The point-charge self-energy in Lee-Wick Theories
aa r X i v : . [ h e p - t h ] M a y The point-charge self-energy in Lee-Wick Theories
F.A. Barone ∗ and G. Flores-Hidalgo † Universidade Federal de Itajub´a, IFQ, Av. BPS 1303,Pinheirinho, cep 37500-903, Itajub´a, MG, Brazil.
A.A Nogueira ‡ IFT - R. Dr. Bento Teobaldo Ferraz, 271, 01140-070, S˜ao Paulo, SP, Brazil.
In this paper we study the ultraviolet and infrared behaviour of the self energy of a point-likecharge in the vector and scalar Lee-Wick electrodynamics in a d + 1 dimensional space time. Itis shown that in the vector case, the self energy is strictly ultraviolet finite up to d = 3 spatialdimensions, finite in the renormalized sense for any d odd, infrared divergent for d = 2 and ultravioletdivergent for d > d ≤
3, and finite, in the renormalized sense, for any d odd. One of the most remarkable features of the so calledLee-Wick electrodynamics is the fact that this theoryleads to a finite self energy for a point-like charge in 3 + 1dimensions [1–5], what has important implications in thequantum context, mainly in what concerns the renor-malizability of the theory [6, 7]. Theories of superiorderivatives for the scalar field has been also consideredin the literature, maily after the propose of the so calledLee-Wick Standard Model (LWSM) [8–20].Among other subjects concerning the Lee-Wick elec-trodynamics, in the work of reference [21] the self en-ergy of a point-like charge in an arbitrary number ofspatial dimensions was discussed. The presented resultswere speculative, not conclusive and indicated that thepoint-like particle self-energy is divergent for space di-mensions higher than 3. That is an important subject inthe context of theories with higher dimensions, in whatconcerns models with superior derivatives, because somewell known results of Lee-Wick theories, which are validin 3 + 1 dimensions, are no longer applicable when thespace has not 3 dime dimensions. ∗ Electronic address: [email protected] † Electronic address: gfl[email protected] ‡ Electronic address: [email protected]
In this paper we show that, by using dimensional regu-larization, the self-energy of a point-like particle is finitewhen the space has an odd dimension and diverges whenthe space has even dimension. We also consider the selfenergy of a point-like source for the Klein-Gordon-Lee-Wick field, where there is two mass parameters involved.Let-us start with the Lee-Wick electrodynamics. It isdescribed by the lagrangian density[4, 5] L A = − F µν F µν − m F µν ∂ α ∂ α F µν − ( ∂ µ A µ ) ξ − J µ A µ , (1)where J µ is the vector external source, F µν = ∂ µ A ν − ∂ ν A µ (2)is the field strength, A µ is the vector potential and m is a parameter with mass dimension. The third termon the right hand side of (1) was introduced in order tofix the gauge and ξ is a gauge fixing parameter. Thecorresponding propagator is [21] D µν ( x, y ) = Z d d +1 p (2 π ) d +1 p − m − p ! (cid:26) η µν − p µ p ν p (cid:20) ξ (cid:18) p m − (cid:19)(cid:21)(cid:27) e − ip ( x − y ) . (3)The energy of the system due to the presence of thesource is given by [21–23] E A = lim T →∞ T Z d d +1 x d d +1 y J µ ( x ) D µν ( x, y ) J ν ( y ) . (4)Now we take the source of a point-like stationarycharge λ placed at position a in a d + 1 space-time J µ = λη µ δ d ( x − a ) , (5)where δ is the Dirac delta function in d dimensions.Replacing above expression in Eq. (4) and performingthe integrals in x , p and y , we have E A = 12 λ Z d d p (2 π ) d (cid:20) p − p + m (cid:21) = 12 λ m Z d d p (2 π ) d p ( p − m ) . (6)Integrating in d − dimensional spherical coordinates, wecan write, E A = λ m (4 π ) d/ Γ( d/
2) lim Λ →∞ Z Λ0 dp p d − p + m . (7)where we used the fact that the integral in the angu-lar variables gives 2 π d/ / Γ( d/ m in the denominatorof the integrand in Eq. (7) and integrate from a givenfinite value P . Then, we get for d = 4, E A ∼ λ m (4 π ) d/ Γ( d/
2) Λ d − − P d − d − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Λ →∞ ∼ λ m (4 π ) d/ Γ( d/
2) Λ d − d − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Λ →∞ , d = 4 (8)For d = 4 we integrate Eq. (7), E A = λ m π ) d/ Γ( d/
2) ln Λ m + 1 !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Λ →∞ ∼ λ m π ) d/ Γ( d/
2) ln Λ m !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Λ →∞ , d = 4 . (9) From Eqs. (8) and (9) we conclude that the self energyis ultraviolet finite for d ≤ d > d = 2.We shall use dimensional regularization in order to reg-ularize the integral in Eq. (6). Rewriting it as E A = 12 λ m µ d − ω Z d ω p (2 π ) ω p ( p − m ) (cid:12)(cid:12)(cid:12)(cid:12) ω → d , (10)where µ is a parameter with mass dimension, integratingin spherical coordinates and using the formula [24] Z ∞ dr r β ( r + C ) α = Γ (cid:16) β (cid:17) Γ (cid:16) α − (1+ β )2 (cid:17) C ) α − (1+ β ) / Γ( α ) , (11)we obtain for the self-energy of a point charge, E A = − λ m ω − µ d − ω ω +1 π ω Γ( ω − − ω )Γ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) ω → d . (12)Above expression can be simplifyed by using theGamma function property Γ( z + 1) = z Γ( z ), E A = − λ m ω − µ d − ω ω +1 π ω Γ(1 − ω ) (cid:12)(cid:12)(cid:12)(cid:12) ω → d . (13)From this expression one can see that the point-chargeself-energy (13) diverges for any d even, d = 2 ω =2 , , , ... and is finite for any value of d odd, with theresult E A = − λ m d − d +1 π d/ Γ(1 − d , d = odd . (14)For instance, we have the following values for d = 1 , , d = 7, E A ( d = 1) = − λ m , E A ( d = 3) = λ m πE A ( d = 5) = − λ m π , E A ( d = 7) = λ m π . (15)In order to split the divergences in the self-energy (13)for even spatial dimensions, we make the substitution2 ω = d − ǫ , where ǫ →
0, what leads to E A = lim ǫ → − λ m d − d +1 − ǫ π ( d − ǫ ) / (cid:16) µm (cid:17) ǫ Γ(1 − d + ǫ . (16)Using the expansions µm ! ǫ ∼ = 1 + ǫ ln( µ/m ) + ǫ [ln( µ/m )] + O ( ǫ ) (17)and [25]Γ( − n + δ ) ∼ = ( − n n ! " δ + Ψ( n + 1) + 12 δ π ( n + 1) − Ψ ′ ( n + 1) ! + O ( δ ) , n = 0 , , , ... (18)where Ψ( s ) = dds ln (cid:16) Γ( s ) (cid:17) , Ψ ′ ( s ) = dds Ψ( s ) and Ψ( n +1) = 1 + + ... + n − γ , with γ standing for the Euler-Mascheroni constant, we rewrite Eq. (16), only for evenvalues of d , in the form E A = ( − d/ λ m d − d +1 π d/ Γ( d/
2) lim ǫ → ǫ + Ψ( d/ µ/m ) ! , d = even , (19)which is explictly divergent.Next we consider the Lee-Wick model for the scalarfield φ , whose lagrangian density is [21] L = 12 ∂ µ φ∂ µ φ + 12 ∂ µ φ ∂ γ ∂ γ m ∂ µ φ − M φ + Jφ , (20)with the corresponding propagator D ( x, y ) = Z d d +1 p (2 π ) d +1 m p − m p + M m exp[ − ip ( x − y )] . (21)From (21) one can show that this model exhibits twomassive poles for momentum square [21], namely m ± = m ± r − M m ! . (22)In order to avoid tachyonic modes, one must take therestriction 0 ≤ M m ≤ . (23)If M = 0, we have a theory similar to the one studiedin the previous case, for the vector field, with one massive mode, with mass m , and a massless one. This case is verysimilar to the one studied previously and has no novelphysical properties.If 0 < M /m < m + and m − , both of themlower than m and M . In this case the propagator can berewritten in the form D ( x, y ) = Z d d +1 p (2 π ) d +1 p − m − p − m − ! × q − M m exp[ − ip ( x − y )] . (24)Once m > M , the self-energy of a point-charge, J = λδ d ( x − a ), is given by E φ = 12 λ q − M m Z d d p (2 π ) d (cid:20) p + m − p + m − (cid:21) = λ m Z d d p (2 π ) d p + m )( p + m − ) . (25)From above expression we conclude, by a power count-ing analysis, that the self-energy is finite for d ≤ d >
3. Following a similar pro-cedure we have done for the vector case, it is not dificultto show that the regularized self-energy is given by E φ = λ ω +1 π ω ( µ d − ω + m ω − − µ d − ω − m ω − − ) q − M m × Γ(1 − ω ) | ω → d , m > M, (26)where µ + and µ − are two arbitrary parameters with massdimension, introduced for each one of the integrals in thefirst line of Eq. (25). It is not difficult to see that theabove expression is finite for d odd. In this case the self-energy reads E φ = λ d +1 π d/ ( m d − − m d − − ) q − M m Γ(1 − d/ ,d = odd , m > M. (27)On the other hand, when d = 2, the integral (25) is finiteand can be solved easily, E φ = λ π q − M m ln m − m + , d = 2 , m > M. (28)When d > ω = d − ǫ in (26) and takingthe limit ǫ →
0. Using the expansions (17) and (18) in(26) we have E φ = lim ǫ → ( − d/ λ d +1 π d/ Γ( d/
2) 1 q − M m " ǫ ( m d − − m d − − )+ m d − ln µ + m + − m d − − ln µ − m + + m d − Ψ( d/ − m d − − Ψ( d/ , d > , d = even , m > M, (29)which is divergent once m > M , what implies m + >m − .Finally we consider the case m = 2 M . Then, the prop-agator (21) reduces to the form D ( x, y ) = Z d d +1 p (2 π ) d +1 m ( p − m / exp[ − ip ( x − y )] (30)and the self energy of a point charge is given by E φ = − λ Z d d p (2 π ) d m ( p + m / , m = 2 M. (31)Again, this integral is finite for d = 1 , , d >
3. Regularizing above integral with similar meth-ods previously employed, we find a finite result for any d odd, E φ = − λ m d − d/ − π d/ Γ(2 − d/ , d = odd , m = 2 M. (32)For even values of d , the divergent part of (31), can besplitted by dimensional regularization as in the previouscases. It is obtained E φ = lim ǫ → − λ m d − d/ − π d/ " ǫ + 2 ln µm + Ψ( d/ − ,d > , d = even , m = 2 M. (33) As in previous cases, it is explicitly divergent.At this point we would like to remark that our resultscould be extendd to the case of charges distributionsalong branes with an arbitrary number of dimensions[21]. For instance, in 3 spatial dimensions, a chargedplate exhibits a finite self energy, while a uniform chargedistribution along a straigth line has a divergent self en-ergy.In summary, in the present work we showed that in theLee-Wick electrodynamics the self energy of a point-likecharge is strictly ultraviolet finite for d = 1 and 3 spatialdimensions. Also, for d > d > d = 2 theself energy is infrared divergent and can not be renor-malized too. For the Klein-Gordon-Lee-Wick theory, wehave a similar situation, with the distinction that when d = 2 the self energy is finite. From the regularized self-energy for any d > m > M and (33) for m = 2 M , we see that there is no way torender such quantities finite by redefining charge or massparameters. Acknowledgments
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