Pauli-Villars Regularization to All Orders in .NET framework

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8-4-2 Pauli-Villars Regularization to All Orders
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We choose to work here with the traditional Pauli-Villars regularization, rather than the dimensional regularization. The latter will, however, be illustrated in the two-loop computation of Sec. 8-4-4. We recall that the Pauli-Villars method amounts to regularizing the fermion loops and photon propagators independently. The photon propagator is replaced in a standard fashion by a superposition of free propagators. Fermions loops, however, are treated as a whole, each one standing now for a sum of contributions corresponding to fermions of different masses, minimally coupled to the electromagnetic current (Fig. 8-16). Explicitly,
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d p (2n)4 tr
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p _ Ms + it; YIlI p + 41 - Ms + it; ... Y1l2.
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with the convention that Co == 1, Mo = m, and ql ... q2n are the momenta entering the loop. We rationalize the denominators, compute the trace, and expand both the numerators and the denominators in powers of M;. We find
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[- f f d p Pn(p2,p qi,ql)+M;Pn- (p2,p qi,ql)+ - s=o Q2n(p2,p.qi,ql)+M;Q2n-l(p2,p qi,qr)+ f C f d4~ [Pn + M; (Pn-l - PnQ~n-l) + ... J (8-90) s=o Q2n Q2n Q2n
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where Pk and Qk are polynomials in p of degree less or equal to 2k. For large 1 I, the coefficient of M;k behaves as 1 1- 2n - 2k. Therefore, if we impose two p p conditions
L Cs=O s=o
CsM; =0
(8-91)
Figure 8-16 A fermion loop.
QUANTUM FIELD THEORY
the integrand of any fermion loop behaves as Ip 6 (we discard vacuum diagrams; hence n ;:::: 1) and is thus superficially convergent. Such conditions may be realized through the introduction of only two auxiliary masses
MI =
+ 2A2 + A2
(8-92)
where A2 is the cutoff (which will ultimately go to infinity). This choice is such that (8-93) The reason for insisting that eland c 2 be integers will soon become clear. As for the photon propagator, a single subtraction 2 G (k) = [_ . (gpa - k pk a/J1 - kpka/J12)] _ [2 2J (8-94) pa,reg 1 k2 _ J12 k2 _ J12 /..1, J1 -+ J11 will make it smooth enough to render all diagrams convergent. It is understood that J1I -+ 00 as A2 -+ 00. All the previous prescriptions follow from a regularized lagrangian SPreg =
J11 - J1
1 [1
--4 (opAa - oaAp)(D
+ J12 + J1i)(oP A" 3_
0" AP) + J1 J11 A2 2
- "2 0 A(AD + J12 + J1i)0 A + s~o ljJs(ii) -
eJ - Ms)ljJs
(8-95)
After integration by parts, the quadratic part in A reads
V SPA = 2( 2 1 2) A pKP"(J1 2)K"v(J1i)A J11 - J1 = i A,K'P(J1 2) [K(J1i) - K(J12)r 1p"K"V(J1i)A v
where the differential operator K pa(J12) is defined as K p,,(J1 2) = gp,,(D + J12) - opo,,(1 - A)
It is then clear that the A propagator, i.e., the inverse of the quadratic form appearing in SPA, is i[K- 1(J12) - K- 1(J1mp". On the other hand, we have introduced in (8-95) three auxiliary fields (ljJo == ljJ), minimally coupled to the electromagnetic field and endowed with the masses M 1 , M2 == M3 satisfying (8-92). Moreover, ljJ1 is considered as an ordinary Fermi field, whereas ljJ2 and ljJ3 are quantized according to the Bose statistics! The effect of these strange rules is, of course, to reproduce the prescription (8-93). Because of the degeneracy between ljJ 2 and ljJ 3 and of the absence of the minus sign in their closed loops, C 2 = - 2. We have reached our aim. The theory has been regularized in a satisfactory way, since SPreg of Eq. (8-95) is gauge invariant (up to the photon mass term and the gauge-fixing transverse terms in o A) and the Ward identities of the previous paragraph are clearly satisfied.
RENORMALIZATION
8-4-3 Renormalization
We now have to show that renormalization may be carried out while preserving the Ward identities provided we choose suitable normalization conditions. We require that
r}f)(p)lp=m == SR1(p)lp=m = or}f)(p) I
op p=m
A~(p, p) p=m
+ wR(k 2)] - gP<TJ12 + AkPk<T}
(8-96a) (8-96b) (8-96c) (8-96d) (8-96e)
=1 = yP
r~<T(k) = _{(k 2gP - kPk<T) [1 <T
These conditions are obviously satisfied to lowest order. Equations (8-96b, c) are in agreement with the identity (8-87), while Eqs. (8-96d, e) incorporate the information derived from (8-79) and (8-82). As for the conditions (8-96a, b), they define the physical mass of the electron, since they guarantee that the complete propagator S has a pole of residue one at p = m. Similarly, condition (8-96c) will lead to a physical definition of the charge as the coupling between an on-shell fermion and a photon of vanishing momentum. However, as already noticed in Chap. 7, the conditions (8-96b, c, e) may no longer be maintained as the photon mass J1 goes to zero. Indeed, the derivative of the self-energy on the mass shell (%p) ICP) Ip=m' and consequently the vertex function, are plagued with infrared divergences as J12 approaches zero. Therefore, if we insist on taking the limit J1 ---+ 0, other normalization conditions must be chosen. To remain on the safe side we keep here J12 small but finite. _ The proof that Ward identities are preserved by renormalization proceeds by induction. We assume that they hold up to a given order h L . In other words, we have determined to this order the bare quantities Z1, Z2, Z3, mo, J15, and Ao in such a way that the lagrangian
.,2"[L]
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