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Gauge Dependence of the Gauge Boson Projector

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Gauge Dependence of the Gauge Boson Projector Tutorial Gauge Dependence of the Gauge Boson Projector Priidik Gallagher, Stefan Groote * and Maria Naeem Institute of Physics, University of Tartu, W. Oswaldi 1, 50411 Tartu, Estonia; [email protected] (P.G.); [email protected] (M.N.) * Correspondence: [email protected] Received: 5 June 2020; Accepted: 24 July 2020; Published: 28 July 2020 Abstract: The propagator of a gauge boson, like the massless photon or the massive vector bosons W± and Z of the electroweak theory, can be derived in two different ways, namely via Green’s functions (semi-classical approach) or via the vacuum expectation value of the time-ordered product of the field operators (field theoretical approach). Comparing the semi-classical with the field theoretical approach, the central tensorial object can be defined as the gauge boson projector, directly related to the completeness relation for the complete set of polarisation four-vectors. In this paper we explain the relation for this projector to different cases of the Rx gauge and explain why the unitary gauge is the default gauge for massive gauge bosons. Keywords: massive gauge fields; Green’s functions; unitary gauge; completeness relations 1. 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, that is, as the vacuum expectation value of the time ordered product of the vector potential at these two locations, mn m n DV (x − y) = h0jT fV (x)V (y)gj0i. (1) However, in order to get to the momentum space representation of this propagator, one needs to use the completeness relation for the polarisation four-vectors. This is not an easy task, as this completeness relation is not given uniquely for a complete set of four polarisation states. As it is well known, a massless vector boson like the photon has two polarisation states. For a massive vector boson (W± or Z), in addition there is a longitudinal polarisation state. However, the addition of a time-like polarisation state is not unique and depends on the gauge we use, as we will show in this paper. In order to get to this point, we construct the propagator of the vector boson in a semi-classical way as Green’s function obeying the canonical equation of motion, derived as Euler–Lagrange equation from the Lagrange density containing a gauge fixing term, 1 m n n m 1 2 m 1 m 2 L = − ¶mVn(¶ V − ¶ V ) + mVVmV − (¶mV ) , (2) 2 2 2xV a result which will be derived in Section5. xV is the gauge parameter in general Rx gauge. The solution of the Euler–Lagrange equation leads to a propagator 4 mn −ik(x−y) m n mn Z d k −iP (k)e mn k k D (x − y) = , P (k) := hmn − (1 − x ) , (3) V ( )4 2 2 V V 2 2 2p k − mV + ie k − xV mV mn mn with a definite second rank tensor structure PV which we call the gauge boson projector. (h ) = diag(1; −1, −1, −1) is the Minkowski metric. Particles 2020, 3, 543–561; doi:10.3390/particles3030037 www.mdpi.com/journal/particles Particles 2020, 3 544 The paper is organised as follows. In Section2 we introduce the gauge boson projector. As a naive extension of the completeness relation for the polarisation vectors fails, we offer a pragmatic solution which will be explained in the following. In Section3 we start with the Lagrange density of the photon and explain why the solution of the corresponding Euler–Lagrange equation needs a gauge fixing term. For a general Rx gauge we solve the equation for the Green’s function. A recourse to historical approaches is needed to understand the occurence of primary and secondary constraints. In Section4 the quantisation of the photon field is continued in a covariant manner. In Section5 we explain the appearance of a mass term via 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 Section6 we explain and give an example for the gauge independence of physical processes. Our conclusions and outlook are found in Section7. For the basics we refer to References [1–5]. 2. The Gauge Boson Projector mn The gauge boson projector as central tensorial object PV (k) in Equation (3) takes the simplest mn mn form PV (k) = h for the Feynman gauge (xV = 1). For Landau gauge xV = 0 one obtains a mn mn m n 2 purely transverse projector PV (k) = h − k k /k , and for the unitary gauge xV ! ¥ one has mn mn m n 2 2 2 PV (k) = h − k k /mV which is transverse only on the mass shell k = mV. But why do we talk about a projector at all? A comparison with the construction of the fermion propagator can help to explain the conceptual approach employed in this paper. 2.1. Construction of the Fermion Propagator As for the gauge boson propagator, there are in principle two ways to construct the fermion propagator. As a Green’s function the fermion propagator has to solve the equation m (4) (ig ¶m − m)S(x − y) = id (x − y), (4) m equivalent to the Dirac equation (ig ¶m − m)y(x) = 0 as the corresponding Euler–Lagrange equation. m In momentum space this equation reads (p/ − m)S˜(p) = i (with p/:= g pm) which can be solved by S˜(p) = i/(p/ − m). Note that the inverse of the matrix (p/ − m) is well defined, since (p/ − m)(p/ + m) = p2 − m2. Back to configuration space one has Z d4 p ie−ip(x−y) S(x − y) = (p/ + m), (5) (2p)4 p2 − m2 + ie where we have added an infinite imaginary shift +ie to obtain a Feynman propagator. On the other hand, the fermion propagator is defined again as two-point correlator, that is, as the vacuum expectation value of the time-ordered product of the spinor and the adjoint spinor, Particles 2020, 3 545 Sab(x − y) = h0jT fya(x)y¯b(y)gj0i 2 Z 3 d p 1 h −ip(x−y) ip(x−y)i = ui(~p )u¯i(~p )e + vi(~p )v¯i(~p )e ∑ ( )3 (~ ) ab i=1 2p 2E p Z 3 d p 1 h m −ip(x−y) m ip(x−y)i = (g pm + m)e + (g pm − m)e (2p)3 2E(~p ) ab Z 3 m d p 1 h −ip(x−y) ip(x−y)i = (ig ¶m + m) e − e ab (2p)3 2E(~p ) Z 3 Z 0 −ip(x−y) m d p dp −e = (ig ¶m + m) ab (2p)3 2pi p2 − m2 + ie Z 4 −ip(x−y) m d p ie = (ig ¶m + m) ab (2p)4 p2 − m2 + ie Z d4 p ie−ip(x−y) = (p/ + m) , (6) (2p)4 ab p2 − m2 + ie where we have started with the field operators 2 Z d3 p 1 y(x) = b (~p )u (~p )e−ipx + b˜†(~p )v (~p )eipx (7) ∑ ( )3 p i i i i i=1 2p 2E(~p ) and y¯(x) = y†(x)g0 with the only non-vanishing antimutators † 0 3 (3) 0 ˜ ˜† 0 3 (3) 0 fbi(~p ), bj (~p )g = (2p) dijd (~p − ~p ), fbi(~p ), bj (~p )g = (2p) dijd (~p − ~p ), (8) where we have used the completeness relations 2 2 m m ∑ ui(~p )u¯i(~p ) = g pm + m, ∑ vi(~p )v¯i(~p ) = g pm − m, (9) i=1 i=1 and, finally, where we have used Cauchy’s theorem to write the integral in a compact four-dimensional form. The result is quite obviously the same as the one obtained via the Green’s function. Still, one might become aware of the central link, given by the completeness relations. A similar construction should work also for the gauge boson propagator. 2.2. Construction of the Gauge Boson Propagator As for the quantisation of the fermion field operator we summed over the spin polarisation states i = 1, 2 (corresponding to up and down spin), it is natural to assume that for quantisation of the gauge boson field operator we have to sum over the polarisations l. Still, the (silent) assumption that the summation runs over all possible (four) polarisation states will have to be looked over again, as it will turn out. Up to that point, we use the summation sign indexed by l without specifying the set of polarisations it runs over. Therefore, starting with Z d3k 1 h i Vm(x) = #m(~k, l)a(~k, l)e−ikx + #m∗(~k, l)a†(~k, l)eikx , (10) ∑ (2p)3 q l 2w(~k ) Particles 2020, 3 546 ~ † ~ 0 0 3 (3) ~ ~ 0 2 ~ ~ 2 2 with [a(k, l), a (k , l )] = (2p) dll0 d (k − k ) and w (k ) = k + mV, the calculation of the two-point correlator leads to mn m n DV (x − y) = h0jT fV (x)V (y)gj0i Z d3k 1 h i = #m(~k, l)#n∗(~k, l)e−ik(x−y) − #n(~k, l)#m∗(~k, l)eik(x−y) . (11) ∑ ( )3 ~ l 2p 2w(k ) However, what kind of completeness relation we can use in this case? We know that there are at least two physical polarisation directions which are orthogonal to each other and at the same time orthogonal to the wave vector~k, ~ ~ ~ ~ 0 k ·~#(k, l) = 0, ~#(k, l) ·~#(k, l ) = dll0 (12) (l, l0 = 1, 2).
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