Ambiguities in Pauli–Villars regularization

The European Physical Journal C, Feb 2015

We investigate regularization of scalar one-loop integrals in the Pauli–Villars subtraction scheme. The results depend on the number of subtractions, in particular the finite terms that survive after the divergences have been absorbed by renormalization. Therefore the process of Pauli–Villars regularization is ambiguous. We discuss how these ambiguities may be resolved by applying an asymptotically large number of subtractions, which results in a regularization that is automatically valid in any number of dimensions.

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Ambiguities in Pauli–Villars regularization

R. H. P. Kleiss 0 T. Janssen 0 0 Institute for Mathematics , Astrophysics and Particle Physics , Radboud University Nijmegen , Heyendaalseweg 135, 6525 AJ Nijmegen , The Netherlands We investigate regularization of scalar one-loop integrals in the Pauli-Villars subtraction scheme. The results depend on the number of subtractions, in particular the finite terms that survive after the divergences have been absorbed by renormalization. Therefore the process of PauliVillars regularization is ambiguous. We discuss how these ambiguities may be resolved by applying an asymptotically large number of subtractions, which results in a regularization that is automatically valid in any number of dimensions. - q2 m2 + i q2 m2 + i a e-mail: b e-mail: thereby reducing the large-q behavior from (q2)1 to (q2)2 and thus improving the integrability properties of diagrams. At the end of the calculation the limit M is implied. With this method, the one-loop diagram from a scalar selfinteracting theory becomes integrable in four dimensions. On the other hand, the two diagrams are not integrable in four and six dimensions, respectively. In that case additional subtractions with a spectrum of fictitious particles are necessary. It is tempting to perform, for a given diagram, precisely so many PV subtractions as are necessary to make the loop integral convergent: once for the diagram of Eq. (2), and twice for those of Eq. (3). But this, of course, runs counter to the idea of quantum field theory, in which Feynman diagrams themselves have no independent status but only their combination into Greens functions counts. If there is even only a single diagram that calls for a double subtraction, say, then all diagrams should undergo the same double subtraction, even if they are already convergent after a single one. An approach in which for each diagram precisely sufficient subtractions are made to make that diagram convergent must be considered incorrect. Another unsatisfactory situation is the fact that the number of necessary subtractions depends on the dimension (with higher dimensions necessitating more subtractions) which appears to be at odds with the dimension independence that we should like to have. This leads to the following question: how do we decide to stop making additional PV subtractions? A priori there is nothing that forbids one from making very many subtractions even if that is, strictly speaking, unnecessary. In what follows, we show that the results of divergent one-loop integrals depend on the number of PV subtractions and are therefore ambiguous, while loop integrals that are themselves convergent do not. We study this dependence, and point to a possible resolution of these ambiguities. As remarked above, depending on circumstances, a single PV subtraction may not be sufficient to regularize loop integrals, and multiple subtractions become necessary. In what follows, we shall use the abbreviations = m2 and = M 2. The k-fold PV subtraction of a propagator is defined as follows: The requirement is that C be independent of s. By first considering the limit s and then s r (assuming the to be all different) we find immediately that j=1 C = j =r j =r ( j By subtracting sufficiently many PV propagators we can achieve any desired high-momentum behavior for the propagator. That the above subtraction involves k different mass scales is of course lacking in elegance. We can therefore take j , and then we obtain s + PV (k) = (s + )(s + ) k 1 1 1 s + PV (k) = s + s + (s + ) 2 k1 A similar observation was made, for instance, in [9]. To understand the relation between Eqs. (4) and (7) it is illustrative to consider the case k = 2. Taking 1 = , 2 = + with small , we see that For higher k analogous expansions result. Moreover, the result is quite independent from the precise way in which the limit j ( j = 1, . . . , n) is reached. Note the following fact: if we keep subtracting without limit, we formally have 1 1 1 s + PV (k) = s + s + j=0 = 0. While, indeed, simply replacing every loop integral by zero makes all loop corrections trivial, this is clearly not what we want. Obviously, we must investigate the dependence of loop-integral results on the number of subtractions. We shall restrict ourselves to one-loop computations in the context of an effective potential. Therefore no external momenta are involved, and all propagators have the form 1/(s + ), where s = q2, q being the loop momentum. The (even) number of dimensions of spacetime dimensions is denoted by 2. After performing the Wick rotation and the angular integral of the loop momentum, the loop integral with n propagators is given by where we have dropped any overall factors. This integral is finite if n > . If this is not the case, we must subtract, although strictly speaking we ought to be allowed to PVsubtract convergent integrals as well. We therefore should replace Eq. (10) by The integral will be convergent for finite We shall be slightly more general and define the generating function h,n(x , s) = (s + )n k1+n (s + ) k = (s + )n(s + ) n(s + x ) From the fact that all the functions h decrease as 1/s2 for large s we see that we have subtracted sufficiently often to make the integrals convergent for finite (and hence PVregularized). For higher values of k, we simply have additional subtractions. The integrals H have series expansions in x : the regularized integral J(k,n) is then given as the coefficient of x nk in H,n(x ). For given and n it is a simple matter to integrate h,n(x , s) over s, where by construction the upper endpoint s = never contributes. The most important point is to note that x ) = log( ) + log(1 x ) + It is already clear that at least some of the log( ) terms will be accompanied by log(1 x ). Below, we give the results for H,n(x ) for the most relevant cases for 2,4, and 6 dimensions. By L we denote log( /) . We have taken / to zero wherever possible, although this has of course to be done more carefully in the case of two-loop computations. 2x2 H3,2(x) = x log(1 x) 1 x L From these we can find the k-fold PV-regularized integrals J(k,n) . These contain the digamma function (z), defined as J3(,k40) q1 1 =1 for integer argument q. The various integrals J now read J2(,k12) = k 1 L + ( (k 1) + E 1), J3(,k13) We can draw the following conclusions. The convergent integrals do not depend on the number of PV subtractions, as was to be expected. In the regularized divergent integrals the only term that does not depend on the number of subtractions, and that can be considered unambiguous, is the logarithmic divergence L. Quadratic ( ) and higher ( 2, . . .) divergences do depend on k and are therefore ambiguous. Their coefficients approach zero with increasing number of subtractions. The finite terms are also ambiguous (as was already remarked in [10]), and increase harmonically in absolute value with k. This is evidenced by the ubiquitous log(1 x )/(1 x ) in Eq. (15). Strictly speaking, therefore, the limit k is not clearly defined. We have checked that these features persist for larger values of . The reason why, in the previous section, we restricted ourselves to 3 is that the scalar 4 theory is renormalizable for = 2 and superrenormalizable for = 1, and the 3 theory is renormalizable for = 3 and superrenormalizable for = 1, 2. We therefore consider the renormalization properties of the integrals J . Obviously, we must apply a sufficient number of PV subtractions to properly regularize these; but there is no obvious recipe for determining when to stop subtracting. The only reasonable approach therefore seems to consider the case of an asymptotically large number of subtractions, i.e. to take k in a sensible way. This has the added advantage of being applicable to theories in any positive dimension. We can use the fact that, asymptotically, so that L (k) log( / k). In this limit we can write the regularized divergent integrals as J1,1 = log J2,2 = log J3,3 = log = This choice contains a certain justice in that when we apply more PV subtractions the subtraction propagators individually become smaller. In that case the higher divergences survive, but the results are still unambiguous: J2,1 = J2,2 = log J1,1 = log J3,1 = J3,2 = J3,3 = log This approach works because the coefficients of n go as kn for large k. We have checked that this persists for larger values of . As an example, we can apply the above strategy to e.g. the calculation of the electron one-loop self-energy and the one-loop vertex correction in QED. In comparison with the standard treatment as given in [11,12] we find that our subtraction scheme results in the following modification (in our notation): in both computations. This shows that the relation between the vertex- and the wavefunction renormalization remains undisturbed; in particular the WardTakahashi identity remains valid for the regularized Green functions. In conclusion, we have extended the original PauliVillars regularization scheme in such a way that it is unambiguous and consistent in any number of dimensions. An application to the one-loop corrections in QED shows that the scheme respects the QED gauge invariance, as it should. The authors are indebted to Prof. W. Beenakker for enlightening discussions. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. Funded by SCOAP3 / License Version CC BY 4.0.


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R. H. P. Kleiss, T. Janssen. Ambiguities in Pauli–Villars regularization, The European Physical Journal C, 2015, 46, DOI: 10.1140/epjc/s10052-014-3240-5