the canonical quantisation, A ( ~x, t ) and (via Coulomb’s law) also ∇ ~A ( ~x, t ) cannot be considered as operators. Using Coulomb gauge ∇ ~A ( ~x, t ) = 0, onehas A ( ~x, t ) = 0 as well, a relation between the Coulomb and axial gauges as two examplesfor noncovariant gauges [14] established by Coulomb’s law for free fields. One obtains A µ ( x ) = Z d k (2 π ) q ω ( ~k ) X λ =1 (cid:16) ε µ ( ~k, λ ) a ( ~k, λ ) e − ikx + ε µ ∗ ( ~k, λ ) a † ( ~k, λ ) e ikx (cid:17) , (32)where the polarisation sum runs over the two physical polarisation states ( λ = 1 ,
2) only. In case of an electric source the right hand side is replaced by ρ ( ~x, t ). A µ ( x ). A much better choicewould be the Lorenz gauge ∂ µ A µ = 0. By using the gauge transformation of the firstkind A µ → A µ + ∂ µ λ , a scalar function λ ( x ) can be found so that after this transformation ∂ µ A µ = 0 is satisfied. Still, the Lorenz gauge does not fix the gauge degree of freedom com-pletely. Indeed, a gauge transformation of the second kind A µ → A µ + ∂ µ f with ∂ f = 0,also called restricted or on-shell gauge transformation, will change the vector potential in away that it still satisfies the Lorenz gauge constraint. The covariant quantisation methodis established by introducing a first type of gauge-fixing term [15], L AF = L A + L GF , L GF = − G ( x ) ∂ µ A µ ( x ) + ξ G ( x ) , ξ ≥ . (33)In this case there is no canonical momentum for G ( x ), and the Euler–Lagrange equationwill lead to the (secondary) constraint ξG ( x ) = ∂ µ A µ ( x ). The Euler–Lagrange equationsfor the vector potential read ∂ µ F µν ( x ) = ∂ A ν ( x ) − ∂ µ ∂ ν A µ ( x ) = − ∂ ν G ( x ), and applyingthe constraint one obtains ∂ A ν ( x ) − − ξ ! ∂ ν ∂ µ A µ ( x ) = 0 . (34)This is the same equation we obtain in case of the Faddeev–Popov approach. Applyingonce more ∂ ν , one obtains ∂ G ( x ) = 0, i.e. G ( x ) is a massless Klein–Gordon field. We continue with the quantisation procedure for the photon field in covariant form. Themanifestly covariant expression for the quantised photon field is given by A µ ( x ) = Z d k (2 π ) q ω ( ~k ) X λ =0 (cid:16) ε µ ( ~k, λ ) a ( ~k, λ ) e − ikx + ε µ ∗ ( ~k, λ ) a † ( ~k, λ ) e ikx (cid:17) . (35)For ξ = 1 (Feynman gauge) we can choose momentum-independent polarisation vectors ε µ ( λ ) = η µλ . Accordingly, these vectors obey the orthogonality and completeness relations η µν ε ∗ µ ( λ ) ε ν ( λ ′ ) = η λλ ′ , X λ,λ ′ η λλ ′ ε µ ( λ ) ε ν ∗ ( λ ′ ) = η µν . (36)Employing the apparatus of canonical quantisation, we are left with the canonical same-time commutators [ A µ ( ~x, t ) , A ν ( ~x ′ , t )] = [ ˙ A µ ( ~x, t ) , ˙ A ν ( ~x ′ , t )] = 0 and[ ˙ A µ ( ~x, t ) , A ν ( ~x ′ , t )] = iη µν δ (3) ( ~x − ~x ′ ) (37)10hich are the same as if the components are independent massless Klein–Gordon fields.However, the sign between the temporal components is opposite to the spatial sector,resulting also in [ a ( ~k, λ ) , a ( ~k ′ , λ ′ )] = [ a † ( ~k, λ ) , a † ( ~k ′ , λ ′ )] = 0 and[ a ( ~k, λ ) , a † ( ~k ′ , λ ′ )] = − η λλ ′ (2 π ) δ (3) ( ~k − ~k ′ ) . (38)As a consequence, states generated by applying a † ( ~k,
0) have a negative norm, h | a ( ~k, a † ( ~k ′ , | i = h | [ a ( ~k, , a † ( ~k ′ , | i = − (2 π ) δ (3) ( ~k − ~k ′ ) . (39)The only possibility to escape this problem is to amend the temporal creation operatorby one of the spatial ones. For instance, the states a † ( ~k, ± ) | i have both zero norm, as a † ( ~k, ± ) | i := (cid:16) a † ( ~k, ± a † ( ~k, (cid:17) / √ D ( ~x, t ) = 0 as an operator condition, as this is incontradiction with the canonical commutation rules. In order to guarantee the validity ofthe Lorenz condition D ( ~x, t ) = 0 at any time, one instead defines a physical state imposingFermi–Dirac subsidiary conditions [15, 16, 17] D ( ~x, t ) | ψ phys i = 0 , ˙ D ( ~x, t ) | ψ phys i = 0 . (40)resulting in a ( ~k, − ) | ψ phys i = 0 and a † ( ~k, − ) | ψ phys i = 0, i.e. both creation and annihilationoperator annihilate the physical state. Using [ a ( ~k, ± ) , a † ( ~k ′ , ∓ )] = − (2 π ) δ (3) ( ~k − ~k ′ ), forthe Hamilton operator one obtains H = − Z d k (2 π ) k X λ,λ ′ =0 η λλ ′ N n a † ( ~k, λ ) a ( ~k, λ ′ ) + a ( ~k, λ ) a † ( ~k, λ ′ ) o = Z d k (2 π ) k X λ =1 a † ( ~k, λ ) a ( ~k, λ ) − a ( ~k, +) a † ( ~k, − ) − a † ( ~k, +) a ( ~k, − ) ! , (41)where N {· · ·} indicates normal ordering with respect to the physical vacuum. Hence thesubsidiary condition makes the last two terms vanish for all physical states. For general R ξ gauges the orthogonality and completeness relations (36) have to be replaced by [12] P µν ( k ) ε ∗ µ ( k, λ ) ε ν ( k, λ ′ ) = η λλ ′ , X λ,λ ′ η λλ ′ ε µ ( k, λ ) ε ν ∗ ( k, λ ′ ) = P µν ( k ) . (42) Note that in contrast to Ref. [12] we integrate over the wave vector instead of summing it. Accordingto the usual agreement for normal ordering, there is no contribution to the vacuum energy soever. .1 The Gupta–Bleuler quantisation Even though the application of both subsidiary conditions leads to the correct physical re-sult, the treatment of (infinite) normalisations of the states dealt with in detail in Sec. 5.4.2of Ref. [12] is exhausting. For processes with at least one particle it is sufficient to imposeonly the first subsidiary condition a ( ~k, − ) | ψ “phys ′′ i = 0 , (43)leading to a pseudophysical state. This condition is the basis of the Gupta–Bleuler ap-proach to Quantum Electrodynamics [18, 19]. Note, however, that for a vacuum energy(for instance in cavities) the nonphysical degrees of freedom are not completely eliminated.In the Faddeev–Popov approach, this vacuum energy contribution will be removed by thenegative vacuum energy contribution of the Faddeev–Popov ghosts.As the operator in the Gupta–Bleuler subsidiary condition (43) contains only thepositive-frequency part, the operator G ( x ) is necessarily a nonlocal operator. On theother hand side, the vacuum state | “phys ′′ i has a unit norm which is an important ad-vantage of the Gupta–Bleuler formalism. However, the main virtue of the Gupta–Bleulerquantisation scheme is that the photon propagator is much simpler than the one obtainedwith the help of a four-dimensional (noncovariant) generalisation of (15), namely (29). Employing again the Green’s function approach, we can get still to another result. Asin Eq. (1), the free photon propagator is given by the vacuum expectation value of time-ordered product of the field operators at spacetime points x and y , D µνA ( x − y ) = h |T { A µ ( x ) A ν ( y ) }| i . (44)Using the invariance of physical quantities under gauge transformations A µ ( x ) → A µ ( x ) + ∂ µ λ ( x ) (45)with some arbitrary scalar function λ ( x ), for the propagator one obtains D µνA ( x − y ) = h |T { A µ ( x ) A ν ( y ) } i → = h |T { A µ ( x ) A ν ( y ) }| i + ∂ µx h |T { λ ( x ) A ν ( y ) }| i ∂ νy h |T { A µ ( x ) λ ( y ) }| i + ∂ µx ∂ νy h |{ λ ( x ) λ ( y ) }| i = D µνA ( x − y ) + ∂ µx D νA ( x − y ) + ∂ νy D µA ( y − x ) + ∂ µx ∂ νy D A ( x − y ) , (46)where D µA ( x − y ) = h |T { λ ( x ) A ν ( y ) }| i and D A ( x − y ) = h |T { λ ( x ) λ ( y ) } i are mixed andscalar propagators. Fourier transformed to momentum space, one obtains˜ D µνA ( k ) → ˜ D µνA ( k ) + k µ ˜ D νA ( k ) + ˜ D µA ( k ) k ν + k µ k ν ˜ D A ( k )= ˜ D µνA ( k ) + k µ (cid:16) ˜ D νA ( k ) + 12 k ν ˜ D A ( k ) (cid:17) + (cid:16) ˜ D µA ( k ) + 12 ˜ D A ( k ) k µ (cid:17) k ν . (47)In a similar way as the gauge field is added in the Lagrange density, replacing the partialderivative by a covariant derivative in order to be able to absorb contributions from localphase transformations of the field operators in transforming according to Eq. (45), thepropagator has to be extended in order to comply with the same transformations (45).The appropriate form of the propagator to comply with this is D µνA ( x − y ) = Z d k (2 π ) − iP µν ( k ) e − ikx k + iǫ , P µν = η µν −
12 ( k µ l ν ( k ) + l µ ( k ) k ν ) , (48)where l µ ( k ) is a four-component function of the wave vector k , the explicit form of whichturns out again to depend on the gauge. Taking for instance l µ ( k ) = k µ /k , one endsup again with the Landau gauge, and for l µ ( k ) = 0 one reaches Feynman gauge. Athird possibility is given by the light cone mirror of the four-vector k + = ( k ; ~k ) = k , l ( k ) = k − / | ~k | with k − = ( k ; − ~k ). Note that the four-component object k − is not acovariant four-vector, called antiscalar in Ref. [12]. We call it light cone mirror of k . Forthe “light-cone form” of the photon projector P µν extracted from Eq. (48) in this case,the two nonphysical polarisation directions are eliminated. This can be seen with a simplecalculation for ε ( ~k,
1) = (0; 1 , , ε ( ~k,
2) = (0; 0 , ,
0) and k ± = | ~k | (1; 0 , , ± η µν − | ~k | ( k µ + k ν − + k µ − k ν + ) = − = − X λ =1 ε µ ( ~k, λ ) ε ν ∗ ( ~k, λ ) . (49)Identifying ε ( ~k, ± ) = k ± / √ | ~k | , one gets back to the Fermi–Dirac (or Gupta–Bleuler)nonphysical modes, concluding that the given combination of momentum vector k + andlight cone mirror k − will eliminate the nonphysical modes from the photon projector.13 The gauge boson propagator
As for the photon field, the vanishing of the temporal component of the canonical momen-tum of the massive gauge boson is a primary constraint. However, there is no secondaryconstraint. Replacing V µ ( x ) → V µ ( x ) + ∂ µ λ ( x ) and inserting this in the Euler–Lagrangeequation of the Lagrange density without gauge fixing L V = − F µν F µν + m V V µ V µ , ∂ µ F µν + m V V ν = (cid:16) ( ∂ + m V ) η µν − ∂ µ ∂ ν (cid:17) V µ = 0 , (50)one obtains m V ∂ µ λ ( x ) = 0 which admits only the constant solution λ ( x ) = λ . At thesame time, the application of ∂ ν to Eq. (50) leads to m V ∂ ν V ν = 0, i.e. the Lorenz gauge bydefault. These two results are closely related to each other as well as to the nonvanishingmass of the gauge boson. The gauge degree of freedom is reduced by one, leaving threeindependent components for the polarisation vector. Actually, for the massive gauge bosonitself the gauge fixing term is not necessary at all, as the operator in the second expressionin Eq. (50) is invertible. This is the reason why the gauge boson projector for the on-shellgauge boson field is given by default by the projector in unitary gauge.As it is convenient to consider the photon as the massless limit of a vector boson,one has to add a gauge fixing term to the Lagrange density to allow for a proper limit.Therefore, with the following consideration we are back to the Faddeev–Popov methodwith a gauge fixing term allowing for a general R ξ gauge. Usually, the gauge bosons (except for the photon) obtain a mass via the spontaneoussymmetry breaking of the scalar Higgs field φ in the framework of the electroweak Glashow–Weinberg–Salam (GWS) theory. At the same time one obtains Goldstone bosons as “scalarpartners” of the gauge bosons. The masslessness of the photon is established due to thefact that the corresponding scalar partner is the Higgs boson which “sets the stage” andkeeps the photon from gaining a mass. A detailed outline of the Higgs mechanism canbe found e.g. in Refs. [4, 5]. Here we only briefly sketch the appearance of the Goldstonebosons and the occurence of mass terms. Given the spontaneously broken Higgs field by ψ = 1 √ h ( x ) + ih ( x ) h + h ( x ) + ih ( x ) ! , (51)14he scalar part of the Lagrange density can be expanded in the fields h i ( x ) to obtain L φ = ( D µ φ ( x )) † ( D µ φ ( x )) + λh φ † ( x ) φ ( x ) − λ (cid:16) φ † ( x ) φ ( x ) (cid:17) = 12 X i =1 ( D µ h i ( x )) ( D µ h i ( x )) − λh h ( x ) + O ( h i ( x ) ) . (52)The second term in this expansion gives a mass m H = h √ λ to the Higgs boson field h ( x ), while the masses of the gauge bosons are obtained from the action of the covariantderivative D µ = ∂ µ − ig B µ − ig ~W µ ~σ (53)at the constant part (proportional to h ) of the Higgs field. One obtains( D µ φ ) † ( D µ φ ) = h h ( g + g ) Z µ Z µ + g ( W + µ W − µ + W − µ W + µ ) i . (54)This has to be compared with the kinetic contributions L W W = − F µν ( U (1)) F µν ( U (1)) − X i =1 F iµν ( SU (2)) F iµν ( SU (2))= − ∂ µ B ν ( ∂ µ B ν − ∂ ν B µ ) − X i =1 ∂ µ W iν ( ∂ µ W iν − ∂ ν W iµ ) . (55)With W ± µ = 1 √ W µ ∓ iW µ ) , W µ = g A µ + g Z µ q g + g , B µ = g A µ − g Z µ q g + g (56)one identifies the masses m A = 0, m Z = h q g + g / m W = h g / D µ φ ( x )) † ( D µ φ ( x )) gives risealso to a mixing of vector and scalar bosons, ih g √ ∂ µ ( h + ih ) W − µ − ih g √ ∂ µ ( h − ih ) W + µ + h q g + g ∂ µ h Z µ = im W ( ∂ µ h + W ) W − µ − im W ( ∂ µ h − W ) W + µ + m Z ( ∂ µ h Z ) Z µ , (57)where it was logical to define h ± W := ( h ± ih ) / √ h Z := h . Using the property thatthe Lagrange density is determined only up to a total derivative, these nonphysical mixingcontributions will finally be cancelled by appropriate additions to the gauge fixings in thegauge fixing terms − ξ A G A − ξ Z G Z − ξ W G ± W G ∓ W , (58)15here G A = ∂ µ A µ , G Z = ∂ µ Z µ − ξ Z m Z h Z , G ± W = ∂ µ W ± µ ∓ iξ W m W h ± W . (59)In addition to the gauge fixing and the cancellation of the boson mixings, we finally ob-tain mass terms also for the Goldstone bosons. The stage is now set for calculating thepropagators both for massive vector gauge bosons and the corresponding scalar Goldstonebosons. As an example we deal with the Z boson and the Goldstone boson field h Z . For the Z boson one obtains a contribution L Z = − ∂ µ Z ν ( ∂ µ Z ν − ∂ ν Z µ ) + 12 m Z Z µ Z µ − ξ Z ( ∂ µ Z µ ) (60)to the Lagrange density. The corresponding equation for the Green’s function reads ∂ η µν − − ξ Z ! ∂ µ ∂ ν + m Z η µν ! D µρZ ( x ) = iη ρν δ (4) ( x ) , (61)and this Proca equation is solved by D µνZ ( x ) = Z d k (2 π ) − ie − ikx k − m Z η µν − (1 − ξ Z ) k µ k ν k − ξ Z m Z ! . (62)For the Goldstone boson field h Z one obtains L h Z = 12 ( ∂ µ h Z )( ∂ µ h Z ) − ξ Z ξ Z m Z h Z , (63)leading to the equation − ( ∂ + ξ Z m Z ) D h Z ( x ) = iδ (4) ( x ) for the Green’s function solved by D h Z ( x ) = Z d k (2 π ) ie − ikx k − ξ Z m Z . (64)Note the ξ Z dependence of the latter Green’s function, also found in the longitudinal partof the corresponding vector boson Green’s function. For the Landau gauge ξ Z = 0 forinstance the mass dependence vanishes in these parts. In this context it is worth notingthat the classical equivalence to the Lorenz gauge is directly seen from Eqs. (59). On theother hand, while for Feynman gauge ( ξ Z = 1) both vector and Goldstone bosons carry16 mass m Z and the propagators are quite similar, for the unitary gauge ( ξ Z → ∞ ) theGoldstone propagator vanishes, and for the vector boson propagator one obtains D µνZ ( x ) (cid:12)(cid:12)(cid:12) ξ Z →∞ = Z d k (2 π ) − ie − ikx k − m Z η µν − k µ k ν m Z ! . (65)This means that for unitary gauge the Higgs boson is the only scalar boson that is propa-gated. This fact makes calculations using the unitary gauge particularly attractive, as thescalar sector is mainly absent. Finally, we obtain the same results also for the W ± bosonand collect our results in Eq. (3) in the Introduction. Even though the gauge boson propagator depends on the R ξ gauge via the gauge parameter ξ , this has no influence on particle processes. In order to understand this, note that massivevector bosons (like W ± and Z ) have to decay into pairs of fermions. Therefore, in exclusiveprocesses the vector boson line is terminated by a fermion line. To continue with the Z boson, as the simplest example we can calculate a Z boson propagator, terminated “on theleft” by a fermion line f and “on the right” by a fermion line f . For our considerations itdoes not matter whether for the particular process the fermion lines constitute a fermion–antifermion pair generated by (or annihilated to) the Z boson, or whether it is a fermion(or antifermion) which emits (or absorbs) the gauge boson. The gauge independence ofthe process can be shown in each of these cases.In momentum space the Z boson propagator reads˜ D µνZ ( k ) = − ik − m Z η µν − (1 − ξ Z ) k µ k ν k − ξ Z m Z ! . (66)It can be easily seen that this propagator can be decomposed into two parts [7],˜ D µνZ ( k ) = − ik − m Z η µν − k µ k ν m Z ! − k µ k ν m Z ik − ξ Z m Z . (67)While the first part is the propagator in unitary gauge, the second part is cancelled bythe propagator of the neutral Goldstone boson field h Z . In order to show this, we replacethe full (gauge-dependent) propagator by the second term only, for this part of the matrixelement obtaining (using the Feynman rules from Appendix A2 of Ref. [5])¯ u ( p ′ ) ieγ µ (cid:16) g − f Λ − + g + f Λ + (cid:17) u ( p ) − k µ k ν m Z ik − ξ Z m Z ! ¯ u ( p ′ ) ieγ ν (cid:16) g − f Λ − + g + f Λ + (cid:17) u ( p )17 e m Z ¯ u ( p ′ ) /k (cid:16) g − f Λ − + g + f Λ + (cid:17) u ( p ) ik − ξ Z m Z ¯ u ( p ′ ) /k (cid:16) g − f Λ − + g + f Λ + (cid:17) u ( p ) (68)with k = p − p ′ = p ′ − p and Λ ± = (1 ± γ ) /
2. The propagator part is now reduced to thepropagator of the neutral Goldstone boson. Inserting the corresponding outer momentumdifferences for k and using the Dirac equations, one obtains em Z ¯ u ( p ′ ) /k (cid:16) g − f Λ − + g + f Λ + (cid:17) u ( p ) = em f m Z ( g − f − g + f )¯ u ( p ′ ) γ u ( p ) ,em Z ¯ u ( p ′ ) /k (cid:16) g − f Λ − + g + f Λ + (cid:17) u ( p ) = − em f m Z ( g − f − g + f )¯ u ( p ′ ) γ u ( p ) . (69)Taking into account that g − f = I f − s W Q f s W c W , g + f = s W Q f c W ⇒ g − f − g + f = I f s W c W (70)with s W = sin θ W , c W = cos θ W the sine and cosine of the Weinberg angle, Q f the electriccharge (in units of the elementary charge e ) and I f the weak isospin of the fermion, thecontribution (68) is indeed cancelled by the process with the Z boson replaced by theneutral Goldstone boson, leaving us with the gauge boson propagator in unitary gauge.Note that in the 1960s and 1970s, the independence of physical processes under gaugetransformations were discussed as an equivalence theorem for point transformations of the S matrix [20, 21, 22, 23, 24]. Also recently there are controversies about whether physicalprocesses including vector bosons are gauge invariant (see e.g. Refs. [25, 26]). As an example for how this cancellation of the gauge dependence works out, we calculatefirst order electroweak corrections to the self energy of a fermion. As particular case wedeal with the first order electroweak self energy to the top quark. The first correctionswhich we denote as baseline corrections are shown in Fig. 1. For the correction (a1) by aphoton one obtains i Π ta = Z d D k (2 π ) D ( − ieQ t γ ν ) i ( q/ + /k + m t )( q + k ) − m t ( − ieQ t γ µ ) − ik η µν − (1 − ξ A ) k µ k ν k ! = − e Q t Z d D k (2 π ) D γ µ ( q/ + /k + m t ) γ µ (( q + k ) − m t ) k − (1 − ξ A ) /k ( q/ + /k + m t ) /k (( q + k ) − m t )( k ) ! . (71)Considering this correction between onshell Dirac states ¯ u ( q ) and u ( q ), for the second partone obtains ¯ u ( q ) /k ( q/ + /k + m t ) /ku ( q ) = (2 qk + k )¯ u ( q ) /ku ( q ) . (72)18 t tA (a1) t t tZ t b tW (a2) (a3) t t t χ Z t b t χ (b2) (b3)Figure 1: top quark self energy diagramsHowever, using principles of dimensional regularisation, one obtains Z d D k (2 π ) D (2 qk + k ) /k (( q + k ) − m t )( k ) = Z d D k (2 π ) D (( q + k ) − m t − q + m t ) /k (( q + k ) − m t )( k ) = ( − q + m t ) Z d D k (2 π ) D /k (( q + k ) − m t )( k ) = 0 . (73)Therefore, for the correction (a1) the gauge dependence drops out, and one obtains i Π ta = − e Q t m t (cid:16) ( D − A ( m t ) + 4 m t B ( m t ; m t , m A ) (cid:17) , (74)where A ( m ) and B ( q ; m , m ) are the one- and two-point functions, A ( m ) = Z d D k (2 π ) D k − m , B ( q ; m , m ) = Z d D k (2 π ) D q + k ) − m ) ( k − m ) , (75)and the photon mass m A is used as regularisator.For the correction (a2) by the Z boson the occurence of a vector boson mass does notallow for the same conclusion. However, a first naive approach can be tried in which the19auge dependence drops out in the sum of the corrections by the Z boson and by thecorresponding Goldstone boson χ Z . In Feynman gauge one obtains i Π ta = Z d D k (2 π ) D ieγ ν (cid:18) g − t − γ g + t γ (cid:19) i ( q/ + /k + m t )( q + k ) − m t × ieγ µ (cid:18) g − t − γ g + t γ (cid:19) − ig µν k − m Z ,i Π tb = Z d D k (2 π ) D em t s W m W γ i ( q/ + /k + m t ) k − m t em t s W m W γ ik − m Z . (76)On the other hand, for unitary gauge there is no Goldstone contribution and one stayswith the correction by the Z boson, i Π t ′ a = Z d D k (2 π ) D ieγ ν (cid:18) g − t − γ g + t γ (cid:19) i ( q/ + /k + m t )( q + k ) − m t × ieγ µ (cid:18) g − t − γ g + t γ (cid:19) − ik − m Z g µν − k µ k ν m Z ! . (77)Looking at the difference i Π ta + i Π tb − i Π t ′ a = e m t m W s W A ( m Z ) (78)one realises that the difference does not vanish. However, as the difference is proportionalto the one-point function A ( m Z ), one might think of tadpole contributions to be taken intoaccount. Tadpole corrections by vector and Goldstone bosons are shown in Fig. 2.For Feynman gauge one obtains i Π tc = 12 (cid:18) − iem t s W m W (cid:19) i − m H Z d D k (2 π ) D iem W g µν c W s W ! − ig µν k − m Z = − De m t c W s W m H Z d D k (2 π ) D k − m Z ,i Π td = 12 (cid:18) − iem t s W m W (cid:19) i − m H Z d D k (2 π ) D − iem H s W m W ! ik − m Z = − e m t s W m W Z d D k (2 π ) D k − m Z . (79)Note the vanishing momentum square for the tadpole tail (Higgs boson). The factor 1 / Z boson is its own antiparticle. As theGoldstone boson is absent for unitary gauge (i.e. does not propagate), the contribution(d2) is obviously the one which compensates the difference on the side of the Feynmangauge. However, once again the contribution (c2) will be different for unitary gauge whereone obtains i Π t ′ c = − e m t c W s W m H Z d D k (2 π ) D k − m Z D − k m Z ! = − ( D − e m t c W s W m H A ( m Z ) . (80)20 tH Z t tHW (c2) (c3) t tH χ Z t tH χ (d2) (d3)Figure 2: top quark self energy tadpole diagramsFinally, this difference will be compensated by the corresponding ghost contributionshown in Fig. 3. For the tadpole with ghost loop u Z one obtains i Π te = − (cid:18) − iem t s W m W (cid:19) i − m H Z d D k (2 π ) D − iem W ξ Z c W s W ! ik − ξ Z m Z , (81)where the minus sign comes from the closed ghost loop. For unitary gauge ( ξ Z → ∞ ) thecontribution remains finite. However, the dependence on the inner momentum k disappearsand, therefore, there is no ghost contribution either. On the other hand, for Feynman gauge( ξ Z = 1) one obtains i Π te = e m t c W s W m H A ( m Z ) . (82)Therefore, taking into account baseline vector and Goldstone corrections as well as tadpolevector, Goldstone and ghost corrections we obtain that the result for Feynman gauge isthe same as the one for unitary gauge.The situation is similar in case of the corrections by W ± , χ ± and u ± . However, notethat in this case i Π ta + i Π tb − i Π t ′ a = e m t | V tb | m W s W A ( m W ) . (83)21 tHu Z t tH u (e2) (e3)Figure 3: top quark self energy tadpole ghost diagramsEven though we take into account only the bottom quark in the loop, the sum in theloop has to run over all down-type quarks. Because of this fact and the unitarity of theCabibbo–Kobayashi–Maskawa matrix, the factor | V tb | will not appear in the final result. As the parts related the two massive vector bosons Z and W ± to the self energy of thefermion show, for the choice of unitary gauge one needs only two instead of five contri-butions, namely the two contributions related to the vector boson itself. Unitary gaugemeans 1 /ξ = 0, i.e. the absence of the gauge fixing term. Indeed, the gauge fixing term isnot necessary at all if the gauge boson carries a mass. The equation (cid:16) − η µν ( k − m V ) + k µ k ν (cid:17) ˜ D µρV ( k ) = iη ρν (84)can be solved again by the ansatz ˜ D µνV ( k ) = ˜ D g η µν + ˜ D k k µ k ν , in this case with the solution˜ D g = − i/ ( k − m V ) and ˜ D k = − ˜ D g /m V , leading to the propoagator in unitary gauge,˜ D µνV ( k ) = − ik − m V η µν − k µ k ν m V ! . (85) The gauge boson projector as the central tensorial object in the propagator of the vectorgauge boson is closely related to the completeness relation for the polarisation vectors.A generalisation of the completeness relation to four-dimensional spacetime is proposedin a pragmatic way. Using this approach, we could identify the polarisation vectors as22etrad fields relating ordinary spacetime to polarisation spacetime (see Eq. (42). Whilethe photon projector could be expressed by mirrors on the light cone (cf. Eq. (48)), theprojector for massive gauge bosons turned out to be expressed in unitary gauge by default.In particular, using the example of first order fermion self energy corrections we could showthat physical processes do not depend on the gauge degree of freedom.From the different treatment of the massless photon and the massive vector bosons wecan draw the conclusion that the photon might not be considered as mass zero limit of thevector boson. Indeed, at least the degree of freedoms in this limit is not continuous. Thisbehaviour is seen also for observables related to the spin of particles, known as spin-flipeffect (see e.g. Refs. [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]). Fundamentally differentLie group structures for massive and massless particles were investigated in Ref. [39], andthe considerations in Ref. [40] allow for a relation of mass and spin. Interesting enough, incombining Refs. [39, 40] a massive particle is constitued by two massless chiral non-unitarystates based on the (massless) momentum vector and the light cone mirror of this, relatingback to the light cone representation of the photon projector. These roughly sketchedrelations will be analysed in detail in a forthcoming publication.
Acknowledgements
We thank J. G. K¨orner for useful discussions on the subject of this paper. The researchwas supported by the European Regional Development Fund under Grant No. TK133, andby the Estonian Research Council under Grant No. PRG356.
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