The massless sunrise diagram with an arbitrary number of loops is calculated
in a simple but formal manner. The result is then verified by rigorous
mathematical treatment. Pitfalls in the calculation with distributions are
highlighted and explained. The result displays the high energy behaviour of the
massive sunrise diagrams, whose calculation is involved already for the
two-loop case.Comment: 10 pages, 1 figure, LATEX, uses kluwer.cls, some references adde

A. Aste

A.I. Davydychev

Andreas Aste

F. Constantinescu

F.A. Berends

F.V. Tkachov

G. Källén

G. Passarino

G. Scharf

G. ’t Hooft

H. Epstein

J. Fleischer

J. Schwinger

K.G. Chetyrkin

M. Caffo

M. Dütsch

P. Post

R.F. Streater

S. Bauberger

S. Groote

S. Laporta

T.-P. Cheng

W. Pauli

English

arXiv

The massless sunrise diagram with an arbitrary number of loops is calculated in a simple but formal manner. The result is then verified by rigorous mathematical treatment. Pitfalls in the calculation with distributions are highlighted and explained. The result displays the high energy behaviour of the massive sunrise diagrams, whose calculation is involved already for the two-loop case

Aste, Andreas

edoc

DOI 10.1007/s11005-006-0090-6Letters in Mathematical Physics (2006) 77:209–218 © Springer 2006Dispersive Calculation of the Massless Multi-loopSunrise DiagramANDREAS ASTEDepartment of Physics and Astronomy, Theory Division, University of Basel,Klingelbergstrasse 82, 4056 Basel, Switzerland. e-mail: andreas.aste@unibas.chReceived: 10 March 2006; revised version: 7 May 2006Published online: 15 June 2006Abstract. The massless sunrise diagram with an arbitrary number of loops is calculatedin a simple but formal manner. The result is then verified by rigorous mathematical treat-ment. Pitfalls in the calculation with distributions are highlighted and explained. The resultdisplays the high energy behaviour of the massive sunrise diagrams, whose calculation isinvolved already for the two-loop case.Mathematics Subject Classification (2000). 81T05, 81T15, 81T18.Keywords. regularization, causality, perturbative calculations.1. IntroductionThe Feynman propagator of a free massless scalar quantum field (x) fulfillingthe wave equation(x)=∂µ∂µ(x)=0 (1)is given in configuration space by the vacuum expectation valueF (x)=−i〈0|T ((x)(0))|0〉=∫d4k(2π)4e−ikxk2 + i0= i4π21x2 − i0 =i4π2P1x2− 14πδ(x2), (2)where T is the time-ordering operator, P denotes principal value regulariza-tion and δ is the one-dimensional Dirac distribution depending on x2 = xµxµ =(x0)2 − (x1)2 − (x2)2 − (x3)2 = x20 − x2. Formally, one could write for the massless(n −1)-loop sunrise diagram210 ANDREAS ASTEkk1kn 1∑i=1kik2kn 1 kFigure 1. The (n −1)-loop sunrise diagram.∫d4x[F (x)]neikx =n(k)=∫d4k1(2π)4. . .d4kn−1(2π)41k21 + i01k22 + i0. . .× 1(k − k1 − k2 −· · ·− kn−1)2 + i0 . (3)This expression above is both ultraviolet and infrared divergent for n2, con-sequently the corresponding expressions in Eq. (3) are all ill-defined. Therefore,[F (x)]n has to be regularized in order to obtain a mathematically well-definedtempered distribution in the dual Schwartz space S ′(R4). The regularization canbe performed by one of the many well-known methods [1–6], which all lead to thesame result. For n =2 one obtainsR2 (k)=R[∫d4k1(2π)41k21 + i01(k − k1)2 + i0]=− i4(2π)2log(−k2 + i0µ2R)=− i4(2π)2log |k2/µ2R|−116π(k2) , (4)where R denotes the regularization procedure with renormalization scale µR and is the Heaviside distribution. Obviously, the regularized expression for the one-loop diagram R2 (k) is defined up to a constant im momentum space or up toa local distribution in configuration space ∼ δ(4)(x). The scaling symmetry of theFeynman propagator F (λx) = λ−2F (x), λ∈R, is spontaneously broken by theregularization procedure, such that a corresponding scaling law R2 (λk) = R2 (k)does not hold as one might expect naively from the formal definition of 2(k) inEq. (3).MASSLESS SUNRISE DIAGRAMS 2112. Formal ApproachIn order to obtain the analytic result for an arbitrary multi-loop diagram, a purelyformal calculation is presented here first. One easily derives formally for n =11(x2 − i0)n =4n(n −1)(x2 − i0)n+1 , (5)and therefore[F (x)]n =− in4n−2(2π)2(n−2)(n −1)!(n −2)!n−2[F (x)]2 . (6)Since the d’Alembert operator  goes over into −k2 in momentum space, oneobtainsRn (k)=−(−i)n(k2)n−24n−2(2π)2(n−2)(n −1)!(n −2)!R2 (k)+n−2∑i=0cRi (k2)i , n2, (7)if the assumption is made that the formal result eq. (6) is also valid for theregularized expressions. The additional terms involving the Fourier transform of(covariant derivatives of) the δ-distribution in configuration space reflect the factthat the regularized Lorentz invariant product of Feynman propagators is onlydefined up to local terms given by derivatives of the δ-distribution, whose ordercan be restricted by the requirement that the scaling behaviour of (7) correspondsto the artificial degree of divergence ω = 2(n − 2) of the (n − 1)-loop sunrise dia-gram. Further restrictions follow from physical symmetry requirements, dependingon the underlying theory in which the diagram plays a role (like in perturbativequantum gravity). As an example, for the 2-loop diagram (n =3) one has the gen-eral resultR3 (k)=−k232(2π)4log(−k2 + i0µ2R)+ cR0 . (8)The simple result given by Eq. (7) is indeed correct and similar formal calculationsas presented above can be found in many textbooks (see, e.g., [7]). The massivetwo-loop case of the sunrise diagram which is already involved from the compu-tational point of view has been studied in detail in numerous papers (see e.g. [8–18] and references therein), and to a certain extent also for the massive multi-loopcase [19–22].3. Properties of Tempered DistributionsA rigorous derivation of (7) is presented below. For the forthcoming discussion,it is advantageous to review some of the useful properties of basic tempered dis-tributions appearing in quantum field theory. A scalar neutral field  can bedecomposed into a negative and positive frequency part according to212 ANDREAS ASTE(x)=−(x)++(x)=∫d3k√(2π)32|k|[a(k)e−ikx +a†(k)eikx]. (9)The commutation relations for such a field are given by the positive and negativefrequency Jordan–Pauli distributions±(x)=−i[∓(x), ±(0)]=−i〈0|[∓(x), ±(0)]|0〉 , (10)which have the Fourier transformsˆ±(k)=∫d4x ±(x)eikx =∓(2π i)(±k0)δ(k2). (11)The fact that the commutator[(x),(0)]= i+(x)+ i−(x)=: i(x) (12)vanishes for spacelike arguments x2 < 0 due to the requirement of microcausali-ty, leads to the important property that the Jordan-Pauli distribution  has causalsupport, i.e. it vanishes outside the closed forward and backward lightcone suchthatsupp(x)⊆ V − ∪ V + , V ± ={x | x20, ±x00} (13)in the sense of distributions. A further crucial observation is the fact that one canintroduce the retarded propagator ret(x) which coincides with (x) on V +−{0},i.e. 〈ret, ϕ〉 = 〈,ϕ〉 holds for all test functions in the Schwartz space ϕ∈S(R4)with support suppϕ ⊂R4 − V −. One might be tempted to write down in configu-ration spaceret(x)=(x0)(x), (14)and to transform this expression into a convolution in momentum spaceˆret(k)=∫d4 p(2π)4ˆ(p)ˆ(k − p) , ˆ(p)=−(2π i)sgn(p0)δ(p2) . (15)The Heaviside distribution (x0) could be replaced by (vx) with an arbitraryvector v ∈ V + = V +− ∂V + inside the open forward lightcone. The Fourier trans-form of the Heaviside distribution (x0) can be calculated easilyˆ(k)= lim→0∫d4x (x0)e−x0eik0x0−i k x = (2π)3ik0 + i0δ(3)(k). (16)For the special case where k is in the forward lightcone V +, one can go to aLorentz frame where k = (k0, 0) such that Eq. (15) becomesˆret(k0, 0)= i2π∫dp0ˆ(p0)k0 − p0 + i0 =i2π∫dtˆ(tk0)1− t + i0 . (17)MASSLESS SUNRISE DIAGRAMS 213Hence, for arbitrary k ∈ V +, ˆret would be given by the dispersion relationˆret(k)= i2π∫dtˆ(tk)1− t + i0 . (18)However, the integral in eq. (18) is undefined. One can circumvent this problem,e.g., by introducing a mass for the field  as a regulator for the -distribution.For the massive Jordan-Pauli distribution m(x) one gets due to the δ-distribu-tion inˆm(k)=−(2π i)sgn(k0)δ(k2 −m2) (19)the expression (k∈V +)ˆretm (k)=∫dtsgn(tk0)δ(t2k2 −m2)1− t + i0=∫dt[δ(t − m√k2)− δ(t + m√k2)]2√k2m(1− t + i0) =1k2 −m2 (k2 =m2) . (20)As a special case of the edge of the wedge theorem [23] it is known that the Fou-rier transform of the retarded distribution ˆret(k) is the boundary value of an ana-lytic function r(z), regular in T + :=R4 + iV +. This way one obtains from r(z)=1/(z2 −m2), z∈T +, and m →0ˆret(k)= 1k2 + ik00 . (21)The analytic expression for the Feynman propagator is then recovered, whichcoincides withˆret(k)= ˆF (k)+−(k)= 1k2 + ik00 (22)for k ∈ V +. The singular part of the Feynman propagator is given byIm (ˆF (k))=−πδ(k2) (23)and can be deduced directly from the causal Jordan–Pauli distribution ˆ in anobvious way.4. Analysis of Causal PropertiesThe causal properties of the Jordan-Pauli distribution and the retarded propaga-tor can be observed in an analogous way in the case of the distributions describ-ing a multi-loop sunrise diagram. The normally ordered product of n free fieldoperators :(x)n : is an operator valued distribution. Therefore, the same is truefor the tensor product : (x)n :: (y)n : [24]. The vacuum expectation value of〈0|:(x)n ::(0)n :|0〉 can be extracted by Wick’s theorem and leads to inn!+(x)n214 ANDREAS ASTE(the combinatorial factor n! will be disregarded in the following), which is indeeda well-defined tempered distribution. Calculating the retarded part of the causaldistribution n(x) which we define for n2 byn(x) := in(+(x)n − (−1)n(−(x)n)= 1n! 〈0|[:(x)n :, :(0)n :]|0〉 , (24)which has causal support, will finally lead to the desired result eq. (7). For thispurpose, the Fourier transform of n(x) has to be calculated. This can be doneinductively. For n =2, we first evaluate the Fourier transform of −2 (x) :=−(x)2given by the convolutionˆ−2 (k)=∫d4q(2π)4ˆ−(q)ˆ−(k −q)=− 1(2π)2∫d4q (−q0)δ(q2)(q0 − k0)δ((k −q)2). (25)The integral vanishes outside the closed backward lightcone due to Lorentz invari-ance and the two -distributions in Eq. (25). Therefore one can go to a Lorentzframe where k = (k0 <0, 0), and using the abbreviation E =√q 2 =|q | and exploit-ing the δ-distribution in Eq. (25) one obtainsˆ−2 (k0<0, 0)=− 1(2π)2∫d3q2Eδ((k0)2 +2k0E)(−E − k0)=− 12(2π)k0∫d|q ||q |δ(k02+|q |)(−|q |− k0)=− 14(2π)(−k0)((k0)2), (26)and for arbitrary k one gets the intermediate resultˆ−2 (k)=−18π(−k0)(k2). (27)According to Eq. (24) we have 2(x)=−(x)2 −+(x)2, and from Eq. (11) fol-lows −(−x)=−+(x), therefore we obtain for the Fourier transform of 2(x)ˆ2(k)= 18π sgn(k0)(k2). (28)It is obvious that ˆ2(k) is related to the real part of Eq. (4).5. Point Splitting Method in Momentum SpaceInstead of introducing a mass term as regulator of the distribution ˆ2(k), we pres-ent here another strategy, namely a point splitting procedure in momentum space,which also leads to the desired result. As was shown in Epstein and Glaser [5] andDu¨tsch et al. [25], the massless causal distribution ˆ2(k) can be split in analogy toMASSLESS SUNRISE DIAGRAMS 215Eq. (18) by computing a retarded part ˆq2,ret(k) in momentum space of 2(x)eiqxin configuration space via the subtracted dispersion relation [26]ˆq2,ret(k)=i2π∫dtˆ2(tk +q)(t − i0)(1− t + i0) (29)for k −q∈V +, q2 <0 and performing a limitˆret2 (k)= limq→0[ˆq2,ret(k)+ cq ] , (30)where cq is a diverging q-dependent constant (i.e., cq is independent of k) whichcan be chosen in such a way that the limit in Eq. (30) exists. Note that ˆ2(k +q)is the Fourier transform of 2(x)eiqx . Therefore, one has to compute (k∈V +)ˆret2 (k)= limq→0{i16π2∫dtsgn(tk0 +q0)((tk +q)2)(t − i0)(1− t + i0) + cq}. (31)The roots of (tk +q)2 = t2k2 +2t (kq)+q2 =0 aret1,2 = 1k2 (−kq ±√N ) (32)with N = (kq)2 − k2q2 >(kq)2 and t1 >0, t2 <0. The integral in (31) becomes{ −i16π2t2∫−∞dt + i16π2∞∫t1dt}{1t− P 1t −1 − iπδ(t −1)}= i16π2(− log |t2|+ log |t2 −1|− log |t1|+ log |t1 −1|− iπ). (33)Since t1,2 → 0 for q → 0 the logarithms log |t1,2 − 1| vanish in this limit and theintegral reduces to− i16π2(log |t1t2|+ iπ)= i16π2(logk2|q2| − iπ). (34)Adding cq = i16π2 log|q2|µ2Rto Eq. (34) leads after analytic continuation to the (covar-iant) splitting solutionˆret2 (k)=i16π2log(−k2 + ik00µ2R), (35)in accordance with the well-known result for R2 given by Eq. (4).216 ANDREAS ASTE6. Results for Higher LoopsWe proceed with the Fourier transform of −(x)3 =:−3 (x)ˆ−3 (k)=i(2π)3∫d4q (−q0)δ(q2)ˆ−2 (k −q)=− i4(2π)4∫d4q (−q0)δ(q2)(q0 − k0)((k −q)2) (36)For k = (k0<0, 0) we obtainˆ−3 (k)=−i4(2π)4∫d3q2E(−E − k0)((k0)2 +2k0E)− i4(2π)3−k0/2∫0d|q ||q |=− i32(2π)3k2 , (37)henceˆ−3 (k)=−i32(2π)3k2(−k0)(k2). (38)Inserting a factor (k −q)2 = (k0)2 + 2k0E in Eq. (37), corresponding to a replace-ment ˆ−n (k) → ˆ−n+1(k) in Eq. (36), leads after a short inductive calculation to(n2)ˆ−n (k)=in(k2)n−24n−1(n −1)!(n −2)!(2π)2n−3(−k0)(k2) (39)orˆn(k)= (k2)n−24n−1(n −1)!(n −2)!(2π)2n−3 sgn(k0)(k2). (40)Keeping in mind that (k2)n−2 in Fourier space corresponds to the local operator(−)n−2 in configuration space, a retarded part of n can be obtained directlyfrom the retarded solution ret2 calculated above for 2 in momentum space,since (k2)n−2ˆret2 (k) is a retarded part of (k2)n−2ˆ2(k). The actual calculationis straightforward and leads after analytic continuation of ˆret2 (k) to the desiredresultRn (k)=−(−i)n+1(k2)n−24n−1(2π)2n−2(n −1)!(n −2)! log(−k2 + i0µ2R)+n−2∑i=0cRi (k2)i (41)for n2.AcknowledgementsThe author wishes to thank Cyrill von Arx for generating the figure. This workwas supported by the Swiss National Science Foundation.MASSLESS SUNRISE DIAGRAMS 217References1. Pauli, W., Villars, F.: On the invariant regularization in relativistic quantum theory.Rev. Mod. 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Dispersive calculation of the massless multi-loop sunrise diagram

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Dispersive calculation of the massless multi-loop sunrise diagram

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