QQuantum fields, periods and algebraic geometry
Dirk Kreimer
Abstract.
We discuss how basic notions of graph theory and associated graphpolynomials define questions for algebraic geometry, with an emphasis givento an analysis of the structure of Feynman rules as determined by those graphpolynomials as well as algebraic structures of graphs. In particular, we discussthe appearance of renormalization scheme independent periods in quantumfield theory.
1. Introduction
In this contribution, we want to review work concerning the structure of localrenormalizable quantum field theories. Our emphasis will be to exhibit the mostrecent developments by way of example, and in particular to stress that at thetime of writing we witness two simultaneous developments: a better understandingof the algebro-geometric underpinning of field theory in four dimensions of spacetime, and also as a consequence the emergence of computational approaches whichsurpass the hitherto established state of the art.
Acknowledgments.
Foremost, I want to thank David Broadhurst, with whommy interest in the periods emerging in QFT started some twenty years ago [ ].Spencer Bloch helped to uncover the mathematics behind it, as did Francis Brown,Christian Bogner, Alain Connes, Dzmitry Doryn, H´el`ene Esnault, Erik Panzer,Oliver Schnetz, and others. It is a pleasure to thank James Drummond for hospi-tality at CERN, Geneva, Feb 24-March 01 2013, as well as the IHES, where partsof this paper were written.
2. Graphs and algebras2.1. Wheels in wheels.
It is the purpose of this section to completely analysean example. We choose wheels with three or four spokes, inserted at most onceinto each other. Results for them are available by methods which were recentlydeveloped [
6, 7, 8, 20 ] and which are presented elsewhere [
18, 19 ]. Mathematics Subject Classification.
Primary 81T15.
Key words and phrases.
Quantum fields, Feynman rules, periods.Author supported by the Alexander von Humboldt Foundation and the BMBF through anAlexander von Humboldt Professorship. a r X i v : . [ h e p - t h ] M a y DIRK KREIMER
We consider the free commutative Q -algebra generated by a sole generator indegree zero, I , which serves as a unit for the algebra. In degree three we putΓ = , whilst the only generator in degree four isΓ = . In degree six we have Γ × Γ andΓ = , whilst in degree seven we have Γ × Γ andΓ = , Γ a = , Γ b = . Note that Γ a , Γ b are the only two different topologies we can obtain by replacingone of the five vertices of Γ by Γ . The four vertices of Γ which are connected toan external momentum all give Γ b (modulo permutations of edge labels), whilstinserting at the internal vertex of Γ gives Γ a . UANTUM FIELDS, PERIODS AND ALGEBRAIC GEOMETRY 3
Finally, in degree eight we only consider Γ × Γ and insertion at the internalvertex: Γ = . At higher degrees, we only allow products of the generators listed so far.We make this into a bi-algebra by setting ∆( I ) = I ⊗ I and∆Γ = Γ ⊗ I + I ⊗ Γ , ∆Γ = Γ ⊗ I + I ⊗ Γ , ∆Γ = Γ ⊗ I + I ⊗ Γ + Γ ⊗ Γ , ∆Γ = Γ ⊗ I + I ⊗ Γ + Γ ⊗ Γ , ∆Γ = Γ ⊗ I + I ⊗ Γ + Γ ⊗ Γ , ∆Γ a = Γ a ⊗ I + I ⊗ Γ a + Γ ⊗ Γ , ∆Γ b = Γ b ⊗ I + I ⊗ Γ b + Γ ⊗ Γ , and ∆( h × h ) = ∆( h ) × ∆( h ).We get a Hopf algebra by setting S ( I ) = I , and recursively S ( h ) = − m H ( S ⊗ P )∆, with P the projection onto elements of positive degree, i.e. the augmentationideal.Define two maps into the augmentation ideal B : H → P H, and B : H → P H by B ( I ) = Γ , B ( I ) = Γ ,B (Γ ) = Γ , B (Γ ) = Γ ,B (Γ ) = Γ , B (Γ ) = 12 (Γ a + Γ b ) , and B i + ( h ) = 0 , i ∈ { , } , else.Then ∀ h ∈ { I , Γ , Γ } and i ∈ { , } ,∆ B i + ( h ) = B i + ( h ) ⊗ I + (id ⊗ B i + )∆( h ) , which ensures that these maps behave as Hochschild one-cocycles in the examplesbelow. Remark . Effectively, we are working in a Hopf algebra of graphs generatedand co-generated by Γ and Γ , a quotient of the full Hopf algebra of graphs. Notethat, for example,∆ B (Γ ) = 0 (cid:54) = B (Γ ) ⊗ I + (id ⊗ B )∆(Γ ) = Γ ⊗ Γ . This is a consequence of restricting to a finite Hopf algebra. It poses no problemsfor our applications below in this finite Hopf algebra.
DIRK KREIMER
Now, let α i : H → C be algebra maps, and let bα i : H → H be defined by bα i ( h ) = m (id ⊗ α i )∆( h ) − α i ( h ) I . Then bα i ( I ) = 0 and bα i (Γ j ) = α i ( I )Γ j , ∀ i, j ∈ { , } . Remark . Were the B i + to provide Hochschild one-cocycles, the bα i wouldprovide co-boundaries. Remark . We choose wheels in wheels as an example as results for themare on the brink of computability at the moment. The methods of Francis Brown[
6, 7 ] combined with [ ] allow to compute the period provided by Γ [
18, 19 ],whilst the periods from a symmetric combination of graphs s as defined beloware realistically in reach by this approch -and this approach only, it seems-, and theeight-loop Γ period remains a challenge. Note that there is an obvious co-radical filtrationand associated grading here, given by the kernels of σ (cid:63)j , with σ := S (cid:63) P = m ( S ⊗ P )∆, i.e. using projections into the augmentation ideal combined with the co-product (see [ ]).We find in grading one the primitives Γ , Γ and, more interestingly, the prim-itive elements p := 2Γ − Γ × Γ , p := 2Γ − Γ × Γ , and in particular p := =: s (cid:122) (cid:125)(cid:124) (cid:123)
12 Γ a + 12 Γ b + Γ − Γ × Γ , which also defines the co-symmetric s and p a − b := Γ a − Γ b . They are all linear combinations of elements in filtration two which combine to giveprimitive elements in the Hopf algebra, hence of co-radical degree 1. Note that s is a co-symmetric element (of co-radical degree two) in the Hopf algebra, whichis the reason why we can subtract the commutative product Γ × Γ to obtain aprimitive element.Let us also define a co-antisymmetric element in degree two, c = 12 Γ a + 12 Γ b − Γ . Then, its reduced co-product ∆ (cid:48) := ( P ⊗ P )∆ delivers∆ (cid:48) c = Γ ⊗ Γ − Γ ⊗ Γ , an element which indeed changes sign when we swap the elements on the lhs andrhs of the tensor product, contrary to∆ (cid:48) s = Γ ⊗ Γ + Γ ⊗ Γ . UANTUM FIELDS, PERIODS AND ALGEBRAIC GEOMETRY 5
Let us now consider the Lie algebra L with gen-erators Z which are Kronecker-dual to the Hopf algebra generators h . Its bracketis determined by (cid:104) Z i ⊗ Z j − Z j ⊗ Z i , ∆( h ) (cid:105) = (cid:104) [ Z i , Z j ] , h (cid:105) , h ∈ H. Here, (cid:104) Z a , Γ b (cid:105) = δ ab is the Kronecker pairing between elements Z a ∈ L and Γ b ∈ H ,and a, b range over the set of subscripts 3 , , , , a, . . . used to denote thegraphs.Consider also the corresponding universal enveloping algebra U ( L ) = QI ⊕ L ⊕ ( L ⊗ S L ) ⊕ · · · . Here, ⊗ S denotes the symmetrized tensor-product, and U ( L ) can be identified,albeit non-canonically, with the symmetric tensor algebra of L .The Lie algebra L itself has a (descending) lower central series decomposition: L := L , L k := [ L , L k − ] , k > . The co-product of an element in H is not co-commutative. It pays to decomposeimages of ∆ (cid:48) and its iterations into symmetric and antisymmetric parts.The idea on which we elaborate in the following is to map elements in theHopf algebra to elements in the above universal enveloping algebra of its dualLie algebra, taking some extra information from physics: we will soon see thatFeynman rules assign to Hopf algebra elements polynomials in a variable L , boundedby the co-radical degree, which respects a decomposition into co-symmetric andco-antisymmetric terms in the Hopf algebra which is particularly illuminating incomparison with the universal enveloping algebra.Concretely, let us consider the following map (extended by linearity) σ : H → U ( L ). We start with primitive elements Γ , Γ , p , p , p , p a − b , which, as we willsee, all evaluate under the Feynman rules to terms linear in L : σ (Γ i ) = Z i ∈ L ⊂ U ( L ) , i ∈ , ,σ ( p ii ) = ∈L (cid:122)(cid:125)(cid:124)(cid:123) Z ii i ∈ , ,σ ( s ) = ∈L (cid:122)(cid:125)(cid:124)(cid:123) Z s ,σ ( p a − b ) = ∈L (cid:122) (cid:125)(cid:124) (cid:123) Z p a − b . Note that under σ these primitives have images ∈ L , but (cid:54)∈ L .Let us now consider non-primitive elements. As we will see under the Feynmanrules, the next two examples give polynomials quadratic in L . This is reflected in σ : σ (Γ ii ) = ∈L (cid:122)(cid:125)(cid:124)(cid:123) Z ii + ∈L ⊗ S L (cid:122) (cid:125)(cid:124) (cid:123) Z i ⊗ Z i , i ∈ , ,σ ( s ) = ∈L (cid:122)(cid:125)(cid:124)(cid:123) Z s + ∈L ⊗ S L (cid:122) (cid:125)(cid:124) (cid:123) Z ⊗ Z + Z ⊗ Z , Note that the second symmetric tensor power shows up here, reflecting the L termin the Feynman rules. DIRK KREIMER
Finally, we have the co-antisymmetric element. It is of co-radical degree two,but is linear in L under the Feynman rules. We map σ ( c ) = [ Z , Z ] ∈ L , [ Z , Z ] (cid:54)∈ L , with [ Z , Z ] = 12 Z a + 12 Z b − Z . Note that the second symmetric tensor power does not show up here due to theco-antisymmetry of c . Nicely, the Feynman rules play along.All others evaluations of σ follow by linearity. Remark . The fact that the Dynkin operator
S (cid:63) Y = m ( S ⊗ Y )∆, -with Y the grading operator multiplying a Hopf algebra element of co-radical degree k by k -, of H vanishes on products very much suggests to construct σ as above. The factthat it maps pre-images σ − of co-symmetric elements in L to primitive elementsof H motivates to look at the lower central series of L for the co-antisymmetricelements. Also, note that pre-images of co-symmetric elements can be generatedfrom I through shuffles of one-cocycles, for example ( B B + B B )( I ) = s .
3. Feynman Rules
We now give the Feynman rules for Hopf algebra elements, next study themin examples provided by our small Hopf algebra, and discuss the induced Feynmanrules on the Lie side at the end.Feynman rules on the Hopf algebra side are provided for scalar fields from thetwo Symanzik polynomials, together with the above Hopf algebra structure. Forgauge fields, a third polynomial [ ] allows to obtain the Feynman rules for gaugetheory from the scalar field rules [ ]. We follow [
8, 9, 12 ]. ψ Γ . For the first Kirchhoff polynomialconsider the short exact sequence(3.1) 0 → H → Q E ∂ (cid:122)(cid:125)(cid:124)(cid:123) → Q V, → . Here, H is provided by a chosen basis for the independent loops of a graph Γ. E = | E Γ | is the number of edges and V = | V Γ | the number of vertices, so Q E is an E -dimensional Q -vectorspace generated by the edges, similar Q V, for the verticeswith a side constraint setting the sum of all vertices to zero.Consider the matrix (see [
2, 3 ]) N ≡ ( N ) ij = (cid:88) e ∈ l i ∩ l j A e , for l i , l j ∈ H .Define the first Kirchhoff polynomial as the determinant ψ Γ := | N | . Proposition . ([ ], see also [ , Prop.2.2]) The first Kirchhoff polynomialcan be written as ψ Γ = (cid:88) T (cid:89) e (cid:54)∈ T A e where the sum on the right is over spanning trees T of Γ. UANTUM FIELDS, PERIODS AND ALGEBRAIC GEOMETRY 7 φ Γ and | N | Pf . To each edge e weassign an auxiliary four-vector ξ e .Let then σ i , i ∈ , , σ = I × the unitmatrix.For the second Kirchhoff polynomial, augment the matrix N to a new matrix N in the following way:(1) Assign to each edge e a quaternion q e := ξ e, σ − i (cid:88) j =1 ξ e,j σ j , so that ξ e I × = q e ¯ q e , and to the loop l i , the quaternion u i = (cid:88) e ∈ l i A e q e . (2) Consider the column vector u = ( u i ) and the conjugated transposed rowvector ¯ u . Augment u as the rightmost column vector to M , and ¯ u as thebottom row vector.(3) Add a new diagonal entry at the bottom right (cid:80) e q e ¯ q e A e .Note that by momentum conservation, to each vertex, we assign a momentum ξ v , and a corresponding quaternion q v . Remark . Note that we use that we work in four dimensions of space-time,by rewriting the momentum four-vectors in a quaternionic basis. This strictly four-dimensional approach can be extended to twistors [ ].The matrix N has a well-defined Pfaffian determinant (see [ ]) with a remark-able form obtained for generic ξ e and hence generic ξ v : Lemma . ([ , Eq.3.12]) | N | Pf = − (cid:88) T ∪ T (cid:88) e (cid:54)∈ T ∪ T τ ( e ) ξ e (cid:89) e (cid:54)∈ T ∪ T A e , where τ ( e ) is +1 if e is oriented from T to T and − T , T are two trees such that their union contains all vertices of thegraph, i.e. T ∪ T is a spanning 2-tree.Note that | N | Pf = | N | Pf ( { ξ v } ) is a function of all ξ v , v ∈ Γ [0] . From the view-point of graph theory, this is the natural polynomial. It gives the second Symanzikpolynomial upon setting the ξ e in accordance with the external momenta p e : Q : ξ e → + p e . Remark . Adding to the second Symanzik polynomial a term ψ Γ (cid:80) e ∈ Γ A e m e allows to treat masses m e . Remark . For γ ⊂ Γ a non-trivial subgraph, and κ ∈ { φ, ψ } we havealmost factorization: κ Γ = κ Γ /γ ψ γ + R κ Γ ,γ , with the remainders R κ Γ ,γ homogeneouspolynomials of higher degree in the sub-graph variables than ψ γ . DIRK KREIMER
In Schwinger parametric form, theunrenormalized Feynman amplitude I Γ (omitting trivial overall factors of powersof π and such) comes from an integrand I Γ (3.2) I Γ = (cid:90) e − φ Γ ψ Γ ψ (cid:124) (cid:123)(cid:122) (cid:125) I Γ (cid:89) e dA e . This form gets modified if we allow for spin and other such complications. Anexhaustive study of how to obtain gauge theory amplitudes from such an integrandis given in [ ]. Remark . A regularized integrand can be obtained by raising the denom-inator 1 /ψ to a noninteger power (dimensional regularization), or multiplicationby non-integer powers of edge variables, together with suitable Γ-functions (ana-lytic regularization). The latter suffices to treat the Mellin transforms as used forexample in [ ] and discussed below. We can render the integrand I Γ inte-grable wrt to the domain σ Γ prescribed by parametric integration by a suitablesum over forests. We define I R Γ := (cid:88) f ∈F Γ ( − | f | I Γ /f I f , where for f = (cid:83) i γ i , I f = (cid:81) i I γ i and the superscript indicates that kinematicvariables are specified according to renormalization conditions.The formula for I R Γ is correct as long as all sub-graphs are overall log-divergent,the necessary correction terms in the general case are given in [ ]. In our examplesbelow, we can always identify the one log-divergent subgraph -if any- with theunique non-trivial forest. Accompanying this integrand is the renor-malized result which can be written projectively:Φ R (Γ) := (cid:90) P | E Γ I | ( R + ) (cid:88) f ∈F Γ ( − | f | ln φ Γ /f ψ f + φ f ψ Γ /f φ /f ψ f + φ f ψ Γ /f ψ /f ψ f Ω Γ , for notation see [
8, 9 ] or [ ]. Let us just mention that for the domain of integrationwe will abbreviate from now on P | E Γ I | ( R + ) = P Γ . Note that this is a well-defined integral obtained from the use of the forest formula.It is obtained without using an intermediate regulator. It is well-suited to analysethe mathematical structure of perturbative contributions to Green functions.Also, combining this approach with [ ], it furnishes a reference point againstwhich to check in a situation where intermediate regulators are spoiling the sym-metries of the theory.Below, we will shortly compare the structure of this integrand to the appaear-ance of analytic regulators provided by anomalous dimensions of quantum fields,wich then define Mellin transforms for the primitives in the Hopf algebra. UANTUM FIELDS, PERIODS AND ALGEBRAIC GEOMETRY 9
Feynman graphs have their external edges labelledby momenta, and internal edges labelled by masses.Renormalized Feynman rules above are therefore functions of scalar products Q i · Q j and mass-squares m e . Equivalently, upon defining a positive definite linearcombination S of such variables, we can write them as functions of such a scale S ,and angles Θ ij := Q i · Q j /S , Θ e := m e /S . We use S , Θ ij , Θ e to specify scale andangles for the renormalization point. A graph which furnishes only a single scalarproduct Q · Q as a scale is denoted a 1-scale graph.Isolating short-distance singularities in 1-scale sub-graphs has many advan-tages, including a systematic separation of angles and scales, and a clean approachto the renormalization group as well as an identification of the freedom provided byexact terms in the Hochschild cohomology, as we discuss below, see also [
16, 17, 11 ].Following [ ], we have the decomposition Theorem . Φ R ( S/S , { Θ , Θ } ) = Φ − ( { Θ } ) (cid:63) Φ R S/S ) (cid:63) Φ fin ( { Θ } ) . Here, the angle-dependent Feynman rules Φ fin are computed by eliminatingshort-distance singularities through the comparison, via the Hopf algebra, with 1-scale graphs evaluated at the same scale as the initial graphs, while the 1-scaleFeynman rules Φ R S/S ) eliminate short-distance singularities by renormalizing1-scale graphs at a reference scale S . Remark . Feynman rules in parametric renormalization allow to treat thecomputation of Feynman graphs as a problem of algebraic geometry, analysing thestructure of two kinds of homogeneous polynomials [
6, 7, 8, 4 ]. Remark . The fact that it is basically the denominator structure which de-termines the computability of Feynman graphs in parametric renormalization makesthis approach very efficient in computing periods in the (cid:15) -expansion of regularizedintegrands.
Remark . We assume throughout that angles and scales are such thatwe are off any infrared singularities, for example by off-shell external momenta.The latter would not be cured by the forest sums which eliminate short-distancesingularities.
4. Examples4.1. Overall finite graphs.
From now on, we write φ Γ = φ Γ (Θ) , φ ≡ φ Γ (Θ ). For a 1-scale graph Γ, we let Γ • be the graph where the two externalvertices of Γ are identified. One has φ Γ = ψ Γ • .Assume we are considering a superficially convergent graph Γ free of subdiver-gences. For example, a graph Γ in four dimensions of space time on n > | Γ | edgesdelivers the integrable form 1 S n − | Γ | (cid:90) ψ (cid:18) ψφ (Θ) (cid:19) n − | Γ | Ω Γ . This is polynomial in the scale dependence, while the angle dependence is di-logarithmic for good reasons [ ]. Inserting logarithmic subdivergences, we get the integrable form (it is integrableas long as external momenta are off-shell such that no infrared singularities arise)1 S n − | Γ | (cid:90) P Γ (cid:88) f ( − | f | ψ /f ψ f (cid:32) ψ Γ /f ψ f φ Γ /f ψ f + φ f ψ Γ /f (cid:33) n − | Γ | Ω Γ . Note that φ ∅ = 0 , ψ ∅ = 1. Remark . Note that for the logarithmic divergent case n = 2 | Γ | , we got alogarithm in the numerator of the renormalized integrand, reflecting the superficialdegree of divergence zero. In the convergent case, the above power of n − | Γ | isthen reflecting the superficial degree of convergence 2( n − | Γ | ). Consider now a logarithmically divergent graph with-out sub-divergences, L = ln S/S . Then,Φ R (Γ) = c L + c (Θ , Θ ) . We have c = (cid:90) P (Γ) Ω Γ ψ ,c (Θ , Θ ) = (cid:90) P (Γ) ln φ Γ φ Ω Γ ψ . The finite part c (Θ , Θ ) can equivalently be expressed in the form of overall finitegraphs. Let P e be the propagator at edge e , P e the same propagator, but with itsexternal momenta evaluated as prescribed by the renormalization condition. Then,1 P e − P e = P e − P e P e P e , where internal loop momenta in edge e drop out in the difference P e − P e .By telescoping we can extend to products of propagators provided by graphs,and hence express the finite part of an overall logarithmically divergent graph asan overall convergent graph, which is an element of a larger Hopf algebra providedby general Feynman integrals. Consider Γ = Γ say,as a generic example. We have ∆ (cid:48) (Γ) = Γ ⊗ Γ .Then Φ R (Γ) = c L + c (Θ , Θ ) L + c (Θ , Θ ) . We have Φ R (Γ ) = (cid:90) P Γ ln SS φ Γ φ ψ − ln SS φ Γ3 ψ Γ4 + φ ψ Γ3 φ ψ Γ4 + φ ψ Γ3 ψ ψ Ω Γ . We then have for the scale independent part c (Θ , Θ ) = (cid:90) P Γ ln φ Γ φ ψ − ln φ Γ3 ψ Γ4 + φ ψ Γ3 φ ψ Γ4 + φ ψ Γ3 ψ ψ Ω Γ , UANTUM FIELDS, PERIODS AND ALGEBRAIC GEOMETRY 11 and for the term linear in L :(4.1) c (Θ , Θ ) = (cid:90) P Γ (cid:32) ψ − φ Γ ψ Γ ψ ψ (cid:2) φ Γ ψ Γ + φ ψ Γ (cid:3) (cid:33) Ω Γ . The term quadratic in L gives c = (cid:90) P Γ (cid:32) φ Γ ψ Γ ψ Γ φ ψ ψ (cid:2) φ Γ ψ Γ + φ ψ Γ (cid:3) (cid:33) Ω Γ . Scaling out from the edge variables of the subgraph one of its variables λ say- andintegrating it, so that Ω Γ → Ω Γ ∧ Ω Γ ∧ dλ (a careful treatment of such changesof variables is in [ ]) gives us c = (cid:90) P Γ3 × P Γ4 (cid:32)(cid:90) ∞ φ Γ (Θ) ψ Γ ψ Γ φ Γ (Θ ) ψ ψ [ φ Γ (Θ) ψ Γ + λφ Γ (Θ ) ψ Γ ] dλ (cid:33) Ω Γ ∧ Ω Γ = (cid:90) P Γ3 ψ Ω Γ (cid:90) P Γ4 ψ Ω Γ , which fully exhibits the desired factorization.One easily checks that ∂ kL vanishes for k greater than the co-radical degree. Let us now consider the primitive p a − b .The two graphs involved are distinguished only by the insertion place into whichwe insert the subgraph Γ . From the previous result is it evident that for p a − b wecould at most find up to a linear term in L Φ R ( p a − b ) = c p a − b L + c p a − b (Θ , Θ ) . We find for this scheme-independent -and hence well-defined- period c p a − b = (cid:90) P Γ34 (cid:32) ψ a − ψ b (cid:33) Ω Γ , where P Γ and Ω Γ are obviously independent of the insertion place.Note that the difference is completely governed by R ψ Γ a , Γ as compared to R ψ Γ b , Γ , while for the term constant in L we also need to consider R φ Γ a , Γ ascompared to R φ Γ b , Γ .From now on we discard the constant terms in L , which we regard as originatingfrom overall convergent integrals. Next, let us look at s whichis of co-radical degree 1. Clearly,Φ R ( s ) = c c L + c s . In general, c s is not a period but rather a complicated function of Θ , Θ .We now assume that we subtract at Θ = Θ . c s could then still be a function of the angles Θ. Instead, it is a constant, asis immediately clear by using Eq.(4.1). This constant is a period which hopefully is known to us soon enough using the methods of [ ]. c s = (cid:90) P Γ (cid:32)
12 1 ψ a + 12 1 ψ b + 1 ψ − φ Γ ψ Γ + φ Γ ψ Γ ψ ψ [ φ Γ ψ Γ + φ Γ ψ Γ ] (cid:19) Ω Γ = (cid:90) P Γ (cid:32)
12 1 ψ a + 12 1 ψ b + 1 ψ − ψ ψ (cid:33) Ω Γ , (4.2)where the notation P Γ , Ω Γ is justified, as edges can be consistently labeled in allterms. Note that the step above from the first to the second line follows as we have φ Γ (Θ ) = φ Γ (Θ) = φ Γ , ∀ Γ. For anti-cocommutative elementslike c angle dependence remains, even if we set Θ = Θ . In such a setting, wefind Φ R ( c ) = c (Θ) L.c (Θ) = (cid:90) P Γ (cid:32)
12 1 ψ a + 12 1 ψ b − ψ − φ Γ ψ Γ − φ Γ ψ Γ ψ ψ [ φ Γ ψ Γ + φ Γ ψ Γ ] (cid:19) Ω Γ . (4.3)In this way, when renormalizing at unchanged scattering angles, angle dependenceis relegated to anti-cocommutativity. Φ − s . In [ ] scale and angle depenence were sepa-rated using 1-scale renormalized Feynman rules Φ R − s . These are massless Feynmanrules which act by choosing two distinct vertices for each subdivergent graph γ ⊂ Γand evaluating the counterterms for this subgraph treating it as a 1-scale graph γ ,so that we have φ γ = ψ γ • .Also, Γ itself allows external momenta only at two distinct vertices.As discussed in [ ], see also [ ] for a detailed example, we can enlarge the setof graphs to be considered by graphs G say so that Γ /γ = G /g , and φ Γ /γ ψ g + ψ γ • ψ Γ /γ is the two-vertex join of co- and subgraph [ ]. In G , edges which connect G − g to g all originate from the two distinct vertices chosen in γ . the reader will haveto consider [ ] for precise definitions.Feynman rules for graphs G are the canonical Φ R subtracting at S = S , andfor example for Γ = Γ ,Φ R ( G ) − Φ R − s (Γ) = (cid:90) P Γ (cid:18) ψ G − ψ (cid:19) Ω Γ , gives us another period. UANTUM FIELDS, PERIODS AND ALGEBRAIC GEOMETRY 13 G in this example is the graph G = , with the understanding that momenta are zero at two of its four marked externalvertices when acted upon by Φ R − s . Its wheel with four spokes subgraph is rendered1-scale upon enforcing a five-valent vertex and hence must be treated in a suitablyenlarged Hopf algebra.The general case is studied in [ ] in great detail. Graphs of the form G can be computed by defining a suitable Mellin transform [
13, 14, 21 ]. This holdseven if the co-graph is not 1-scale, the important fact being that the subgraph is.This Mellin transform is defined by raising a quadric Q ( e ) for an internal edge e , or a linear combination of such quadrics, to a non-integer power1 Q ( e ) → Q ( e ) ρ , in a cograph which has no subdivergences. This defines a Mellin transform (stayingin the above example) M Γ ( ρ, L ) = e − ρL f ( ρ ) , where f ( ρ ) has a first order pole in ρ at zero with residue 6 ζ (3) = c .Also, f ( ρ ) = f (1 − ρ ). We set f ( ρ ) = 6 ζ (3) ρ (1 − ρ ) (1 + d (Θ) ρ + O ( ρ )) . We can compute Φ R ( G ) asΦ R ( G ) = − ζ (5) (cid:122)(cid:125)(cid:124)(cid:123) c ∂ ρ ( e − ρL − f ( ρ ) . One hence finds that c G = 60 ζ (3) ζ (5) and c G = 120 ζ (3) ζ (5) − ζ (5) d (Θ).Can we confirm this structure from the parametric approach?We first note that | ψ G / Γ | = | φ G / Γ | − , | ψ G / Γ | = | ψ G − Γ | + 1 . Returning to affine coordinates, scaling out a subgraph variable and integratingit, using R ψG , Γ = ψ Γ • ψ G − Γ and returning to P Γ × P Γ delivers c G = (cid:90) P Γ4 ψ Ω Γ (cid:124) (cid:123)(cid:122) (cid:125) ζ (cid:90) P Γ3 ψ γ (cid:18) − ln ψ G − Γ φ Γ ψ (cid:19) Ω Γ (cid:124) (cid:123)(cid:122) (cid:125) ζ − d , (4.4)as desired. Finally, afurther angle independent period is furnished by the 1-scale version c , − s of theanti-cocommutative c . c c , − s = (cid:90) P Γ (cid:32) ψ − ψ a − ψ b − ψ Γ • ψ Γ − ψ Γ • ψ Γ ψ ψ ( ψ Γ • ψ Γ + ψ Γ • ψ Γ ) (cid:33) Ω Γ , where it is understood that all graphs have their subgraphs as 1-scale subgraphs asin G . By the previous result, this can be decomposed into two separate projectiveintegrals. We have seen that when a sub-graph γ is 1-scale, the evaluation of the full graph factorizes the period c γ . This isthe crucial fact which allows to use co-boundaries to alter the Taylor coefficients ofMellin transforms [
11, 16, 17 ].For example, with B now effecting a 1-scale insertion, B (Γ ) = G and bα (Γ ) = α ( I )Γ :Φ R (( B + bα )(Γ ) = Φ R (Γ ) + 20 ζ (5) α ( I ) L, where we are free to choose α ( I ) to modify d (Θ), a useful fact in light of themanipulations in [
21, 22 ].
5. Feynman rules from a Lie viewpoint
The map σ : H → U ( L ) can be combined with projectors T k into the k -thsymmetric tensorpower of L . Let then σ k := T k ◦ σ. Then, σ takes values in L , σ takes values in L ⊗ s L , and so on.The map σ : H → L is such that an element h ∈ H in the k -th co-radicalfiltration (so that ∆ (cid:48) k ( h ) (cid:54) = 0) has contributions in L k at most, for example theco-radical degree two c fulfils this bound as it maps to [ Z , Z ] ∈ L .In general, a co-radical degree k element has a non-vanishing component in L k if and only if ∆ (cid:48) k ( h ) contains corresponding anti-symmetric elements.The symmetric parts in ∆ (cid:48) k ( h ) map under σ to an element l ( h ) ∈ L say, l ( h ) (cid:54)∈ L , so that the Dynkin operator S (cid:63) Y maps the pre-image σ − ( l ) to aprimitive element in H , Φ R ( S (cid:63) Y σ − ( l ( h ))) = c l L, while all other terms in c h come from the pre-images of elements in L k , k > σ k provide the contributions to order L k similarly, in fullaccordance with the co-radical filtration and the renormalization group [ ]. References [1] S. Bloch, talk at Spring School: Feynman Graphs and Motives, Bingen, march 2013, .[2] S. Bloch, H. Esnault, D. Kreimer,
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Renormalization, Hopf algebras and Mellin transforms , ∼ maphy/panzer.pdf , these proceedings.[18] E. Panzer, talk at Spring School: Feynman Graphs and Motives, Bingen, March 2013, .[19] E. Panzer, On the analytic computation of massless propagators in dimensional regulariza-tion,
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Dept. of Physics and Dept. of Mathematics, Humboldt University, Unter den Linden6, 10099 Berlin, Germany
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