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PauliVillars Regularization to All Orders in .NET framework
842 PauliVillars Regularization to All Orders PDF 417 Recognizer In .NET Framework Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET framework applications. Encoding PDF417 In .NET Using Barcode creation for .NET framework Control to generate, create PDF 417 image in .NET framework applications. We choose to work here with the traditional PauliVillars regularization, rather than the dimensional regularization. The latter will, however, be illustrated in the twoloop computation of Sec. 844. We recall that the PauliVillars 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. 816). Explicitly, Read PDF417 In .NET Framework Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET framework applications. Making Barcode In VS .NET Using Barcode creation for Visual Studio .NET Control to generate, create bar code image in VS .NET applications. f (1 Decoding Bar Code In Visual Studio .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Draw PDF 417 In Visual C# Using Barcode encoder for Visual Studio .NET Control to generate, create PDF417 2d barcode image in VS .NET applications. d p (2n)4 tr
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EAN 128 Maker In .NET Framework Using Barcode drawer for .NET framework Control to generate, create GS1 128 image in .NET applications. Painting 2D Barcode In VS .NET Using Barcode encoder for VS .NET Control to generate, create 2D Barcode image in .NET framework applications. (889) GS1 RSS Generation In Visual Studio .NET Using Barcode generation for .NET framework Control to generate, create GS1 DataBar14 image in Visual Studio .NET applications. I2/5 Creator In Visual Studio .NET Using Barcode maker for .NET Control to generate, create ANSI/AIM ITF 25 image in .NET framework applications. 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 Barcode Reader In VB.NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. Matrix 2D Barcode Encoder In Java Using Barcode generation for Java Control to generate, create 2D Barcode image in Java applications. [ f f d p Pn(p2,p qi,ql)+M;Pn (p2,p qi,ql)+  s=o Q2n(p2,p.qi,ql)+M;Q2nl(p2,p qi,qr)+ f C f d4~ [Pn + M; (Pnl  PnQ~nl) + ... J (890) s=o Q2n Q2n Q2n EAN 128 Creation In ObjectiveC Using Barcode generator for iPad Control to generate, create EAN / UCC  14 image in iPad applications. UPC A Drawer In Visual Studio .NET Using Barcode printer for Reporting Service Control to generate, create UPC Code image in Reporting Service applications. (2n)4 UPCA Printer In .NET Using Barcode generator for ASP.NET Control to generate, create UPCA Supplement 2 image in ASP.NET applications. Reading Code 39 Extended In VB.NET Using Barcode decoder for .NET Control to read, scan read, scan image in Visual Studio .NET applications. (2n) Encoding EAN13 In None Using Barcode drawer for Online Control to generate, create EAN / UCC  13 image in Online applications. Draw DataMatrix In ObjectiveC Using Barcode creator for iPad Control to generate, create Data Matrix image in iPad applications. 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 (891) Figure 816 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
(892) where A2 is the cutoff (which will ultimately go to infinity). This choice is such that (893) 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 (894) 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
(895) 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 (895) three auxiliary fields (ljJo == ljJ), minimally coupled to the electromagnetic field and endowed with the masses M 1 , M2 == M3 satisfying (892). 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 (893). 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. (895) is gauge invariant (up to the photon mass term and the gaugefixing transverse terms in o A) and the Ward identities of the previous paragraph are clearly satisfied. RENORMALIZATION
843 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} (896a) (896b) (896c) (896d) (896e) =1 = yP
r~<T(k) = _{(k 2gP  kPk<T) [1 <T
These conditions are obviously satisfied to lowest order. Equations (896b, c) are in agreement with the identity (887), while Eqs. (896d, e) incorporate the information derived from (879) and (882). As for the conditions (896a, 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 (896c) will lead to a physical definition of the charge as the coupling between an onshell fermion and a photon of vanishing momentum. However, as already noticed in Chap. 7, the conditions (896b, c, e) may no longer be maintained as the photon mass J1 goes to zero. Indeed, the derivative of the selfenergy 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]

