One-Loop Radiative Corrections to the QED Casimir Energy
aa r X i v : . [ h e p - t h ] M a y One-Loop Radiative Corrections to the QED Casimir Energy
Reza Moazzemi ∗ and Amirhosein Mojavezi Department of Physics, University of Qom, Ghadir Blvd., Qom 371614-611, I.R. Iran
In this paper, we investigate one-loop radiative corrections to the Casimir energy in the presenceof two perfectly conducting parallel plates for QED theory within the renormalized perturbationtheory. In fact, there are three contributions for radiative corrections to the Casimir energy, upto order α . Only the two-loop diagram, which is of order α , has been computed by Bordag et.al (1985), approximately. Here, up to this order, we consider corrections due to two one-loopterms, i.e., photonic and fermionic loop corrections resulting from renormalized QED Lagrangian,more precisely. Our results show that only the fermionic loop has a very minor correction and thecorrection of photonic loop vanishes. I. INTRODUCTION
The Casimir effect is a physical manifestation of changes in the zero point energy of a quantum field for differentconfigurations. The zero point configuration refers to one in which there does not exist any on-shell physical excitationof the field.In 1948 Casimir predicted the existence of this effect as an attractive force between two infinite parallel unchargedperfectly conducting plates in vacuum [1]. This effect was observed experimentally by Sparnaay [2] and Arnold etal [3] (for a general review on the Casimir effect, see Refs.[4, 5]). Similar measurements have been done for othergeometries, and their precisions have been greatly improved [6–11]. The manifestations of the Casimir effect havebeen studied in many different areas of physics. For example, the magnitude of the cosmological constant has beenestimated using the Casimir effect [12–14]. This effect has been also studied within the context of string theory [15]. Ithas been investigated in connection with the properties of the spacetime with extra dimensions [16–18]. The majorityof the investigations related to the Casimir effect concerns with the calculation of this energy or the consequenceforces for different fields in different geometries, such as parallel plates [1, 19], cubes [20–28], cylinders [27, 29–31],and spherical geometries [27, 32–34].Although the Casimir effect has been known for nearly 70 years, the question of what are the leading radiativecorrections to this effect is still a subject of discussion. The first endeavors to compute the radiative corrections tothe Casimir energy were reported in a paper by Bordag, Robaschik, and Wieczorek (BRW) [35]. There exist manyworks on the radiative corrections to the Casimir energy for various cases (see for example [35–45]). In the case of areal massive scalar field, Next to Leading Order (NLO) correction to the energy has been computed in [4, 28, 41–50].Moreover, the two-loop radiative corrections for some effective field theories have been investigated in [37–39]. Bordagand his collaborators have approximately calculated radiative correction to the Casimir energy due to one of the threerelated terms of order of α , , in the presence of two perfectly conducting parallel plates for QED theory. In thisviewpoint, the photon propagator satisfies boundary conditions on the plates, while the plates are transparent to theelectrons. They found the correction E (1) = π α ma to the popular leading term of Casimir energy (per unit area) E (0)em = − π a , where a is the distance between plates and m is the electron mass. In 1998 this result with anotherapproach has been reported [36]. Although they postulate no boundary conditions for the electron field because suchconditions would lead to additional contributions in zeroth order which have not been observed, the fermionic term[51], E (0)fermion = − m π a P n =1 − j [2 K (2 amj ) − K (4 amj )] is exist (Here, K n ( x ) is the modified Bessel function of order n .) However, due to its Yukawa form for the large mass case, E (0)fermion ∼ − m π a p πma e − ma , at distances much largerthan a few Compton wave length of electron it has really too small value to be observed.In the context of perturbation theory we need the renormalization to compute loop diagrams. There are twocompletely equivalent methods for renormalization; first, bare perturbation theory: working with the bare parametersand relate them to their physical values at the end of calculations, second, renormalized perturbation theory: usingcounterterms at the starting point to absorb unphysical part of parameters. Both of them need renormalizationconditions to fix the infinities in certain conditions. The differences between two renormalization procedures are ∗ Electronic address: [email protected] purely a matter of bookkeeping. In the framework of renormalized perturbation theory for QED there are threevacuum bubbles of order of α . Up to now, all the papers on the Casimir effect, that we are aware of, have not beencalculated two of those, namely: photonic loop resulting from electromagnetic field and fermionic looprelated to the spinor field. Note that, although according to the common understanding we use the same countertermsfor two different situations (with and without plates), the difference of vacuum energies may still be nonzero due tothe difference of boundary conditions applying on loop propagators.The main purpose of this paper is to directly calculate radiative correction to the Casimir energy resulting from one-loop corrections namely: one-loop photon and one-loop fermion, in the framework of the renormalized perturbationtheory for QED theory. These corrections are of order α . In order to do this, we use Green’s functions in the presenceof plates for Electromagnetic field with Dirichlet boundary condition and for spinor field with MIT bag boundarycondition as propagators. Our main regularization is dimensional regularization.Our approach for the calculation of radiative corrections to Casimir energy is the most direct one. In this way wesubtract two infinite energies: one relate to presence and the other without presence of two plates. We adjust bothof their regulators in such a way that the divergences removed and the physical result is obtained.To have a throughout complete correction, up to order α , one must also compute two-loop term once the fermionicfield is submitted to MIT bag boundary conditions. However, it is notable that almost all the Casimir forces forvarious massive fields, which precisely calculated in the literature, have Yukawa asymptotic forms (usually K n ( ma ))even for leading term in different dimensions. Therefore, here we only calculate the one-loop diagram which seemsmore important than two-loop one that has two fermion propagators.We organized the paper as follows: In Section II we briefly review the renormalization of quantum electrodynamics.In Section III using analogies between an electromagnetic field and a massless scalar field, photonic loop correction isconsidered. We use the Dirichlet boundary condition on the two plates. In Section IV we directly calculate radiativecorrection to the Casimir energy resulting from fermionic loop where MIT bag boundary condition, as constraints onboth of the plates, is considered. In Section 5 we summarize our results and state our conclusions. II. RENORMALIZATION OF QUANTUM ELECTRODYNAMICS: A BRIEF REVIEW
In this section we briefly review systematics of renormalization for QED theory (see for complete details [52]). Theoriginal QED Lagrangian is L QED = −
14 ( F µν ) + ¯ ψ ( i∂/ − m ) ψ − e ¯ ψγ µ ψA µ . (1)By replacing ψ = z ψ r and A µ = z A µr , it becomes L QED = − z ( F rµν ) + z ¯ ψ r ( i∂/ − m ) ψ r − e z z ¯ ψ r γ µ ψ r A µr , (2)where e is the bare electric charge and z and z are the field-strength renormalizations for ψ and A µ respectively.We define a scaling factor z as follows: ez = e z z . (3)We can split each term of the Lagrangian into two pieces as follows: L QED = −
14 ( F rµν ) + ¯ ψ r ( i∂/ − m ) ψ r − e ¯ ψ r γ µ ψ r A µr − δ ( F µνr ) + ¯ ψ r ( iδ ∂/ − δ m ) ψ r − eδ ¯ ψγ µ ψ r A µr , (4)where δ = z − δ = z − δ m = z m − m and δ = z − e e ) z z − m and e arethe physical mass and physical charge of the electron which measured at large distances. Now, the Feynman rules forthis Lagrangian are µ = − ieγ µ (5) µ = − ieδ γ µ (6) νµ k = − ik + iǫ (cid:18) g µν − (1 − ξ ) k µ k ν k (cid:19) (7) νµ = − i ( g µν k − k µ k ν ) δ (8) p = ip (cid:30) − m + iǫ (9)= i ( p (cid:30) δ − δ m ) . (10)Each of the four counterterms must be fixed by renormalization conditions. For QED theory these conditions are (seeplease [52]) 1PI = − i Σ( p ) (11) νµ i Π µν ( q ) = i ( g µν k − k µ k ν )Π( k ) (12) µ amputated = − ie Γ µ ( p ′ , p ) . (13)In the above equations − i Σ( p ) denotes the sum of all one-particle irreducible (1PI) diagrams with two externalfermion lines. By pretending that the photon has a small nonzero mass µ to control the infrared divergences, up toleading order in α , the one-loop diagram contributing to − i Σ( p ) becomes − i Σ( p ) = O ( α ) − e Z dx Z d l (2 π ) − xp (cid:30) + 4 m [ l − x (1 − x ) p − xµ − (1 − x ) m ] . (14)One can evaluate the diagrams in dimensional regularization. If fact, we compute them as an analytic function of thedimensionality of spacetime d . The final expression for any observable quantity should have a well-defined limit as d →
4. Up to leading order in α , i Σ( p ) becomes − i Σ( p ) = − i e m (4 π ) d Z dx Γ(2 − d ) (cid:18) (1 − x ) m + xµ − x (1 − x ) p (cid:19) − d [(4 − ǫ ) m − (2 − ǫ ) xp ] . (15)with ǫ = 4 − d . Since we prefer to work with dimensionless parameters we convert this formula as − i Σ( p ) = − i e a d − (4 π ) d Z dx Γ(2 − d ) (cid:18) (1 − x ) ˜ m + x ˜ µ − x (1 − x )˜ p (cid:19) − d [(4 − ǫ ) ˜ m − (2 − ǫ ) x ˜ p ] , (16)where ˜ l = la, ˜ p = pa, ˜ µ = µa, ˜ m = ma . Here 1 /a is an arbitrary scale with mass dimension 1 (in the problem ofCasimir effect a can be the plates separation.)Moreover, i Π( k ) defines the sum of all 1PI insertions into the photon propagator and up to order α becomesΠ( k ) = − e a d − (4 π ) d Z dx Γ(2 − d ) (cid:18) ˜ m − x (1 − x )˜ k (cid:19) − d x (1 − x ) , (17)where ˜ k = ka . In Eq.(13), Γ µ ( p ′ , p ) denotes the sum of vertex diagrams. More accuratelyΓ µ ( p ′ , p ) = γ µ F ( k ) + iσ µν k ν m F ( k ) , (18)where F and F are unknown functions of k called form factors and σ µν = i [ γ µ , γ ν ]. To lowest order, F = 1 and F = 0, we have Γ µ = γ µ . By using Eqs. (17),(16) and (18), up to leading order in α , the counterterms are derivedas follows: δ = − e a d − (4 π ) d Z dx Γ(2 − d )( ˜ m ) − d x (1 − x ) , (19) δ m = ˜ mδ a d − − e ˜ ma d − (4 π ) d Z dx Γ(2 − d )[(1 − x ) ˜ m + x ˜ µ ] − d (4 − x − ǫ (1 − x )) , (20) δ = − e a d − (4 π ) d Z dx Γ(2 − d )[(1 − x ) ˜ m + x ˜ µ ] − d (cid:20) (2 − ǫ ) x − ǫ x (1 − x ) ˜ m (1 − x ) ˜ m + x ˜ µ (4 − x − ǫ (1 − x )) (cid:21) , (21) δ = − e a d − (4 π ) d Z dz (1 − z ) (cid:26) Γ(2 − d )((1 − z ) ˜ m + z ˜ µ ) − d (2 − ǫ )
2+ Γ(3 − d )[(1 − z ) ˜ m + z ˜ µ ] − d [2(1 − z + z ) − ǫ (1 − z ) ] ˜ m (cid:27) . (22)According to the above discussion three vacuum bubbles contribute to the Casimir energy: , , .Two first diagrams arise from Eqs. (8) and (10). Bordag et al. have computed only the last one, though approximately.In the next two sections we will consider the effect of the other two vacuum bubbles. III. PHOTONIC LOOP
In this section, we calculate NLO radiative correction to the Casimir energy due to the photonic loop. We useDirichlet boundary condition on the two parallel perfectly conducting plates in (3+1) dimensions. Although electro-magnetic field cannot be submitted to Dirichlet boundary conditions itself, one can describe the TE and TM modes ofthe electromagnetic field in the presence of the conducting plates as two scalar fields submitted to Dirichlet boundaryconditions. Obviously in the presence of the two plates, propagators automatically incorporate the boundary condi-tions and are position dependent. The contribution of one-loop photon to the vacuum energy in the interval (cid:0) − a , a (cid:1) is ∆ E Ph = Z a/ − a/ d x h Ω |H I | Ω i = i Z a/ − a/ d x + O ( α ) , (23)using Eq.(8) it becomes ∆ E (1)Ph = i Z a/ − a/ d x D B ( x, x )[ − i ( g µν k − k µ k ν ) δ ] , (24)where D B ( x, x ′ ) is the propagator of electromagnetic field in the bounded space. For overall consistency, we usedimensional regularization to control ultraviolet divergences, and a photon mass µ to control infrared divergences.Using analogies between an electromagnetic field and a massless scalar field, photon propagator is considered as D B ( x, x ′ ) = − ig µν a Z dω π Z d d − k ⊥ (2 π ) d − X n e − iω ( t − t ′ ) e − ik ⊥ . ( x ⊥ − x ′⊥ ) sin( k n ( z + a )) sin( k n ( z ′ + a )) ω − k ⊥ − k n + µ . (25)Here k ⊥ and k n denote the parallel and the perpendicular momenta to plates (in z -direction), respectively. Notethat, both contributions related to TE mode and TM mode are considered to be the same, hence the final energyshould become twice. After the usual Wick rotation and using Eqs. (19) and (25), with x = x ′ , and carrying out theintegration over the space then over solid angle in the d -dimensional Euclidean space we have∆ E (1)Ph = 12 Sδ π d − (2 π ) d − Γ( d − ) Z dk E k d − E X n k E + k n k E + k n + µ , (26)where S is the area of the planes, k E = ω + k ⊥ and k n is obtained using the Dirichlet boundary condition on thewalls, k n = nπa , n = 1 , , , . . . . (27)The one-loop photon correction to the vacuum energy in free space is∆ E ′ (1)Ph = i Z d x D F ( x, x )[ − i ( g µν k − k µ k ν ) δ ] , (28)where D F ( x, x ′ ) is the propagator of electromagnetic field in free space in Feynman gauge ( ξ = 1). We can use atrivial periodic boundary condition on the walls located at − L/ L/
2. Carrying out the space integrations gives∆ E ′ (1)Ph = 12 SLδ π d − (2 π ) d Γ( d − ) Z dk E k d − E Z dk k E + k k E + k + µ . (29)To get a vacuum energy comparable with the volume between plates, we should multiply the above energy by a factor aL then take the limit L → ∞ . We carry out k E integration and import δ from Eq. (19). Here we have two types ofregulators, d and µ , to control the ultraviolet and infrared divergences, respectively. We first work with d to eliminatesome of divergences and derive a result for a photon with mass µ , finally we will approach µ to zero. As d → d correspondingto free and bounded cases. Then we can perform the integration of x parameter of δ to get∆ E PhCas. = ∆ E (1)Ph − ∆ E ′ (1)Ph + O ( α ) = 2 αSµ ′ a (cid:20) X n =1 p n + µ ′ (cid:16) γ − p n + µ ′ (cid:17) (30) − Z ∞ dk ′ p k ′ + µ ′ (cid:16) γ − p k ′ + µ ′ (cid:17) (cid:21) + O ( α ) , where γ is the Euler-Mascheroni number and we have changed the variables as k ′ = ak π and µ ′ = aµ π . Now, we canuse the Abel-Plana Summation Formula (APSF)[53], which basically converts our summation into an integration, ∞ X n =1 f ( n ) = − f (0)2 + Z ∞ f ( x ) dx + i Z ∞ dte πt − f ( it ) − f ( − it )] . (31)Apply this formula for Eq.(30) yields (see Appendix for details)∆ E PhCas. = 2
Sαµ ′ a " − µ ′ (cid:0) γ − µ ′ (cid:1) (32)+2 Z ∞ µ ′ dte πt − p t − µ ′ (cid:16) γ − p t − µ ′ (cid:17) + O ( α ) . (33)It is obvious that as µ tends to zero, E PhCas. approaches zero, up to order α :lim µ → ∆ E PhCas. = 0 . (34)Therefore, the photonic loop does not contribute to O ( α ) radiative correction to the Casimir energy. IV. FERMIONIC LOOP
In this section, we calculate NLO radiative correction to the Casimir energy due to fermionic loop . We usethe MIT bag boundary condition on the plates. According to MIT bag boundary condition there is no flux of fermionsthrough the boundary, this means that n µ j µ = 0 , (35)where j µ indicates the current of the Dirac field and n µ is the normal unit vector to the boundary, or more strictlyit implies to complete confinement of the spinor field. Note that, ideal conductor boundary condition for the elec-tromagnetic field and bag boundary conditions for the spinor field can go together. This can be seen from the fieldequations (Maxwell equations) written in the form ∂ µ F µν = e ¯ ψγ ν ψ, (36)after multiplying with the normal vector n ν ∂ µ n ν F µν = e ¯ ψn ν γ ν ψ. (37)Dirichlet boundary condition on the walls vanishes the left side, so that we can use the bag boundary condition. ThenMIT bag boundary condition turns out to be [55–57][1 + i (ˆ n . γ )] ψ ( x ) = 0 , (38)which is satisfied on the boundary, more accurate on the plates. Applying this condition to Dirac spinor field, onecan derive pa cot( pa ) = − ma, (39)which determines quantized modes. Two limits are interesting to calculate; small mass and large mass limits. As amatter of fact, the mass is small (large) in comparison with the distance a , i.e. ma ≪ ma ≫ p n = ( n + 12 ) πa with n = 0 , , , . . . (40)where p n denotes the parallel momenta to the plates (in z -direction). Now, again for the bounded space we have∆ E F = Z a/ − a/ d x h Ω |H I | Ω i = i Z a/ − a/ d x + O ( α )= i Z a/ − a/ d x Tr[ S B ( x, x ) i ( p (cid:30) δ − δ m )] + O ( α ) , (41)where S B ( x, x ′ ), the Feynman propagator of spinor field between plates, is S B ( x, x ′ ) = ia Z dω π Z d p ⊥ (2 π ) X n =0 p (cid:30) + mω − p ⊥ − p n − m + iǫ e − iω ( t − t ′ ) e − ip ⊥ ( x ⊥ − x ′⊥ ) e − ip n ( z − z ′ ) . (42)Here p ⊥ and p n indicate the parallel and the perpendicular momenta to the plates, respectively. Converting theintegrals into dimensionless form in d spacetime dimensions we have S B ( x, x ′ ) = ia d − Z d ˜ ω π Z d d − ˜ p ⊥ (2 π ) d − X n =0 ˜ p (cid:30) + ˜ m ˜ ω − ˜ p ⊥ − p n − ˜ m + i ˜ ǫ e − i ˜ ωa ( t − t ′ ) e − i ˜ p ⊥ a ( x ⊥ − x ′⊥ ) e − i ˜ pna ( z − z ′ ) . (43)After the usual Wick rotation and carrying out the integration, one can obtain∆ E (1)F = 32 S (2 π ) d − a d − π d − Γ( d − ) Z d ˜ p E ˜ p d − E X n =0 (cid:18) ˜ p E + ˜ p n ˜ p E + ˜ p n + ˜ m δ − ˜ m ˜ p E + ˜ p ⊥ + ˜ p n + ˜ m δm (cid:19) , (44)where ˜ p E = ˜ ω + ˜ p ⊥ . Similarly, for the free space we have∆ E ′ (1)F = i Z d x Tr[ S F ( x, x ) i ( p (cid:30) δ − δ m )] . (45)Using Eq.(9) for the free propagator and after integration we obtain∆ E ′ (1)F = 32 S (2 π ) d − a d − π d − Γ( d − ) Z d ˜ p E ˜ p d − E Z d ˜ p π (cid:18) ˜ p E + ˜ p ˜ p E + ˜ p + ˜ m δ − ˜ m ˜ p E + ˜ p + ˜ m δ m (cid:19) . (46)For the small mass case the radiative correction to Casimir energy corresponding to the fermionic loop becomes∆ E FCas. = ∆ E (1)F − ∆ E ′ (1)F + O ( α )= 32 Sπ d − (2 π ) d − Γ( d − ) π d − a d − Z dp ′ E p ′ d − E (cid:20) δ (cid:18) X n =0 ( n + ) + p ′ E ( n + ) + p ′ E + m ′ − Z dp ′ p ′ + p ′ E p ′ + p ′ E + m ′ (cid:19) − am ′ δ m π (cid:18) X n =0 n + ) + p ′ E + m ′ − Z dp ′ p ′ + p ′ E + m ′ (cid:19)(cid:21) + O ( α ) , (47)where we use the change of variables as p ′ = ˜ p/π , p ′ E = ˜ p E /π and m ′ = ˜ m/π . Integrating of p ′ E yields∆ E FCas. = 32 Sπ d − d − a d − Γ( d − ) 12 sec (cid:18) dπ (cid:19) π ( δ m ′ (cid:20) ∞ X n =0 [( n + 1 / + m ′ ] d − − Z ∞ [ p ′ + m ′ ] d − dp ′ (cid:21) + am ′ δ m π (cid:20) ∞ X n =0 [( n + 1 / + m ′ ] d − − Z ∞ [ p ′ + m ′ ] d − dp ′ (cid:21)) + O ( α ) . (48)Here we need another type of APSF to convert the sum into integral, ∞ X n =0 f ( n + 12 ) = Z ∞ f ( x ) dx − i Z ∞ dte πt + 1 [ f ( it ) − f ( − it )] . (49)We can use the following formula to calculate the branch cut integral: if f ( z ) = ( z n + α m )) p/ i Z ∞ f ( it ) − f ( − it ) e πt + 1 dt = − (cid:16) pnπ (cid:17) Z ∞ α m/n ( t n − α m ) p/ e πt + 1 dt. (50)In addition we know that 1 e πt + 1 = ∞ X j =1 ( − j +1 e − πjt . (51)We use these formulae, and import δ m and δ from Eqs. (20) and (21), respectively, into Eq.(48). Again, similarto the procedure adopted in photonic loop, which lad to Eq. (30), we first expand the expression about d = 4 thentake the limit d →
4. No divergent term remains due to the usual subtraction in Casimir effect. We then, do the x integration. Finally, taking the limit µ → E F+PhCas. = ∆ E FCas. + ∆ E PhCas. = ∞ X j =1 − ( − j Sαm j π a (cid:26) K (2 jam ) + 2 jamK (2 jam ) (cid:20)
145 ln( ma ) + N j (cid:21)(cid:27) , (52)where N j = γ − ln( jπ ) − . This is the final result of the radiative correction to the Casimir energy due tofermionic loop, for the small mass case.The other interesting limit is the large mass limit. In this case, Eq.(39) turns out to be ma tan( pa ) = − pa. (53)Now, the solutions are p n = nπa with n = 1 , , . . . . (54)We follow the similar way for obtaining Eq. (48), but now we should apply APSF Eq. (31) and need the followingrelation 1 e πt − ∞ X j =1 e − πjt . (55)Finally, our the radiative NLO correction to Casimir energy in this case becomes∆ E F+PhCas. = ∆ E FCas. + ∆ E PhCas. = ∞ X j =1 Sαm j π a (cid:26) K (2 jam ) + 2 jamK (2 jam ) (cid:20)
145 ln( ma ) + N j (cid:21)(cid:27) (56)The first term, in Eq. (31) (i.e. + f (0) /
2) turns out to be independent of distance between plates a . Therefore thisterm has no impact on the physics of problem and we ignore it. For the large mass case which is also equivalent tothe large distances, Eq. (56) takes the form∆ E F+PhCas. ∼ Sα π − / m / a − / ln( am ) e − am . (57)The pressure on the plates related to this term is∆ P F+PhCas. ∼ α π − / m p m/a ln( am ) e − am , (58)which clearly shows the exponentially damping structure. In figures 1 and 2 we compare our result with the leadingterms of the Casimir energy for electromagnetic and fermion fields, respectively. Fig. 1 shows that the computedcorrection is negligible even in very small separations. In Fig. 2 we see that the impact of this correction increases inlarge separations. FIG. 1: The ratio between the one-loop correction derived here and the leading term of electromagnetic Casimirenergy E F+PhCas. /E (0)Cas. , vs the plates separation ( λ e denotes the Compton wavelength of electron.) Solid (dashed) line showsthe large (small) mass limit.FIG. 2: The ratio between the one-loop correction derived here and the leading term of fermionic Casimir energy E F+PhCas. /E (0)Fermion ,vs the plates separation ( λ e denotes the Compton wavelength of electron.) Solid (dashed) line shows the large (small) masslimit. V. CONCLUSIONS
We have calculated one-loop radiative correction to the Casimir energy due to photonic and fermionic countertermswithin the renormalized perturbation theory for QED theory. The topology considered here is two perfectly conductingparallel plates in (3+1) dimensions. We have used Dirichlet boundary condition for Electromagnetic field and MITbag boundary condition for electron. To control ultraviolet divergences we have used dimensional regularization anda photon mass µ also is used to control infrared divergences. It is found that photonic loop does not have anycontribution up to order α . The force per unit area related to fermionic and photonic loops, up to this order, at largedistances have been obtained as ∆ P Cas.F ∼ − α π − / m √ ma ln( am ) e − am . We illustrate our result in Figs. 1 and02 and compare it with the related leading Casimir energy of electromagnetic and fermion field. Acknowledgements
It is a great pleasure for us to acknowledge the useful discussion and comments of S.M. Fazeli and M.M. Ettefaghi.This research was supported by the office of research of the University of Qom.
Appendix: Calculation Of The Branch-Cut Terms
In this Appendix we calculate two types of branch-cut terms which appear in Eq. (30). Regardless of someconstants, this equation is of the form X n =1 p n + b (cid:16) C + ln p n + b (cid:17) − Z ∞ dx p x + b (cid:16) C + ln p x + b (cid:17) (59)In the APSF ∞ X n =1 f ( n ) = − f (0)2 + Z ∞ f ( x ) dx + i Z ∞ dte πt − f ( it ) − f ( − it )] , Assuming f ( x ) = √ x + b (cid:0) C + ln √ x + b (cid:1) we can write f ( it ) − f ( − it ) = C ( p b + ( it ) − p b + ( − it ) )+ (cid:16)p b + ( it ) ln p b + ( it ) − p b + ( − it ) ln p b + ( − it ) (cid:17) , (60)Choosing, b = | b | e iθ b , t = | t | e iθ t , we have for the first term p b + ( it ) − p b + ( − it ) = q | b | e i θ b + e iπ | t | e i θ t − q | b | e i θ b + e − iπ | t | e i θ t = (cid:18) e iπ e iθ t − e − iπ e iθ t (cid:19)q | b | e i (2 θ b + π − θ t ) + | t | = 2 i sin (cid:16) π (cid:17) p t − b (61)where one should note that e i (2 θ b + π − θ t ) = − t > | b | . For t < | b | this term is exactly zero. Similarlyfor second term, we have p b + ( it ) ln p b + ( it ) − p b + ( − it ) ln p b + ( − it ) = p b + e iπ t ln p b + e iπ t − p b + e − iπ t ln p b + e − iπ t = p b + e iπ t ln( e iπ/ p e − iπ b + t )) − p b + e − iπ t ln( e − iπ/ p be iπ + t )= i π hp b + e iπ t + p b + e − iπ t i + hp b + e iπ t − p b + e − iπ t i ln p t − b . (62)Now, the first of the last line is similar to similar to 61 but with plus sign between its terms, so we get p b + e iπ t + p b + e − iπ t = 2 p t − b cos π . (63)For the second term, using Eq.(61) and Eq.(63), we have for t < | b | p b + e iπ t ln( b + e iπ t ) − p b + e − iπ t ln( b + e − iπ t ) = 2 i p t − b ln p t − b . (64)For t < | b | this term is exactly zero. Therefore, our final result derived as follows: i Z ∞ dte πt − f ( it ) − f ( − it )] = 2 Z ∞ b dte πt − p t − b (cid:16) C − ln p t − b (cid:17) . (65)1 [1] H. B. G. Casimir and D. Polder, The Influence of Retardation on the London-van der Waals Forces, Phys. Rev. (1948)360.[2] M. J. 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