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mesonic lagrangians are often used to describe the decays and the spectral functions of resonances listed in the pdg @xcite . such lagrangians make use of some symmetries of the underlying qcd ( such as flavor or chiral symmetry ) , e.g. chiral perturbation theory @xcite and ( extended ) sigma models @xcite . in particular , in the case of some enigmatic resonances , such as the light @xmath2 meson addressed in this work , the role of mesonic loops turns out to be extremely important , see e.g. refs . @xcite and refs . therein . yet , mesons are not elementary particles but are extended objects with a radius of about @xmath4 fm . some type of form factor is needed . already in the @xmath5 model , e.g. ref . @xcite , a form factor reducing the decay for increasing phase space is present . this form factor is also useful when calculating quantum fluctuations ( loops ) , since all contributions become finite ( see sec . it is possible to include a form factor directly in the lagrangian by using nonlocal interaction terms , e.g. refs . in many studies of mesons , a 3d form factor is employed for simplicity . this is usually regarded as a breaking of covariance . here we discuss how to introduce nonlocal terms which deliver the desired 3d cutoff ( sec . 3 ) . interestingly , an extension which preserves covariance is possible @xcite . implications for the scalar kaonic sector and conclusions are discussed in sec . in general , effective lagrangians contain both derivative and nonderivative interaction terms @xcite . a prototype of such a lagrangian in which a ( scalar ) state @xmath6 interacts with two ( pseudoscalar ) particles @xmath7 and @xmath8 reads : @xmath9 the scalar kaonic sector is obtained upon identifying @xmath10 ( in first approximation corresponding to @xmath1 ) and @xmath11 and @xmath12 ( other isospin combinations are here neglected ) . the tree - level decay width @xmath13 as function of the ` running ' mass @xmath14 of @xmath6 reads:@xmath15   ^{2}\text { , } % \ ] ] where @xmath16 is the mass of @xmath7 and @xmath17 the mass of @xmath8 . the quantity @xmath18 is the modulus of the three - momentum of ( one of the ) outgoing particle(s):@xmath19 the actual value of the tree - level decay width is obtained by setting @xmath14 to the tree - level ( nominal ) mass @xmath20 of the field @xmath6 , @xmath21 the lagrangian ( [ laglocal ] ) as it stands is not suitable for loop calculations . one can regularize the theory by considering an ` ad hoc ' modification of both vertices via a form factor:@xmath22 the quantity @xmath23 is often called ` cutoff ' ( a smooth cutoff here ) . the choice of a gaussian is part of the modelling and is obviously not unique . anyway , there is clear physical motivation behind @xmath23 : it is the energy scale which takes into account that mesons are extended objects . the dimension of the system is roughly given by @xmath24 numerically @xmath23 lies between @xmath4 and @xmath25 gev @xcite . the ` local ' limit is recovered for @xmath26 as a consequence of eq . ( [ abmod ] ) , the decay width changes as @xmath27 and is similar to the form factor implemented in the @xmath5 model ( see e.g. @xcite and the ` quark model ' review in  ref . @xcite ) . the contribution of the loop @xmath28 in which the particles @xmath7 and @xmath8 circulate as calculated from the original local lagrangian ( [ laglocal ] ) reads@xmath29   ^{2}}{\left [   k_{1}^{2}-m_{1}^{2}% + i\varepsilon\right ]   \left [   k_{2}^{2}-m_{2}^{2}+i\varepsilon\right ]   } \text { , } % \ ] ] where the constraint @xmath30 is understood and @xmath31 is the momentum of the unstable particle @xmath32 in its reference frame @xmath33 as mentioned above , this loop contribution is divergent ( with @xmath34 ) . the substitution ( [ abmod ] ) makes it convergent thanks to the form - factor:@xmath35   ^{2}f_{\lambda } ^{2}(\vec{k}_{1}^{2})}{\left [   k_{1}^{2}-m_{1}^{2}+i\varepsilon\right ] \left [   k_{2}^{2}-m_{2}^{2}+i\varepsilon\right ]   } \text { . } \label{looplambda}%\ ] ] at this point , one may object that the form factor breaks covariance , since it depends on the three - momentum only . we will show in the next section that this is not necessarily the case . once the form factor is introduced , the full propagator of the particle @xmath6 is calculated as @xmath36 ^{-1}$ ] where @xmath37 is the bare mass of @xmath6 and @xmath38 at the one - loop level , the breit - wigner mass of @xmath6 is defined as @xmath39 while the pole mass is obtained by solving the equation @xmath40 in the complex @xmath41-plane , see e.g. ref . @xcite for the scalar @xmath42 and @xmath43 sectors . the real part of the loop can be also obtained as @xmath44 where @xmath45 is an additional constant due to a subtlety of qft containing derivatives , see ref . @xcite for details . for @xmath46 , @xmath47 scale as @xmath48 then a three - time subtracted dispersion relation would be needed . this is quite cumbersome and reinforces the viewpoint that -for the particular case of hadronic physics- a physical cutoff is a meaningful procedure . the modification of eq . ( [ abmod ] ) and , consequently , the form factor in eqs . ( [ gammalambda ] ) and ( [ looplambda ] ) have been introduced as an ` ad hoc ' change of the feynman rules . yet , it is possible to modify the lagrangian in order that the modified equations automatically follows from it .    * nonlocal extension 1 * : the easiest way ( see refs . @xcite ) is to consider the following nonlocal lagrangian ( for simplicity we discuss here only the @xmath49-term in eq . ( [ laglocal ] ) , but the @xmath50-term is very similar):@xmath51 the feynman rule at the @xmath49-vertex is modified ( upon defining @xmath52 @xmath53 , hence : @xmath54 ) : @xmath55 for @xmath56 one reobtains the local limit . a covariant form factor requires a dependence @xmath57 , hence @xmath58 follows . quite interestingly , if the masses of @xmath7 and @xmath8 are equal and on - shell , @xmath59 , in agreement with eq . ( [ gammalambda ] ) upon setting @xmath60 . however , the loop is different from  eq . ( [ looplambda ] ) , since it involve @xmath61 into the integral [ and not simply @xmath62 . a generalization in case of unequal masses is discussed in  refs . @xcite . the choice @xmath63 leads to the desired result @xmath64 of eq . ( [ abmod ] ) and also of eqs . ( [ gammalambda ] ) and ( [ looplambda ] ) , but it explicitly breaks covariance . this result is anyway valuable since it shows that it is possible to get the desired 3d cutoff form , but should be used only in the reference frame of the decaying particle . * nonlocal extension 2 * : we aim to determine the lagrangian which generates the vertex function ( [ abmod ] ) in a covariant manner @xcite . we start from the general nonlocal expression:@xmath65 where the vertex - function @xmath66 in position space has been introduced . the case @xmath66 @xmath67 delivers the local limit ( [ laglocal ] ) . the vertex function in momentum space is given by:@xmath68 here we _ assume _ that @xmath66 is such that @xmath69 it respects covariance because the final form factor is a function of lorentz - products . nevertheless , in the rest frame of @xmath6 one recovers the desired dependence : @xmath70 note , in order to get the desired expression , @xmath71 out of which @xmath72 with @xmath73 and @xmath74 this line of reasoning shows that it is -at least in principle- possible to reconcile covariance with a 3d form factor in the rest frame of the decaying particle . in this work we have discussed form factors entering in mesonic interaction terms . we have started from a local lagrangian and we have modified the interaction vertex ` ad hoc ' by introducing a function of the three - momentum of one outgoing particle . then , we have presented two nonlocal extensions of the lagrangian that deliver the desired expressions without the need of modifying by hand the feynman rules . the first nonlocal extension is relatively simple but delivers the desired form factor at the price of breaking covariance . the second extension is more involved but delivers a 3d cutoff in the rest frame of the decaying particle by respecting covariance . recently , the expressions of the propagator described in sec . 2 ( which are then justified by our study of  sec . 3 ) were used to study the positions of the poles in the complex plane in both the isodoublet- and isovector - scalar sectors . in both cases , the lagrangians contains only one seed state , corresponding to the quark - antiquark resonances @xmath75 and @xmath76 respectively . yet , the loops ( with the necessary presence of a form factor ) are strong enough to generate two additional resonances : @xmath77 and @xmath78 these are companion poles , and hence dynamically generated states . moreover , these states disappear in the large-@xmath3 limit , confirming their non - conventional nature . in particular , @xmath2 is not yet confirmed in the pdg @xcite . the study based on the formalism described in these proceedings unequivocally finds a pole : @xmath79 gev @xcite . this is in agreement with other works , e.g. ref . @xcite and refs . therein , and reinforces the need of accepting this state in the pdg .    in both refs . @xcite one obtains a similar values of the cutoff : @xmath80 gev . moreover , in  ref . @xcite different form factors have been tested . it is found that they can not fit the pion - kaon scattering data so well as the gaussian form factor . thus , even if there is in principle no fundamental reason behind a gaussian function , it nevertheless seems to be the best choice in order to describe hadronic phenomenology . d. parganlija , p. kovacs , g. wolf , f. giacosa and d. h. rischke , phys . rev . * d87 * , 014011 ( 2012 ) arxiv:1208.0585 [ hep - ph ] ; s.  janowski , f.  giacosa and d.  h.  rischke , phys . d * 90 * ( 2014 ) 11 , 114005 [ arxiv:1408.4921 [ hep - ph ] ] .                  y.  v.  burdanov , g.  v.  efimov , s.  n.  nedelko and s.  a.  solunin , phys . d * 54 * ( 1996 ) 4483 [ hep - ph/9601344 ] ; f.  giacosa , t.  gutsche and a.  faessler , phys . c * 71 * ( 2005 ) 025202 [ hep - ph/0408085 ] ; r.  d.  bowler and m.  c.  birse , nucl . a * 582 * ( 1995 ) 655 [ hep - ph/9407336 ] ; f.  giacosa , glueball phenomenology within a nonlocal approach , phd thesis , tuebingen ( 2105 ) . a.  faessler , t.  gutsche , v.  e.  lyubovitskij and y.  l.  ma , phys .  rev . d * 76 * ( 2007 ) 014005 [ arxiv:0705.0254 [ hep - ph ] ] . a.  faessler , t.  gutsche , v.  e.  lyubovitskij and y.  l.  ma , phys . d * 76 * ( 2007 ) 114008 [ arxiv:0709.3946 [ hep - ph ] ] .
mesons are extended objects , hence their interaction can be described by utilizing form factors . at the lagrangian level 
 , one can use nonlocal interaction terms . here 
 we describe two possible nonlocal lagrangians leading to a 3d form factor : the first one is simple but does not fulfill covariance ( if one insists on a 3d cutoff ) , the second extension is more involved but guarantees covariance . 
 such form factors are useful when calculating mesonic loops . as an important example 
 , we discuss the scalar kaonic sector , @xmath0 . 
 the lagrangian contains a single scalar kaon ( the well - establish state @xmath1 ) , but through loops @xmath2 emerges as a dynamically generated companion pole ( which disappears in the large-@xmath3 limit ) .