arXiv:hep-ph/0301090v1 14 Jan 2003 xenlln oetm r nterms shell, mass their on are momentums external-line ihtefil eomlzto constants renormalization field the with hmdw.Tebr emo edi eomlzda follows as le renormalized Firstly is [3]. field wrc. fermion Fo their bare The in diagrams. considered down. Feynman them be of should part functions imaginary self-energy give the we for purpose Finally especially the elements. precisely, for S-matrix matrices of two calculations into the simplify wrc. thus fermion fer legs, a the scheme unstable this change renormalization for we different of formula between Next layout difference reduction The the LSZ consta discuss [3]. extended renormalization we field mixing the renor of fermions introduce field introduction having the we the discard of from also case We arising the problem 2. section in compar in choice question discuss suitable at will a we as is [2], resem scheme scheme region renormalization resonant mass the on-shell in the propagator particle’s unstable unstab the for of constant(wrc.) renormalization function wave proper with Z ihteoelo eomlzdself-energy renormalized one-loop the with ¯ 2 1 fw nrdc h edrnraiaincniin httefermion the that conditions renormalization field the introduce we If eaeitrse nteLZrdcinfrua[]i h aeo unst of case the in [1] formula reduction LSZ the in interested are We ln ihteipoeeto h rcso feprmn,teradia the experiment, of precision the of improvement the with Along + 1 = γ L and a 1 2 e eomlzto ceeo emo il nSadr Mo Standard in Field Fermion of Scheme Renormalization New A δ nttt fTertclPyis hns cdm fScien of Academy Chinese Physics, Theoretical of Institute 11.h 15.s 12.15.Lk S-matr 11.55.Ds, of 11.10.Gh, off-d calculations of of contributions convenience the the include for to legs reno order external wave-function in fermion matrices two the to obtaining compl comparis After and The renormalization discussed. mass renormaization. on-shell on-s field between adopted without amplitude and W of fermion region, framework unstable resonant the for the formula in reduction formula LSZ Breit-Wigner with consist Z γ ¯ R hsteivrefrinpoaao sa olw teletters (the follows as is propagator fermion inverse the Thus . nodrt banpoe aefnto eomlzto co renormalization wave-function proper obtain to order In h et n ih-addpoeto prtr ihnoelo a one-loop Within operator. projection right-handed and left- the Σ ˆ ij ( p = ) Z 2 1 p/ 2 = 1 (Σ δ Z Z ¯ ij L ij L u ¯ ( − ( 1 2 p Z i S p/ ( γ 2 iS p ij − 2 1 ) − L Ψ ′ γ 1 ij − ) and + L ( S m 0 1 p ij ( − Σ + = ) Z .INTRODUCTION I. j p/ u 1 + ) ( = ) R Z ( j Z ¯ p ( 2 1 ij R 1 2 p ′ 1 2 Jn1,2003) 10, (Jan γ ) 2 1 ) ( ogZhou Yong Ψ | R [3] | p p ( p/ p p/ ′ 2 2 , 2 , − ) = = − γ m m m R 1 m Z Ψ j 2 i 2 ¯ ¯ Σ + ) i 0 i ) 0 = 0 = 1 2 ) δ = δZ ij = efrin nti rcs ehv eaddteform the demanded have we process this In fermion. le a + Ψ ij S ij ¯ ) Z lsterltvsi ri-inrfrua Therefore formula. Breit-Wigner relativistic the bles ¯ ( Z xelements. ix n h ag eedneo hsclamplitude. physical of dependence gauge the and s L ¯ Σ e,PO o 75 ejn 000 China 100080, Beijing 2735, Box P.O. ces, − ˆ p i inadoti h rprfrinwc.Next wrc.. fermion proper the obtain and mion mlzto osat,w xedthem extend we constants, rmalization 1 2 2 2 1 ij xpl asrnraiainhsbeen has renormalization mass ex-pole aoa w-on igasa fermion at diagrams two-point iagonal δm 6= )( γ t ntecs fhvn emosmxn.Next mixing. fermions having of case the in nts ficuigtecretoso emo external fermion of corrections the including of dwt h ope-oems renormalization mass complex-pole the with ed , ( sitouesm oaini re owrite to order in notation some introduce us t ntbepril h mgnr ato their of part imaginary the particle unstable r R aeasmda xeso fthe of extension an assumed have e elms eomlzto ceein scheme renormalization mass hell p/ m no ag eedneo physical of dependence gauge of on . j ) i aiaincntnswihhv enproven been have which constants malization + sat o ntbefrinand fermion unstable for nstants bepril,frteproeo bann a obtaining of purpose the for particle, able i δ , iecretosne ob acltdmore calculated be to need corrections tive u conclusion. our w-on ucin r ignlwe the when diagonal are functions two-point γ ij Z L ¯ . R tcei sflos is edsusthe discuss we First follows. as is rticle + 2 1 ,j i, m γ L j crc ehave we ccuracy eoetefml indices) family the denote γ . R )+ Z 1 2 del + 1 = 1 2 δZ and (4) (5) (1) (2) (3) withu ¯i and uj the Dirac spinors, we will encounter a bad thing that there is no solution for the above equations if 1 1 we keep the ”pseudo-hermiticity” relationship between Z¯ 2 and Z 2 [3]
1 0 1 0 Z¯ 2 = γ Z 2 †γ (6)
L R S This is because of the existence of the imaginary part of the fermion two-point functions, which makes Σij ,Σij and Σij un-Hermitian [4]. As we know the relationship of Eq.(6) comes from the hermiticity of Lagrangian, which guarantees the S-matrix is unitary, thus cannot be broken. A method to solve this problem is to introduce two sets of field renormalization constants, one set for the external- line particles, the other set for the inner-line particles in a Feynman diagram [3]. But it brings some complexity and cannot explain why we couldn’t find a consistent field renormalization constant for all particles.
II. WAVE-FUNCTION RENORMALIZATION CONSTANTS
Another method to solve the above problem is to abandon field renormalization, since particle fields don’t belong to physical parameters thus don’t need to be renormalized. One effect of this method is that the calculations of Feynman diagrams becomes simpler than before. The other effect will be seen in section 3. Since there aren’t field renormalization constants, we need to find a way to determine the fermion wrc. We demand the propagator of unstable particle has the following form when the momentum is on mass shell
1 (7) ∼ p2 m2 + imΓ(p2) − where mΓ(p2) is a real number which represents the imaginary part of the self-energy function times the particle’s wrc.. This form resembles the relativistic Breit-wigner formula, thus will be a reasonable assumptions. Under this assumption only the on-shell mass renormalization scheme is suitable, the other scheme, complex-pole mass renormalization scheme, cannot satisfy Eq.(7). We can see this point in the boson case. The mass definition in complex-pole scheme is based on the complex-valued position of the propagator’s pole [2]:
2 s¯ = M0 + A(¯s) . (8) with A(p2) the self-energy function of boson. Writings ¯ = m2 im Γ , the mass and width of the unstable particle 2 − 2 2 may be identified with m2 and Γ2, respectively. Given m2 and Γ2, there are two other definitions:
m = m2 +Γ2 , Γ = m1 Γ , 1 2 2 1 m2 2 s¯ = (m i Γ3 )2 . p3 − 2 The form of the propagator in resonant region is
2 1 1 1 (1 A′(m ))− = − p2 M 2 A(p2) ∼ m2 M 2 A(m2) + (p2 m2)(1 A (m2)) p2 m2 + B − 0 − − 0 − − − ′ − 2 2 2 2 with B = (m M0 A(m ))/(1 A′(m )). Through simple calculations we find all of these three definitions of mass cannot make− B −a pure imaginary− number. So the complex-pole mass renormalization scheme isn’t suitable for the assumption of Eq.(7). Because the unstable fermion’s propagator near resonant region must contain Dirac matrices, Eq.(7) isn’t enough. We further assume it has the following form:
1 1 2 2 2 ¯ 2 i p mi iZi (p/ + mi + ix)Zi → 2 2 2 . (9) p/ m + Σˆ (p/) −−−−−−→ p mi + imiΓi(p ) − i ii − 1 1 2 ¯ 2 where Zi and Zi are the fermion wrc., x is a quantity which has nothing to do with our present calculations. The LSZ reduction formula which relates the Green functions to S-matrix elements may have such form:
2 dx4dx4eip1x1 eip2x2 < Ω T ψ¯(x )ψ(x ) Ω > 1 2 | 1 2 · · · | (10) 2 2 R 1 1 p m 2 ¯ 2 i i Z1 (p/1+m1+ix1) (p/2+m2+ix2)Z2 → 2 2 2 < ψ¯(p ) S ψ(p ) > 2 2 2 . p m +im1Γ1(p ) 1 2 p m +im2Γ2(p ) −−−−−−→ 1− 1 1 · · · | | · · · 2− 2 2
here we have assumed the Heisenberg field ψ¯(x1) can be transformed into the out-state <ψ¯(p1) with an additional 1 | 2 Z1 (p/1+m1+ix1) coefficient 2 2 2 , and the Heisenberg field ψ(x2) can be transformed into the in-state ψ(p2) > with an p1 m1+im1Γ1(p1) − 1 | ¯ 2 (p/2+m2+ix2)Z2 additional coefficient 2 2 2 . According to Eq.(9), we can sum up the corrections to each external legs in a p2 m2+im2Γ2(p2) Green function and analyse− the Green function into the amputated part times the corresponding propagators. That is to say
dx4dx4eip1x1 eip2x2 < Ω T ψ¯(x )ψ(x ) Ω > 1 2 | 1 2 · · · | (11) 2 2 R 1 1 1 1 p m 2 ¯ 2 2 ¯ 2 i i Z1 (p/1+m1+ix1)Z1 Z2 (p/2+m2+ix2)Z2 → 2 2 2 < Ω T ψ¯(p )ψ(p ) Ω > 2 2 2 . p m +im1Γ1(p ) 1 2 amp p m +im2Γ2(p ) −−−−−−→ 1− 1 1 | · · · | 2− 2 2 where the subscript amp refers to the amputated Green function. Thus the LSZ reduction formula with two fermions at external legs will be:
p1 pn ··· n+m−2 1 1
= Z 2 u¯(p )Z¯ 2 Amp. Z 2 u(k ) . (12) 1 n| | 1 m 1 f f 1
k1 ··· km 1 1 ¯ 2 2 with Zf and Zf the fermion wrc. and Z the boson wrc.. The fermion spinorsu ¯(p1) and u(k1) have been added to the right hand side of the above equation, as are demanded. In actual calculations, the fermion propagator needs to be transformed into the form of Eq.(9). To start with, we introduce the renormalized fermion self-energy functions as follows
Σˆ = p/(ΣL(p2)γ +ΣR(p2)γ )+ΣS,L(p2)γ +ΣS,R(p2)γ δm . (13) ii ii L ii R ii L ii R − i Although the self-energy functions contain only the contributions of one-particle-irreducible (1PI) diagrams in usual sense, it isn’t denied in principle to include the contributions of reducible diagrams in the self-energy functions. In the present scheme the non-diagonal two-point functions aren’t zero when external momentum is on shell, since there aren’t the conditions of Eqs.(5). Thus the diagonal self-energy functions will contain the contributions of the reducible diagrams:
i i i k i i k l i i k l m i + + + + (14) k k,l ··· k, l, ,m ··· P P P··· k, l, ,m = i ··· 6 where the disc is the 1PI diagram. The condition k, l, ,m = i avoids the reduplication in actual calculations. Now we write the inverse fermion propagator as follows ··· 6
1 iS− (p/)= p/ m + Σˆ (p/)= p/(aγ + bγ )+ cγ + dγ (15) − ii − i ii L R L R with a = 1+ΣL(p2) , b = 1+ΣR(p2) , ii ii (16) c = ΣS,L(p2) m δm , d = ΣS,R(p2) m δm . ii − i − i ii − i − i and therefore after some algebra [3]
1 p/(aγL + bγR) dγL cγR iS− (p/)= − − (17) − ii p2ab cd − That’s to say
3 i i(m + δm + p/γ (1+ΣL)+ p/γ (1+ΣR) γ ΣS,L γ ΣS,R) = i i L ii R ii − L ii − R ii . (18) 2 L R S,L S,R p/ mi + Σˆ ii(p/) p (1+Σ )(1 + Σ ) (m + δm Σ )(m + δm Σ ) − ii ii − i i − ii i i − ii Adopting the on-shell mass renormalization scheme1, the mass renormalization condition is
Re[m2(1+ΣL(m2))(1 + ΣR(m2)) (m + δm ΣS,L(m2))(m + δm ΣS,R(m2))] = 0 (19) i ii i ii i − i i − ii i i i − ii i 2 2 Expanding the denominator of Eq.(18) at p = mi according to Eq.(9), we obtain
2 2 L 2 R 2 S,L 2 S,R 2 i p mi i(mi + δmi + p/γL(1+Σii(mi )) + p/γR(1+Σii (mi )) γLΣii (mi ) γRΣii (mi )) → 2 2 2 − − (20) p/ m + Σˆ (p/) −−−−−−→ (p mi + imiΓi(p ))A − i ii − with
L R L R 2 ∂ L R L R A = Re[1 + Σii +Σii +ΣiiΣii + mi ∂p2 (Σii +Σii +ΣiiΣii )+ S,L S,R (21) ∂Σii S,R ∂Σii S,L 2 (m + δm Σ )+ 2 (m + δm Σ )] ∂p i i − ii ∂p i i − ii
1 m Γ (p2)= Im[p2(ΣL(p2)+ΣR(p2)+ΣL(p2)ΣR(p2)) + (m + δm )(ΣS,L(p2)+ΣS,R(p2)) ΣS,L(p2)ΣS,R(p2)] i i A ii ii ii ii i i ii ii − ii ii (22)
Noted we have discarded the regular terms in Eq.(20). From Eq.(9) and (20) we obtain
1 1 1 1 L 2 ¯L 2 R 2 R 2 ¯R 2 L 2 Zi Zi = (1+Σii (mi ))/A, Zi Zi = (1+Σii(mi ))/A , L 1 R 1 R 1 L 1 (23) Z 2 Z¯ 2 = (m + δm ΣS,R(m2))/(A(m + ix)) ,Z 2 Z¯ 2 = (m + δm ΣS,L(m2))/(A(m + ix)) . i i i i − ii i i i i i i − ii i i In order to solve these equations, we firstly obtain a equation about x
(m + ix)2(1+ΣL(m2))(1 + ΣR(m2)) = (m + δm ΣS,L(m2))(m + δm ΣS,R(m2)) . (24) i ii i ii i i i − ii i i i − ii i
Replacing mi + ix in Eqs.(23) by this relationship, we obtain
L 1 L 1 R m2 R 1 R 1 L m2 2 ¯ 2 1+Σii ( i ) 2 ¯ 2 1+Σii( i ) Zi Zi = A ,Zi Zi = A , ZL m δm − S,R m2 R m2 i ( i+ i Σii ( i ))(1+Σii ( i )) ZR = m δm − S,L m2 L m2 , (25) i ( i+ i Σii ( i ))(1+Σii( i )) Z¯L m δm − S,L m2 R m2 i ( i+ i Σii ( i ))(1+Σii ( i )) Z¯R = m δm − S,R m2 L m2 . i ( i+ i Σii ( i ))(1+Σii( i ))
Because we haven’t introduced the fermion field renormalization constants, which contain Dirac matrices thus may change the Lorentz structure of the fermion’s self-energy functions, the self-energy functions in our scheme will keep the Lorentz invariant structure:
Σˆ (p/)= p/(ΣL (p2)γ +ΣR (p2)γ )+ΣS (p2)(m γ + m γ ) δm δ . (26) ij ij L ij R ij 0,i L 0,j R − i ij
where m0,i and m0,j the bare fermion masses. It is obvious that
ΣS,R = ΣS,L m ΣS (27) ii ii ≡ i ii Therefore Eqs.(25) can be simplified as follows
1 There are many articles discussing the gauge dependence of the on-shell mass renormalization scheme [5], we will discuss this problem in section 4
4 1 1 1 1 L 2 ¯L 2 R 2 R 2 ¯R 2 L 2 Zi Zi = (1+Σii (mi ))/A Zi Zi = (1+Σii(mi ))/A , (28) L R R 2 L 2 ¯L ¯R R 2 L 2 Zi /Zi = (1+Σii (mi ))/(1+Σii(mi )) , Zi /Zi = (1+Σii (mi ))/(1+Σii(mi )) . Because the four equations aren’t independent, we need to find an additional condition. From the strong symmetry ¯ ¯L L ¯R R between Z and Z manifested in the above equations, a reasonable assumption is to set Zi = Zi , Zi = Zi , which is also the case of stable fermion. Therefore the final results are ¯L L R 2 Zi = Zi = (1+Σii (mi ))/A , ¯R R L 2 (29) Zi = Zi = (1+Σii(mi ))/A . and
δmi L 2 R 2 S 2 2 S 2 = Re[(1 + Σii(mi ))(1 + Σii (mi ))] + Im[Σii(mi )] + Re[Σii(mi )] 1 (30) mi − q It is very easy to give the explicit one-loop level results. The wrc., mass counterterm and decay rate are listed below:
L L L 2 2 ∂ L 2 R 2 S 2 R 2 Z¯ = Z =1 Re[Σ (m )+ m 2 (Σ (p )+Σ (p )+2Σ (p )) 2 2 ]+ iIm[Σ (m )] , i i ii i i ∂p ii ii ii p =mi ii i ¯R R − R 2 2 ∂ L 2 R 2 S 2 | L 2 (31) Z = Z =1 Re[Σ (m )+ m 2 (Σ (p )+Σ (p )+2Σ (p )) p2=m2 ]+ iIm[Σ (m )] . i i − ii i i ∂p ii ii ii | i ii i
m δm = i Re[ΣL(m2)+ΣR(m2)+2ΣS (m2)] (32) i 2 ii i ii i ii i
2 L 2 R 2 S 2 miΓi = mi Im[Σii(mi )+Σii (mi )+2Σii(mi )] (33) Another meaningful result concerning x (which appears in Eq.(9)) is:
x = Γ /2 (34) − i Until now we haven’t discussed the Optical Theorem. It is a very complex problem, so we want to leave it for the future work. Here we only point out that Eq.(33) is really the fermion decay rate at one-loop level [3].
III. COMPARISON BETWEEN DIFFERENT RENORMALIZATION SCHEMES
Adopting which mass renormalization scheme and whether introducing field renormalization constants determine the difference of the renormalization schemes. Here we will list three other schemes and compare their results. Firstly we consider the renormalization scheme which adopts the complex-pole mass renormalization scheme and doesn’t introduce field renormalization constants. In this case Eq.(9) isn’t satisfied, but we can assume Γi is a complex number and independent of the propagator’s momentum. Thus all of the results associated with fermion wrc. in previous section keep unchanged, except for removing the operator Re from A, because in this case it is no sense to treat the real part and the imaginary part of the denominator of the fermion propagator differently. Therefore we obtain ¯L L R 2 Zi = Zi = (1+Σii (mi ))/A1 ¯R R L 2 (35) Zi = Zi = (1+Σii(mi ))/A1 with ∂ ∂ΣS,L ∂ΣS,R A =1+ΣL +ΣR +ΣLΣR + m2 (ΣL +ΣR +ΣLΣR)+ ii (m + δm ΣS,R)+ ii (m + δm ΣS,L) 1 ii ii ii ii i ∂p2 ii ii ii ii ∂p2 i i − ii ∂p2 i i − ii (36)
On the other hand, δmi and Γi are different from the previous results very much. But at one-loop level they are the same as Eq.(32) and (33). Next we study the renormalization scheme which adopts the complex-pole mass renormalization scheme and has introduced field renormalization constants. We still assume Γi is a complex number and independent of the propaga- tor’s momentum. Because the field renormalization constants have been introduced, the fermion self-energy functions will contain their contributions. We can write down the fermion self-energy functions in such form:
5 i i L R S,L S,R = Σiip/γL +Σii p/γR +Σii γL +Σii γR , × × 1 1 (37) 2 L R S,L S,R 2 Z¯ (Σ′ p/γ +Σ′ p/γ +Σ′ γ +Σ′ γ )Z . ≡ ii ii L ii R ii L ii R ii where denotes the field renormalization constants. This form can be easily understood if we regard it as an × application of the LSZ reduction formula Eq.(12) at two-point Green functions. The new self-energy function Σ′ is called ”pure” self-energy function by us. It is easy to obtain from the above equation that
1 1 1 1 L ¯L 2 L 2 L R ¯R 2 R 2 R Σii = Zii Zii Σii′ , Σii = Zii Zii Σii′ , 1 1 1 1 (38) S,L ¯R 2 L 2 S,L S,R ¯L 2 R 2 S,R Σii = Zii Zii Σii′ , Σii = Zii Zii Σii′ . We can write down the inverse fermion propagator in the form of Eq.(15) with
1 1 1 1 ¯L 2 L 2 L 2 ¯R 2 R 2 R 2 a = Zii Zii +Σii(p ) , b = Zii Zii +Σii (p ) , R 1 L 1 L 1 R 1 (39) c = ΣS,L(p2) (m + δm )Z¯ 2 Z 2 , d = ΣS,R(p2) (m + δm )Z¯ 2 Z 2 . ii − i i ii ii ii − i i ii ii 2 2 According to Eq.(17), the condition that in the limit p mi the chiral structure in the numerator must be canceled out leads to →
1 1 1 1 ¯L 2 L 2 L 2 ¯R 2 R 2 R 2 Zii Zii +Σii(mi ) = Zii Zii +Σii (mi ) , R 1 L 1 L 1 R 1 (40) ΣS,L(m2) (m + δm )Z¯ 2 Z 2 = ΣS,R(m2) (m + δm )Z¯ 2 Z 2 . ii i − i i ii ii ii i − i i ii ii On the other hand the unit residue renormalization condition amounts to requiring ∂ ∂ ∂ ∂ a = ab + m2a b + m2b a c d d c (41) i ∂p2 i ∂p2 − ∂p2 − ∂p2
Using Eq.(39) and Eqs.(38), we have
1 1 ′L ′S,R ′S,L L L 1+Σ mi+δmi Σ S,L mi+δmi Σ S,R ¯ 2 2 L 2 ∂ L ii ∂ R − ii ∂ − ii ∂ 1= Zii Zii [1 + Σii′ + mi ( 2 Σii′ + ′R 2 Σii′ )+ ′R 2 Σii′ + ′R 2 Σii′ ] (42) ∂p 1+Σii ∂p 1+Σii ∂p 1+Σii ∂p Like the case in section 2 we need an additional condition to determine the field renormalization constants. If we ¯L L ¯R R adopt the same assumption as the ones in the previous section, Zii = Zii , Zii = Zii , we will obtain ¯L L R 2 Zii = Zii = (1+Σii′ (mi ))/A2 , ¯R R L 2 (43) Zii = Zii = (1+Σii′ (mi ))/A2 . with
L R L R 2 ∂ L R L R A2 = 1+Σii′ +Σii′ +Σii′ Σii′ + mi ∂p2 (Σii′ +Σii′ +Σii′ Σii′ )+ ′S,L ′S,R (44) ∂Σii S,R ∂Σii S,L 2 (m + δm Σ′ )+ 2 (m + δm Σ′ ) ∂p i i − ii ∂p i i − ii
A2 is the same as A1, except for replacing the self-energy functions with ”pure” self-energy functions. But at one-loop level they are the same, so the one-loop results of Eqs.(43) are [3]
L L L 2 2 ∂ L 2 R 2 S 2 Z¯ = Z =1 Σ (m ) m 2 (Σ (p )+Σ (p )+2Σ (p )) 2 2 , i i ii i i ∂p ii ii ii p =mi ¯R R − R 2 − 2 ∂ L 2 R 2 S 2 | (45) Z = Z =1 Σ (m ) m 2 (Σ (p )+Σ (p )+2Σ (p )) p2=m2 . i i − ii i − i ∂p ii ii ii | i Finally we consider the renormalization scheme which adopts the on-shell mass renormalization scheme and has introduced field renormalization constants. In this scheme, the self-energy functions can still be written down in the form of Eqs.(37) and the renormalization conditions of Eqs.(40) keep unchanged. The only variation comes from the unit residue condition. In on-shell scheme we should treat the real part and the imaginary part of the denominator of the fermion propagator in different ways. Thus the unit residue condition becomes ∂ ∂ ∂ ∂ a = Re[ab + m2a b + m2b a c d d c] (46) i ∂p2 i ∂p2 − ∂p2 − ∂p2 after some algebra we have
6 L 1 L 1 2 2 ¯ 2 2 L mi ∂ L mi ∂ R 1 = Zii Zii (1+Σii′ )[1 + Re[ ′L 2 Σii′ + ′R 2 Σii′ + 1+Σii ∂p 1+Σii ∂p ′S,R ′S,L (47) mi+δmi Σii ∂ S,L mi+δmi Σii ∂ S,R ′L − ′R 2 Σii′ + ′L − ′R 2 Σii′ ]] (1+Σii )(1+Σii ) ∂p (1+Σii )(1+Σii ) ∂p ¯L L ¯R R If we adopt the consistent assumption: Zii = Zii , Zii = Zii , we will have ¯L L L 2 Zii = Zii = 1/(A3(1+Σii′ (mi ))) , ¯R R R 2 (48) Zii = Zii = 1/(A3(1+Σii′ (mi ))) . with 2 2 S,R S,L mi ∂ L mi ∂ R mi + δmi Σii′ ∂ S,L mi + δmi Σii′ ∂ S,R ′ ′ A3 = 1+ Re[ L 2 Σii′ + R 2 Σii′ + L − R 2 Σii + L − R 2 Σii ] 1+Σii′ ∂p 1+Σii′ ∂p (1+Σii′ )(1 + Σii′ ) ∂p (1+Σii′ )(1 + Σii′ ) ∂p (49) which are different from all of the previous results. At one-loop level, Eqs.(48) becomes
L L L 2 2 ∂ L 2 R 2 S 2 Z¯ = Z =1 Σ (m ) m 2 Re[Σ (p )+Σ (p )+2Σ (p )] 2 2 , i i ii i i ∂p ii ii ii p =mi ¯R R − R 2 − 2 ∂ L 2 R 2 S 2 (50) Z = Z =1 Σ (m ) m 2 Re[Σ (p )+Σ (p )+2Σ (p )]p2=m2 . i i − ii i − i ∂p ii ii ii i which are different from Eqs.(31) and (45). Through these comparison we find that whether to introduce field renormalization constants or not may change the fermion wrc.(comparing Eqs.(50) with Eqs.(31)). On the other hand, introducing field renormalization constants will enhance the complexity of calculating S-matrix elements. As it’s very known, ”bare” field isn’t a physical quantity thus doesn’t need to be renormalized. So the best way to renormalize a quantum field theory is to discard field renormalization. It not only simplifies the calculations of S-matrix elements, but also avoids the possible deviation arising from the introduction of field renormalization constants.
IV. GAUGE DEPENDENCE OF RENORMALIZATION SCHEMES
In this section we will discuss the gauge dependence of the renormalization schemes we have mentioned. For convenience we only discuss two typical renormalization schemes. The first scheme is the scheme we have adopted, the second scheme is the scheme which adopts the complex-pole scheme and contains field renormalization constants. At one-loop level, the results of δmi and Γi are the same in these two schemes. The difference comes from the imaginary part of the fermion’s wrc., if you compare Eqs.(31) with Eq.(45). So we only need to consider the imaginary part of the fermion self-energy functions in the following calculations. We have selected two physical processes to test the gauge dependence of the physical amplitude in these two renormalization schemes. One process is W gauge boson decays into a lepton and an anti-neutrino. For our purpose it is sufficient to only consider the dependence of W boson gauge parameter ξW . Thus there are only two Feynman diagrams which contribute to the imaginary part of the lepton self-energy functions, as shown in Fig.1
G W
ei ei
ei ei
νi νi
FIG. 1. lepton’s self-energy diagrams which have contributions to the ξW -dependent imaginary part of lepton’s self-energy functions
Using the cutting rules [6], we can calculate the imaginary part of the lepton’s wrc.. In the first scheme we have: (restrained ξW > 0) e2(m2 M 2 ξ )2θ[m2 M 2 ξ ] L l,i − W W l,i − W W Im[Zi ] = 2 2 2 (51) 64πMW sW ml,i with ml,i the lepton’s mass, MW the W gauge boson mass, sW the sine of the weak mixing angle θW and θ the Heaviside function.
7 νi νi G Z
ei νi W W γ G ei ei
νi νi W Z
ei νi W W γ W ei ei − FIG. 2. irreducible one-loop diagrams for W → eiν¯i which have contributions to the ξW -dependent imaginary part of this amplitude
In Fig.2 we show the irreducible diagrams that contribute to the ξW -dependent imaginary part of the one-loop W − e ν¯ amplitude. Using the cutting rules we obtain → i i L Im[T1] = ALFL + BLGL + ALIm[Zi ]e/(2√2sW ) (52) with A =u ¯(p )ǫ/γ ν(p ) ,B =u ¯(p ) ǫ.p1 γ ν(p ) . L 2 L 1 L 2 MW L 1 (53) and the form factors
3 3 2 3 2 2 2 2 2 e (ξ 3ξ +2)θ[1 ξW ] e (ml,i MW ξW ) θ[ml,i MW ξW ] F = W − W − − − , L 48√2πs 64√2πM 2 s3 m2 − W − W W l,i 3 3 (54) e (ξW 1) ml,iMW θ[1 ξW ] GL = − 2 2− . 32√2πsW (M m ) W − l,i Thus we have
3 3 2 3 2 2 2 2 2 e (ξ 3ξ +2)θ[1 ξW ] e (ml,i MW ξW ) θ[ml,i MW ξW ] Im[T ] = A ( W − W − − − )+ 1 L 48√2πs 128√2πM 2 s3 m2 − W − W W l,i 3 3 (55) e (ξW 1) ml,iMW θ[1 ξW ] BL − 2 2− 32√2πsW (M m ) W − l,i
We find that the imaginary part of the one-loop W − eiν¯i amplitude is gauge dependent in our scheme. On the other hand, the second scheme gives →
e2(m2 M 2 ξ )2θ[m2 M 2 ξ ] L l,i − W W l,i − W W Im[Zii ] = 2 2 2 (56) 32πMW sW ml,i and the imaginary part of the one-loop W − e ν¯ amplitude (according to the above results) is → i i 3 3 2 3 3 e (ξW 3ξW + 2)θ[1 ξW ] e (ξW 1) ml,iMW θ[1 ξW ] Im[T ′] = AL − − + BL − − (57) 1 − 48√2πs 32√2πs (M 2 m2 ) W W W − l,i also a gauge dependent quantity. At one-loop level Eq.(55) and (57) don’t contribute to the modulus square of the physical amplitude. But at two-loop level they will have the contributions: Im[T1]†Im[T1] or Im[T1′]†Im[T1′]. At the limit m 0 they have the same contributions as follows l,i → 6 2 2 2 2 e MW (ξW 1) (ξW 2ξW 2) θ[1 ξW ] Im[T1]†Im[T1]ml,i 0 = Im[T1′]†Im[T1′]ml,i 0 = − −2 2 − − (58) → → 2304π sW
8 Of course there will be no such gauge-dependent terms if we adopt the unitary gauge. We hope that the contributions from the two-loop diagrams of the process W − eiν¯i may cancel these gauge-dependent terms. Another process we have studied is Z gauge boson→ decays into a pair down-type quarks. The calculations are more complex than the previous ones. The imaginary parts of the down-type quark’s wrc. are the same in these two schemes in the limit m 0 (m is the down-type quark’s mass). They are as follows d,i → d,i ¯ 1 3 2 3/2 2 Im[M(Z didi)]md,i 0 = AL[ 384πc3 s3 e (1 4cW ξW ) θ[1 4cW ξW ]+ → → − W W − − 1 3 2 4 2 2 4 2 (59) 192πc3 s e ((ξW 1) cW 2(ξW 5)cW + 1) (ξW 1) cW 2(ξW + 1)cW +1 θ[1/cW √ξW 1]] W W − − − − − − − p The contribution of this quantity to the modulus square of the Z d d¯ amplitude at two-loop level is → i i ¯ ¯ 1 6 2 2 3 2 Im[M(Z didi)]†Im[M(Z didi)]md,i 0 = 2 8 6 e MW (4cW ξW 1) θ[1 4cW ξW ]+ → 73728π cW sW → 1 6 2 → 2 4 − 2 − − 18432π2c8 s4 e MW ((ξW 1) cW 2(ξW 5)cW + 1)θ[1/cW √ξW 1] W W − − − − − (60) (s2 ((ξ 1)4c8 4(ξ 2)(ξ 1)2c6 + (6ξ2 20ξ 18)c4 4(ξ 2)c2 + 1) W W − W − W − W − W W − W − W − W − W − (1 4c2 ξ )3/2 (ξ 1)2c4 2(ξ + 1)c2 + 1) − W W W − W − W W There are also gauge-dependentp terms. But we don’t know how they can be cancelled in the full modulus square of the Z did¯i amplitude at two-loop level. This problem will be left for our next work. Through→ these discussions we can draw a conclusion that based on the present knowledge we cannot judge which scheme is the better.
V. INTRODUCTION OF OFF-DIAGONAL WAVE-FUNCTION RENORMALIZATION CONSTANTS
In order to simplify the calculations of the corrections of fermion external legs in a Feynman diagram we can change the fermion wrc. to two matrices, which will contain the contributions of the off-diagonal two-point diagrams at fermion external legs. In order to simplify this definition we need to define two new self-energy functions as follows:
j i j k i j k l i j k l m i iΣij = + + + + k k,l ··· k, l, ,m ··· P P P··· k, l, ,m = j ··· 6 (61) j i j k i j k l i j k l m i iΣ¯ ij = + + + + k k,l ··· k, l, ,m ··· P P P··· k, l, ,m = i ··· 6 and the corresponding renormalized self-energy functions (after adding proper counterterms to the above diagrams):
L R S,L S,R Σˆ ij = p/(Σ γL +Σ γR)+Σ γL +Σ γR , i = j , ij ij ij ij 6 (62) Σ¯ˆ = p/(Σ¯ L γ + Σ¯ R γ )+ Σ¯ S,Lγ + Σ¯ S,Rγ , i = j . ij ij L ij R ij L ij R 6 We will see in a second that the new self-energy functions have no other meanings except for bringing about great convenience in actual calculations. 1 1 ¯ 2 2 Now we introduce the fermion wrc. matrices Zij and Zij as follows:
1 1 ¯ 2 2
1 1 2 iδi6=j ˆ 2 Z u(mj ) = (δij + iΣij (p/ ))Z u(mj ) p2=m2 , ij p/j mi j j j j 1 1− | (64) 2 2 ˆ iδi6=j u¯(mi)Z¯ =u ¯(mi)Z¯ (δij + iΣ¯ ij (p/ ) ) 2 2 . ij i i p/ mj pi =mi i− |
9 after some algebra we have
1 1 1 1 1 1 ¯L 2 L 2 δi6=j 2 L 2 ¯ L 2 R 2 ¯ R 2 R 2 ¯ S,L 2 L 2 ¯ S,R 2 Zij = Zi δij + 2 2 [mi Zi Σij (mi )+ mimj Zi Σij (mi )+ miZi Σij (mi )+ mj Zi Σij (mi )] , mj mi 1 1 − 1 1 1 1 (65) ¯R 2 R 2 δi6=j L 2 ¯ L 2 2 R 2 ¯ R 2 R 2 ¯ S,L 2 L 2 ¯ S,R 2 Zij = Zi δij + m2 m2 [mimj Zi Σij (mi )+ mi Zi Σij (mi )+ mj Zi Σij (mi )+ miZi Σij (mi )] , j − i and
1 1 1 1 1 1 L 2 L 2 δi6=j 2 L 2 L 2 R 2 R 2 L 2 S,L 2 R 2 S,R 2 Zij = Zj δij + 2 2 [mj Zj Σij (mj )+ mimj Zj Σij (mj )+ miZj Σij (mj )+ mj Zj Σij (mj )] , mi mj 1 1 − 1 1 1 1 (66) R 2 R 2 δi6=j L 2 L 2 2 R 2 R 2 L 2 S,L 2 R 2 S,R 2 Zij = Zj δij + m2 m2 [mimj Zj Σij (mj )+ mj Zj Σij (mj )+ mj Zj Σij (mj )+ miZj Σij (mj )] . i − j At one-loop level these results agree with Eqs.(3.3) and Eqs.(3.4) in Ref. [3]. Using these wrc. matrices we can calculate S-matrix elements without considering any of the corrections of fermion external legs in a Feynman diagram. Noted such matrices are also suitable for gauge bosons which have mixing between each other.
VI. CONCLUSION
In this paper we have attempted to extend LSZ reduction formula for unstable fermion. We adopted the assumption of Eq.(7), which resembles the relativistic Breit-Wigner formula thus can describe the behavior of unstable particle’s propagator in the resonant region. Under this assumption only the on-shell mass renormalization scheme is suitable (see section 2). In consideration of the difficulty of introducing fermion field renormalization constants [3] we have abandoned field renormalization. We assume the unstable fermion’s propagator has the form of Eq.(9) in the limit 2 2 p mi . Thus we can get the LSZ reduction formula Eq.(12) for unstable fermion and obtain the proper fermion wrc.→ Eqs.(29). Next we change our assumption and compare the difference between different renormalization schemes. There are four basic renormalization schemes, corresponding to whether or not to introduce field renormalization constants and to adopt which mass renormalization scheme, on-shell scheme or complex-pole scheme [2]. Through comparison we find that whether to introduce field renormalization constants or not will affect the values of fermion wrc. and demonstrate its difference in the imaginary part of physical amplitude, and so does the choice of mass renormalization scheme (see section 3 and 4). This discrepancy is gauge dependent and can manifest itself at the modulus square of physical amplitude at two-loop level. But we haven’t found strong evidence to judge which scheme is better that the others. Next we change the fermion wrc. to two matrices, in order to contain the corrections of fermion external legs in a Feynman diagram. The new fermion wrc. matrices, according to Eqs.(63), will simplify the calculations of physical amplitude, because we don’t need to calculate the corrections of off-diagonal self-energy diagrams at fermion external legs. We note that the concept of wrc. Matrix can be extended to bosons provided they have mixing in themselves. In conclusion, we recommend the renormalization scheme which adopts the on-shell mass renormalization scheme and abandons field renormalization, because this renormalization scheme is very simple and clear. We will study this point further in the future.
Acknowledgments The author thanks professor Xiao-Yuan Li for his useful guidance and Dr. Xiao-Hong Wu for his sincerely help.
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