Charged massive vector boson propagator in a constant magnetic field in arbitrary ξ -gauge obtained using the modified Fock-Schwinger method
aa r X i v : . [ h e p - t h ] O c t Charged massive vector boson propagatorin a constant magnetic field in arbitrary ξ -gaugeobtained using the modified Fock-Schwinger method Iablokov S. N. ∗ P.G. Demidov Yaroslavl State Univeristy, Yaroslavl, RussiaA.A. Kharkevich Institute for Information Transmission Problems, Moscow, Russia
Kuznetsov A. V.
P.G. Demidov Yaroslavl State Univeristy, Yaroslavl, Russia (Dated: October 13, 2020)We applied a recently published modified Fock-Schwinger (MFS) method to find the exact solutionof the propagator equation for a charged vector boson in the presence of a constant magnetic fielddirectly in the momentum space as a sum over Landau levels in arbitrary ξ -gauge. In contrast to thestandard approaches for finding propagators, MFS method demonstrated several improvements interms of computational complexity reduction and revealed simple internal structures in intermediateand final expressions, thus allowing to obtain new useful representations of the propagator. I. INTRODUCTION
Analysis of elementary particle loop processes in ex-treme conditions, such as strong magnetic fields, requiresa knowledge of particle propagators where the field effectsare taken into account exactly.There exist at least two naturally arising scales ofstrong magnetic fields. The first one corresponds tothe so-called critical, or Schwinger value B e = m e /e ≃ . × G, which is the strength of the quantizingfield for an electron (hereafter, we use the Planck units: ~ = 1 , c = 1). The fields of the order of B e are connectedwith the concept of magnetars, i.e. neutron stars whichevolution is driven largely by magnetic fields [1]. Otherexamples when such strong (and even stronger) magneticfields could possibly manifest themselves include the ex-periments at modern colliders, e.g. with non-central colli-sions of heavy ions [2], and high-intensity electromagneticwaves generated by a system of lasers [3–5].The second scale is defined by the mass of the gaugeboson m W : B W = m W /e ≃ . × G [6]. In thiscase, there arises a question of applicability of the Stan-dard Model in these conditions, namely, the stability ofelectroweak vacuum at B → B W . As it was shown inRef. [7], the radiation corrections act to prevent the in-stability of the electroweak vacuum in such strong fields.A knowledge of the vector-boson propagator at thescale of B W expanded over the Landau levels can be help-ful for investigations of processes in the early Universe.An example of possible influence of the quantizing effectof the strong magnetic field on the W propagator wasconsidered in Ref. [8]. A model was used of dynamicalgeneration of the primordial magnetic field in the earlyUniverse by ferromagnetic domain walls, see [9] and thereferences cited therein. Due to this effect, the decay ∗ [email protected] width of ν e → e − W + has a “sawtooth” profile, thus lead-ing to the significant decrease of neutrino mean free pathat some neutrino energies. If the lepton-antilepton asym-metry (induced by the CP violation in the lepton sector)has arisen before the electroweak phase transition, lead-ing to overabundance of neutrinos over antineutrinos inthe Universe, then the considered mechanism would pro-vide an overabundance of W + over W − inside domainwalls. The subsequent decay of the W boson by dom-inant quark channels could have influenced the baryonasymmetry in the early Universe.Quantum field propagators can be either constructedas time-ordered correlation functions of the field opera-tors or found from the propagator equation provided bythe path integral formalism. The first approach requiresto solve the corresponding field equation in order to con-struct a quantum field operator. The solutions should benormalized and, for fields with spin, orthogonalized withrespect to some spin operator. As an intermediate com-putational step, the spin parts should be multiplied andsummed over. However, these spin-related manipulationsare absent in the second approach. The solution of thepropagator equation already contains all the spin partssummed over, and the δ -function in the right-hand sideensures the correct normalization. Therefore, the latterapproach seems to require less computational effort toobtain a propagator.Additional difficulties arise when some external field,e.g. electromagnetic, is present. In this case, one can nolonger rely on the translational invariance when apply-ing Fourier transform. In some scenarios (e.g. constantexternal magnetic field), this could be remedied throughthe use of the Fock-Schwinger (FS) proper-time method[10, 11]. However, this approach leads to the expressionfor the propagator as an integral over the proper-time pa-rameter. In order to get the expression in the momentumspace as a sum over Landau levels, which is convenient forthe calculation of scattering amplitudes, one can applyintegration techniques described in [12].In this paper, we applied a recently published modi-fied Fock-Schwinger (MFS) approach [13] to the solutionof the massive vector boson propagator equation in thepresence of a constant magnetic field in arbitrary ξ -gauge.It allowed to obtain the propagator expression directly inthe momentum space as a sum over the Landau levels.The paper is structured as follows. First, we briefly men-tion known results and approaches for finding quantumfield propagators in external electromagnetic fields, andprovide a known expression for the case of W-boson, how-ever, obtained using the East-Coast metric convention.Next, we discuss the main steps of the MFS method anduse it to get the expression for the massive vector bosonpropagator in the same metric convention. Finally, weapply the above-mentioned integration techniques [12] tothe proper-time representation expression of the propa-gator to show that, in East-Coast metric convention, thetransformed expression coincides with the result obtainedusing the MFS approach. In the end, we also derive thepropagator equation in the West-Coast metric convention(commonly used in modern particle physics) and providethe corresponding solution. II. KNOWN APPROACHES AND RESULTS
A history of calculations of charged particle propaga-tors in a magnetic field is rather long. The exact expres-sion for the electron propagator in a constant uniformmagnetic field was first obtained by J. Schwinger [10] inthe Fock-Schwinger [14] proper time formalism. Thereexist a number of papers where another forms of thepropagator are derived. For example, the case of a su-perstrong magnetic field was analyzed in Ref. [15] wherethe contribution from the ground Landau level to theelectron propagator was obtained. In Ref. [16], the prop-agator was transformed from the Schwinger form [10] toa series over an integer number n , where the poles in theexpansion terms corresponded to Landau levels. An ex-act proof for the propagator [16] to be an expansion overthe Landau levels, was presented in Ref. [17], where theelectron propagator in a constant uniform magnetic fieldwas obtained in the same form using the exact solutionsof the Dirac equation. A misprint in the formula for thepropagator [16] was later corrected in Refs. [18, 19], but without any comments. In Ref. [19], the expansion of theelectron propagator as a power series of the intensity ofa magnetic field was presented.The formula for the propagator of a charged scalarparticle expanded over Landau levels was obtained forthe first time in Ref. [20].The propagator equation (and the corresponding so-lution) for W -boson in a constant magnetic field foran arbitrary ξ -gauge was previously obtained in [21](see also [22]) using the East-Coast metric convention g µν = ( − , + , + , +), and was given by: H µν G νρ ( X, X ′ ) = δ µρ δ (4) ( X − X ′ ) , (1)where H µν = (cid:0) ΠΠ + m (cid:1) δ µν − ieQF µν + (cid:18) ξ − (cid:19) Π µ Π ν . (2)The following standard notations are also assumed: X µ = ( t, x, y, z ) , X µ = g µν X ν = ( − t, x, y, z ) ,∂ µ = ( ∂ t , ∇ ) , ∂ µ = ( − ∂ t , ∇ ) ,D µ = ∂ µ + i eQA µ , Π µ = − i D µ , ΠΠ = Π ν Π ν . Here, Q is a dimensionless charge of the W ( − ) particle,which is equal to −
1, and e > F µν = ∂ µ A ν − ∂ ν A µ (3)for a constant magnetic field along the z -axis is given by F = − F = B , with the rest of components beingequal to 0. The solution of Eq. (1) was obtained usingthe Fock–Schwinger method and, in proper-time repre-sentation, reads: G µν ( X, X ′ ) = φ ( X, X ′ ) Z d p (2 π ) e i( p ( X − X ′ )) G µν ( p ) , (4)with the translationally non-invariant phase factor φ ( X, X ′ ) = exp (cid:18) − i eQ X ′ ρ F ρσ X σ (cid:19) , (5)and the Fourier transform of the translationally invariantpart G µν ( p ) = i Z ∞ d s cos( βs ) e − i s ( p k + p ⊥ tan( βs ) βs ) (cid:26) e − i s ( m − i ε ) h δ µ k ν + δ µ ⊥ ν cos(2 βs ) − Q ϕ µν sin(2 βs ) i + (6)+ 1 m (cid:16) e − i s ( m − i ε ) − e − i s ( ξm − i ε ) (cid:17) (cid:20)(cid:18) p µ − Q ( ϕp ) µ tan ( βs ) (cid:19)(cid:18) p ν − Q ( pϕ ) ν tan ( βs ) (cid:19) −− i β (cid:18) Q ϕ µν + δ µ ⊥ ν tan( βs ) (cid:19)(cid:21)(cid:27) . Here, the subscript ⊥ stands for the components or-thogonal to the direction of the magnetic field, namely,belonging to the plane ( x, y ) for the field directed alongthe z -axis, while the subscript k stands for t and z compo-nents in this case. We also introduced the dimensionlessmagnetic field tensor ϕ µν = F µν /B and an auxiliary no-tation β = eB . The tensor indices of four-vectors andtensors standing inside the parentheses are contractedconsecutively, e.g. ( ϕp ) µ = ϕ µλ p λ .Alternatively, in Ref. [23], the W-boson propagatorwas constructed in the Feynman gauge ( ξ = 1) as a time-ordered product of the field operators that were expandedas a series over the solutions of the corresponding wave-equation. Finally, in Ref. [24] (see also [12]), the proper-time parameter s in (6) was integrated out, giving theFourier transform that coincided (for ξ = 1) with theresult of Ref. [23].Knowledge of different representations of the chargedparticle propagators in an external magnetic field isimportant because it allows to consider the conditionsof their applicability. There exist several precedentswhen misunderstanding of such conditions led to incor-rect studies. For instance, a calculation of the neu-trino self-energy operator in a magnetic field was per-formed in Refs. [25, 26] by analyzing the one-loop di-agram ν → e − W + → ν . The authors restricted them-selves by the contribution to the electron propagator fromthe ground Landau level. As it was shown in Ref. [27],in that case the contribution from the ground Landaulevel did not dominate due to the large electron virtual-ity, and contributions from other levels were of the sameorder. Ignoring such a fact led the authors [25, 26] toincorrect results. Another example of this kind was anattempt to reanalyze the probability of the neutrino de-cay ν → e − W + in an external magnetic field in the limitof ultra-high neutrino energies, calculated via the imag-inary part of the one-loop amplitude of the transition ν → e − W + → ν . Initially, the result was obtained inRef. [28]. Later, the calculation was repeated in [29]where authors insisted on another result. The third inde-pendent calculation [30] confirmed the result of Ref. [28].The most likely cause of the error in Ref. [29] was thatthe authors used only linear terms in the expansion ofthe W -boson propagator over the electromagnetic tensor F µν whereas the quadratic terms were essential as well.The modified Fock-Schwinger approach, developed in[13] and discussed below, provides additional useful rep-resentations of the W-boson propagator. III. OUTLINE OF THE MODIFIEDFOCK-SCHWINGER APPROACH
Here we present a brief overview of the modified Fock-Schwinger (MFS) approach. We are to solve the followingpropagator equation: H ( ∂ X , X ) G ( X, X ′ ) = δ (4) ( X − X ′ ) . (7) As in the original Fock-Schwinger (FS) method (see, e.g.,[11]) one should, first, switch to the following integralrepresentation: G ( X, X ′ ) = i Z ∞ d s U ( X, X ′ ; s ) . (8)Considering U ( X, X ′ ; s ) as some sort of an evolution op-erator satisfying a Schr¨odinger-type equationi ∂ s U ( X, X ′ ; s ) = H ( ∂ X , X ) U ( X, X ′ ; s ) (9)with the appropriate boundary conditions U ( X, X ′ ; ∞ ) = 0 , (10) U ( X, X ′ ; 0) = δ (4) ( X − X ′ ) , one obtains the following result: U ( X, X ′ ; s ) = e − i s [ H ( ∂ X ,X ) − i ε ] δ (4) ( X − X ′ ) , (11) G ( X, X ′ ) = H − ( ∂ X , X ) δ (4) ( X − X ′ ) , (12)where H − ( ∂ X , X ) = i Z ∞ d s e − i s [ H ( ∂ X ,X ) − i ε ] (13)is the inverse of H .The i ε prescription was added in order to satisfy theboundary conditions (10). From now on, we will skipwriting it explicitly, always assuming its presence.Expressions, such as (11), considered in the frameworkof the distribution theory, make perfect sense due to theinfinite differentiability of the δ -function. In the origi-nal FS method, one reduces the task of finding U to asolution of a special differential equation. The MFS ap-proach, however, consists in the direct evaluation of theexponential operator action on the δ -function. In orderto do so, an appropriate representation of the δ -functionshould be chosen: δ (4) ( X − X ′ ) = X Z ψ λ ( X ) ψ λ ( X ′ ) , (14)where ψ λ ( X ) is an eigenvector of the H operator: H ( ∂ X , X ) ψ λ ( X ) = H ( λ ) ψ λ ( X ) . (15)Therefore, Eq. (12) simplifies to: G ( X, X ′ ) = i Z ∞ d s X Z e − i s H ( λ ) ψ λ ( X ) ψ λ ( X ′ ) . (16)Next, the exponential part is integrated out: G ( X, X ′ ) = X Z ψ λ ( X ) ψ λ ( X ′ ) H ( λ ) . (17)In many cases, it is not H itself that satisfies (15), butrather a part of it: H = H + H ,H ( ∂ X , X ) ψ λ ( X ) = H ( λ ) ψ λ ( X ) . (18)If H commutes with H , that indeed is the case for theproblem discussed below, the exponential operator de-composes into two parts and the solution takes the fol-lowing form: G ( X, X ′ ) = i Z ∞ d s X Z e − i s H e − i s H ( λ ) ψ λ ( X ) ψ λ ( X ′ ) . (19)Further simplifications highly depend on the exact formof H and H . However, in some cases it is possible totransform corresponding expressions such that the de-pendence on s is accounted through the exponential fac-tor as in (16), which allows for straightforward evaluationof the integral over s . IV. W -BOSON PROPAGATOR IN ACONSTANT MAGNETIC FIELD(EAST-COAST METRIC) Let’s apply MFS method to the equation (1) usingthe East-Coast metric convention ( − , + , + , +) through-out this section. First, we notice that the left-hand sideoperator H consists of three parts ( H = H + H F + H ξ ):( H ) µν = (cid:0) ΠΠ + m (cid:1) δ µν , (20)( H F ) µν = − ieQF µν , (21)( H ξ ) µν = (cid:18) ξ − (cid:19) Π µ Π ν . (22)There exist several useful commutation relations for thecase of a constant electromagnetic field F :[Π µ , Π ν ] = − i eQF µν , (23)[Π µ , ΠΠ] = − eQF µν Π ν , [Π µ Π ν , ΠΠ] = − eQ (cid:0) F µρ Π ρ Π ν − Π µ Π ρ F ρν (cid:1) . Only one of three commutators between parts of H isvanishing ([ H , H F ] = 0). Two others are not equal tozero separately, however, their sum is. This leads to [ H + H F , H ξ ] = 0, thus, allowing for a step-by-step separationof e − i s ( H + H F + H ξ ) :e − i s ( H + H F + H ξ ) = e − i sH ξ e − i s ( H + H F ) =e − i sH ξ e − i sH F e − i sH . (24)Let’s briefly discuss the anatomy of so constructed prop-agator: G ( X, X ′ ) = i Z ∞ d s e − i sH ξ e − i sH F e − i sH δ (4) ( X − X ′ ) . (25) The H part is the basic building block, which representsthe propagation of a scalar particle. The H F part addssome additional structure due to the spin properties of avector boson, similar to the case of electron’s propagator[13]. However, for ξ = 1 there is yet another layer ofcomplexity due to the choice of ξ -gauge.In order to proceed with further calculations, wechoose the Landau gauge for the electromagnetic poten-tial A µ : A µ = (0 , , Bx, . (26)Making a standard change of variables η = p β (cid:18) x + Q p y β (cid:19) , η ′ = p β (cid:18) x ′ + Q p y β (cid:19) , (27)we consider the following δ -function representation (thesame as in [13]): δ (4) ( X − X ′ ) = p β ∞ X n =0 Z d p q ,y (2 π ) e i ( p ( X − X ′ ) ) q ,y V n V ′ n . (28)Here, V n = V n ( η ) [ V ′ n = V n ( η ′ )] is a shorthand notationfor the n -th level quantum harmonic oscillator (QHO)eigenfunction: V n ( η ) = e − η / H n ( η ) p n n ! √ π , (29)where H n is a Hermite polynomial. Accounting for theaction of the operator (20) on the e i( px ) -type expressions,we obtain:( H ) µν = (cid:16) p k + m − β ( ∂ η − η ) (cid:17) δ µν . (30)This form of H , therefore, justifies the δ -function rep-resentation (28) due to the following equation for QHOeigenfunctions:( ∂ η − η ) V n = − (2 n + 1) V n . (31)This being said, we evaluate the action of the first expo-nential operator in Eq. (25): (cid:0) e − i sH (cid:1) µν δ (4) ( X − X ′ ) = p β ∞ X n =0 Z d p q ,y (2 π ) e − i s [ p k + m +(2 n +1) β ]e i ( p ( X − X ′ ) ) q ,y V n V ′ n δ µν . (32)Expanding the exponential series for the operator H F , see Eq.(21), one obtains the following expression: (cid:2) e − i sH F (cid:3) µν = (cid:2) e − Qβsϕ (cid:3) µν = δ µ k ν + cos(2 βs ) δ µ ⊥ ν − Q sin(2 βs ) ϕ µν = δ µ k ν + e i2 βs δ µ ⊥ ν + i Qϕ µν ) + e − i2 βs δ µ ⊥ ν − i Qϕ µν ) . (33)Next, we shift the summation index over the Landau levels, such that the expression in the exponent stays the samefor all terms in Eq. (32). This gives the expression for the consecutive action of two exponential operators on the δ -function: (cid:0) e − i sH F e − i sH (cid:1) µν δ (4) ( X − X ′ ) = p β ∞ X n = − Z d p q ,y (2 π ) e i ( p ( X − X ′ ) ) q ,y e − i s [ p k + m +(2 n +1) β ] d µν , (34) d µν = δ µ k ν V n V ′ n + 12 ( δ µ ⊥ ν + i Qϕ µν ) V n +1 V ′ n +1 + 12 ( δ µ ⊥ ν − i Qϕ µν ) V n − V ′ n − . (35)Finally, the exponential operator for the gauge-dependent part H ξ simplifies to:e − i s ( ξ − ) Π µ Π ν = δ µν + Π µ e − i s ( ξ − ) ΠΠ − ν . (36)In the MFS approach, the gauge-dependent part de-composes into the sum of two terms. The first termrepresents the choice of Feynman gauge ( ξ = 1) andleads to the expression given by Eqs. (34), (35). Thesecond one accounts for additional complexity in thecase ξ = 1.In order to evaluate the expression (cid:20) Π µ e − i s ( ξ − ) ΠΠ − ν (cid:21) e − i sH F e − i sH δ (4) ( X − X ′ )(37)let’s introduce some standard notations: a + = 1 √ η − ∂ η ) , a − = 1 √ η + ∂ η ) , (38) ∂ η = 1 √ a − − a + ) , η = 1 √ a − + a + ) . (39)The following auxiliary vector v ρ = (cid:18) , i √ , Q √ , (cid:19) (40) is also useful for further computations due to its proper-ties: v ρ ( δ ρ ⊥ ν + i Qϕ ρν ) = 0 , v ∗ ρ ( δ ρ ⊥ ν + i Qϕ ρν ) = 2 v ∗ ν ,v ∗ ρ ( δ ρ ⊥ ν − i Qϕ ρν ) = 0 , v ρ ( δ ρ ⊥ ν − i Qϕ ρν ) = 2 v ν . (41)Therefore, the action ofΠ ρ = p k ρ + p βv ρ a + + p βv ∗ ρ a − (42)on d ρν is given by:Π ρ d ρν = p k ν V n V ′ n + (43)+ p nβ v ν V n V ′ n − + p ( n + 1) β v ∗ ν V n V ′ n +1 . We notice that all QHO eigenfunctions that depend on η have the same index n , thus, justifying the followingsubstitution in expression (37):ΠΠ → p k + (2 n + 1) β . (44)Therefore, the middle part of [...] operator in Eq. (37)can be moved away safely as a c -number, even prior to theevaluation of Π µ action on expression (43). This resultsin the following intermediate formula: (cid:0) e − i sH (cid:1) µν δ (4) ( X − X ′ ) = (cid:0) e − i sH ξ e − i sH F e − i sH (cid:1) µν δ (4) ( X − X ′ ) = p β ∞ X n = − Z d p q ,y (2 π ) × (45) × δ µρ + e − i s ( ξ − )[ p k +(2 n +1) β ] − p k + (2 n + 1) β Π µ Π ρ ! e i ( p ( X − X ′ ) ) q ,y e − i s [ p k + m +(2 n +1) β ] d ρν . Now that the dependence on s is brought to the exponents, one could easily integrate it out: G µν ( X, X ′ ) = i Z ∞ d s e − i sH δ (4) ( X − X ′ ) = (46)= p β ∞ X n = − Z d p q ,y (2 π ) e i ( p ( X − X ′ ) ) q ,y p k + m + (2 n + 1) β d µν + ξ − p k + ξm + (2 n + 1) β f µν ! ,f µν ≡ Π µ Π ρ d ρν = p µ k p k ν V n V ′ n + (47)+ p µ k (cid:16)p βn v ν V n V ′ n − + p β ( n + 1) v ∗ ν V n V ′ n +1 (cid:17) + (cid:16)p βn v ∗ µ V n − V ′ n + p β ( n + 1) v µ V n +1 V ′ n (cid:17) p k ν ++ β p n ( n + 1) (cid:0) v µ v ν V n +1 V ′ n − + v ∗ µ v ∗ ν V n − V ′ n +1 (cid:1) + β ( n + 1) v µ v ∗ ν V n +1 V ′ n +1 + βn v ∗ µ v ν V n − V ′ n − . Among many explicit representations of the propaga-tor, expression (46) is not the most useful one. It isasymmetric with respect to x, y coordinates. In order tosymmetrize it, let’s evaluate the integral over d p y . Wenotice that the integrand depends on p y not just throughthe exponential factor but also through η and η ′ in QHOfunctions, see Eq. (27). Therefore, the following integralsfor different n and n ′ should be calculated: I n,n ′ = Z d p y e i p y ( y − y ′ ) V n ( η ) V n ′ ( η ′ ) . (48)First, we make a change of the integration variable: u = Q p y √ β + √ β x + x ′ ) − i Q ( y − y ′ )] . (49)This leads to: I n,n ′ = e iΦ( X,X ′ ) √ n + n ′ n ! n ′ ! π p β e − β ( X − X ′ ) ⊥ ˜ I n,n ′ , (50) whereΦ( X, X ′ ) = − Qβ x + x ′ )( y − y ′ ) , (51)˜ I n,n ′ = Z ∞−∞ d u e − u H n ( u + a ) H n ′ ( u + b ) , (52)with the following substitutions: a = √ β x − x ′ ) + i Q ( y − y ′ )] , (53) b = − √ β x − x ′ ) − i Q ( y − y ′ )] . The phase (51) can be shown to be in agreement withthe one in Eq. (5), see e.g. Ref. [12]. Secondly, accordingto Ref. [31], the ˜ I n,n ′ integral evaluates to:˜ I n,n ′ = 2 n ′ √ π n ! b n ′ − n L ( n ′ − n ) n ( − ab ) == 2 n ′ √ π n ! b n ′ − n L ( n ′ − n ) n (cid:18) β Z ⊥ (cid:19) [ n ≤ n ′ ] , (54)where the functions L ( m ) n are the Laguerre polynomialsand Z µ = X µ − X ′ µ . The symmetrized representationof the propagator then reads: G µν ( X, X ′ ) = β π e iΦ ∞ X n = − Z d p q (2 π ) e i( pZ ) k e − βZ ⊥ / p k + m + (2 n + 1) β ˜ d µν + ξ − p k + ξm + (2 n + 1) β ˜ f µν ! , (55)˜ d µν = δ µ k ν L n + 12 δ µ ⊥ ν (cid:18) L n +1 + L n − (cid:19) + i Q ϕ µν (cid:18) L n +1 − L n − (cid:19) , (56)˜ f µν = (cid:20) p µ k p k ν − βQ (cid:18) p µ k ( Zϕ ) ν + ( ϕZ ) µ p k ν (cid:19) + (cid:18) (2 n + 1) β − β Z ⊥ (cid:19) δ µ ⊥ ν + i Qβ ϕ µν (cid:21) L n + (57)+ i β (cid:20)(cid:18) p µ k Z ⊥ ν + Z µ ⊥ p k ν (cid:19) + i δ µ ⊥ ν − Qβ Z ⊥ ϕ µν (cid:21)(cid:18) L (1) n + L (1) n − (cid:19) − β ( ϕZ ) µ ( Zϕ ) ν L (2) n − . In (56) and (57), all the Laguerre polynomials L ( m ) n have βZ ⊥ / L ( m ) n = L ( m ) n ( βZ ⊥ / t, z coordinates). However,the established symmetry allows to calculate the remaining Fourier transforms (with respect to x, y coordinates). Inorder to do this, one should switch to polar coordinates in the x − y plane and make use of the integral identities [31]: Z π d ϕ e − i ζcos ( ϕ ) ± i mϕ = ( − i) m πJ m ( ζ ) , (58) Z ∞ d ζ ζ λ/ e − pζ J λ ( b p ζ ) L ( λ ) n ( cζ ) = (cid:18) b (cid:19) λ ( p − c ) n p λ + n +1 e − b p L ( λ ) n (cid:18) b c pc − p (cid:19) , (59)where J m are the Bessel functions of order m . After some simple but rather cumbersome rearrangements andsubstitutions, one finally arrives to the following result: G µν ( X, X ′ ) = e iΦ Z d p (2 π ) e i ( p ( X − X ′ ) ) G µν ( p ) , (60) G µν ( p ) = ∞ X n = − − n e − p ⊥ /β p k + m + (2 n + 1) β ≈ d µν + ξ − p k + ξm + (2 n + 1) β ≈ f µν ! , (61) ≈ d µν = δ µ k ν L n − δ µ ⊥ ν ( L n +1 + L n − ) − i Q ϕ µν ( L n +1 − L n − ) , (62) ≈ f µν = (cid:18) p µ p ν − i Qβ ϕ µν (cid:19) L n − ( ϕp ) µ ( pϕ ) ν (cid:16) L n + 4 L (2) n − (cid:17) + (63)+ i Q (cid:18) p µ ( pϕ ) ν + ( ϕp ) µ p ν + i Qβ δ µ ⊥ ν (cid:19) (cid:16) L (1) n + L (1) n − (cid:17) . In (62) and (63), all the Laguerre polynomials L ( m ) n have 2 p ⊥ /β as their arguments: L ( m ) n = L ( m ) n (2 p ⊥ /β ). V. COMPARISON OF TWO WAYS OF OBTAINING THE W -BOSON PROPAGATOR Let’s transform Eq. (4) according to the recipe from [16] (see also [12] and [19]) and then compare the result withEqs. (60) - (63). The Fourier transform of the translationally invariant part of the W boson propagator (4) in the ξ -gauge is presented in Eq. (6). Let us rewrite it in a more convenient form: G µν ( p ) = i β ∞ Z d v e − i ρv (cid:20) δ µ k ν F ( v ) + δ µ ⊥ ν F ( v ) − Q ϕ µν F ( v ) (cid:21) + i βm ∞ Z d v (cid:0) e − i ρv − e − i ρ ξ v (cid:1) ×× (cid:20)(cid:18) p µ p ν − i Qβ ϕ µν (cid:19) F ( v ) − (cid:18) Q p µ ( pϕ ) ν + Q ( ϕp ) µ p ν + i β δ µ ⊥ ν (cid:19) F ( v ) + ( ϕp ) µ ( pϕ ) ν F ( v ) (cid:21) , (64)where the functions are introduced: F ( v ) = 1cos v exp( − i α tan v ) , (65) F ( v ) = cos(2 v )cos v exp( − i α tan v ) , (66) F ( v ) = sin(2 v )cos v exp( − i α tan v ) , (67) F ( v ) = tan v cos v exp( − i α tan v ) = i ∂∂α F ( v ) , (68) F ( v ) = tan v cos v exp( − i α tan v ) = − ∂ ∂α F ( v ) , (69)and v = βs , ρ = ( m + p k ) /β , ρ ξ = ( ξ m + p k ) /β , α = p ⊥ /β .Since all the functions F j ( v ) ( j = 1 ...
5) satisfy the relation F j ( v + π n ) = ( − n F j ( v ), let usdivide the integration domain (0 , ∞ ) into intervals(0 , π ) , ( π, π ) , . . . , ( nπ, [ n + 1] π ) , . . . . Making in each seg-ment the change of variable, v → v + nπ , we can write: I j = ∞ Z d v exp( − i ρv ) F j ( v ) == ∞ X n =0 ( − n exp( − i ρnπ ) A j = (70)= 11 + exp( − i ρπ ) A j , where A j = π Z d v exp( − i ρv ) F j ( v ) . (71)The details of calculations of the functions A j can befound in Refs. [12, 24]. One finally obtains: A = − (cid:0) − i ρπ (cid:1) ∞ X n =0 ℓ n − ( α ) ρ + 2 n − , (72) A = − i (cid:0) − i ρπ (cid:1) ∞ X n =0 ℓ n ( α ) + ℓ n − ( α ) ρ + 2 n − , (73) A = − (cid:0) − i ρπ (cid:1) ∞ X n =0 ℓ n ( α ) − ℓ n − ( α ) ρ + 2 n − , (74) A = 2 (cid:0) − i ρπ (cid:1) ∞ X n =0 ℓ ′ n − ( α ) ρ + 2 n − , (75) A = 2 i (cid:0) − i ρπ (cid:1) ∞ X n =0 ℓ ′′ n − ( α ) ρ + 2 n − , (76)where the auxiliary functions were introduced to writethe resulting expressions in a more compact form: ℓ n ( α ) = ( − n e − α L n (2 α ) . (77)After substituting the integrals (72)–(76) into the ex-pression for the propagator and performing simple ma-nipulations with Laguerre polynomials, we observe that(64) transforms exactly to (61) - (63). VI. W-BOSON PROPAGATOR IN ACONSTANT MAGNETIC FIELD(WEST-COAST METRIC)
Expressions (60) – (63) were obtained in the East-Coast metric convention g µν = ( − , + , + , +), however, inmodern particle physics literature the West-Coast metricconvention g µν = (+ , − , − , − ) prevails. In this section, the following standard notations are assumed: X µ = ( t, x, y, z ) , X µ = g µν X ν = ( t, − x, − y, − z ) ,∂ µ = ( ∂ t , ∇ ) , ∂ µ = ( ∂ t , −∇ ) ,D µ = ∂ µ + i eQA µ , Π µ = i D µ , ΠΠ = Π ν Π ν . In order to get the expression for the propagator, we,first, consider relevant terms in the Standard Model La-grangian: L = L W + L W W A + L W W AA + L gauge , (78) L W = − W † µν W µν + m W † µ W µ , L W W A = i eQ (cid:20) F µν W † µ W ν − W † µν A µ W ν + W µν A µ W † ν (cid:21) , L W W AA = e (cid:20) A µ W † µ A ν W † ν − A µ A µ W † ν W ν (cid:21) , L gauge = − ξ (cid:18) D µ W µ (cid:19) † (cid:18) D ν W ν (cid:19) . The last term accounts for gauge fixing, in the sameway as in Ref. [21]. Relative signs in the StandardModel Lagrangian were chosen according to the conven-tion adopted by the majority of modern textbooks (see,Ref. [32, 33]). This implies the choice of η = 1 in W iµν = ∂ µ W iν − ∂ ν W iµ − ηgε ijk W jµ W kν . (79)However, in Ref. [21] this choice was made according toRef. [34] with η = −
1, thus leading to different rela-tive signs in the Lagrangian and the corresponding waveequation.Next, from the given Lagrangian we derive a fieldequation for W-boson in external electromagnetic field.Adding δ ’s to the right-hand side results in the corre-sponding propagator equation: H µν G νρ ( X, X ′ ) = δ µν δ ( X − X ′ ) , (80) H µν = (cid:0) ΠΠ − m (cid:1) δ µν − ieQF µν + (cid:18) ξ − (cid:19) Π µ Π ν . Finally, using the same procedure as in section IV, oneobtains the expression for the propagator in the West-Coast metric convention: G µν ( X, X ′ ) = e − iΦ Z d p (2 π ) e − i ( p ( X − X ′ ) ) G µν ( p ) , (81) G µν ( p ) = ∞ X n = − − n e − p ⊥ /β p k − m − (2 n + 1) β ≈ d µν + ξ − p k − ξm − (2 n + 1) β ≈ f µν ! , (82) ≈ d µν = δ µ k ν L n − δ µ ⊥ ν ( L n +1 + L n − ) + i Q ϕ µν ( L n +1 − L n − ) , (83) ≈ f µν = (cid:18) p µ p ν − i Qβ ϕ µν (cid:19) L n − ( ϕp ) µ ( pϕ ) ν (cid:16) L n + 4 L (2) n − (cid:17) − (84) − i Q (cid:18) p µ ( pϕ ) ν + ( ϕp ) µ p ν + i Qβ δ µ ⊥ ν (cid:19) (cid:16) L (1) n + L (1) n − (cid:17) . VII. DISCUSSION
We applied the modified Fock-Schwinger (MFS)method to obtain the exact analytical solution for thepropagator of the W-boson in external constant mag-netic field in arbitrary ξ -gauge. Commutation relations(23) allowed us to study the solution’s internal structureby considering the consecutive action of exponential op-erators inside the inverse operator (25) for the propaga-tor equation (1). Each of the exponential operators ac-counted for the contribution to the final expression due tothe (i) propagation of charged particle, (ii) its spin inter-action with magnetic field and (iii) the choice of ξ -gauge.A particular form of δ -function decomposition reducedthe action of these exponentials either to the eigenvaluesubstitution or to the use of the QHO ladder operators.Several straightforward manipulations with special func-tions allowed us to make a transition from partial Fouriertransform (with respect to t , y and z coordinates) to thefull Fourier transform, however, for the translationallyinvariant part of the propagator.In contrast to the nowadays standard approaches forfinding propagators, the MFS approach demonstratesseveral improvements in terms of computational com-plexity reduction and reveals simple internal structuresin intermediate and final expressions. First of all, it doesnot require the full knowledge of exact analytical solutionfor the corresponding wave equation, but only a form of the solution for its scalar part ( H ). The remaining com-plexity (e.g., correct normalization) is hidden inside theappropriate decomposition of the δ -function, while thecommutation relations (23) justify the use of the samedecomposition for the full problem (with spin and gauge-dependent parts). Secondly, if compared with the orig-inal Fock–Schwinger approach followed by applicationof integration techniques discussed in section IV, MFSmethod requires significantly less effort to evaluate theintegral over the proper-time parameter, therefore, lead-ing directly to a (partial) Fourier transform. Finally, theintroduction of QHO eigenfunctions and ladder opera-tors on early computational stages allows us to use theconvenient relations from the QHO problem.Provided that the form of the solution for the H partis known and the corresponding commutation relationsfor the remaining parts of H are satisfied, MFS approachhas a potential for its efficient use in higher dimensionsdue to the structure of the inverse operator, where eachexponential operator accounts for a specific layer of com-plexity. This makes it promising for further developmentof quantum field theory methods in external fields. ACKNOWLEDGMENTS
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