University of Amsterdam
Report Bachelor Project Physics and Astronomy
Institute for Theoretical Physics Amsterdam
The Hopf algebraic structure of
imaginary parts of Feynman diagrams
Author:
Robert Beekveldt
Student number:
10669809
Size of project:
15 EC
Supervisor:
Prof. Dr. Eric Laenen
Second assessor:
Dr. Juan Rojo
Conducted between:
3-4-2017 and 1-7-2017
1
Abstract
It is necessary to calculate multi-loop Feynman diagrams to be able to do precise predictions in high-energy collisions. These multi-loop Feynman diagrams contain integration over internal momenta which have been very problematic to compute. Intense study of multi-loop Feynman diagrams has lead to constraints on functions computed from these diagrams. These expressions contain generalizations of polylogarithms, the multiple polylogarithms (MPLs). Starting this the-sis with the discussion of the properties of MPLs leads to the mathematical construction of the Hopf algebra. From the Hopf algebra the coproduct is derived which enables us to manipulate difficult MPLs and find functional equations among them.
The use of unitary cuts to calculate imaginary parts of Feynman integrals have been helpful in calculations of multi-loop Feynman integrals. Using Schwarz reflection we derived a relationship between these imaginary parts and thus the unitary cuts with the discontinuity of functions inside the Feynman expressions. The Hopf algebra reviewed in this thesis uses an asymmetry in the coproduct to quickly calculate the discontinuities of the Feynman expressions. The relationship between unitary cuts and the Hopf algebraic structure of imaginary parts of Feynman integrals can be used in the computation of multi-loop Feynman diagrams. In this thesis this relationship is verified on the case of a scalar self-energy diagram.
2
Populaire samenvatting(dutch)
In de natuurkunde willen we graag te weten komen hoe de kleinste deeltjes zich gedragen. Een voorbeeld van een klein deeltje is het proton, in de bekende deeltjesversneller LHC in Gen`eve wor-den deze op elkaar geschoten. Bij deze botsingen staan grote meetapparatuur om de producten van de botsing te analyseren. De metingen kunnen de theorieen van wetenschappers testen. Het mooiste zou zijn om een theorie op te stellen en deze vervolgens te bevestigen met metingen. Een buitengewoon succesvolle theorie is het standaard model dat alle interacties tussen fundamentele deeltjes beschrijft. In deze theorie worden grafische weergaves van interacties omgezet in formules die kunnen worden getest in de LHC. Deze grafische weergave wordt een Feynman diagram ge-noemd. Een voorbeeld van een Feynman diagram is de annihilatie van een elektron(e−)en zijn tegengestelde deeltje het positron(e+), in het figuur hieronder zie je links de inkomende elektron
en positron.
Ze raken elkaar en annihileren tot een foton, vervolgens ontstaat uit het foton weer een positron en elektron. Er zijn echter ook moeilijkere interacties te bedenken zoals het onderstaande diagram.
Hier vindt weer annihilatie plaats van een elektron positron paar echter wordt daarna niet gelijk weer datzelfde paar gecre¨eerd. Er wordt nu tussentijds een onbekend paar deeltjes gevormd dit wordt gerepresenteerd door de cirkel. Vervolgens annihileren deze deeltjes weer en wordt er weer een elektron positron paar gevormd. De tijdelijke vorming van de onbekende deeltjes vormt een groot probleem in de huidige natuurkunde. Omdat deze deeltjes niet gemeten worden in de detector moet er rekening gehouden worden met veel scenario’s. In formules betekent dit dat we moeilijke integralen moeten berekenen. In deze scriptie bekijken we een relatie tussen twee manieren om deze integralen op te lossen. Door een link tussen de twee manieren te leggen zijn natuurkundige beter in staat om de berekeningen te voltooien.
Contents
1 Abstract 1 2 Populaire samenvatting(dutch) 2 3 Introduction 4 3.1 Overview thesis . . . 5 4 Multiple polylogarithms 6 4.1 Shuffle algebra . . . 7 4.2 Stuffle algebra . . . 85 Mathematical structure of the polylogarithmic algebra 10 5.1 Algebra formalism . . . 10
5.2 Coalgebra formalism . . . 11
5.3 Hopf algebra . . . 12
6 The coproduct applied to MPLs 13 6.1 Regularised version of MPLs . . . 16
6.2 Zeta problem . . . 17
6.3 Asymmetry in discontinuity operator . . . 17
7 Functional equations of MPLs 19 8 Optical theorem 21 8.1 Cutting equation . . . 22
9 Relationship between the coproduct and discontinuities inside the Hopf algebra 25 10 Calculation of self-energy scalar diagram 26 10.1 Wick Rotation . . . 27
10.2 Evaluating the Feynman parameter integral . . . 29
10.3 Calculation of the discontinuity . . . 30
10.4 Evaluation cut diagram . . . 31
11 Conclusion 32 12 Acknowledgments 33 13 Appendix 34 13.1 I conversion of generic base point to a zero base point . . . 34
13.2 Behaviour of the bilinear tensor product . . . 34
13.3 Basic properties of the coproduct . . . 34
13.4 Discontinuity of Lin . . . 35
13.5 Veltman rules . . . 35
13.6 Feynman rules for scalar particles . . . 35
3
Introduction
The predictions of quantum field theory(QFT), especially the standard model have been tested and verified up to high precision in particle accelerators such as the LHC. Approaching higher energies new predictions may be tested or new physics may be discovered. QFT makes use of so called Feynman diagrams.
Figure 1: A first order Feynman diagram for electron-proton scattering.
The lowest order diagram of electron proton scattering contains two vertices and is drawn in the above figure. The electron and proton scatter under exchange of a photon (γ). Using the Feynman rules we can convert the diagram into an integral. The answer of the integral contributes to M the Feynman amplitude. With M we can calculate cross sections (loosely speaking the chance of interaction) of the incoming particles. As an example the differential cross-section of the process e−p → e−p can be calculated with,
dσ dΩ = 1 64π2( 1 mp+ E1− E1cos θ )2|Mf i|2
with E1 the energy of the incoming electron and θ the scattering angle of the electron.
Higher order Feynman diagrams, containing more vertices, contribute to amplitude M as well. Depending on the energy the particles higher order diagrams can be neglected.
At higher energies, more and more Feynman diagrams start to contribute to the amplitude M. These diagrams consist of multi-loop integrals in which integration over all internal momenta has to be performed. This means we have to integrate functions often containing complicated branch cut structures. The first approach in the literature is to compute these integrals with rather ’brute force’ involving dimensional regularization. The increasing difficulty of these integrals forces us to look at other techniques to compute these Feynman diagrams.
Intense study of Feynman integrals leads to certain constraints on the computed expressions. It is conjectured that certain functions cannot appear in Feynman integrals, moreover functions like the (poly)logarithm appear often in the calculation. As an example a one-loop triangle integral leads to the expression,
T (p21, p22, p23) = 2
λ[Li2(z) − Li2(˜z) − log(z ˜z) log( 1 − z 1 − ˜z)]
involving dilogarithms and ordinary logarithms. One can expect more difficult generalizations of polylogarithms in multi-loop computations. A study of the algebraic structure (Hopf algebraic structure) of polylogarithms has achieved to reduce a multi-loop expression, from multiple pages to a couple of lines. The Hopf algbra combined with the known technique of unitary cuts can be combined to calculate difficult diagrams.
3.1
Overview thesis
This thesis consists of two parts. Part one(Ch.4-7): how the shuffle multiplication among MPLs leads to the Hopf algebraic structure of MPLs. Followed by the description of finding functional equations among MPLs.
Part two(Ch. 8-11): How the optical theorem leads to unitary cuts on Feynman diagrams. Ex-plaining the relationship between unitary cuts and the discontinuities of the expression inside Feynman integral computations. Followed by explaining how to take advantage of the discontinu-ity operator inside the Hopf algebra. Finally concluding the thesis by verifying of this relationship on a scalar self-energy diagram.
4
Multiple polylogarithms
In particle physics ’classical’ polylogarithms and zeta values appear in the computation of one-loop Feynman integrals. One can expect that more complicated structures of logarithmic functions appear in multi-loop integrals.It is therefore likely needed to introduce a generalization of the polylogarithm, namely the multiple polylogarithms(MPLs). The classical polylogarithm has a iterated integral definition
Lin =
Z z
0
dt
t Lin−1(t). (1)
The MPLs can be recursively defined in integral form as well [3]. The recursion relation starts (by definition) with G(; z) = 1. G(a1, a2, ..., an; z) = Z z 0 dt t − a1 G(a2, ..., an; t), (n ≥ 0) (2)
with an ∈ C being a constant and z ∈ C being a complex variable. One of the properties of
the MPLs is that it will be divergent if a1 = z. The input of n complex numbers in MPL
G(a1, a2, ..., an; z) can be thought of as an input of an arbitrary vector −→a = (a1, a2, ..., an), called
the vector of singularities. The number of components of this vector is called the weight of the corresponding MPL.
The MPL should contain special cases such as the classical polylogarithm and the ordinary log(z). Powers of ordinary logarithms can be found by inserting the vector −a→m= (a, a, ..., am) with a = 0,
G(−0→m; z) =
1 m!log(z)
m (3)
other special cases can derived such as the classical polylogarithm
G(−→0m−1, 1; z) = −Lim(z). (4)
In mathematical literature the MPLs are defined in a more general way,
I(a0; a1, ..., an; an+1) = Z an+1 a0 dt t − an I(a0; a1, ..., an−1; t) (5)
starting the iteration at I(a0; ; a1) = 1. This integral form enables us to choose arbitrary
bound-aries of integration. Representing the same function eq. 2 and 5 are related by,
G(an, ..., a1; an+1) = I(0; a1, ..., an; an+1) (6)
notice the need of 0 as the starting point of the I integration. So to convert I to G we need the zero base point, in the appendix 13.1 one can find the conversion from a MPL with an arbitrary base point I(a0; a1, ..., am; an+1) to a MPL with a zero base point I(0; b1, ..., bm; bm+1)(Appendix
9.1).
The classical polylogarithm has also a power series representation, generalizing this power series to [5],
this series converges for |zj| <1. The number of arguments i is called the depth of the MPL, so
Li3,5(z1, z2) has depth 2 and weight 8. Eq. 7 can be rewritten into a nested summation,
Lim1,m2,...,mi(z1, ..., zi) = ∞ X ni=1 zni i nmi i ni−1 X ni−1=1 ... n2−1 X n1=1 zn1 1 nm1 1 (8)
both generalizations, the integral representation (eq. 2) and the power series representation (eq. 16) represent the same function. As a result these two representations are related by
Lim1,m2,...,mi(z1, ..., zi) = (−1) iG(0, .., 0 | {z } mi−1 , 1 zi, ..., 0, ..., 0 | {z } m1−1 , 1 z1...zi; 1). (9)
4.1
Shuffle algebra
The shuffle algebra can be generated through the iterated integral definition (2). The product of two iterated integrals which share the same boundaries of integration can always be decomposed into a linear combination of iterated integrals (Furbini’s theorem). Starting with the simplest case, the product of two MPLs of weight one,
G(a; z)G(b; z) = Z z 0 dy y − a Z z 0 dx x − b = x square dydx (y − b)(x − a) (10)
by making use of Furbini’s theorem we have rewritten the iterated integral into an area integral over a square.
We can divide the area of the square into an area of two triangles,
= x 0≤x≤y≤z dydx (y − a)(x − b)+ x 0≤y≤x≤z dydx (y − a)(x − b) (11)
the two iterated integrals left are known MPLs,
= Z z 0 dy y − a Z y 0 dx x − b+ Z z 0 dx x − b Z x 0 dy y − a (12) = G(a, b; z) + G(b, a; z) (13)
A more difficult example is the product of MPLs of weight one and weight two. The MPL of weight two represents an iterated integral over a triangular area, the MPL of weight one represents a integral over a side, if we multiple these two they will form a volume integral over half a cube. To put it in a more formal way,
G(a, b; t)G(c; t) = Z t 0 dz z − a Z z 0 dy y − b Z t 0 dx x − c = y half cube dzdydx (z − a)(y − b)(x − c) (14)
decomposing the volume integral, using the same analogy as before, into three iterated volume integrals over different pyramids. These three iterated integrals correspond to three MPLs of weight three, = y 0≤x≤y≤z≤t dzdydx (z − a)(y − b)(x − c)+ y 0≤y≤x≤z≤t dzdydx (z − a)(y − b)(x − c) + y 0≤y≤z≤x≤t dzdydx (z − a)(y − b)(x − c) = G(a, b, c; t) + G(a, c, b; t) + G(c, a, b; t). (15)
notice the important fact that the iterated integral G(a, b; t) preserves the order of (a, b) inside all three found MPLs after decomposition.
The generalized strategy arising is to rewrite the product of two MPLs of weight n and weight m into a n + m dimensional integral (hypercube), whereupon we decompose this object along its diagonals into MPLs of weight n + m. As a result we found a simple rule to determine a product of two MPLs which does not involves the explicit calculation of integrals The rule states that any product between two MPLs of weight n and m can be written as a linear combination of MPLs of weight n + m,
G(a1, ..., an; z)G(an+1, ..., an+m) =
X
P ∗
G(a1, .., an, an+1, ..., an+m) (16)
where we sum over all permutations of the arguments of both MPLs but preserving the order inside both vectors of singularities. Illustrating the preservation of order and the conservation of the weight,
G(a, b, c)G(d, e) = G(a, b, c, d, e) + G(a, b, d, c, e) + G(a, b, d, e, c) + G(a, d, b, c, e) + G(a, d, b, e, c) + G(a, d, e, b, c) + G(d, a, b, c, e) + G(d, a, b, e, c) + G(d, a, e, b, c) + G(d, e, a, b, c)
(17)
in which the shuffeling of arguments is clearly visible, the multiplication satisfies the mathematical shuffle multiplication which is commutative and associative.
4.2
Stuffle algebra
Another relationship among MPLs is the stuffle relationship and looks much like the shuffle rela-tionship. The stuffle relationship follows from the power series representation of the MPLs. The derivation is rather straightforward and will be given by a ’simple’ example,
Li1,4(z1, z2)Li5(z3) = ∞ X n2=1 zn2 2 n4 2 n2−1 X n1=1 zn1 1 n1 ∞ X n3=1 zn3 3 n5 3 = ( X n3>n2>n1≥1 + X n2>n3>n1≥1 + X n2>n1>n3≥1 + X n2=n3>n1≥1 + X n2>n1=n3≥1 )z n1 1 z n2 2 z n3 3 n1n42n53 = Li1,4,5(z1, z2, z3) + Li1,5,4(z1, z3, z2) + Li5,1,4(z3, z1, z2) + Li1,9(z1, z2z3) + Li6,4(z1z3, z2) (18)
the summation of the first line gets decomposed into five smaller sums. Looking at the first three sums of line 2, summing over all possible permutations of n3 > n2 > n1 keeping in mind that
this permutation is restricted by the nested summation of Li1,4(z1, z2), which means that only
summations with n2 > n1 survive. Hence the summation n3 > n1> n2 ≥ 1 does not appear in
the computation.
The last two summations of line 2 contain the terms left out in the first three sums, these are terms involving equal summation indices(example: n2= n3 > n1 ). Again only summations restricted
by the nested summation are allowed (n2> n1).
To put it simply; this algebra created looks like shuffling the indices around and is therefore called ’quasi shuffle algebra’ or stuffle algebra. More complicated products of MPLs can be tackled with stuffle algebra in a similar way, to clarify the properties of the stuffle product lets consider a general example,
Lim1,m2,m3,m4(z1, z2, z3, z4)Lim5(z5) = Lim1,m2,m3,m4,m5(z1, z2, z3, z4, z5)
+ Lim1,m2,m3,m5,m4(z1, z2, z3, z5, z4) + Lim1,m2,m5,m3,m4(z1, z2, z5, z3, z4)
+ Lim1,m5,m2,m3,m4(z1, z5, z2, z3, z4) + Lim5,m1,m2,m3,m4(z5, z1, z2, z3, z4)
+ Lim1,m2,m3,m4+m5(z1, z2, z3, z4z5) + Lim1,m2,m3+m5,m4(z1, z2, z3z5, z4)
+ Lim1,m2+m5,m3,m4(z1, z2z5, z3, z4) + Lim1+m5,m2,m3,m4(z1z5, z2, z3, z4)
it is clear that the weight of the MPLs is conserved. The depth of the MPLs on the other hand is not conserved, a product of Lim1,..,mj(z1, ..., zj)Lin1,..,ni(x1, .., xi) creates terms with j ≤depth≤ j + i with j being the largest depth of the two (equal depth: j ≤depth≤ 2j).
We have discussed the shuffle product arising from the integral definition and the stuffle product derived from the power series representation. Both products have the important property of pre-serving the weight of MPLs. These products between MPLs are important for the basic structure of the upcoming Hopf Algebra.
5
Mathematical structure of the polylogarithmic algebra
5.1
Algebra formalism
We need a mathematical framework in order to find functional equations and get more insight into the mathematical structure of the multiple polylogarithms. We start with the following axioms 1. All MPLs of weight n form a vectorspace called An over the field Q. ’Special’ vectorspaces are
A0 = Q containing rational functions and A1containing ordinary logarithms.
2. All MPLs form a sumspace A = ⊕∞n=0An. Which implies A0∩ An = {0} leading to the
conjucture; ’All MPLs are transcendental’.
As mentioned before examining MPLs we found a multiplication (shuffle/stuffle product) which turned A into an associative (graded) algebra i.e. a vectorspace equipped with a bilinear map: α : A × A → A which assigns to each pair of elements a, b ∈ A the multiplication: α(a, b) = ab. Using the bilinear map β : A×A → A⊗A we can turn this multiplication into the more convenient linear map: µ : A ⊗ A → A with µ(a ⊗ b) = ab, the mapping are related by α = µ ◦ β . The pair (a,b) expressed as a ⊗ b follows the bilinear rules due to the bilinear map β (appendix 13.2). The algebra is graded because the shuffle and stuffle multiplications preserve the weight: AnAm∈
Am+n.
The assocativity a(bc) = (ab)c of the shuffle multiplication gives rise to the following equality,
a · (b · c) = µ(a ⊗ (b · c)) = µ(a ⊗ µ(b ⊗ c)) = µ(Id ⊗ µ)(a ⊗ b ⊗ c), (19) (a · b) · c = µ((a · b) ⊗ c)) = µ(µ(a ⊗ b) ⊗ c) = µ(µ ⊗ Id)(a ⊗ b ⊗ c), (20) leading to the identity:µ(Id⊗µ) = µ(µ⊗Id) which will become crucial information in the following section. The assocativity can be represented in the following commutative diagram representing the morphism between all objects.
Figure 2: The commutative diagram shows that by composition µ(Id ⊗ µ) = µ(µ ⊗ Id).[3] The commutative diagram shows graphically that both paths from A ⊗ A ⊗ A to A are commuta-tive i.e. by composition µ ◦ (Id ⊗ µ) = µ ◦ (µ ⊗ Id). The fact that Q, containing the unit element 1, is part of the vectorspace A turns A into a unital associative algebra.
5.2
Coalgebra formalism
Coalgebras are dual to algebras. So to introduce the ´algebra - coalgebra´ structure one can make the analogy with the ´vector space - dual vector space´ structure. As used in quantum mechanics lets consider the vector space V and W with a linear map l : V → W . We then know that there exists another linear map l† between the dual spaces of V and W l† : W∗→ V∗.
To create l†we have used three steps; replace each vector space with its dual vector space, replace each linear map with its hermitian conjugate and reverse the arrows in the mapping. If we apply the same rules for an algebra A with a linear map µ : A ⊗ A → A we create a coalgebra i.e. a vector space A∗= C with a linear map ∆ = µ†: C → C ⊗ C. We can again make a commutative diagram of the coalgebra by turning all arrows around in figure 2,
Figure 3: The commutative diagram shows that by composition (∆ ⊗ id)∆ = (id ⊗ ∆)∆.[3] the diagram makes clear that the associativity of the algebra A causes the coalgebra C to be ’(co)associative’ as well i.e. (∆ ⊗ id)∆ = (id ⊗ ∆)∆. Eq. 19 shows that the linear map µ assigns to every tensor product a ⊗ b ∈ A ⊗ A the product ab ∈ A and the order in which way we multi-plicate multiple elements is of no consequence. The linear map ∆ corresponds to the ’opposite’, it decomposes an element c ∈ C into a tensor productP
ic (1)
i ⊗ c
(2)
i ∈ C ⊗ C and the order in which
you decompose multiple elements is again of no consequence.
The dual behaviour of the algebra can be explained by making again the analogy to quantum mechanics and its bra-ket notation. Recall that V∗and V are dual spaces, the bra ∈ V∗represents a linear functional which maps a ket ∈ V representing a vector to the underlying field. Using the same form we can say something about the co-multiplicative map µ†,
hA|a ∗ (b ∗ c)i = hA|µ(a ⊗ µ(b ⊗ c))i = hµ†(A)|a ⊗ µ(b ⊗ c)i =X i hA1 i ⊗ A 2 i|a ⊗ µ(b ⊗ c)i =X i hA1 i ⊗ µ†(A2i)|a ⊗ b ⊗ ci = h(id ⊗ µ†)µ†(A)|a ⊗ b ⊗ ci
From this derivation (line 2) the form of the proposed mapping µ† = ∆(c) :P
ic (1) i ⊗ c
(2)
i becomes
clear. The appearance of the co-multiplication in the bra form (as the linear functional in bra form) loosely explains the duality of the algebra and coalgebra.
To clarify this last statement lets look at the decomposition of ∆(c) =P
ic (1)
i ⊗ c
(2)
i into three,
(∆ ⊗ id)∆(c) =X i,j c(1,1)ij ⊗ c(1,2)ij ⊗ c(2)i (21) or decomposing c(2)i creating, (id ⊗ ∆)∆(c) =X i,j c(1)i ⊗ c(2,1)ij ⊗ c(2,2)ij (22)
as we know from the commutative diagram these two decompositions are equal and hence the decomposition is always uniquely defined.
As a result the iteration of the coproduct can be written as
C−∆→ C ⊗ C−(∆⊗id)−−−−→ C ⊗ C ⊗ C. (23)
5.3
Hopf algebra
We conjecture that A is a Hopf algebra i.e. a vector space over Q which is a unital associative algebra and a coalgebra in which the multiplication and comultiplication are compatible. It will require the following equality for these two to be compatible
∆(ab) = ∆(a)∆(b). (24)
The basic properties of the coproduct are listed in appendix 13.3. This appendix contains the review of a basic example of the coproduct, the mapping ∆i1,..,ik and primitive elements of the coproduct.
Actually, we need an antipode to conclude that A is a Hopf algebra but this does not change anything in the upcoming sections and thus skip this subtlety.
6
The coproduct applied to MPLs
In order to find functional equations among MPLs we must define the coproduct on MPLs. The coproduct of the MPLs will be taken in I-notation for reasons which will be clear in the coming examples. The coproduct is defined in the following abstract way [3]
∆[I(a0; a1, ...., an; an+1)] =
X
0=i1<i2<...<ik<ik+1=n
I(a0; ai1, ..., aik; an+1) ⊗ [ k Y p=0 I(aip; aip+1, ..., aip+1−1; aip+1)]. (25)
This coproduct can in fact be calculated easily through a graphical representation.
1. Start by drawing a semicircle which represents the MPL of weight n. Then add two dots, one on the left corner representing the left boundary of integration and one on the right representing the right boundary of integration. Followed by adding n arguments (a1, a2, ..., an) in a clockwise
manner on the curved part between the boundaries of integration.
2. The base of the semicircle containing only the two boundaries of integration can be graphically connected with m points, 0 ≤ m ≤ n, forming convex polygons. Each polygon found represent an individual term on the left side of the tensor product in eq. 25.
3. After choosing m dots to form a convex polygon with m + 2 vertices; a0, ai1, ai2..., aik, an+1. The first possible complementary polygon consists of the vertices a0, a1, ..., ai1, note: if ai1 = a1 then no polygon can be formed so we must be continue to the case; ai1, ai1+1, ..., ai2, again if there are no available dots between ai1and ai2we will continue to ai2, ai2+1, ..., ai3 etc. The found complementary polygons represent the terms on the right side of the tensor product.
In advance I like to emphasize that the ’polygon’ appears on the left side of the coproduct and ’complementary polygons’ appear on the right side of the coproduct. The ’polygon’ is always colored black and the ’complementary polygons’ are not black.
To illustrate the coproduct lets look at an example of the coproduct on a MPL of weight 5. We start with writing the two trivial solutions, choosing all points on the semicircle thus creating: a polygon of 7 vertices and no complementary polygon (left side of figure 4) or choosing no points on the semicircle creating: only a complementary polygon of 7 vertices(right side of figure 4).
. Figure 4: Represented in this figure are the two trivial coproducts on a MPL of weight 5. On the left side all points on the semicircle are chosen creating a polygon of 7 vertices (colored black). On the right side no points on the semicircle are chosen creating only a complementary polygon of 7 vertices (colored brown).
We continue the computation by forming five polygons with three vertices a0, ai(0 < i < 6), a6.
Showing two of these polygons, a0, a1, a6with one complementary polygon a1, a2, a3, a4, a5, a6(brown)
and polygon a0, a2, a6with two complementary polygons a0, a1, a2(brown) and a2, a3, a4, a5, a6(red).
. Figure 5: Represented in this figure are two examples of choosing one point on the semicircle creating ’polygons’ with three vertices. On the left side there is one possible complementary polygons, on the right side there are two possible complementary polygons
Moving on by forming polygons with four vertices a0, a1, ai(1 < i < 6), a6. Showing again two of
these polygons, a0, a1, a2, a6 with one complementary polygon (brown) and polygon a0, a1, a3, a6
with two complementary polygons (brown and red).
. Figure 6: Represented in this figure are two examples of choosing two points on the semicircle cre-ating ’polygons’ with four vertices. On the left side there is one possible complementary polygons, on the right side there are two possible complementary polygons
Concluding the coproduct with the last example of forming polygons with again four vertices but with a0, a2, ai(2 < i < 6), a6. The polygons drawn are a0, a2, a3, a6 with two complementary
polygons and the special case of this coproduct a0, a2, a4, a6 with three complementary polygons
(colored brown,red and green),
. Figure 7: Represented in this figure are two examples of choosing two points on the semicircle creating ’polygons’ with four vertices. On the left side there are two possible complementary polygons, on the right side there are three possible complementary polygons
we can iterate this approach to form all possible polygons with 4 vertices (total of 10) a0, aj, ai(j <
i < 6), a6. To complete the whole coproduct one must also find all possible 5 (total of 10) and
6 vertex-polygons (total of 5) in a similar way as illustrated above, in total one can construct 32 different polygons and thus 32 different terms in eq. 25 at weight 5.
6.1
Regularised version of MPLs
The first problem the arises calculating the coproduct of certain non-generic MPLs is creating divergent tensor products out of convergent MPLs. As an example lets consider the coproduct of the covergent MPL: ∆[I(0; 0, 1, 1; z)],
.
Figure 8: In this figure the coproduct is applied to the convergent MPL I(0; 0, 1, 1; z). As can be seen, one of the terms consist of a divergent quantity I(1; 1; z).
one term in the sum of eq. 25 equals I(0; 0, 1; 1) ⊗ I(1; 1; z). The right side of this tensorproduct equals G(1; z) − G(1; 1) which is divergent (use appendix 13.1 to rewrite I).
To solve this flaw we are going to regularize our MPLs with the shuffle algebra, we rewrite every divergent MPL inside our product with shuffle algebra into convergent MPLs excluding the diver-gent MPLs of the form G(z, z, ..., z; z) which contain no converdiver-gent MPLs. The regularization is finished by setting al divergent quantities of this form left to be zero.
The mathematical framework from chapter 3 still holds in the presence of the regularized definition. All algebraic properties remain the same, the unregularized convergent MPLs obviously equal the regularized MPLs. The shuffle and stuffle algebra will(less obviously) hold as well. As an example lets look at a regularized product of MPLs,
[G(z, a; z)G(b, c; z)]reg= [G(z, a, b, c; z) + G(z, b, a, c; z) + G(b, z, a, c; z) + G(z, b, c, a; z) + G(b, z, c, a; z) + G(b, c, z, a; z)]reg = Greg(z, a, b, c; z) + Greg(z, b, a, c; z) + Greg(b, z, a, c; z) + Greg(z, b, c, a; z) + Greg(b, z, c, a; z) + Greg(b, c, z, a; z) = −G(a, z, b, c; z) − G(a, b, z, c; z) − G(a, b, c, z; z) − G(b, a, z, c; z) − G(b, a, c, z; z) − G(b, c, a, z; z)
= −G(a, z; z)G(b, c; z) = Greg(z, a; z)Greg(b, c; z)
(26)
as can be deduced from line 1 and line 5; the regularized product of two MPLs equals the product of the two regularized MPLs. Moreover, line 2 and line 5 show that shuffle algebra also holds for regularized MPLs.
With our gathered information we can now safely apply the coproduct on classical polylogarithms leading to the general formula
∆[Lin(z)] = 1 ⊗ Lin(z) + n−1 X k=0 Lin−k(z) ⊗ logk(z) k! . (27)
Solving the divergence problem with regularization and obtaining eq. 27 leaves us with a final but crucial problem. When this eq. is applied to even zeta values we get a contradiction. The solution
6.2
Zeta problem
The zeta values are Riemann zeta functions evaluated at integer values or Lin(1),
ζn = ∞ X m=1 1 nm (28)
zeta values are convergent if n > 1. The weight and depth of zeta values are defined in the same way as for MPLs.
As mentioned in the previous section taking the coproduct of zeta values creates another contra-diction. According to eq. 27 ordinary zeta values (zeta values of depth one) are primitive, but recalling that ordinary even zeta values can always be written as a power of ζ2 leads to a different
expression than eq. 27 predicts, ∆(ζ6) = 8 35∆(ζ 3 2) = 8 35(1 ⊗ ζ 3 2+ ζ 3 2⊗ 1 + 3ζ 2 2 ⊗ ζ2+ 3ζ2⊗ ζ22) (29)
which is obviously not primitive. This problem does not arise for odd zeta values because they cannot be decomposed in any way, when considering odd ordinary zeta values of weight n; ζn is
always a basis vector.
We can solve this problem by using the ideas of Brown [2], who states that one must define ∆(iπ) = iπ ⊗ 1 which solves the problem of even zeta functions. This definition has of course consequences for our previously formed mathematical framework it changes our previously defined coproduct to: ∆A → A ⊗ H with A = Q[iπ] ⊗ H with Q[iπ] being the ring of polynomials in iπ with still rational coefficients.
In simple language the changes come down to a simple ’rule’ of setting all terms proportional to power of iπ to zero except for the most left terms of the tensor product. As a result eq. 29 turns into, ∆(ζ6) = ζ6⊗ 1 = 8 35ζ 3 2⊗ 1 = 8 35∆(ζ 3 2) (30)
which solves our contradiction.
6.3
Asymmetry in discontinuity operator
The ideas of brown solve our problems but create an asymmetry between the terms inside a tensor product and this affects the properties of certain operators. The discontinuity operator is one of them. The asymmetry is very important in our main focus to link imaginary parts of Feynman diagrams to the discontinuity of the functions arising from these diagrams (Ch.8).
Looking at the behaviour of the discontinuity operator ’Disc’ which takes the discontinuity of a function over its branch cut. Its conjectured by Duhr [3] that the discontinuity operator now only acts on the left most component of the coproduct. Which leads to the following property
∆[Disc(I)] = (Disc ⊗ id)∆(I). (31)
To illustrate this property by a concrete example we will first look at the discontinuity of polylog-arithms in general (appendix 13.4). Verifying eq.(31) on Li3 starting with the left side of the eq.
and using the compatibility condition,
∆[Disc(Li3)] = ∆(2πi)∆(
1 2log
2
(z)) = πi log2(z) ⊗ 1 + 2πi log(z) ⊗ log(z) + 2πi ⊗1 2log
2
continuing on to the right side of eq. (31) and confirming the asymmetrical behaviour of the ’Disc’ operator
(Disc ⊗ id)∆(Li3) = (Disc ⊗ id)(1 ⊗ Li3+ Li3⊗ 1 + Li2⊗ log(z) + Li1⊗
1 2log
2
(z)) = πi log2(z) ⊗ 1 + 2πi log(z) ⊗ log(z) + 2πi ⊗1
2log
2
(z).
This section concludes the review of the Hopf algebraic structure of MPLs. We have defined a product between MPLs and discussed its importance in our algebraic structure. We then defined the coproduct on MPLs and solved two apparent contradictions when applied to certain concrete examples.
The upcoming part of this thesis contains the discussion of two applications of the Hopf algebra, first we are going to apply our knowledge of the coproduct to find functional equations between (multiple)polylogarithms.
Followed by the even more important application, our main focus; the asymmetrical behaviour of the Disc operator which will be vital in our approach to combine unitary cuts with the Hopf algebraic structure of imaginary parts of Feynman diagrams.
7
Functional equations of MPLs
We previously defined the mathematical structure and explained the coproduct on multiple poly-logarithms, it is conjectured by [3] that if
∆0(Fw) = ∆0(Gw), (32) then Fw= Gw+ X i ciPw,i (33)
where Pw,i are primitive elements of weight w and ci are rational coefficients. The primitive
ele-ments are: (iπ)w and ζ
w [4]. We can use eq. 32 to rewrite a difficult function Fw into a simpler
function Gw. Eq. 32 implies that if all possible mappings ∆i1, .., ik of Fn are equal to the
corre-sponding mapping of Gn then equation 33 holds.
The mapping of maximum iteration: ∆1,...,1[Fn] has the special property,
∆1,...,1[Fn] mod iπ = S(F ) (34)
where S is the so called ’Symbol Map’ which is often used to calculate equations in high energy physics.
Eq. 32 and 33 imply that we can use the coproduct to decompose a polylogarithm of a certain weight into a tensor product of lower weights. This allows us to use known identities at lower weight to ultimately prove identities at higher weight. As an example we look at the inversion relationship for Li3(x1) (x being a positive variable).
We can map Li3(z) in three ways; ∆1,1,1, ∆2,1and ∆1,2(so we are excluding the primitive elements).
To obtains the maps ∆2,1,∆1,2 ∈ A ⊗ A we only have to apply the coproduct once. In order to
obtain the symbol map(∆1,1,1 ∈ A ⊗ A ⊗ A) of Li3(z) we have to apply the coproduct twice i.e.
(∆ ⊗ id)∆(Li3). Using eq. 27 for ∆[Li3(z)],
[1 ⊗ Li3+ Li3⊗ 1 + Li2(z) ⊗ log(z) + Li1(z) ⊗
1 2log
2
(z)] ∈ A ⊗ A (35)
the underlined terms are appearing in the map:∆2,1= Li2(z)⊗log(z) and ∆1,2= Li1(z)⊗12log2(z).
Mapping this product to A ⊗ A ⊗ A by applying (∆ ⊗ id) on eq. 35,
(∆ ⊗ id)[1 ⊗ Li3+ Li3⊗ 1 + Li2(z) ⊗ log(z) + Li1(z) ⊗
log2(z)
2 ] =
∆(1) ⊗ Li3(z) + ∆Li3(z) ⊗ 1 + ∆Li2(z) ⊗ log(z) + ∆Li1(z) ⊗
log2(z)
2 =
∗
Li2(z) ⊗ 1 ⊗ log(z) + Li1(z) ⊗ log(z) ⊗ log(z) + 1 ⊗ Li2(z) ⊗ log(z)
(36)
the underlined term in line 2 is the only term that contributes to the ∆1,1,1 map and thus all
other terms in line 2 are redundant. The underlined term in the last line appears in the map: ∆1,1,1= Li1(z) ⊗ log(z) ⊗ log(z) . We have now reached full decomposition of a weight 3 element
and we are left to manipulate the decomposed polylogs in order to find functional relationships. Starting with the ∆1,1,1map to manipulate products of (only!)the form F1⊗ F1⊗ F1where Fnis
∆1,1,1Li3( 1 x2) = Li1( 1 x) ⊗ log( 1 x) ⊗ log( 1 x)
= − log(1 − x) ⊗ log(x) ⊗ log(x) + log(x) ⊗ log(x) ⊗ log(x) − iπ ⊗ log(x) ⊗ log(x) = ∆1,1,1(Li3(x) + 1 6log 3(x) −iπ 2 log 2(x)) (37)
We must look at the image of ∆1,2 as well to include terms which will be decomposed to tensor
products of type F1⊗ F2. In order to find the missing terms in our inversion relationship we
subtract the already found terms of 37,
∆1,2[Li3( 1 x) − Li3(x) − 1 6log 3(x) +iπ 2 log 2(x)] =1 2Li1( 1 x) ⊗ log 2 (1 x) + 1 2log(1 − x) ⊗ log 2 (x) −1 2log(x) ⊗ log 2 (x) +iπ 2 ⊗ log 2 (x) =1
2[− log(1 − x) + log(x) − iπ] ⊗ log
2
(x) +1
2[log(1 − x) − log(x) + iπ] ⊗ log
2
(x) = 0
(38)
the last term of eq. 35 already gave us the image of Li3(1x) and Li3(x), looking at the integral
defi-nition of functions of the form logn!n(x) : I(0;−0→n; z) using the coproduct (think about the polygon) on
it always creates functions of the formPn
m=0I(0; −−−→ 0n−m; z)⊗I(0; −→ 0m; z) =Pnm=0log n−m(x) (n−m)! ⊗ logm(x) m! .
Hence ∆1,2 of 16log3(x) = 12log(x) ⊗ log2(x). The last term of eq. 38 can be calculated using this
similar technique and making use of the compatibility of the product and coproduct.
The fact that eq. 38 equals zero means that the image of ∆1,2does not provide us with additional
information. Looking at the last image:∆2,1to include elements which are decomposed into tensor
products of the form F2⊗ F1,
∆2,1[Li3( 1 x) − Li3(x) − 1 6log 3 (x) +iπ 2 log 2 (x)] = Li2( 1 x) ⊗ log(x) + [−Li2(x) − 1 2log 2
(x) + iπ log(x)] ⊗ log(x) = −[−Li2(x) −
1 2log
2
(x) + iπ log(x) + 2ζ2] ⊗ log(x) + [−Li2(x) −
1 2log
2
(x) + iπ log(x)] ⊗ log(x) = −2ζ2⊗ log(x)
= ∆2,1[−2ζ2log(x)]
(39) in line two there is made use of the inversion relationship of Li2(1x) = −Li2(x) − 12log2(x) +
iπ log(x) + 2ζ2. The fact that the previous image is non zero gives us the final missing term:
−2ζ2log(x) which is indeed 0 if you act on it with ∆2,1. Thus by making use of the Hopf algebra
we can conclude that,
Li3( 1 x) = Li3(x) + 1 6log 3(x) − iπlog 2 (x) 2 − 2ζ2log(x) + c1ζ3+ c2iπ 3 (40)
where ζ3and iπ3are the primitive elements at weight 3.
8
Optical theorem
In this second part of this thesis we will look at similarities between cutting Feynman diagrams (by unitarity) and using the asymmetrical behaviour of the disc operator in the Hopf algebra. It appears that from basic statement in quantum mechanics one can obtain a powerful method to simplify difficult Feynman diagrams. We know from quantum mechanics that the transformation from a initial (normalized) state |ii into a final state |f i can be mathematically expressed by a linear transformation i.e. by a (scattering) matrix S. The basic statement ’Conservation of probability’ which corresponds to hf |f i = hi|ii = 1, in which the inner product can be rewritten into,
hf |f i = hi|S†S|ii = 1 (41)
which means that to obey conservation of probability S†S = 1 and therefore S has to be unitary.
In the literature the scattering matrix S is often written as S = I + iT , I represents the case in which nothing happens to the initial state, T represents the transition matrix which changes the initial state in some final state. The unitarity of S causes T to be ’constraint’ as well,
I = S†S = (I + iT )†(I + iT ) (42)
I = (I − iT†)(I + iT ) (43)
i(T − T†) = −T†T (44)
the last line is the constraint on T which is known as the ’Optical theorem’.
In QED M is needed to calculate cross sections of scattering processes it is therefore necessary to extent the optical theorem even further to constraints on Mi→f = hf |T |ii. Looking at the case
of ’forward scattering’, manipulating both sides of eq. 42,
i hi|(T − T†)|ii = − hi|T†T |ii i(Mi→i− M∗i→i) = −
X
k
hi|T†|ki hk|T |ii 2Im[Mi→i] =
X
k
|Mi→k|2
(45)
inserting a complete set of eigenstates P
k|ki hk| = 1 between the operator T† and T . We can
calculate the imaginary part of a forward-scattering Feynman diagram by ’cutting’ the diagram and calculating all possible final state tree diagrams. The physical interpretation of the right side of eq. 45 can be found by looking at the cutting equation mentioned in the following section.
8.1
Cutting equation
To find a relationship between the optical theorem and the physical interpretation of these ’cuts’ we need to introduce the largest time equation [7].
Looking at Feynman rules (appendix 13.6) for the scalar particle case, with propagator:
∆F(z) = 1 (2π)4i Z d4k e ikz k2+ M2− i (46)
with k being the momentum of the propagator, z = x−y with x, y being two space-time coordinates connected by the propagator and M is the mass of the propagator. This propagator formula can be written into two parts which are the complex conjugate of each other,
∆F(z) = θ(z0)∆+(z) + θ(−z0)∆−(z), (47) with ∆±(z) = 1 (2π)3 Z d4keikzθ(±k0)δ(k2+ m2) (48)
Now consider a propagator with two vertices at space-time coordinates x1and y1 if x1> y1then
z0 is positive and the step function reduces eq.48 to ∆+(z). The propagator reduces to ∆−(z) if
x1< y1. This technique can be represented in diagrams as well, this leads to some ’new’ Feynman
rules which can be found in appendix 13.5.
Considering the easiest example of this technique first, namely the diagram containing only one propagator, using the new Feynman rules one obtains:
. Looking at the diagram a simple but crucial rule emerges; starting a diagram with n vertices circled then by removing or adding a circle around the point of largest time (in this case the point y) will result in minus the original diagram. This rule is called the largest time equation. We can extend this rule to more difficult diagrams such as the self-energy diagram which will be important for this article. Using our recently found largest time equation on a self energy diagram one can conclude the following,
using the same orientation of the space-time coordinates (right vertex has the largest time) by the Feynman rules diagram (1) cancels (2) and diagrams (3) cancels (4).
The largest time equation applied to a diagram with n vertices can be represented mathematically by,
X
circlings
F (x1, x2, ..., xn) = 0 (49)
where F (x1, x2, ..., xn) with spacetime coordinates x1, .., xnrepresents a diagram with n uncircled
vertices, F (x1, x2, .., xn) represents a diagram with only x2 circled etc, we sum over all possible
circling.
The ’physical information’ of the largest time equation is in the θ(±k0) functions inside ∆(z). The
θ(±k0) function causes the positive energy to flow (positive energy flowing forward in time) from
a uncircled vertex to a circled vertex, there are no restriction for positive energy flow between two circled or two uncircled vertices. We can conclude quickly that diagrams containing a circled vertex which is only connected to uncircled vertices and is not connected to external lines are zero. In this case all postive energy flows to the circled vertex and hence energy conservation is violated see figure 8.1(1).
Diagram (2) is zero by violation of energy as well because all positive energy flows out of the un-circled vertex. In practice diagrams are drawn with shadowed and unshadowed lines, respectively representing connected sets of circled points and connected sets of uncircled points. To illustrate this new representation see figure 8.1 containing two identical diagrams.
in diagram (1) is the energy flow drawn with arrows indicating the direction. As become clear from the arrows in diagram (1) energy can only escape a ’shadowed’ region via external lines. In
order to satisfy conservation of energy a shadowed area must have at least one external line, as is the case for a unshadowed region. With our new representation eq. 49 turns into
X
Cuts
F = 0. (50)
This equation can become useful when one wants to calculate the imaginary part of a diagram quickly.By extracting the diagram which is fully shadowed and the diagram which is fully unshad-owed, noticing that these two diagrams are there complex conjugate, one obtains the more useful equation
−X
Cuts0
F = F + F = 2Re[F ] (51)
A careful reader may have noticed that F + F = 2 Re[F ] and not 2 Im[F ] as written in eq. 45. We don’t have a problem because it is conventional to extract a factor i out of the Feynman integral expression which means that eq. 51 turns into.
−X
Cuts0
F = F + F = 2i iIm[ ˜F ] (52)
Note that F = i ˜F i.e. ˜F now means the functions inside the Feynman integral of which a factor i has been extracted.
Recall eq. 45 where there is summed over all possible intermediate states. This can be interpreted as the summation over all possible cuttings of a diagram.
Using unitary cuts we can calculate ’imaginary’ parts of Feynman diagrams. To calculate the real part of the corresponding Feynman diagram one usually use dispersion relationships. These dispersion relationships will not be part of this paper.
Somehow we want to use the discontinuity operator inside the polylogarithmic algebra in our advantage, using the Schwarz reflection we can rewrite the imaginary part of F into a discontinuity of F . This will be the topic of the upcoming section.
9
Relationship between the coproduct and discontinuities
inside the Hopf algebra
We want to take advantage of the properties of the Hopf Algebra in order to quickly calculate Feynman integrals. One way of simplifying this computation was to first calculate the imaginary part of the diagram using unitary cuts(eq. 51) followed by the use of a disperion relationships to calculate the real part. The calculation of imaginary parts through unitary cuts appears to be complicated as well.
To take advantage of our knowledge of the polylogalgebra we need to couple the imaginary part of Feynman diagrams to discontinuities of the functions inside those diagrams. We can achieve that by using the Schwarz Reflection[6]:F (z) = F (z), the discontinuity: Disc[F ] = F (x+i)−F (x−i) = F (x + i) − F (x + i) = 2iIm[F ]. This turn eq. 52 into,
−X
Cuts0
F = i Disc[ ˜F ] (53)
this means that if ˜F has no brachcuts Disc[ ˜F ]=0. According to [1] the discontinuity of a Feynman diagram can be calculated with,
Discs[ ˜Fn] = µ[(Discs⊗ id)∆1,n−1F˜n] (54)
where Discs is the discontinuity of a function along a certain momentum channel s. With eq. 54
we have introduced the polylogarithmic algebra and we can use the coproduct in our advantage. Combining the cuts and the coproduct into an equation which will be tested on the self-energy case
−X
Cuts0
10
Calculation of self-energy scalar diagram
A scalar self-energy diagram can be represented in the following Feynman diagram,
Figure 9: Represented in the figure is a self-energy diagram for scalar particles. With incoming 4-momentum Q and internal 4-momenta p+Q and p.
applying the Feynman rules to this diagram (appendix 13.6) gives us the equation,
F = Z
d4q
1
((q + Q)2+ M2− i)(q2+ M2− i) (56)
which goes at large q like: R d4q1
q4 =R q3dqdΩ
1
q4 and hence is logarithmically divergent. I would like to illuminate the (old) convention used to calculate this integral. The usage of the four-momentum pµ = (−→p , iE) where E is the fourth component. To label the fourth component of a
4-vector we use p4 and to label only E (without the imaginary i) we use p0, so p4 = ip0. Using
the Feynman trick which states that, 1 AB = Z 1 0 1 (Ax + B(1 − x))2 (57)
we can rewrite eq. 56 into,
F = Z 1 0 dx Z d4q 1 [((q + Q)2+ M2− i)x + (q2+ M2− i)(1 − x)]2 F = Z 1 0 dx Z d4q 1 [(q + Q)2x + q2(1 − x) + M2− i]2 = Z 1 0 dx Z d4q 1 [q2+ 2xqQ + xQ2+ M2− i]2 (58)
using the substitution k = q + xQ which changes the variable of integration to d4k which leaves
us with, F = Z 1 0 dx Z d4k 1 [k2+ x(1 − x)Q2− i]2 (59)
the reason for the change of variable is to eliminate the linear variable q inside the denominator of eq. 58, leaving only k2 in the denominator of eq. 58 which is easier to integrate.
10.1
Wick Rotation
Our integrationR d4k is defined as
R∞
−∞dk0R d3k which means we integrate over the energy and
momentum-space. In order to calculate the integral we must first change our integration from Minkowski space to Euclidean space (for reasons which will become clear in a moment). We can accomplish this by using the ’Wick rotation’. The Wick rotation makes use of contour integration, therefore we must investigate the denominator of eq. 62 and find the poles. Expanding the denominator using k2=−→k2− k2 0 into, − → k2− k2 0+ A − i (60)
which has poles (assuming−→k2+ A is positive) at k 0= ±
q−→
k2+ A − iexpand= ±[
q−→
k2+ A − i] ,
with A = x(1 − x)Q2+ M2. The locations of these two singularities are represented with red dots
in the following figure.
Figure 10: Represented in the figure is the contour and singularities (red dots) in the complex k0 plain. The contour integration is used to change the integration along the real axis to the
imaginary axis(wick rotation).
We now perform the closed contour integral represented in figure 10 for R goes to infinity. It is clear that there are no singularities inside the contour so our closed integral equals zero. On the two curved paths the absolute value of F ∝ 1
k4 0
for large R. Using the ML-inequality on both curved paths which are a quarter of a circle we know that |RsF (k0)dk| < 4RπR4, which obviously goes to zero as R goes to infinity. Concluding that both circle segments are zero we can immediately see that we are left only with integration along both axis i.eR∞
−∞F dk0 = −
R−i∞
i∞ F dk0 and we have
now ’Wick-Rotated’ the integral from the real axis to the imaginary axis.
The Wick Rotation enables us to integrate along the real axis in eucledian space by using the substitution k4= ik0. If done so the following equality holds −R
−i∞
i∞ F dk0= i
R∞
−∞F dk4 which
also changes the K2 in the denominator of eq. 62 from k2 =−→k2− k2
0 into the Euclidean k 2 =
− →
k2+ k24. We are now left with the integral in Euclidean space
F = i Z 1 0 dx Z d4k 1 [k2+ x(1 − x)Q2− i]2. (61)
We can evaluate the momentum integral using spherical coordinats in 4-dimensions; Z d4k =y sin2(ψ) sin(θ)dψdθdφ Z ∞ 0 dp p3
immediately integration over all angular variables and using the more elegant upper limit Λ = ∞
F = i Z 1 0 dx Z Λ=∞ 0 2π2p3dp [p2+ x(1 − x)Q2− i]2. (62)
To finish the computation of the integral using the substitution:z = p2to reduce the degree of the variable inside the denominator.
F = π2i Z 1 0 dx Z Λ2 0 zdz [z + A − i]2 = π2i Z 1 0 dx Z Λ2 0 ( 1 [z + A − i]− A − i [z + A − i]2)dz = π2i Z 1 0 dx[log(z + A − i) Λ 0 + A − i (z + A − i) Λ2 0 ] = π2i Z 1 0
dx[log(Λ2+ A − i) − log(A − i) − A − i
(Λ2+ A − i) − 1]
neglecting all functions behaving like Λ12 and neglecting A − i compared to Λ
2
π2i[log(Λ2) − 1 − Z 1
0
10.2
Evaluating the Feynman parameter integral
Starting with the computed expression of a scalar self-interaction diagram
F = iπ2[log(Λ2) − 1 − Z 1
0
log(x(1 − x)Q2+ M2− i)dx]. (64)
By manipulating the argument of the logarithm making the integral part easier to compute . Rewriting the quadratic equation inside the logarithm x(1 − x)Q2+ M2 into,
log[−Q2(x −1 2 − 1 2 s 1 +4M 2 Q2 )(x − 1 2 + 1 2 s 1 +4M 2 Q2 )] (65)
which enables us to split the logarithm in three parts. What follows is a brute force calculation of the integral of the form,
I = Z 1 0 [log(x −1 2− 1 2C) + log(x − 1 2 + 1 2C) + log(−Q 2)]dx with C = s 1 + 4M 2 Q2 (66)
for the convenience of the reader the first logarithm will be worked out in most detail.
Z 1 0 log(x −1 2− 1 2C)dx = [x −1 2 − 1 2C] log(x − 1 2− 1 2C) − x + 1 2+ 1 2b 1 0 = [1 2 − 1 2C] log( 1 2 − 1 2C) + [ 1 2+ 1 2C] log(− 1 2 − 1 2C) − 1 (67)
the second integral computed exactly alike,
Z 1 0 log(x −1 2 + 1 2C)dx = [1 2 + 1 2C] log( 1 2 + 1 2C) + [ 1 2 − 1 2b] log(− 1 2 + 1 2C) − 1 (68)
the third and final integral equals log(−Q2). The three integrals consist of terms which are of the
form: C log and log. To compute the sum of the integral lets start by summing the terms of the form C log coming from integral one and two,
−1 2C log( 1 2− 1 2C) + 1 2C log(− 1 2 − 1 2C) + 1 2C log( 1 2+ 1 2C) − 1 2C log(− 1 2+ 1 2C) = 1 2C[log(−( 1 2+ 1 2C) 2) − log(−(−1 2 + 1 2C) 2)] = 1 2C log( (C + 1)2 (C − 1)2) = C log( C + 1 C − 1) (69)
1 2log( 1 2− 1 2C) + 1 2log(− 1 2 − 1 2C) + 1 2log( 1 2 + 1 2C) + 1 2log(− 1 2 − 1 2C) = 1 2[log(( 1 2 + 1 2C) 2(−1 2 + 1 2C) 2)] = log(1 4[C 2− 1]) (70)
Now the total integral becomes: I = C log(C + 1 C − 1) log( 1 4[C 2− 1]) + log(−Q2) − 2 = C log(C + 1 C − 1) + log( M2 Q2) + log(−Q 2) − 2 = s 1 + 4M 2 Q2 log( q 1 + 4MQ22 + 1 q 1 + 4MQ22 − 1 ) + log(M2) − iπ − 2 (71)
The final result of Feynman integral is
F = iπ2[log(Λ 2 M2) + iπ + s 1 +4M 2 Q2 log( q 1 +4MQ22 − 1 q 1 +4MQ22 + 1 ) + 1].
10.3
Calculation of the discontinuity
Applying eq. 54 to the self-energy diagram, we have extracted (by convention) a factor of π2 out of ˜F which we will add after the evaluating of the coproduct. Calculating the discontinuity of the π2 extracted ˜F with eq. 54,
µ[Disc ⊗ id]∆1,0[C log(
C − 1 C + 1)] = µ[Disc ⊗ id][C log(C − 1
C + 1) ⊗ 1] = µ[Disc[C log(C − 1 C + 1)] ⊗ 1] = 2iπC = 2iπ s 1 + 4M 2 Q2 (72)
adding the extracted π2 and multiplying by i to compute the right hand sight of eq. 55 we get,
iπ2µ[(Discs⊗ id)∆1,n−1F˜n] = −2π3
s
1 +4M
2
10.4
Evaluation cut diagram
The evaluation of the cut diagram will be handled shortly.
Figure 11: Represented in the figure is a self-energy diagram which has been cut (red line). The diagram has incoming 4 momentum Q and internal 4 momenta p+Q and p.
Using the Veltman rules (appendix 13.5) one obtains the integral,
i(2π)4i(2π)4 1 (2π3) 1 (2π3) Z d4p δ(p2+ M2)θ(p0)δ((p + Q)2+ M2)θ(p0+ Q0) (74)
we can rewrite this into, −4π2Z d
4p
Z
d4kδ4(Q + p − k)
| {z }
trivial use of deltaf unction
θ(p0)δ(p2+ M2)θ(k0)δ(k2+ M2) (75)
with the introduction of the trivial use of a δ4distribution. With this introduction we can rewrite
this integral into a two-body decay integral by performing the integrating over k0 and q0,
−4π2 Z d3p Z d3kδ4(Q + p − k) Z dp0θ(p0)δ(p2+ m2) | {z } Appendix 13.7 Z dk0θ(k0)δ(k2+ m2) | {z } Appendix 13.7 (76)
using appendix 9.7 to evaluate the two underlined integrals,
−π2 Z d3p Z d3k δ4(Q + p − k) p0k0 (77) which is indeed a two-body decay integral. This familiar integral can be computed and found in the literature [7] −π2Z d 3p Z d3k δ4(Q + p − k) p0k0 = −2π3 s 1 +4M 2 Q2 . (78)
Comparing this result with eq.(73) we see that two results are the same and hence the relationship between unitary cuts and the discontinuity operator inside the coproduct (eq. 55) is verified.
11
Conclusion
There are many techniques in the literature to tackle multi-loop Feynman integrals. One of these techniques is the unitary cut which enables us to calculate imaginary parts of Feynman diagrams. Another technique, reviewed in this thesis, is the use of the Hopf algebraic structure of imaginary parts of Feynman diagrams. Starting with acknowledging the need of a generalization of polylog-arithms to describe multi-loop Feynman expressions leads to the creation of MPLs. The MPLs have a multiplication among them which is called the shuffle product. From the shuffle product and the vector properties of the MPLs we can create a Hopf algebra.
One of the applications of the Hopf algebraic structure is the use of the coproduct to find func-tional equations between (multiple)polylogarithms, the coproduct allows us to decompose a MPL into lower weight tensor products of MPLs and use known relationships among these lower MPLs to ultimately find relationships between the higher weight MPLs.
Even a more important feature of the Hopf algebra is its relationship with unitary cuts. As is shown in the thesis unitary cuts calculate the imaginary parts of Feynman diagrams and are re-lated to the discontinuities of these diagrams by the Schwarz reflection. The coproduct creates a simpler tensor product of functions, in which the discontinuity operator behaves asymmetrically. This asymmetry allows us to quickly calculate discontinuities of complicated Feynman expressions. We have taken the self-energy case of scalar particles to show the relationship between the two techniques. Starting with the explicit calculation of the self-energy Feynman diagram leads to an expression containing a logarithm. Followed by applying the discontinuity operator inside the coproduct to calculate the ’discontinuity’ of the self-energy diagram. After that the imaginary part of the self-energy diagram is calculated by unitary cuts and the relationship between unitary cuts and the Hopf algebraic structure of the imaginary parts of Feynman expressions is verified. The relationship between these two techniques can be of vital importance in the computation of more difficult multi-loop integrals. One should realize that for more difficult diagrams sequential cuts have to be made to these diagrams.
12
Acknowledgments
I have enjoyed working on this thesis with Prof. Dr. Laenen. I would like to thank him for taking time from his busy schedule to meet and help me with this thesis, you have been a source of great inspiration. I would like to thank Dr. Rojo as well for being present at my presentation and for reviewing my thesis.
13
Appendix
13.1
I conversion of generic base point to a zero base point
Weight 1:
I(a0; a1; a2) = I(0; a1; a2) − I(0; a1; a0) (79)
Weight 2: I(a0; a1, a2, a3) = Z a3 a0 dt t − a2 I(a0; a1; t) = Z a3 a0 dt t − a2 [I(0; a1; t) − I(0; a1; a0)]
= I(0; a1, a2, a3) − I(0; a1, a2, a0) − I(0; a1; a0)[I(0; a2; a3) − I(0; a2; a0)]
(80)
13.2
Behaviour of the bilinear tensor product
a, b, c ∈ A ⊗ A and c ∈ Q follow the following bilinear axioms a ⊗ (b + c) = a ⊗ b + a ⊗ c (a + b) ⊗ c = a ⊗ c + b ⊗ c (ca) ⊗ b = c(a ⊗ b) = a ⊗ (cb)
(81)
13.3
Basic properties of the coproduct
In this section we clarify the compatibility condition, a primitive element and the ∆i1,..,ik maps. Consider the set of three elements of weight one [a, b, c] ∈ A. Where A is graded bialgebra, the num-ber of letters in a word defines the weight. There exists a linear (coproduct) map ∆ : A → A ⊗ A. On elements x ∈ A : ∆(x) = 1 ⊗ x + x ⊗ 1
∆(1) = 1 ⊗ 1 (82)
Using compatibility condition (eq. 24):
∆(ab) = ∆(a)∆(b) = (1 ⊗ a + a ⊗ 1)(1 ⊗ a + a ⊗ 1)
= 1 ⊗ ab + b ⊗ a + a ⊗ b + ab ⊗ 1 (83)
∆(abc) = ∆(ab)∆(c) = (1 ⊗ ab + b ⊗ a + a ⊗ b + ab ⊗ 1)(c ⊗ 1 + 1 ⊗ c)
(c ⊗ ab + bc ⊗ a + ac ⊗ b + abc ⊗ 1 + 1 ⊗ abc + b ⊗ ac + a ⊗ bc + ab ⊗ c) (84) ∆i1, ..., ik picks out the components of weight (i1, ..., ik) from the tensor product.
∆2,1(abc) = (bc ⊗ +ac ⊗ b + ab ⊗ c)
∆1,2(abc) = (c ⊗ ab + b ⊗ ac + a ⊗ bc)
(85) The reduced coproduct is defined as
13.4
Discontinuity of Li
nWe know that the discontinuity of log(x) caused by its branch cut equals 2πiθ(x). This can be easily found using residue integration. To find the discontinuity of Lin we can use a strategy which
will be shown on Li2 but can be applied to any classical polylogarithm. For x > 1 the following
closed integral holds,
Li2(x + i) − Li2(x − i) = − Z x+i 0 dt t log(1 − t) + Z x−i 0 dt t log(1 − t) (87)
with a branch cut along the positive x-axis from 1 to infinity. This integral can be reduced to a contour around the branch cut,
Li2(x + i) − Li2(x − i) = − Z x+i 1+i dt t log(1 − t) + Z x−i 1−i dt t log(1 − t) + Z 1+i 1−i dt t log(1 − t) (88) where the last integral is a semicircle around the point (1,0). This integral vanishes if lim→0.
Leaving us with the two integrals left which can be rewritten using change of variables v = t ± i,
− Z x+i 1+i dt t log(1 − t) + Z x−i 1−i dt t log(1 − t) = − Z x 1 dv v + ilog(1 − v + i) + Z x 1 dv v − ilog(1 − v − i) (89) with lim→0 we can write v±idv ≈dvv leaving us with one integral,
Z x
1
1
v(log(1 − v + i) − log(1 − v − i))dv (90)
the logarithms inside the integral gives us the known discontinuity 2πiθ(−x + 1) leading us to the discontinuity of Li2,
Z x
1
dv
v 2πi = 2πi log(x)θ(−x + 1). (91)
This method can be repeated for polylogarthims of higher weight leading to Disc(Lin) = 2πilog
n−1z
(n−1)!
with a branch cut along the real axis [1, ∞].
13.5
Veltman rules
Anticipating complex conjugation we will give a circled vertex a minus sign. A line connecting two uncircled vertices corresponds to a propagator ∆F(z)
A line connecting a uncircled to a circled corresponds to a propagator ∆+(z) A line connecting a circled to a uncircled corresponds to a propagator ∆−(z) A line connecting two uncircled corresponds to a propagator ∆∗F(z)
∆F(y − x) + ∆+(y − x)(−1)
(−1)∆−(y − x) + (−1)∆∗(y − x)(−1)
13.6
Feynman rules for scalar particles
External lines: √ 1 2V k0 Propagator: (2π)14i 1 k2+m2−i Shadowed region: (2π)−14i 1 k2+m2−i Cut propagator: (2π)1 3θ(k0)δ(k2+ M2)
13.7
Integrals
Here theR dk0θ(k0)δ(k2+ m2) integral is evaluated (used twice in paper). Making use of
Z
dx g(x)δ(f (x) − y) = X
all xiwhich satisf y: f (xi)=y g(xi)
|f0(x i)|
. (92)
Starting the computation at,
Z dk0θ(k0)δ(k2+ m2) = Z dk0θ(k0)δ( − → k2− k20+ m 2) = Z dk0θ(k0)δ( − → k2− k2 0+ m2) = 1 2k0 = 1 2 q−→ k2+ m2 (93)
References
[1] S. Abreu. From multiple unitarity cuts to the coproduct of feynman integrals. 2014. [2] F. Brown. On the decomposition of motivic multiple zeta values. 2011.
[3] C. Duhr. Hopf algebras, coproducts and symbols an application to higgs boson amplitudes. 2012.
[4] C. Duhr, H. Gangl, and J. Rhodes. From polygons and symbols to polylogarithmic functions. 2012.
[5] A. Goncharov. Multiple polylogarithms, cyclotomy and modular complexes. 1998. [6] M. Peskin and D. Schroeder. An introduction to Quantum Field Theory. 1995. [7] M. Veltman. Diagrammatica - The Path to Feynman Diagrams. 1994.
