PoS(LL2016)035. Cutkosky Rules from Outer Space. Dirk Kreimer Humboldt Univ.

Transcription

1 Cutkosky Rules from Outer Spe Humoldt Univ. E-mil: We overview reent results on the mthemtil foundtions of Cutkosky rules. We emphsize tht the two opertions of shrinking n internl edge or putting internl lines on the mss-shell re nturl opertion on the uil hin omplex studied in the ontext of geometri group theory. This together with Cutkosky s theorem regrded s theorem whih informs us out vritions onneted to the monodromy of Feynmn mplitudes llows for systemti pproh to norml nd nomlous thresholds, dispersion reltions nd the optil theorem. In this report we follow [] losely. PoS(LL06)05 Loops nd Legs in Quntum Field Theory 4-9 April 06 Leipzig, Germny Speker. The results presented here grew out of ollortion with Spener Bloh. Also, mny thnks to Dvid Brodhurst for stimulting disussions on the sujet. Copyright owned y the uthor(s) under the terms of the Cretive Commons Attriution-NonCommeril-NoDerivtives 4.0 Interntionl Liense (CC BY-NC-ND 4.0).

2 Cutkosky Rules from Outer Spe. Motivtion Understnding of the nlyti struture of the ontriution of grph to Feynmn mplitude, time-honored prolem [], is relted to n nlysis of its redued grphs nd the grphs in whih internl edges re on the mss-shell. The former se reltes to grphs in whih internl edges shrink. The ltter se reltes to grphs with ut edges. The set of ut edges is uniquely determined y the hoie of spnning forest for the grph: suh spnning forest defines unique set of edges onneting distint omponents of the forest. It is those edges we will put on the mss-shell. Pirs of grphs nd hosen ordered spnning tree or forest deliver the uil hin omplex []. A given ordering of the edges of the spnning tree T defines sequene of spnning forests F, nd to ny pir (Γ,F) for fixed Γ we n ssoite: - redued grph Γ F otined y shrinking ll edges of Γ to length zero whih do not onnet different omponents of the spnning forest - ut grph Γ F where ll those edges onneting different omponents re put on-shell, so re mrked y Cutkosky ut, -the set of grphs G F = Γ E ΓF otined from Γ y removing the edges whih onnet distint omponents of the spnning forest. Suh dt define ell-omplex. With it they define set of lower tringulr mtries, one for eh ordering of the edges in T, whih llow to nlyse grph mplitude from its redued grphs nd the vritions otined y putting internl edges on-shell.. Results A sequene of uts (edge sets ε i determines from i-omponent forests, i ) ε ε ε vγ PoS(LL06)05 will shift the norml threshold s 0 (ε ) ssoited with hosen ut ε to nomlous thresholds s 0 (ε ) s (ε ) s vγ (ε vγ ). The resulting sequene of nomlous thresholds s i (ε i+ ), i > 0 is sequene of vlues for hnnel vrile s defined y ε. They re omputed from the divisors ssoited to ε i+. The ltter re funtions of ll kinemtil vriles. For exmple, for the one-loop tringle the divisor in C ssoited to ε is simple funtion of λ(p, p, p )= p.p p p = p.p p p = p.p p p, p + p + p = 0. The three representtions of λ(p, p, p ) llow to ompute s (ε ) for the three hoies of hnnel vrile s= p or s= p or s= p respetively. As result, to grph Γ we n ssign olletion of lower tringulr mtries Mi Γ with the following properties: i) All entries in the mtrix orrespond to well-defined integrle forms under on-shell renormliztion onditions. ii) Anomlous thresholds s i re determined from properties of grph polynomils. They provide

3 Cutkosky Rules from Outer Spe lower oundries for dispersion integrls ssoited to these integrle forms. iii) Along the digonl in the mtries Mi Γ we find leding threshold entries: ll qudris for ll edges in grph re on the mss-shell. iv) The vrition of olumn in Mi Γ wrt to given hnnel is given y the olumn to the right. v) The sudigonl entries (Mi Γ) k,k re determined from the digonl entries (Mi Γ) k,k nd (Mi Γ) k,k vi dispersion integrl. This gives (k ) two-y-two mtries eh of whih hs n interprettion vi the optil theorem. This hene determines the first sudigonl. vi) Continuing, ll sudigonls nd hene the whole mtrix (Mi Γ) r,s is determined vi iterted dispersion. This nswers the question how to ontinue the optil theorem eyond two-point funtions.. The uil hin omplex We follow []. Consider pir (Γ,T) of ridge free grph Γ nd hosen spnning tree T for it. Assume T hs k edges. Consider the k-dimensionl unit ue. It hs origin (0,,0) nd k unit vetors (,0,,0),..., (0,,0,) form its edges regrded s -ells. A hnge of ordering of the edges of T permutes those edges. The origin is deorted y rose on Γ petls, nd the orner(,,,) deorted y(γ,v Γ ), with k= V Γ, nd we regrd V Γ s spnning forest onsisting of k+ distint verties. The omplex is est explined y ssigning grphs s in the following exmple. 4 PoS(LL06)05 The ell is two-dimensionl s eh of the five spnning trees of the grph Γ, the dune s p grph, in the middle of the ell hs length two. We hve hosen spnning tree T provided y the edges e nd e, indited in red. The oundry of our two-dimensionl ell hs four one-dimensionl edges, ounded y two of the four 0-dimensionl orners eh.

4 Cutkosky Rules from Outer Spe To these lower dimensionl ells we ssign grphs s well s indited. The spnning tree hs length two nd so there re =! orderings of its edges, nd hene two lower tringulr mtries Mi Γ whih we n ssign to this ell. They look s follows: 4 4 M Γ M Γ These squre mtries re lower tringulr. Note tht we hve uts whih seprte the grph into two omponents determining norml threshold whih ppers lredy in redued grph on the digonl, nd in the lower right orner ut into three omponents, whih determines n nomlous threshold. All these uts determine vritions, s stted in Cutkosky s theorem [].. Cutkosky s theorem PoS(LL06)05 We quote from [] where you find detils. For grph Γ nd hoosen spnning forest F we let the quotient grph Γ -the redued grph- e the grph otined y shrinking ll edges e E of Γ whih do not onnet distint omponents of F, so E = E Γ E, nd E ll edges of Γ whih do onnet distint omponents of F. Assume the reduedgrph Γ hs physil singulrity t n externl momentum point p, i.e. the intersetion e E Q e of the propgtor qudris ssoited to edges in E hs suh singulrity t point lying over p. Let p e n externl momentum point for Γ lying over p. Then the vrition of the mplitude I(Γ) round p is given y Cutkosky s formul vr(i(γ))=( πi) #E e E δ + (l e ) e E l e. 4. Anomlous thresholds Let us ome k to generi grph Γ. We wnt to determine nomlous thresholds. With their help, dispersion reltions n e estlished when rel nlyity in kinemtil vriles n e estlished.

5 Cutkosky Rules from Outer Spe We nlyse the Lndu singulrities of Γ in terms of Γ/e, where e is suh n edge. To ompletely nlyse the grph, we hve to onsider ll possiilities to shrink it edge fter edge (the generliztion to multiple edges is in []). We hve for the seond Symnzik polynomil Φ =:X =:Y {}}{ {{}}{ Φ(Γ)= Φ(Γ/e)+A e Φ(Γ e) m e ψ(γ/e) } Z {}}{ A e m eψ(γ e). Solving Φ(Γ) = 0 for A e is qudrti eqution with oeffiients X,Y,Z. Note tht Z > 0 is independent of kinemtil vriles, while X,Y depend on moment nd msses. In prtiulr, for hosen hnnel vrile s we n write X = sx s + N, with X s independent of kinemtis nd N onstnt in the hnnel vriel s. It depends on other kinemti vriles though. In terms of prmetri vriles, X,Y re funtions of the prmetri vriles of the redued grph. The ove qudrti eqution hs disriminnt D=Y +4XZ, nd we find physil Lndu singulrity for positive Y nd vnishing disriminnt D = 0. Define Y 0 := Y(p Γ/e A,{Q,M}) to e the evlution of Y y evluting prmetri vriles t the point of the Lndu singulrity for the redued grph. The ondition D = 0 llows to determine the nomlous threshold from minimizing over prmetri vriles A e 0. s({a},{q,m})= Y 4ZN 4ZX s, Let T Γ s e the set of ll ordered spnning trees T of fixed grph Γ whih llow for the sme ssoited hnnel vrile s. We hve the following result. i) A neessry nd suffiient ondition for physil Lndu singulrity is Y 0 > 0 with D=0. ii) The orresponding nomlous threshold s F for fixed msses nd moment {M,Q} is given s the minimum of s({,},{q,m}) vried over edge vriles {,}. It is finite (s F > ) if the minimum is point inside p P e Γ in the interior of the integrtion domin A i > 0. If it is on the oundry of tht simplex, s F =. iii) If for ll T T Γ s nd for ll their forests(γ,f) we hve s F >, the Feynmn integrl Φ R (Γ)(s) is rel nlyti s funtion of s for s<min F {s F }. iv) For Y > 0 nd X < 0, oth zeroes of Φ(Γ) = 0 pper for A e > 0. For Y > 0 nd X > 0, only one zero is inside the domin of integrtion. As result for X = 0 orresponding to the threshold provided y the redued Γ/e we hve disontinuity. PoS(LL06)05 5. Exmple We onsider the tringle grph. In ft, we ugment it with one of its three possile spnning 4

6 Cutkosky Rules from Outer Spe trees, sy on edges e,e, so E T ={e,e }. The orresponding ell in the uil hin omplex is For the Cutkosky ut we hoose two of the three edges, sy ε = {e,e }. This defines the hnnel s= p nd the mtrix M. The other ut in tht mtrix is the full ut seprting ll three verties. We give M in the following figure: M = (5.) PoS(LL06)05 We now lulte: Φ = =Φ Γ/e {}}{ p A A (m A + m A )(A + A ) +A ((p m m )A +(p m m )A ) A m, so = Φ = Φ /e + A Y A m {}}{ ψ e, 5

7 Cutkosky Rules from Outer Spe s nnouned: =:l {}}{{}}{ X = Φ /e, Y = (p m m )A + (p m m )A, Z = m. We hve Y 0 = m l + m l, nd need Y 0 > 0 for Lndu singulrity. Solving Φ( /e ) = 0 for Lndu singulrity determines the fmilir physil threshold in the s= p hnnel, leding for the redued grph to =:l We hve p Q : s 0 =(m + m ), p A : A m = A m. (5.) We let D= Y + 4XZ e the disriminnt. For Lndu singulrity we need Φ = m ( D=0. A Y + D m )( A Y D m ), (5.) where Y,D re funtions of A,A nd m,m,m,s, p, p. Note tht t D=0 we hve m A = A l + A l, whih determines o-dimension one ( line) hypersurfe of P. Finding the nomlous thresold determines point on this line (it fixes the rtio A /A ), nd hene the nomlous threshold determines point in P. We n write 0=D= Y + 4Z(sA A N), with N =(A m + A m )(A + A ) s-independent. This gives s(a,a )= 4ZN (A l + A l ) 4ZA A =: A ρ + ρ 0 + A ρ. A A (5.4) Define two Kllen funtions ρ = λ = λ(p,m,m ) nd ρ = λ = λ(p,m,m ). Both re rel nd non-zero off their threshold or pseudo-threshold. Then, for ρ > 0, ρ > 0, PoS(LL06)05 we find the threshold s t s =(m + m ) + 4m ( λ m λ m ) ( λ l + λ l ) 4m. (5.5) λ λ On the other hnd for the oeffiients of ρ < 0 nd/or ρ < 0 we find minimum long the oundries A = 0 or A = 0. s =, (5.6) 6

8 Cutkosky Rules from Outer Spe The domins Y > 0,X < 0 nd Y > 0,X > 0 determine the domins of prmetri integrtion for the vrition presried y Cutkosky s theorem, whilst the norml nd nomlous threshold (when finite) determine the lower oundries of the dispersion integrls needed to reonstrut the funtion from its vrition. Let us now disuss the tringle in more detil. It llows three spnning trees on two edges eh, so we get six mtries Mi, i=,...,6 ltogether, y hving two possiilities to order the two edges for eh spnning tree. The six mtries Mi ome in groups of two for eh spnning tree. For eh of the three spnning trees we get ell s in (5.). The oundry opertor for suh ell in the uil ell omplex of [] is the ovious one stemming from o-dimension one hypersurfes t 0 or with suitle signs. So the squre populted y the tringle in (5.) hs four oundry omponents, the edges populted y the four grphs s indited. Those four edges re the ovious oundry of the squre. If we now onsider ll grphs in (5.) s evluted y the Feynmn rules, we n onsider for given ell oundry opertor whih reples evlution t the x e = 0-hypersurfe y shrinking edge e, nd evlution t the x e = -hypersurfe y setting edge e on the mss-shell. Then, to hek tht this is oundry opertor for the mplitudes defined y the grphs in (5.) we need to hek tht the mplitudes for the four grphs t the four orners re uniquely defined from the mplitudes of the grphs t the djent edges: for exmple, the imginry prt of the mplitude of the grph on the left vertil edge is relted to the mplitude of the grph t the upper left orner: This imginry prt must e lso otined from shinking edge e in the grph on the upper horizontl edge y setting A to zero in the integrnd nd integrting over the hypersurfe A = 0 of the integrtion simplex σ. This is indeed the se, nd similr heks work for ll other orners. In summry, the nlyti struture of Feynmn mplitudes relized the struture of the uil hin omplex. The ltter is highly non-trivil. Its further study in the onext needed for physis will inform our understnding of mplitudes onsiderly. Future work will e dedited in understnding the reltion etween the monodromy of physil singulrities nd the fundmentl group underlying Outer Spe s used in []. PoS(LL06)05 Referenes [] S. Bloh, D. Kreimer, Cutkosky rules nd Outer Spe, rxiv: [hep-th]. [] F. Phm, Introdution à l Etude Topologique des Singulrités de Lndu, Gutheir-Villrs (967). [] A. Hther, K. Vogtmnn, Rtionl Homology of Aut(F n ), Mth. Reserh Lett. 5 (998)