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in the past few years , quantum theory on a cone has attracted much attention .
one of the motivations , besides theoretical interest in @xmath0 dimensional field theory , is the application to quantum field theory around cosmic strings .
an idealized cosmic string , which is infinitely thin , can be represented by a conical singularity of space - time .
quantum effect around cosmic strings has been studied by many authors including dowker @xcite , frolov @xcite , linet @xcite and smith.@xcite    the scattering by a cosmic string or a conical singularity has also been investigated recently.@xcite this non - trivial scattering is often dubbed as the gravitational aharonov - bohm ( ab ) effect .    it is interesting to consider `` radiative '' corrections to the gravitational ab scattering , which arise due to quantum fluctuations of all interacting field around cosmic strings .    in the present paper ,
we consider vacuum fluctuations of the electromagnetic field .
then , charged particles scattered by a cosmic string undergo quatum corrections even if the cosmic string has no charge and no classical field strength .
the organization of this paper is as follows : in section 2 , the vacuum polarization of gauge field is obtained . in section 3 , we derive the quantum - mechanically corrected klein - gordon equation for a charged scalar field . the scattering amplitude is calculated in section 4 .
section 5 is devoted to conclusion .
in this section , we compute the vacuum value of @xmath1 around a infinite straight cosmic string , where @xmath2 is the electromagnetic field .
connection to an effective wave equation for a charged scalar field is presented in the succeeding section .
we first anticipate that @xmath3 , since there is no dimensional quantity other than the distance from the string .
a comment on gauge invariance is in order .
classically the presence of the cosmic string does not break the gauge invariance . at the one - loop level , however , one can see the breakdown of gauge symmetry near the cosmic string , due to the mass - like term @xmath4 in the scalar qed ; here the vacuum expectation value @xmath5 depends on the distance from the cosmic string .
thus the non - zero expectation value of @xmath6 does not give rise to a real conflict .
note also that we often encounter the expression @xmath7 in the discussion on the coleman - weinberg potential ( or radiative gauge - symmetry breaking ) in scalar qed at zero and finite temperature.@xcite    we assume the following line element around the idealized cosmic string which lies along the @xmath8 axis : @xmath9 where @xmath10 is related to the mass density of the string @xmath11 by @xmath12 .
working in the coulomb gauge , we can denote the gauge field in the form of normal - mode expansion , such as @xmath13 mode functions @xmath14 are the solutions for the maxwell equations . in the coordinate system represented by eq .
( [ 2.1 ] ) , we find the following set of mode functions : @xmath15 where @xmath16 is the bessel function , @xmath17 is positive integer and @xmath18 .    using the mode functions
, we can perform the calculation of @xmath19 by mode - summation .
after regularization , we find that a finite portion of @xmath19 is given by @xmath20    the quantity has been regularized so as to become zero if @xmath21 .
( note that this is twice of the real - scalar contribution.@xcite ) the mode expansion is also useful to investigale properties of charged fields around a cosmic string.@xcite    in the next section , we consider the modification of field equation due to the vacuum field @xmath19 .
suppose that a minimally - coupled scalar field @xmath22 is governed by klein - gordon equation , where the space - time derivative is replaced by the covariant derivative including the gauge field .    in this paper , we assume that the cosmic string has no classical electromagnetic field .
although general cosmic strings in guts have fluxes in their core , their fluxes are not magnetic fluxes we know , but are associated with gauge fields of broken symmetry . for a superconducting cosmic string @xcite
the analysis ought to be done by some other ways , since the metric is more or less deformed by electric currents .
therefore , we treat the case with normal cosmic strings throughout this paper .    however , we take into account the vacuum expcctation value of the electromagnetic field .
only the second moment @xmath23 appears and the field equation is expressed as @xmath24 where @xmath25 is the coupling constant and @xmath26 is the mass .
the term @xmath27 behaves as a potential term .    if we use the metric ( [ 2.1 ] ) and substitute ( [ 2.2 ] ) into eq .
( [ 3.1 ] ) , we obtain @xmath28 where @xmath29 .    a solution for the modified equation ( [ 3.2 ] ) is written in the form @xmath30 with @xmath31 and @xmath32 .    in the following section
, we study the scattering problem by use of this solution .
we set @xmath33 in the following analysis in this paper .
to treat the scattering by an infinite string , we consider two - dimensional scattering , taking the @xmath8-component of the momentum ( @xmath34 in ( [ 3.3 ] ) ) to be zero . we have set @xmath33 .
we first review the scattering in two dimensions.@xcite the wave function behaves asymptotically , @xmath35 when a target sits at the origin . here
@xmath36 is a scattering amplitude , and a differential cross section ( for a cosmic string , per unit length ) is given by @xmath37 .
the phase in front of @xmath36 is chosen to simplify the expression of the optical theorem in two dimensions @xcite ( see later ) . in order to find the scattering amplitude , we compare the asymptotic form of the solution of the wave equation with ( [ 4.1 ] ) .
we determine @xmath38 by matching the coefficient of the in - going `` spherical '' wave ( @xmath39 ) .
the asymptotic form of the plane wave going in the @xmath40-direction is given by @xmath41 further if we assume that we can take the following asymptotic behavior of the solution for the wave equation : @xmath42 then we find @xmath43 where @xmath44 is called as a phase shift .
now , we will turn back to the present problem . the wave equation to be considered
( [ 3.2 ] ) with @xmath33 .
we regard @xmath45 as an independent parameter of @xmath10 for a later use .
the mode solution ( [ 3.3 ] ) behaves asymptotically as @xmath46 where @xmath47 .
thus one can find @xmath48    if @xmath49 is independent of @xmath10 , @xmath44 is negative when @xmath21 and @xmath50 .
therefore we can say that the `` potential '' @xmath51 gives rise to a repulsive force . as studied and stated in ref .
@xcite , the scattering amplitude for @xmath52 contains delta functions .
namely , for @xmath53 , one can find @xcite @xmath54 this peculiarity originates from the conical space generates `` long - range force '' in some naive sense .
we discuss the amplitude in which such divergences are removed.@xcite    we will calculate @xmath55 where @xmath56 and @xmath57 ( @xmath17 : integer ) .
we first compute the scattering amplitude for @xmath21 and @xmath58 ( we take @xmath45 as an independent parameter . ) .
this is used for a check of general results .
the result obtained by numerical calculation is shown in fig .  1 .
we examine some approximation schemes .     and
@xmath58 . ( a )
@xmath59 , ( b ) @xmath60 .
the solid line stands for @xmath61 and the broken line for @xmath62.,title="fig:",width=226 ]   and @xmath58 . ( a )
@xmath59 , ( b ) @xmath60 . the solid line stands for @xmath61 and the broken line for @xmath62.,title="fig:",width=226 ]    the @xmath63-wave ( @xmath64 ) contribution in the formula ( [ 4.4 ] ) gives @xmath65 and these seem to be a good estimation as the lowest order .    the optical theorem @xmath66 is applicable for @xmath21 and one can confirm this at the order in using the @xmath63-wave result ( [ 4.9 - 1 ] , [ 4.9 - 2 ] ) . note that the amplitude does not contain divergences for @xmath21 ( for instance , delta functions cancel one another in ( [ 4.7 ] ) when @xmath21 . ) .
although the potential has a short - range nature , the born approximation is not applicable because the singularity near the origin is too strong .
next we consider the case for a cosmic string . here
we recover the relation @xmath67 .
we define @xmath68 by the difference in the scattering amplitudes for @xmath67 and @xmath53 ; i.e. , the contribution of ( [ 4.7 ] ) is subtracted .
the numerical results are given in fig .
the value of @xmath69 in the vicinity of @xmath70 is not indicated because of the poor convergence in the sum of the oscillating series .
the value seems almost constant .     and @xmath53 in the case of @xmath71 .
we set @xmath72 .
the solid line stands for @xmath73 and the broken line for @xmath74.,width=226 ]    the real part of @xmath68 exhibits a logarithmical divergence at @xmath75 .
the behavior of the amplitude is approximately given by : @xmath76 @xmath77 for a realistic cosmic string , @xmath78 may be too small for us to detect the deviation in the scattering amplitude unless the strength of the coupling become sufficiently large .
in this paper we have derived the lowest - order vacuum quantum space electromagnetic corrections to the scattering amplitude in a conical background space - time .
inclusion of the quantum effect of non - abelian gauge fields around a cosmic string is an important extension of the present work : it may be connected with non - abelian ab effect @xcite and the baryon decay / genesis mediated by cosmic strings.@xcite it is also important to treat the effect of self - interaction in a conical space for the yang - mills case .
( self - interacting scalar fields around a cosmic string have been studied in ref .
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it is known that the vacuum polarization of zero - point field arises around a conical singularity generated by an infinite , straight cosmic string . in this paper 
 we study quantum electromagnetic corrections to the gravitational aharonov - bohm effect around a cosmic string . 
 we find the scattering amplitude from a conical defect for charged klein - gordon field .