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the effective potential in quantum field theory plays a crucial role in connection with the problem of the spontaneous symmetry breaking . in this field there are three classic papers @xcite . coleman and weinberg@xcite were the first ones to calculate the higher - order effective potential of a scalar field at one loop level by summing up an infinite number of feynman graphs . jackiw@xcite has used the feynman path - integral method to obtain a simple formula for the effective potential . he has succeeded in representing each loop order containing an infinite set of conventional feynman graphs by finite number of graphs using this algebraic method which can formally be extended to the arbitrary higher - loop order . in ref . @xcite the functional integral is explicitly evaluated using the steepest descent method at two - loop level . higher - loop calculations with this method are very difficult . the purpose of this paper is to show that there is a missing portion in the two - loop effective potential of the massless @xmath0 @xmath2 theory obtained by jackiw @xcite . in this paper we employ the dimensional regularization method @xcite instead of the cutoff regularization method used in ref . @xcite and for the sake of brevity we confine ourselves to the case of single component theory ( @xmath3 ) . the lagrangian for a theory of a self - interacting spinless field @xmath4 is given as ( ( x))&=&1+z2_^ -m^2+m^22 ^ 2-+4!^4 , [ lg ] where the quantities @xmath4 , @xmath5 , and @xmath6 are the renormalized field , the renormalized mass , and the renormalized coupling constant respectively , whereas @xmath7 , @xmath8 , and @xmath9 are corresponding ( infinite ) counterterm constants . we will confine ourselves to the massless theory ( @xmath10 ) . the effective potential is most suitably defined , when the effective action @xmath11)$ ] , being the generating functional of the one - particle - irreducible ( 1pi ) green s functions ( @xmath12 ) , is expressed in the following local form ( the so - called derivative expansion ) : & = & d^4x , [ loc ] where @xmath13 is the vacuum expectation value of the field operator @xmath14 in the presence of an external source . by setting @xmath13 in @xmath15 to be a constant field @xmath16 , we obtain the effective potential @xmath17 v_eff()(_cl(x))|__cl(x)=.[ve ]    following the field - shift method of jackiw @xcite for the calculation of the effective potential , we first obtain the shifted lagrangian with the constant field configuration @xmath16 ( ; ( x))&=&1+z2_^ -12(m^2++2 ^ 2)^2 + & -&+6 ^ 3-+4!^4 . [ slg ] the feynman rules for this shifted lagrangian are given in fig .  1 .    without introducing any new loop - expansion parameter , which is eventually set to be unity , we will use @xmath18 as a loop - counting parameter @xcite . this is the reason why we have kept all the traces of @xmath18 s in the feynman rules above in spite of our employment of the usual `` god - given '' units , @xmath19 . in addition to the above feynman rules , fig .  1 , which are used in constructing two- and higher - loop vacuum diagrams , we need another rule ( fig . 2 ) solely for a one - loop vacuum diagram which is dealt with separately in jackiw s derivation of his prescription and is essentially the same as that of coleman and weinberg @xcite from the outset .    using the rules , fig .  1 and fig .  2 , and including the terms of zero - loop order , we arrive at the formal expression of the effective potential up to two - loop order : v_eff()&=&+ + + .[onn ] the last three ( bracketed- ) terms on the right - hand side in the above equation appear in fig .  3 .    for the purposes of renormalization we first expand the counterterm constants in power series , beginning with order @xmath18 : m^2&=&m_1 ^ 2+^2m_2 ^ 2 + , + &=&_1+^2_2 + , + z&=&z_1+^2z_2 + . in what follows we will use the following notation for the effective potential up to the @xmath20-loop order : v_eff^[l]()=_i=0^l ^i v_eff^(i ) ( ) . the zero - loop part of the effective potential is given as v_eff^(0)()=4!^4 . [ ve0 ] the one - loop part of the effective potential is readily obtained as v_eff^(1)()&=&m_1 ^ 22 ^ 2+_14 ! ^4 -^2 ^ 48(4)^2 + ^2 ^ 4(4)^2,[ve1 ] where @xmath21 is the usual euler constant and @xmath22 is an arbitrary constant with mass dimension . the @xmath23 poles in this equation are readily cancelled out by choosing the counterterm constants @xmath24 and @xmath25 as follows : m_1 ^ 2=a_1 ,  _1=3^2(4)^2+b_1 , [ 11 ] where @xmath26 and @xmath27 are unspecified but finite constants at this stage . one may put @xmath26 ( and @xmath28 below ) to be zero from the beginning because the theory is massless . in our dimensional regularization scheme the pole part of @xmath24 vanishes , but this is not the case in the cutoff regularization method . besides @xmath24 and @xmath25 , there is another counterterm constant . it is @xmath29 . in jackiw s calculation , @xmath29 is set to be zero . this is matched to the standard condition for the defining the scale of the field    afflicted by the infrared singularity , as remarked by coleman and weinberg @xcite . ( in fact , this singularity can not be seen in @xmath30 , the one - loop order contribution to @xmath31 . the infrared singularity appears for the first time in the two - loop order @xcite . ) now let us determine @xmath29 so as to meet the following modified condition which avoids the infrared singularity : |_^2=m^2=1 . to this end , we use the following relation @xcite |_^2=m^2= ~^(2)_(p^2)p^2|_p^2=0,^2=m^2 , [ mc ] where @xmath32 is the ( momentum - conserving ) 1pi two - point green s function in the shifted theory . the right - hand side of eq . ( [ mc ] ) is calculated as 1+6(4)^2+z_1 , from which we find  z_1=-6(4)^2c_1.[c1 ] note that this wave function renormalization constant @xmath29 is free of @xmath23 singularity . but in a higher - loop order the wave function renormalization constant @xmath33 may have the @xmath23 singularity . the two - loop part of the effective potential is obtained as v_eff^(2)()&=&m_2 ^ 22 ^ 2+_24!^4 -a_1^22(4)^2 -3^3 ^ 4 8(4)^4 ^ 2 + ^3 ^ 48(4)^4 -b_1^44(4)^2 + c_1^2 ^ 44(4)^2 + & + & a_1^2 ^ 2(4)^2+b_1^4(4)^2 + & -&c_1^2 ^ 4(4)^2 + & + & ^3 ^ 4(4)^4 .[ve2 ] in the above equation , @xmath34 is a constant whose value is defined in eq . ( [ ab ] ) . notice that the so - called `` dangerous '' pole terms such as @xmath35\ln^n [ \l\fh^2/(4\p m^2)]$ ] , @xmath36 @xmath37 , in the above equation , which can not be removed by terms of counterterm constants ( @xmath38 and @xmath39 ) , have been completely cancelled out among each other . the counterterm constants @xmath40 and @xmath41 are determined as m_2 ^ 2&=&a_1(4)^2+a_2 , + _2&=&9^3(4)^4 ^ 2 -3^3(4)^4 + 6b_1(4)^2 -6c_1^2(4)^2+b_2 , [ 22 ] where @xmath28 and @xmath42 are also unspecified but finite constants .    in the massive @xmath0 @xmath2 theory , the renormalization conditions ~^(2 ) ( 0)=-m^2 , ~^(4 ) ( 0)=- , are respectively translated into |_=0=m^2 ,   d^4 v_eff()d^4|_=0= .    in our massless theory , however , we encounter the infrared singularity in the defining condition for a coupling constant . to avoid this difficulty we follow coleman and weinberg @xcite and require    @xmath26 , @xmath28 , @xmath27 , and @xmath42 are determined order by order as follows : a_1&=&a_2=0 , + b_1&=&-^2(4)^2 , + b_2&=&^3(4)^4 + & & + c_1^2(4)^2.[1122 ] after disposing successfully all divergent terms in eqs . ( [ ve1 ] ) and ( [ ve2 ] ) by the counterterm constants in eqs . ( [ 11 ] ) and ( [ 22 ] ) , we eventually arrive at our new result satisfying the conditions in eq . ( [ rc ] ) : v_eff^[2]()&= & + ^2 ^ 4(4)^2+^2^3 ^ 4(4)^4 + & + & .[2p ] our result differs from that of jackiw ( see eq . ( 3.17 ) in ref . @xcite ) just by the underlined square - bracket term in eq . ( [ 2p ] ) which is the missing portion in his calculation of the two - loop effective potential . after substituting the value of @xmath43 of eq . ( [ c1 ] ) , we have the final form of the effective potential up to the two - loop order as follows : v_eff^[2]()&= & + ^2 ^ 4(4)^2+^2^3 ^ 4(4)^4.[f2p ] let us now apply a renormalization condition @xmath44 to the two - loop effective potential with most general form of @xmath45 as an assumed series solution to the renormalization group equation : v_eff^[2]()&= & + ^2 ^ 4(4)^2+^2^3 ^ 4(4)^4,[im ] where @xmath46 , @xmath47 , @xmath48 are constants . then we readily obtain |_^2=m^2&=&+ .[rrc ] the boxed term ( @xmath6-cubic term ) on the right - hand side of the above equation is an unwanted term . thus it should vanish . next let us require the parametrization invariance of the theory . the renormalization mass , @xmath22 , is indeed an arbitrary parameter , with no effect on the physics of the problem . if we pick a different mass , @xmath49 , then we define a new coupling constant =d^4v_eff^[2]()d^4|_^2=m^2=+ p_1^2+p_2^3,[rp ] where p_1&=&32(4)^2(m^2m^2 ) , + p_2&=&^2(4)^4 . ( [ rp ] ) is readily inverted iteratively as  = -p_1^2+(2p_1 ^ 2-p_2)^3+o(^4 ) . we now substitute this @xmath6 into eq . ( [ i m ] ) . then the two - loop effective potential is given in terms of @xmath50 and @xmath49 as follows : v_eff^[2]()&= & + ^2 ^ 4(4)^2 + & & + ^2^3 ^ 4(4)^4+o(^4).[uw ] the parametrization invariance requires that the boxed term in eq . ( [ uw ] ) should vanish . from this and the vanishing boxed term of eq . ( [ rrc ] ) we obtain v_eff^(2)()&=&^2^3 ^ 4(4)^4.[2lp ] this shows us that even if one has an arbitrary value of @xmath47 , the parametrization invariance still holds . this is the reason why the jackiw s result ( eq .  ( 3.17 ) in ref . @xcite ) is safe from the check of the parametrization invariance . in the above equation @xmath47 is fixed not by the parametrization invariance but by the correct two - loop calculation of the effective potential .    in summary , jackiw used a wrong renormalization condtion , eq . ( [ wzc ] ) , in the massless @xmath0 @xmath2 theory and obtained such an incorrect value of @xmath47 as @xmath51 , but the correct value of @xmath47 is @xmath52 as given by our eq . ( [ f2p ] ) . this work was supported in part by ministry of education , project number bsri-97 - 2442 and one of the authors ( j.  -m . c. ) was also supported in part by the postdoctoral fellowship of kyung hee university . in this appendix , the momenta appearing in the formulas are all ( wick - rotated ) euclidean ones and the abbreviated integration measure is defined as _ k = m^4-n , where @xmath53 is the space - time dimension in the framework of dimensional regularization @xcite and @xmath22 is an arbitrary constant with mass dimension . for the sake of completeness , we simply list one - loop and two - loop integrals needed in our calculations though they are well known . for the two - loop integrations one may refer to ref . @xcite . s_1&&_k(1+^2k^2+^2 ) = -(^2+^2)^2(4)^2 ( ^2+^24m^2)^-/2(2 - 2 ) + , + s_2&&_k1k^2+^2 = ^2(4)^2(^24m^2 ) ^-/2(2 - 1 ) , + s_3 & & _ k1(k^2+^2)^2=1(4)^2(^24m^2 ) ^-/2(2 ) , + s_4&&_k , p1(k^2+^2 ) ( p^2+^2)[(p+k)^2+^2 ] = ^2(4)^4(^24m^2)^- ^2(1+/2)(1-)(1-/2).[www ] in the above equation , @xmath21 is the usual euler constant , @xmath54 , and the numerical value of the constant @xmath34 in eq . 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we point out that there is a missing portion in the two - loop effective potential of the massless @xmath0 @xmath1 theory obtained by jackiw in his classic paper , phys . 
 rev . 
 d 9 , 1686 ( 1974 ) . 
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