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it has been known for some time that cu benzoate is a quasi-1d @xmath0 ( heisenberg ) antiferromagnet @xcite . in a recent neutron scattering experiment @xcite in a magnetic field ,
the existence of field - dependent incommensurate low energy modes was established .
the incommensurability was found to be consistent with the one predicted by the exact solution of the heisenberg model in a magnetic field . however , the system exhibited an unexpected excitation gap induced by the applied field .
these findings were quite surprising as the coexistence of a gap and a magnetization is inconsistent with rotational symmetry around the direction of the magnetization @xcite .
the resolution of this puzzle goes as follows : dender _ et al _ @xcite suggested that the gap could be due to the staggered magnetic field generated by the alternating @xmath1-tensor in cu benzoate ; oshikawa and affleck @xcite then showed that this mechanism ( in particular the generation of a staggerd field _
perpendicular _ to the direction of the applied magnetic field ) , together with the dzyaloshinskii - moriya interaction present in cu benzoate , indeed can account for the experimental findings .
they proposed that cu benzoate is described by the hamiltonian h=_i j _ i_i+1- h s_i^z + h ( -1)^i s^x_i  , [ hamil ] where @xmath2 .
the low energy effective theory of is obtained by abelian bosonization and is given by a sine - gordon model with lagrangian density @xcite = ( _ ) ^2 + (  )  . [ lagr ] here @xmath3 is the dual field and fulfils @xmath4 , @xmath5 .
this implies that @xmath6 and thus is a sine - gordon model for the dual field .
we note that all of our analysis is based on the model and so that the results carry over to any material described by this theory ( with appropriate fit of @xmath7 to experiment ) .
the coupling constant @xmath8 is a function of the applied field @xmath9 and is related to the `` dressed charge '' ( see @xcite and references therein ) of the isotropic heisenberg chain in a magnetic field by =  . for further convenience
we define the quantity =  .
[ xi ] the dressed charge can be determined exactly from the bethe ansatz solution .
it is given by @xmath10 , where @xmath11 is a solution of the integral equation @xcite z()+_-a^a d  z ( ) = 1  .
the integration boundary @xmath12 is a function of the magnetic field @xmath9 and is determined by the condition @xmath13 , where ( ) + _ -a^a d  ( ) = -  .
these integral equations can easily be solved numerically . in fig .
[ fig : beta ] we plot @xmath14 of ( where @xmath8 is the coupling constant of the sine - gordon model ) as a function of the applied magnetic field @xmath9 for cu benzoate .
we note that at zero field @xmath15 .
the spectrum of the sine - gordon theory is well known @xcite ( see also @xcite ) : it consists of a soliton - antisoliton doublet of mass @xmath18 and their bound states which are called `` breathers '' . for cu benzoate
the soliton mass as a function of the applied magnetic field was determined by oshikawa and affleck @xcite m 1.85 ( ) ^1/(2-^2/4)j  , [ solitonmass ] where the induced staggered field @xmath19 is determined as a function of the applied field @xmath9 ( and the dzyaloshinskii - moriya interaction in cu benzoate ) in @xcite .
in addition there are @xmath20 $ ] ( here @xmath21 $ ] denotes the largest integer smaller than @xmath22 ) breathers , denoted by @xmath23 , with masses m_n = 2 m  n/2  ;   n=1 ,  , .
[ mass ] the mass spectrum as a function of magnetic field for cubenzoate is plotted in fig .  [
fig : breathermass ]      we see that for small fields there are _ three _ breathers , one of which is just below the soliton - antisoliton continuum starting at @xmath24 . for higher fields further breathers split off this continuum .
however they appear at energies higher than the threshold of the @xmath25 two breather continuum . for small fields
the dependence of the mass ratios @xmath26 on @xmath9 is m_n~2 m , where @xmath27 .
the lagrangian ( [ lagr ] ) can be brought to standard sine - gordon form by performing the duality transformation @xmath28 .
the bosonized expressions of the spin operators change under this duality transformation as well .
we find @xcite @xmath29 ,    \(x ) & = & ( x ) + ( -1)^n ( x )  , j^z&=&_x  , j^+&= & ( -i)(-2x ) , n^x ( x ) & = & ( ( x)),n^y ( x ) & = & ( ( x)),n^z ( x ) & = & ( ( x)-2x ) .
[ boso ] here @xmath30 are the components of the staggered magnetization , @xmath31 are the current operators , @xmath32 and @xmath33 is a nonuniversal coefficient .
the quantity @xmath34 is the difference between the fermi momenta for up spins and down spins .
it is easily determined from the exact solution of the xxx heisenberg chain in a magnetic field ( see @xcite and references therein ) but is not important for the present work
.    the dynamical magnetic susceptibilities for wavevectors close to the antiferromagnetic wave vector @xmath35 ( @xmath36 for the longitudinal part @xmath37 ) are then given by ^xx(,q)&&_-^dx_0^dt  e^i ( + i)t - i(q-)x   & & ^yy(,q)&&_-^dx_0^dt  e^i ( + i)t - i(q-)x   & & ^zz(,q)&&_-^dx_0^dt  e^i ( + i)t - i(q-)x   & & -20pt .
[ suscept ]    the transverse susceptibilities @xmath38 and @xmath39 can be determined exactly through the formfactor approach to quantum correlation functions @xcite ( our discussion follows @xcite ) .
the longitudinal susceptibility involves the dual field which is a nonlocal operator and can not be treated in the same way .
we therefore restrict our analysis to the transverse susceptibilities . as the longitudinal modes become soft at the incommensurate wavevectors @xmath36 rather than at the neel wave vector @xmath35 ( where the transverse modes are soft ) it is possible to separate @xmath37 from @xmath40 experimentally by performing constant wavevector scans around @xmath41 and @xmath42 respectively .
the quantities @xmath43 in are the formfactors of the operator @xmath44 and have been calculated exactly @xcite .
the expansion is in terms of multiparticle states . in order to make contact with the neutron scattering experiments we are interested in the imaginary parts of the dynamical susceptibilities , which are proportional to the neutron scattering cross sections .
because all elementary excitations in the sine gordon theory have a gap , @xmath45-particle states will contribute to @xmath46 ( @xmath47 ) only at energies larger than the @xmath45-particle gap .
in other words , by taking into account only the first few terms in the expansion we obtain an exact expression for @xmath48 for energies below a threshold that depends on the omitted terms .
the formfactor expansion is now readily performed using the results of @xcite and the bootstrap procedure to determine breather formfactors . using that @xmath44 is even under charge conjugation @xcite we find m ^xx(,q ) & & 2_n=1^[1/]z_2n  ( s^2 - m_2n^2 ) & & + & & + 2 & & +  [ cos ] where ( m_1,m_2,s ) = arccosh ( )  . here
the dependence of @xmath51 on the applied field is given by and fig .
[ fig : breathermass ] . the first terms in correspond to single - particle breather states .
the ( squares of the ) breather formfactors are given by z_2&= &  & & , z_4&= & & & .
the soliton - antisoliton formfactor is & & |f^cos_+-()|^2= & & , and the @xmath25 breather - breather formfactor is & & |f^cos_11()|^2=^4 & & , where = 2/2   . the next most important contribution ( _ i.e. _ the term with the lowest threshold ) in the expansion comes from @xmath52 breather - breather states .
it will contribute at energies larger than @xmath53 , where @xmath54 are given by .    in fig .
[ fig : chixx35 ] we plot our result for @xmath55 as a function of @xmath56 ( where @xmath18 is given by ) for an applied field of @xmath57 tesla , which is one of the values studied experimentally in @xcite .
we see that there is one coherent mode at @xmath58 with weight @xmath59 . at @xmath60
the @xmath25 continuum appears and at @xmath61 the soliton - antisoliton continuum .
the susceptibility is regular at both thresholds .
[ fig : chixx7 ] shows @xmath55 as for @xmath62 tesla .
the picture is qualitatively similar to the one at @xmath63 tesla although the soliton - antisoliton threshold has become more pronounced .
the weight of the delta - function corresponding to the breather @xmath64 has diminished to @xmath65 .
as the field increases a singularity develops at the @xmath61 threshold until at @xmath66 tesla the breather @xmath67 splits off the continuum .
the situation at @xmath68 tesla is shown in fig .
[ fig : chixx14 ] .
the weight of @xmath64 is @xmath69 whereas the fourth breather has only a very small spectral weight of @xmath70 .
m^yy(,q ) & & 2_n=1^[1/]z_2n-1  ( s^2 - m_2n-1 ^ 2 ) & & + 2&&+ 2 & & +  [ sin ] here the breather formfactors are given by z_1&= &  & & , z_3&= & & & .
the soliton - antisoliton formfactor is found to be & & |f^sin_+-()|^2= & & .finally we take into account the @xmath73 breather - breather state which has a formfactor of & & |f^sin_12()|^2=      in figs .
[ fig : chiyy35]-[fig : chiyy14 ] we plot @xmath76 for three different values of the applied field , namely @xmath77 tesla .
the emerging picture remains qualitatively unchanged for all three values of @xmath9 .
there are two coherent modes corresponding to the breathers @xmath78 and @xmath79 . at @xmath61
the soliton - antisoliton continuum starts and at @xmath80 we observe the onset of the @xmath73 breather - breather continuum . the latter one exhibits a singularity as a function of @xmath81 .
the weight of the delta functions corresponding to the breather @xmath78 decrease with increasing field : @xmath82 , @xmath83 and @xmath84 . for @xmath79
we find a decrease with increasing field after an initial increase : @xmath85 , @xmath86 , @xmath87 .
note that the spectral weight of the heavier @xmath79 particle is always significantly smaller .
let us now compare our results with the experimental findings .
our results for the imaginary part of the dynamical susceptibilities are consistent with the experiment at @xmath88 tesla ( fig .
3 ( c ) of @xcite ) : as the beam was unpolarized the experiment observed @xmath89 ( the contribution from @xmath37 emerges only at energies greater than @xmath90 ) .
the observed peaks at @xmath91 , @xmath92 and @xmath93 correspond to the three breathers @xmath78 , @xmath64 and @xmath79 and agree very well with our prediction for the mass spectrum and ( the first two peaks were already discussed in @xcite ) . in order to compare our calculated weights of the delta - functions in the susceptibilities to the experimentally observed intensities one needs to convolve with the instrumental resolution . convolving our exact results with a gaussian
we obtain a fit to the experimental data shown in fig .
[ fig : fit ] .          we are grateful to g. aeppli , c. broholm , r. coldea , r. cowley , a. millis and l.p .
regnault for important discussions .
f.h.l.e . was supported by the eu under human capital and mobility fellowship grant erbchbgct940709 .
recent experiments on the quasi 1-d antiferromagnet cu benzoate revealed a magentic field induced gap coexisting with ( ferro)magnetic order . 
 a theory explaining these findings has been proposed by oshikawa and affleck . in the present work 
 we discuss consequences of this theory for inelastic neutron scattering experiments by calculating the dynamical magnetic susceptibilities close to the antiferromagnetic wave vector by the formfactor method .    
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