The corolla polynomial: a graph polynomial on half-edges

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1 : grph polynomil on hlf-edges Depts. of Physis nd Mthemtis Humoldt University, Berlin E-mil: The study of Feynmn rules is muh filitted y the two Symnzik polynomils, homogeneous polynomils sed on edge vriles for given Feynmn grph. We review here the role of reently disovered third grph polynomil sed on hlf-edges whih filittes the trnsition from slr to guge theory mplitudes: the oroll polynomil. We review in prtiulr the use of grph homology in the onstrution of this polynomil. Loops nd Legs in Quntum Field Theory (LL2018) 29 April My 2018 St. Gor, Germny Speker. Thnks to Johnnes Blümlein nd Peter Mrqurd for orgnizing our workshop. 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 1. Introdution The omputtion of guge theory suffers the enormous numer of integrls one is onfronted with. This results from the ft tht the numer of Feynmn grphs ontriuting to given mplitude is muh igger thn in slr field theory, nd from the ft tht the spin struture of the Feynmn rules gives muh more omplited tensoril struture. Hene, the redution to mster integrls is muh more umersome undertking. Here we report on reent progress onneting the integrnd for onneted mplitude in non-elin guge theory to the mplitude for slr onneted 3-regulr mplitude: ll verties re ui. 2. Slr vs Guge Theory mplitudes The strting point is the mplitude for Feynmn grph in slr ui theory whih is given (in D=4 dimensions sy) through the two grph- or Symnzik-polynomils Φ Γ,ψ Γ, where ( ) Φ Γ = φ Γ + A e m 2 e ψ Γ, e E Γ nd ψ Γ,φ Γ re the lssi first nd seond grph polynomils: ψ Γ = spnningtrees T e T A e, φ Γ = Q(T 1 ) Q(T 2 ) A e, spnningtwo trees T 1 T 2 e T 1 T 2 with Q(T i ) the sum of the externl moment tthed to verties of T i. There exists then oroll differentil D Γ suh tht the Feynmn integrnd I Γ for onneted 3-regulr grphs Γ, Φ Γ I Γ = e ψ Γ ψγ 2, gives rise, when summed over onneted 3-regulr grphs Γ to the totl guge theory mplitude, using D Γ I Γ. For n exmple onsider the onneted one-loop tringle grphs. The slr grphs re, + 1

3 whilst in the guge theory se we hve more: We hve internl qurk- nd ghost-loops, nd 4-vlent gluon verties. If we reple first ll edges in the 3-regulr slr se y guge-oson edges there re two steps remining: to either shrink internl oson edges to generte 4-vlent oson verties, or else to reple internl guge-oson loops y ghost or fermion loops. It is remrkle tht in this proess, the rnks of the utomorphism groups of grphs ply long (mening tht symmetry ftors ply long). This is due to n underlying doule omplex of two grph homologies sed on either shrinking edges or mrking losed yles in the grph [2]. Both homologies n e implemented using new grph polynomil on hlf-edges, the oroll polynomil [1]. 3. Grph Homology Let e e n edge onneting two 3-gluon verties in grph Γ, χ+ e e the opertor whih shrinks edge e, extend χ+ e to zero when ting etween ny other two verties. Let S, with S 2 = 0, e the orresponding grph homology opertor. Then, for guge theory mplitude r: Let 0x; jgl e the sum of ll 3-regulr onneted grphs, with j ghost loops, nd with externl legs determined y r nd loop numer n, weighted y olour nd symmetry, let /x, jgl e the sme llowing for 3- nd 4-vlent verties. We hve i) : e χ + 0 x; jgl = /x; jgl, ii) : Se χ + 0x; jgl = 0. This theorem shows we n generte ll 4-vlent ouplings y grph homology s studied in Vogtmnn s pper [4] sed on study of grph homology initited y Kontsevih. Let δ C + e the opertor whih mrks yle C through 3-vlent verties nd unmrked edges, extend 2

4 δ+ C to zero on ny other yle. Let T, with T 2 = 0, e the orresponding yle homology opertor. Then: Let jx;0gl e the sum of ll onneted grphs with j 4-verties ontriuting to mplitude r nd loop numer n nd no ghost loops, weighted y olour nd symmetry, jx;/gl e the sme llowing for ny possile numer of ghost loops. We hve i) : e δ + jx;0gl = jx;/gl, ii) : Te δ + jx;0gl = 0. This theorem ensures tht we n insert ghost loops (or fermion loops for tht mtter) using yle homology on grphs. This homology ws introdued in [2]. These two opertions re omptile [2, 5]: i) We hve [S,T]=0 (S+T) 2 = 0 nd Te δ ++χ + 0x;0gl = 0, Seδ ++χ + 0x;0gl = 0. ii) Together, they generte the whole guge theory mplitude from 3-regulr grphs: e δ ++χ + 0x;0gl = /x;/gl =:., the series over ll grphs ontriuting to physil mplitude, eh grph weighted y its symmetry ftor, is the only non-trivil element in the iomplex of yle- nd grph-homology [2, 5]. This indeed is BRST homology grph-theoretilly. To interprete BRST homology y grph homologies is n pproh very muh in line with n erly nlysis of QCD provided for exmple in Predrg Cvitnovi s leture notes [6]. 4. The oroll polynomil nd differentil The next step is to use the ove strutures to find n effiient trnsition from slr to guge theory mplitudes. This is filitted y the oroll polynomil given in [1] whose definition we follow: It is polynomil sed on hlf-edge vriles v, j ssigned to ny hlf-edge (v, j) determined y vertex v nd n edge j of grph Γ. We hve For vertex v V let n(v) e the set of edges inident to v (internl or externl). For vertex v V let D v = j n(v) v, j. Let C e the set of ll yles of Γ (yles, not iruits). This is finite set. 3

5 For C yle nd v vertex in V, sine Γ is 3-regulr, there is unique edge of Γ inident to v nd not in C, let v C e this edge. For i 0 let C i = C 1,C 2,...C i C C j pirwise disjoint (( i v,vc j=1 v C j ) ) D v v C 1 C 2 C i Then C= ( 1) j C j j 0 C = C Γ is the oroll polynomil. Hving this polynomil t our disposl, we n reple ny of its hlf-edges y differentil opertor. This defines oroll differentil whih ts on the slr integrnd. It ts on uxiliry externl moment ξ e (whih re set to the physil moment only fter the tion) whih we supplement in the slr integrnd for eh internl edge e. D g (h) := 1 2 gµ h+ µ h ( ε h+ 1 A e(h+ ) ξ(h + ) µh ε h 1 A e(h ) ), ξ(h ) µh for ny hlf-edge h. Here, hlf-edge h determines t 3-regulr vertex two other hlf-edges h +,h using tht we n orient eh oroll y theorem in [4] muh used in [2] (whih in tun determines the signs ε h± ), nd eh suh hlf-edge furthermore determines n edge e = e(h) to whih it elongs. e(h ) h e(h) h h+ e(h+) Doule differentils wrt the sme hlf edge generte the Feynmn rules for 4-vlent vertex vi Cuhy s residue formul: differentiting twie, the Leiniz rule ensures the emergene of poles with residues whih re the ontriutions of grpgs with 4-vlent gluon verties [2]. 5. Results Finlly, we get the Feynmn integrnds in the unrenormlized nd renormlized se for guge theory mplitude r from 3-regulr onneted grphs of slr fields. The full Yng Mills mplitude Ū Γ for grph Γ n e otined y ting with oroll differentil opertor on the slr integrnd U Γ ({ξ e }) for Γ, setting the edge moment ξ e = 0 fterwrds. 4

6 Moreover, Ū Γ gives rise to differentil form JŪΓ Γ nd there exists vetor H Γ suh tht the unrenormlized Feynmn integrnd for the sum of ll Feynmn grphs ontriuting to the onneted k-loop mplitude r is Φ(X r,k )= Γ =k,res(γ)=r olour(γ) sym(γ) JŪΓ Γ, H Γ The renormlized nlogue is given y writing Ū R Γ insted of Ū Γ [2]. This ws ll generlized to the full Stndrd Model y Dvid Prinz [3], showing tht even spontneous symmetry reking respets the struture of grph homologies nd n e pprohed y the oroll polynomil, with suitle doptions for the mtter nd ghost ontent of tht model. 6. Outer Spe Struture of Guge Theory There re growing signs tht the prmetri pproh to Feynmn digrms hs deep onnetion to the struture of Outer Spe [8, 9], in prtiulr when investigting monodromies of mplitudes. For the se of guge theories where verties of vlene higher thn four vnish, this genertes rther interesting Outer Spes where ells orresponding to o-dimension two hypersurfes re missing. A simple exmple is the one-loop tringle grph. The o-dimension one edges re oundries populted y grph with one three-vlent nd one four-vlent vertex (unions like in the figure). The zero-dimensionl o-dimension two ells re points deorted y tdpoles whih would hve foridden 5-vlent vertex. This is to e removed Note tht when suh Outer Spes with missing ells dopted to guge theories hve een onstruted, Green Funtion is n integrl over the whole suh Outer Spe - pieewise liner sum of integrls [10] over the volume of ll ells of the still llowed odimensions. 7. Outlook The two Kirhhoff polynomils re distinguished s unique polynomils on edge vriles hving reusive ontrtion deletion properties. 5

7 The oroll polynomil is similrly distinguished mongst hlf-edge polynomils hving reursive hlf-edge deletion properties [1]. This llows to onstrut the renormlized integrnd for onneted mplitude in guge theory from the slr mplitudes for onneted 3-regulr grphs [2]. Dvid Prinz hs generlized this pproh to mplitudes to the full SM [3]. Mrel Golz is turning this into very effiient lgorithm for QED mplitudes [7]. Wht is the oroll polynomil for spin 2 osons nd hene for quntum gvity? Cn we omine the redution to mster integrls (Lport s lgorithm) with this trnsition to guge theory? Referenes [1] D. Kreimer, K. Yets: Properties of the oroll polynomil of 3-regulr grph, El.J.Com.20, Issue 1 (2013) Pper no.p41 (rxiv: [mth.co]). [2] D. Kreimer, M. Srs, W.D. vn Suijlekom: Quntiztion of guge fields, grph polynomils nd grph homology, Annls Phys.336 (2013) (rxiv: [hep-th]). [3] D. Prinz: The Coroll Polynomil for spontneously roken Guge Theories, Mth.Phys.Anl.Geom.19 (2016) no.3, 18 (rxiv: [mth-ph]). [4] J. Connt, K. Vogtmnn: On theorem of Kontsevih, Alger.Geom.Topol.3 (2003) (rxiv:mth/ v2 [mth.qa]). [5] André Knispel, Comintoril BRST homology nd grph differentils, mster thesis, Humoldt U.2017, /wp-ontent/uplods/knispel.pdf. [6] P. Cvitnović, Field Theory,ChosBook.org/FieldTheory, Niels Bohr Institute (Copenhgen 2004). [7] M. Golz, New grph polynomils in prmetri QED Feynmn integrls, Annls Phys. 385 (2017) 328 doi: /j.op [rxiv: [mth-ph]], Contrtion of Dir mtries vi hord digrms, rxiv: [mth-ph], nd in preprtion. [8] S. Bloh, D. Kreimer, Cutkosky Rules nd Outer Spe, rxiv: [hep-th]. [9] D. Kreimer, Cutkosky Rules from Outer Spe, PoS LL 2016 (2016) 035 doi: / [rxiv: [hep-th]]. [10] M. Berghoff, Feynmn mplitudes on moduli spes of grphs, rxiv: [mth-ph]. 6