pqcd at High Temperature

Transcription

1 pqcd at High Temperature Diplomarbeit an der Fakultät für Physik Universität Bielefeld vorgelegt von Ervin Bejdakic 3. März 006

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3 Contents Introduction 5 QCD at high temperature 9. Finite Temperature (scalar) Field Theory QCD Partition Function Nested Sums 5 3. Introduction Harmonic sums, Euler-Zagier sums Harmonic Polylogarithms Methods of Computation 7 4. Difference Equations Differential Equations x-space Evaluation Mellin-Barnes Method Applications and Loop Loop Loop Discussion and Outlook 53 A Example Codes 55 B Additional Results 6 C Acknowledgments 63 3

4 4 CONTENTS

5 Chapter Introduction There are four forces in nature: gravity, electromagnetism and the strong and weak nuclear forces. The strong interaction has up to now been successfully described by Quantum Chromodynamics (QCD), a non-abelian gauge theory. Its theoretical foundations was established in the early 970 s [],[],[3]. QCD allows one to calculate the interactions of quarks and gluons, which carry color charge. Because of the strongly interacting nature, the quarks and gluons are confined into hadrons, so that no free quarks can be seen. The non-abelian structure of QCD leads to non-linear equations of motion for gluons, which unlike photons in QED, interact with each other. Yet, exactly this property of gluons leads to an astonishing feature of QCD, asymptotic freedom [],[3], which leads to a deconfinement at small distances, or high energies, of hadrons into a weakly interacting quark-gluon plasma. Numerical computations suggest that at a critical temperature T c = 70 MeV this phase transition takes place [5]. In the energy region above T c, one can, since the running coupling g is small, apply the perturbation theory and compute amplitudes through Feynman diagrams (diagram and graph will be used as synonyms). Some of these graphs lead to infinities, and need to be regularized and/or renormalized, something which can be done by so-called dimensional reduction [6]. One computes the integrals in d = 4 ɛ (or in d = 3 ɛ in three dimensional theory) dimensions and expands in ɛ. The infinities now become poles in ɛ, which cancel each other when one sums all diagrams that contribute to a physical quantity. This thesis is about the analytic calculation of typical scalar integrals that appear in Feynman diagrams. We will take a closer look at several teqniques that are used in computation of Feynman graphs and apply these methods 5

6 6 CHAPTER. INTRODUCTION on the set of the so-called master integrals (MI s) of the SU(N c ) + Adjoint Higgs Theory, which is an effective theory for QCD at high temperatures. The thesis is organized as follows. In the second chapter we introduce perturbative quantum field theory (pqft) and perturbative QCD (pqcd) at high temperature and derive the generic Feynman rules. There we argue that even at temperatures above T c one cannot apply the simple perturbative expansion in the gauge coupling g since at high temperature the free energy or pressure gets contributions from different momentum scales T, gt and g T, so that in reality we have a multi-scale system. The perturbative expansion breaks also at g 6 order for other reasons [7]. What one has to do is to apply the approach of constructing effective three-dimensional theories [8],[9], that can be computed by analytical and numerical methods. The resulting electrostatic QCD (EQCD) can be used to compute the pressure or free energy up to g 5 order [], [0], at g 6 order one has to take into account the contribution of g T scale, which enters the stage. The contribution of g T scale can be described by the magnetostatic QCD (MQCD), which is entirely non-perturbative. Still one has to compute the EQCD contribution up to four loops for the g 6 order. The set of all graphs of EQCD up to four loops can be reduced by general integration-by-parts (IBP) method to a set of master graphs which were given in [39]. The ɛ-expansion of these Feynman graphs is expressible in terms of so called S/Z-sums. In chapter 3 we introduce the concept of nested sums and their algebra, which will be used as a natural language for the results of the expansion of Feynman graphs. Two special cases of nested sums, the harmonic sums and harmonic polylogarithms or polylogs for short, are treated in more detail. The following chapter deals with four methods of computing Feynman diagrams, namely via difference equations, differential equations, the x-space method and the Mellin-Barnes method. Some simple examples will be given to illustrate the methods. There will appear diagrams which can be computed with several methods, like, where the solid lines stand for scalar massive propagators and dotted lines for massless propagators. Unfortunately, there will be diagrams which cannot (yet) be solved with any method presented here. These are and. In the fifth chapter we apply these methods to the MI s of SU(N c ) + Adjoint Higgs Theory. Since some of the master integrals of SU(N c ) + Adjoint Higgs Theory also appear in computations in various models we try to solve all graphs in d dimensions if possible, if not then in 3 dimensions for EQCD or in 4 dimensions for Standard Model (SM). At one loop level there is only

7 7 one graph, and there are no two loop integrals. At three loop level there are two master integrals, and. The first is computed solving the corresponding difference equation, the second by using a relation to a master integral which can be computed using differential equation method from [37]. In [37] also three four loop integrals, and could have been computed using the same method. The master integrals, and can be solved using difference equations, but can only be expanded in d = 4 ɛ. The two master integrals and couldn t have been solved, because essentially the result of BB4(x) for all x is necessary and we were not been able to compute it. In the last chapter we discuss the methods used and take an outlook on possible future developments. In the appendix we collect some sample programs.

8 8 CHAPTER. INTRODUCTION

9 Chapter QCD at high temperature. Finite Temperature (scalar) Field Theory Let us first start with the statistical Quantum Mechanics, and generalize it then to Field Theory (we follow in this chapter basically [8]). The most important entity in statistical physics is the (grand-canonical) partition function Z(T ) which is defined as: [exp( βĥ) ] Z(T ) = T r (.) where β = T, T being the temperature of the system and Ĥ is the Hamiltonian. From Z(T ) one can compute all thermodynamic quantities through fundamental thermodynamic relations, e.g. free energy F = T ln(z(t )) or entropy S = F T + E T. It will be useful to write the partition function as a functional integral over the eigenstates of the system: Inserting p e ɛ Ĥ Z(T ) = + dx x e βĥ x (.) ipx dp π h p p = and using the relations p x = e h h x = e ipx ɛ h H+O(ɛ) we get: Z(T ) = lim N [ N i= ] dp i dx i e π h { h and ( )} N j= ɛ p j m +ip x j+ x j j +V (x ɛ j ) (.3) where x N+ x and ɛ = β h N. Performing the limes and integrating over p i, since it is Gaussian integral, we can write formally the path integral: Z(T ) = x(0)=x(β h) Dxe h 9 β h 0 dτl E ( dx dτ, x(τ)) (.4)

10 0 CHAPTER. QCD AT HIGH TEMPERATURE where L E = H pẋ is the Euclidean Lagrangian which is connected with the one in Minkowski space by τ = it. Now let us interpret the argument x for a given τ in the above path integral as an internal degree of freedom of a general function, called φ(τ, x) = φ(x). We get then the path integral representation of a partition function for a scalar field theory: Z SF T (T ) = φ(0, x)=φ(β h, x) Dφ(x)e h β h 0 dτ d 3 xl E (.5) For the fermions analog way leads to fermionic partition function for a scalar field theory: Z ferm. (T ) = = ψ(0, x) = ψ(β h, x) ψ(0, x) = ψ(β h, x) ψ(0, x) = ψ(β h, x) ψ(0, x) = ψ(β h, x) D ψ(x)dψ(x)e h D ψ(x)dψ(x)e h β h dτ d 3 x ψ[γ E 0 µ µ+m]ψ β h 0 dτ d 3 xl E (.6) with Euclidean γ-functions with properties {γ E µ, γ E µ } = δ µν, and (γ E µ ) = γ E µ, γ E 0 = γ0, γ E i = iγ i.. QCD Partition Function The Euclidean Lagrangian is given for QCD by: L QCD = 4 F a µνf a µν + ψγ µ D µ ψ (.7) where F a µν µ A a ν ν A a µ+gf abc A b µa c ν is the field strength tensor and D µ µ iga µ µ iga a µt a the covariant derivative, the T a, a =,..., N are generators of the fundamental representation of SU(N) and f abc are the structure coefficients of SU(N) given by [T a, T b ] = if abc T c. The QCD Lagrangian is invariant under: A µ A a µt a G A µ G + i g ( µg )G ψ G ψ where G exp[igt a α a ] (.8)

11 .. QCD PARTITION FUNCTION with α a being a smooth function. The partition function for QCD is: Z QCD = DA a µ D ηdη D ψdψ periodic periodic antiperiodic { } β 0 e dτ d 3 x[ 4 F µν a F µν a + ψ[γ µ D µ +m]ψ+ ξ ( µa a µ νa a ν )+ ηa ( δ ab +gf abc A c µ µ)η b ] = DA a µ D ηdη D ψdψ e S 0+S INT periodic periodic antiperiodic (.9) where η, η are the Faddeev-Popov ghosts, which have the same boundary conditions as the gauge fields. The QCD Lagrangian, split in free and interaction part, gives us the Feynman rules. From the free (quadratic) part of the action, S 0 = β 0 dτ d 3 x{ Aa µ ( δ µν ( ξ ) µ ν ) A a ν + ψ(γ µ µ m)ψ + η η} (.0) one gets the propagators in momentum space by a Fourier transformation, which are: = [iγ µp µ + m] p + m (.) = p (.) = [ p δ µν ( ξ) p ] µp ν (.3) The vertex functions can be obtained from the interaction part, β S INT = g dτ d 3 x{ g 4 f abc f ade A b µa c νa d µa e ν + f abc ( µ A a ν)a b µa c ν 0 i ψa/ψ + f abc η a A c µ µ η b } (.4) and are given in momentum space by: ( b = g {f eab f ecd (δ αγ δ βδ δ αδ δ βγ ) + β c γ d )} δ = ig {δ αβ (p q) γ + δ βγ (q r) α + δ αγ (r p) β } p = gγ α (T a ) ij = igf abc p α (.5)

12 CHAPTER. QCD AT HIGH TEMPERATURE However, although at high temperature the coupling constant is small, the straightforward perturbation theory in powers of g fails. This is due to the fact that at high temperature we are dealing with a multi-scale system. The reason for this is that the non-static modes gain effective masses that grow linearly with increasing temperature and then decouple, leaving the zero-modes of the gauge fields A a µ(x) = T exp[iωnτ]a b a µ,n(x) where ω b n = nπt are the Matsubara frequencies, (.6) as true degrees of freedom contributing. These modes can be described by an electrostatic scalar field A a 0 (x) and magnetostatic gauge field Aa i (x) of a three dimensional effective theory, called electrostatic QCD (EQCD), with the Lagrangian: with L EQCD = T rf ij + T r[d i, A 0 ] + m ET ra 0 + ig3 3π µ f T ra λ () E (T ra 0) + λ () E T ra4 0 + higher order operators f (.7) Fij a = i A a j j A a i + g E f abc A b ia c j (.8) D i = i ig E A i (.9) Using this Lagrangian the partition function, or to use a physical quantity, minus the grand canonical energy density (that is the pressure), can be written (valid for sufficiently high energy) as: p QCD (T ) = p E (T ) + T V ln DA a i DA a 0exp{ S E } (.0) where p E = p EQCD is a parameter of the effective theory computable in perturbative full QCD [0]. With this theory one is able to compute the pressure of the full theory to the order g 5 [0]. To be able to compute higher orders the g 5, one must make further separations since there are still two dynamical scales gt and g T [8]. The non-perturbative scale g T which enters in the computation at order g 6, originates from the magnetostatic sector, that is from the fields A a i, so that we can write: p QCD = p EQCD + p MQCD + T V ln DA a i exp{ S MQCD } (.)

13 .. QCD PARTITION FUNCTION 3 where L MQCD = T rf a ij F a ij = i A a j j A a i + g M f abc A b ia c j (.) g M = g MQCD is, analogous to p E, computable through perturbative expansion of EQCD. The next goal in computation of p QCD is g 6 order (for a review see [44]) and that means in particular that one needs to go for four loop Feynman graphs in EQCD. The set of all four loop graphs has been given in [39] and after applying the general partial integration identities [7] this set is further reduced to the set of so called master integrals [3], which have to be computed in d = 3 ɛ. Again, since most of the master integrals are also needed in other theories, we will try to solve them all in d = 4 ɛ as well.

14 4 CHAPTER. QCD AT HIGH TEMPERATURE

15 Chapter 3 Nested Sums 3. Introduction The Z-sums are defined recursively by Z(n) = { : n 0 0 : n < 0 n x i Z(n; m,..., m k ; x,..., x k ) = i m Z(i ; m,..., m k ; x,..., x k ) (3.) i= i where k is called the depth and w = m + m m k the weight of the Z-sum. Equivalent definition can be given by Z(n) = n i >i >...>i k >0 x i i m xi k k i m k k (3.) Analogous definition can be given for the S-sums or S(n) = S(n; m,..., m k ; x,..., x k ) = S(n) = { : n > 0 0 : n 0 n i= n i i... i k > x i i m i S(i; m,..., m k ; x,..., x k ) (3.3) 5 x i i m... xi k k i m k k (3.4)

16 6 CHAPTER 3. NESTED SUMS Notice that the difference between the S- and Z-sums is the upper summation boundary, (i-) for Z- and (i) for S-sums. With the help of the following formula, one can easily convert Z-sums into S-sums and vice versa S(n; m,... ; x,...) = n x i i x i i i= i m S(i ; m 3,... ; x 3,...) i = + S(n; m + m,... ; x x, x 3,...) Z(n; m,... ; x,...) = n i= x i i m i i = x i i m Z(i ; m 3,... ; x 3,...) Z(n; m + m,... ; x x, x 3,...) (3.5) Since, as we will see, Z-sums form an algebra, the product of two Z-sums with the same upper summation, that is the same argument, can be written in terms of single Z-sums = + + Z(n; m,..., m k ; x,..., x k ) Z(n; m,,..., m, l ; x,,..., x, l ) n x i Z(i ; m,..., m k ; x,..., x k ) i = n i = n i= i m x,i i m, Z(i, m,,..., m, l ; x,,..., x, l ) Z(i ; m,..., m k ; x,..., x k ) Z(i, m,,..., m, l ; x,,..., x, l ) (x x, )i i m +m, Z(i ; m,..., m k ; x,..., x k ) Z(i, m,,..., m, l ; x,,..., x, l ) (3.6) This formula can be applied recursively on the RHS and since per definition it has an ending, it is a well defined recursion which at the end leaves us with single Z-sums. For example: Z(n; m, m ; x, x ) Z(n; m 3 ; x 3 ) = Z(n; m, m, m 3 ; x, x, x 3 ) + Z(n; m, m 3, m ; x, x 3, x ) +Z(n; m 3, m, m ; x 3, x, x ) + Z(n; m, m + m 3 ; x, x x 3 ) +Z(n; m + m 3, m ; x x 3, x ) (3.7)

17 3.. INTRODUCTION 7 i i i i i i i i Figure 3.: An intuitive, geometric picture for the multiplication of two Z-sums in eq. (3.6) Similarly the product of two S-sums simplifies to sum of single sums: = + S(n; m,..., m k ; x,..., x k ) S(n; m,,..., m, l ; x,,..., x, l ) n x i S(i ; m,..., m k ; x,..., x k )S(i, m,,..., m, l ; x,,..., x, l ) i = n i = n i= i m x,i i m, S(i ; m,..., m k ; x,..., x k )S(i, m,,..., m, l ; x,,..., x, l ) (x x, )i i m +m, S(i; m,..., m k ; x,..., x k )S(i, m,,..., m, l ; x,,..., x, l ) (3.8) The proof for the equation (3.6) uses the triangle relation (see Fig. 3.): n n n i n j n a ij = a ij + a ij + a ii (3.9) i= j= i= j= j= i= i= The equation (3.6) actually states that the Z-sums form a so called Hopf Algebra. To see this we first introduce some notation: : A A A (m, x ) (m, x ) = (m + m, x x ) (3.0) where is a multiplication of the set of letters, called the alphabet A, a pair (m i, x i ) is called the letter of the alphabet and m i is called the degree of the letter. A word is then a concatenation, that is an ordered sequence, of letters, e.g. W = (m, x ), (m, x ),..., (m k, x k )

18 8 CHAPTER 3. NESTED SUMS S sums multiple polylogs Euler Zagier sums harmonic polylogs multiple zeta values Nielsen polylogs classical polylogs Figure 3.: Inheritance diagram for Z-sums A word of length zero is called e. We define the quasi-shuffle algebra A as a vector space of words with: e W = W e = W ((m, x ), W ) ((m, x ), W ) = (m, x ), (W (m, x ), W ) +(m, x ), (((m, x ), W ), W ) + ((m, x ) (m, x )), (W W ) (3.) where is multiplication of letters and is product of algebra A. One can see that the equation (3.6) is actually defining a quasi-shuffle algebra so Z-sums form a quasi-shuffle algebra. Now from [4] one knows that it is sufficient to show that some objects form a quasi-shuffle algebra and a general theorem proves that they also form a Hopf algebra. (The proof that a quasi-shuffle algebra forms a Hopf algebra will not be given here, for details see [4].) The Z/S-sums are a fairly general object, in a lots of cases it wont be necessary to consider these general objects, but instead some simpler ones (see fig. 3.). If one for example takes the index n in Z-sums to be infinity, one ends with the so called multiple polylogarithms of Goncharov [5]: Z( ; m,..., m k ; x,..., x k ) = Li m,...,m k (x,..., x k ). (3.) If, in addition to n = one also sets x = = x k = then one gets

19 3.. HARMONIC SUMS, EULER-ZAGIER SUMS 9 multiple Zeta-values [6]: Z( ; m,..., m k ;,..., ) = ζ(m,..., m k ) (3.3) By taking only x = = x k = and leaving n general, we get Euler-Zagier sums ([7] [8]): Z(n; m,..., m k ;,..., ) = Z m,...,m k (n) (3.4) On the other hand, the S-sums for values x = = x k = and m i > 0 reduce to harmonic sums []: S(n; m,..., m k ;,..., ) = S m,...,m k (n) (3.5) Multiple polylogs, in turn contain as a subset the classical polylogs Li n (x), Nielsen s generalized polylogs [9]: S n,p (x) = Li,...,,n+ (,...,, x) (3.6) }{{} p and harmonic polylogs introduced by Vermaseren and Remiddi [] H m,...,m k (x) = Li mk,...,m (,...,, x) (3.7) }{{} k In this work we will specially use harmonic sums and harmonic polylogs, therefore we will take a closer look of these two subclasses in the following section. 3. Harmonic sums, Euler-Zagier sums As we have seen, the harmonic sums (or Euler-Zagier sums) are a special case of general S(/Z)-sums (or speaking in historical terms, the S(/Z)-sums are a generalization of harmonic sums (Euler-Zagier sums), namely when in the S(/Z)-sums the arguments are x =... = x k =. So starting from S-sums and setting all x i =, we can define harmonic sums as (we follow the convention of []): S m (n) = S m (n) = n i= n i= i m ( ) i i m, m > 0 (3.8)

20 0 CHAPTER 3. NESTED SUMS Recursively, one can write higher harmonic sums, that is sums with bigger weight, by: n S m,j,...,j p (n) = i m S j,...,j p (i) (3.9) i= Similarly, the so called Euler-Zagier sums are defined the same way, only starting from Z-sums instead of S-sums. Or in other words, the relation between harmonic sums and S-sums is analogous to the relation between Euler-Zagier sums and Z-sums, only difference being the upper summation limit. Here we will therefore only consider harmonic sums. One example would be n i j ( ) k S,3, (n) = i j 3 k (3.0) i= The above formula is a well defined recursion, which can be easily implemented in computer algebra system, like FORM [30], where harmonic sums are implemented as j= k= S i,...,i m = S(R(i,i,...,im),n) What we will be interested in, is to sum over a product of harmonic sums and possible denominators, which will occur as formal solutions of difference equations in next chapter. Since harmonic sums inherit the algebra from S- sums it is possible to write product of two harmonic sums as sum of single, higher harmonic sums. Example similar to equation (3.8) would be: + n j = j m S m,...,m k (n) S m,,...,m, l (n) = n S m,...,m k (j )S m, (j ),...,m, l j = n j m j m +m, j= S m,...,m k (j )S m,,...,m, l (j ) S m,...,m k (j)s m,,...,m, l (j) (3.) In the case that the arguments are not the same, one has to synchronize the harmonic sums first, that means to bring the arguments, possible denominators and/or factorials to the equal form, for example : n i= S (i + ) i = = n i= n i= S (i) i S (i) i + + n i= n i= (i + )i n i i= i + (3.)

21 3.. HARMONIC SUMS, EULER-ZAGIER SUMS This of course doesn t work if the argument is a difference of symbolic values. In that case one can use recursively one of the formulas: S j,r,...,r s (i + m) i S j,r,...,r s (i) i + m = S j,r,...,r s (i + m ) i = S j,r,...,r s (i + ) i + m + S r,...,r s (i + m) i(i + m) j S r,...,r s (i + ) (i + m)(i + ) j (3.3) On the right hand side one can use the general partial fractioning formula i + a i + b = δ a,b (i + a) + (Θ (a b) + Θ (b a)) b a ( i + a i + b ) (3.4) where Θ (n) is zero if n 0 and one if n > 0. Applying the above formulas recursively, (since it has a defined ending, it is a well defined recursion), one gets synchronized harmonic sums over which one can perform summation. This synchronization and summation is in FORM implemented in the package called summer. An example code for the typical expression: n j = S (j )S, (j ) j + (3.5) is given in the appendix, together with the result. The harmonic/euler- Zagier sums can be used to expand Gamma functions in ɛ around integer valued numbers, through the formula: and Γ(a + bɛ) = Γ( + bɛ)γ(a)( + (bɛ) j Z,..., (a )) (3.6) j= }{{} j Γ(a + bɛ) = = Γ( + bɛ)γ(a) ( + (bɛ) j Z,..., (a )) j= }{{} j Γ( + bɛ)γ(a) ( + ( bɛ) j S,..., (a )) j= }{{} j (3.7) for a N and a > 0. For negative a one uses the analytic continuation formula aγ(a) = Γ(a + )

22 CHAPTER 3. NESTED SUMS Using this property and the fact that harmonic/euler-zagier sums form an algebra, one can expand in ɛ the so called hypergeometric functions JF J, which are defined by: = i=0 JF J ({A,..., A k }, {B,..., B k }, x) = (A ) i (A k ) i x i (A ) i (A k ) i x i = + (B ) i (B k ) i i! (B ) i (B k ) i i! = = + Γ(B ) Γ(B J ) Γ(A ) Γ(A J ) i= i= Γ(A + i) Γ(A J + i) Γ(B + i) Γ(B J + i ) x i Γ(i + ) (3.8) where (A) i = A(A + ) (A + i ) = Γ(A+i) Γ(A) ) is the Pochhammer symbol. Here we will briefly sketch the algorithm from [] which allows to compute expressions like the one above (There it is called Algorithm A). For algorithms for other, similar expressions see []. Let us write down the most general expression which one can solve with this algorithm: n i= x i (i + c) m Γ(i + a + b ɛ) Γ(i + a k + b k ɛ) Γ(i + c + d ɛ) Γ(i + c k + d k ɛ) Z(i + o ; m,..., m l ; x,..., x l ) (3.9) First one has to expand the Γ functions to Z-sums (in our case it suffices to expand in Euler-Zagier sums, but we keep the algorithm description general) according to eq. (3.6)(3.7), and synchronize the remaining Z-sum Z(i + o ; m,..., m l ; x,..., x l ). Then we have terms like: n i= x i (i + c) m Z(i, m...) c 0 (3.30) The only thing one has to do now is to reduce the offset c to zero, since then one would per definition have another Z-sum. To do that, use recursively the formula: n i= x i (i + c) m = x n i= x i (i + c ) m c m + x n (n + c) m (3.3) in case that the depth of the Z-sum in (3.30) is zero, or: n i= x i (i + c) m Z(i ; m...) = x n i= x i (i + c ) m Z(i ; m...)

23 3.3. HARMONIC POLYLOGARITHMS 3 n i= x i x i (i + c) m i m Z(i ; m x n... ; x...) + (n + c) m Z(n ; m...) (3.3) if the depth of the Z-sum is not zero. At the end one has only terms of the form: n x i i m Z(i ; m...) (3.33) i= which is of course again a Z-sum. Notice that in the formula (3.8), in the definition of the hypergeometric function, the upper limit of the sum is infinity. In that case, as we already mentioned in eq. (3.), we get from the equation ( ) generalized polylogarithms. If further the Z-sum is a Euler-Zagier sum, then we get multiple zeta values (eq. (3.3)). Of course the last term in the equations (3.3) and (3.3) respectively do not contribute. The above algorithm, and others, are available in FORM package XSUMMER [3], or in GINAC [4] nestedsums library [5]. One important remark should be made here, namely, from the algorithm described above, especially from reduction formula of the offset c to zero (3.3),(3.3) one can immediately see, that one can expand only hypergeometric functions with integer valued coefficients. If our c, or equivalently the offset o in Z-sum in equation (3.9), would be a rational number, then the algorithm would not work, since the offset is recursively reduced by one until it reaches zero and similarly synchronization of the subsum also works the same way, so that in case of a rational number the algorithm would never reach zero. This means that hypergeometric functions, like F ( + ɛ, ɛ; + ɛ; x) cannot be expanded with the algorithm described. There is however a generalization of the above algorithm which includes rational numbers [6], but these rational number up to now have to have certain form. The general algorithm for no matter what distribution of rational numbers has not been found. In the appendix we give an example of a FORM program that expands hypergeometric function F (ɛ, ɛ; ɛ; x). 3.3 Harmonic Polylogarithms As we have seen in the first section, harmonic polylogarithms (HPL) are a special case of multiple polylogarithms in eq. (3.7). On their own, one

24 4 CHAPTER 3. NESTED SUMS can define HPL s recursively as following: H(0,..., 0; x) }{{} = n! logn x n H(a, a,..., a k ; x) = x 0 f a (t)h(a,..., a k ; x)dt (3.34) for general vector of length,or weight n, where a i =, 0, and functions f a (x) are f (x) = + x, f 0 (x) = x, f (x) = x (3.35) The beginning of the recursion also has to be given, in this case that would be the lowest weight: H(; x) = H(0; x) = H( ; x) = x 0 x 0 x 0 dt t = ln( x) dt t = ln(x) dt = ln( + x) (3.36) + t An alternative definition would be: d dx H(a, a,..., a k ; x) = f a (x)h(a,..., a k ; x) (3.37) From the equation (3.8), it is easy to see that HPL s are a generalization of Nielsen polylogarithms [9]. Historically, that was the reason for their introduction []. HPL s also form an algebra, so one can write, just like in case of S/Z-sums or harmonic/euler-zagier-sums, the product of two HPL s (with the same argument of course) as a sum of single HPL s of higher weight. For example: H(a, a ; x)h(b, b ; x) = H(a, a, b, b ; x) + H(a, b, a, b ; x) + H(a, b, b, a ; x) + H(b, a, a, b ; x) + H(b, a, b, a ; x) + H(b, b, a, a ; x) (3.38) Notice, that in the above formula the relative order of the elements of a vector a = (a, a ) and b = (b, b ) respectively, is preserved. This is due to shuffle algebra. The general formula is then: H(a,..., a k ; x)h(b,..., b k ; x) = H(c,..., c k +k ; x) (3.39) > c i a i bi

25 3.3. HARMONIC POLYLOGARITHMS 5 where the symbol > stands for the fact mentioned earlier, namely that the internal order of the elements a i and b i resp. is preserved. The HPL s can be Mellin transformed and Taylor expanded. Since we wont need Mellin transforms and the Taylor expansion of HPL s, we refer the interested reader to original literature []. What we will need are the HPL s with argument x=. These are actually nothing else then harmonic/euler-zagier sums at infinity, which are nothing else then multiple zeta values (MZV) for positive a s, or colored MZV for arbitrary a s. H(a; ) = ζ(a), a > 0 H( a; ) = ( a )ζ(a), a > 0 H(a,..., a k ; ) = ( ) #(a i<0) ζ(ā,..., ā k ), k > (3.40) where ζ s are: ζ(a,..., a k ) = i i i k k sgn(a j ) i j j= i a j j (3.4) and vector ā = (a, sgn(a )a,..., sgn(a i )a i,..., sgn(a k )a k ). The MZV s themselves possess an algebra, which means that they can be expressed in terms of a few mathematical constants, like powers of π, ζ- functions and certain polylogarithms. For the relations see for example [3], [3]. The HPL s are implemented in FORM package called HARMPOL [], and also in MATHEMATICA package HPL from [33]. We will use HPL s to calculate, or integrate, differential equations, therefore we will need to evaluate expressions like: x 0 H(, 0, ; t)h(, ; t) (3.4) + t The solution of the above formula using FORM package HARMPOL is given in the appendix.

26 6 CHAPTER 3. NESTED SUMS

27 Chapter 4 Methods of Computation 4. Difference Equations Difference equations or recurrences occur from the so called integration-byparts (IBP) method [34], [35]. Feynman integrals F (q,..., q n ; d) = d d k d d k n L l= D F,l (p l ) (4.) where k i are internal momenta, q i external momenta, L the number of lines, or loop momenta p which are certain linear combinations of external momenta q i and n = L V +, where V is number of vertices, and n is the number of loops, and D F,l (p l ) = iz l (p) (p l m l + i0), (4.) where Z l (p) is a polynomial in p, have the property that in dimensional regularization a derivative of an integral of the form (4.), with respect to mass or a momentum equals the corresponding integral of the derivative. As a direct consequence of this fact one deduces: d d k i k µ i L l= D F,l (p l ) = 0 }{{} surfaceterm, i =... n (4.3) Application of this formula gives many relations of the given integral as a linear combination of some other integrals with one pinched, that is contracted, line, and/or denominators risen to a different power then the 7

28 8 CHAPTER 4. METHODS OF COMPUTATION original integral. Since IBP relations give an under-determined, homogeneous, linear system of equations whose unknowns are the integrals, there exist some integrals, called master integrals, whose values cannot be determined by IBP relations. They have to be calculated by other means. The masters form so to speak the basis in which the other integrals can be expressed. Since one can by a linear transformation always transform a given basis into another one, it is not unique. So instead of solving all Feynman integrals, one only needs to solve the master integrals. In the literature there exists an approach which automatizes the procedure of IBP of a particular integral. We illustrate the method on the most simple example possible, the massive tadpole: J(x) = d d p (π) d (p + m ) x (4.4) Here we will denote the denominator to the power x by a dot on a line in. Now we take the derivative of mass times mass m m (could also take impulse): m mj(x) = = d d p (π) d mm (p + m ) x = d d p (π) d (p + m ) x m x d d p (π) d (p + m ) x+ = = J(x) m xj(x + ) (4.5) On the other hand by rescaling p p m and pulling m out of the integral and again taking derivative with respect to m times m, we get: m m d d p (π) d md x (p + ) x = ( + d x)m d x d d p (π) d (p + ) x = ( + d x)j(x) (4.6) Setting the two equations (4.5),(4.6) equal, one gets: m xj(x + ) = (d x)j(x) or J(x + ) = d x m J(x) (4.7) x Now this is a homogeneous, linear difference equation (or recurrence) first order. In [7] there is general method, which describes how to solve difference

29 4.. DIFFERENCE EQUATIONS 9 equations numerically with high precision. We will however concentrate only on analytic calculations. Let us look at a general, inhomogeneous, difference equation of first order (see for example [9],[0]) : y(x + ) = a(x) y(x) + g(x), y(x 0 ) = y 0, x x 0 0. (4.8) The initial values can be found in our case easily, namely the initial value at x 0 = 0 means that we take our Feynman graph and pinch [see fig. 5.] the dotted line which corresponds to a propagator to the power x. To find a boundary value is also easy, in [7] it is said that for x, the generic Feynman graph U(x) is given by: [ lim U(x) = x ] (p + ) x [g(0)] () x x d g(0) (4.9) where g(0) is the original graph with the dotted line cut away (not pinched). One can see a difference equation as a discrete differential equation, and just like one can formally solve, or integrate, an ordinary differential equation of first order, one can formally solve, or sum, the first order difference equation, with the result: x y(x) = a(i) y 0 + i=x 0 x x r=x 0 i=r+ a(i) g(r) This solution is valid for all x Z +. Applied to our equation for the tadpole, we get with a(x) = d x m x : J(x) = [ x i= ] d i m J() i which is (4.0) J(x) = (m ) x ( d ) x J() (4.) Γ(x) where (x) n = Γ(x+n) Γ(x) is the so called Pochhammer symbol. From the solution of the difference equation one can see that any J(x) for x >, can be expressed in terms of J(), which is the only master integral. It can be solved analytically: J() = d d p (π) d p + m = Γ( d ) (4π) d/ (m J (4.) ) d/

30 30 CHAPTER 4. METHODS OF COMPUTATION For a n-th order difference equation, like for n-th order differential equation, there is no formal solution. But since there is more knowledge in the field of differential equations, it might be useful to have a general way to transform difference into differential equations. 4. Differential Equations One way to obtain differential equations is to use IBP relations and additional derivatives of the integrals [40]. Here we will use a general approach which uses the recurrences obtained by the IBP relations and transforms them by means of Laplace transform [0] into a differential equation. We will demonstrate this method on a second order difference equation, since no third or higher order equations will appear in this thesis. A homogeneous -nd order recurrence would be: a 0 (x)y(x) + a (x)y(x + ) + a (x)y(x + ) = 0 (4.3) a i s are polynomials in x of maximum degree p. To Laplace transform the equation we substitute : y(x) = dt t x v(t) (4.4) l where l is a line of integration suitably chosen and v(t) is a solution of a certain differential equation which will be derived below. We rewrite the coefficients a i (x) in the form: a (x) = A p (x + )(x + 3) (x + p + ) A (x + )(x + 3) + A (x + ) + A 0 a (x) = B p (x + )(x + ) (x + p) B (x + )(x + ) + B (x + ) + B 0 a 0 (x) = C p x(x + ) (x + p ) C x(x + ) + C x + C 0 (4.5) where A p 0 and C p 0. Defining: Φ p (t) = A p t + B p t + C p and Φ i (t) = A i t + B i t + C i, i = 0,... p (4.6)

31 4.. DIFFERENTIAL EQUATIONS 3 one can show [0] that (4.4) provides a solution of the difference equation (4.3), if v(t) is a solution of the differential equation t p Φ p (t) dp dt p v(t) tp Φ p (t) dp dt p v(t) +... ( )p Φ 0 (t)v(t) = 0 (4.7) and a line of integration l is chosen so that p I(x, t) = v(t) k=0 d k p dt k [tx+k Φ k+ (t)] v (t) k=0 d k dt k [tx+k+ Φ k+ (t)] ( ) p v (p ) (t)[t x+p Φ p (t)] (4.8) has the same value at each endpoint. One can immediately see that for a second order difference equation with coefficients of degree p, one gets a p-th order differential equation with coefficients of degree +p. For a non-homogeneous difference equation N N a i (x)y(x + i) = b i (x)z(x + i) (4.9) i=0 i=0 where b i (x) are polynomials and z(x) is a solution of some difference equation that has already been solved before, one can analogously say that: y NH (x) = dt t x v NH (t) and l z(x) = dt t x w(t) (4.0) l (where w(t) is a solution of a differential equation associated with z(x) by the method already described) are solutions of (4.9), if v NH (x) and w(x) satisfy p i=0 Φ i (t)( t) i v (i) p NH (t) = i=0 Ψ i (t)( t) i w (i) NH (t). (4.) As an illustration we give an example from [38] of massive two loop graph S(x). Its difference equation is: S(x) J 0 = 3(x + )S(x + ) + (x + 3 d)s(x + ) + (x + d)s(x) Γ(x + d/)(d ) Γ(x + )Γ( d/) (4.)

32 3 CHAPTER 4. METHODS OF COMPUTATION with boundary conditions: S(0) = (4.3) S(x ) = Γ(x d/) d/ Γ(x) Γ( d/) d/ x d/ Γ( d/) (4.4) Simple manipulations (eq. ( )) lead to the differential equation: t( d + ( d)t + 3t )v(t) t( t)( + 3t)v (t) = ( d) Γ( d )Γ( d )t d/ ( t) d/ (4.5) The homogeneous solution of the equation is v H (t) = c H t d ( t) d 3 ( + 3t) d 3 (4.6) which does not satisfy the x boundary condition because it grows like x d, so c H (t) = 0, and the inhomogeneous solution is formally obtained by varying the constant, leading to: S(x) = d Γ( d )Γ( d ) t 0 dtt x+ d ( t) d 3 d 3 ( + 3t) dzz d ( z) ( + 3z) d (4.7) This integral is convergent for x > in four dimensions. Using the relation S(3) = ( 36 3(d ) + (d 8)(d 3)S() ) which follows from the difference equation by setting x = 0, one can compute at least numerically the S(). In the appendix the analytical result up to second order in ɛ is given. 4.3 x-space Evaluation In this section we present a calculation method for a class of so called sunset graphs, which is performed in coordinate space [4]. One writes in coordinate space the massive propagator according to: d d p (π) d e ip x d p + m = (m) (π) d x d K d (mx) (4.8) where K µ (ax) is a modified Bessel function of third order or MacDonald function. The massless propagator to arbitrary power is: d d p e ip x (π) d (p ) ν = Γ( d ν) Γ(ν) x ν d 4 ν π d (4.9)

33 4.3. X-SPACE EVALUATION 33 Analytically, at least in terms of hypergeometric functions, one can solve all sunrise topology n-loops diagrams with at most three massive lines. The reason for this constraint of number of massive lines is that in coordinate space one has to perform only one integration for sunrise-type graphs, and it is known that [4]: 0 = α 4 dxx α K λ (ax)k µ (bx)k ν (cx) = {A(α, µ) + A(α, µ) + A( α, µ) + A( α, µ)}, cα for [Re α > Re λ + Re µ + Re ν ; Re(a+b+c) > 0], where ( ) a α ( ) b µ A(α, µ) = [Γ( λ) + Γ( µ) + Γ( α + λ + µ ν ) + c c +Γ( α + λ + µ + ν )] ( α + λ + µ ν F 4, α + λ + µ + ν ) ; λ +, µ + ; a c, b c, (4.30) where the F 4 is Appell s hypergeometric function of two variables which has the form: F 4 (a, b; c, d; z, z ) = j =0 j =0 (a) j +j (b) j +j (c) j (d) j z j zj j!j! ; (4.3) The above formula simplifies in case of only two massive propagators to: 0 dxx α K µ (bx)k ν (cx) = A α µ,ν, for [Re (b+c) > 0; Re α > Re µ + Re ν], where A α µ,ν = α 3 α µ bµ c Γ(α) [Γ(α + µ + ν ) + Γ( α + µ ν ) + Γ( α µ + ν ) +Γ( α µ ν ( α + µ + ν )] F, α µ + ν ) ; α; b c (4.3) For only one massive propagator the formula is: 0 dxx α K µ (bx) = α 3 b α Γ(α + µ )Γ( α µ ) (4.33)

34 34 CHAPTER 4. METHODS OF COMPUTATION A simple example for one massive (m=) propagator would be [36]: = 0 ( x d Γ( d ) 4π d ) 3 x d K d (x) (π) d = Γ(4 3d )Γ( d)γ ( d ) (π) 4d (4.34) Let us now apply the method to the two massive lines graph more explicitly, since it is a master integral. It has two massive lines, with the same mass, which we set to one, and three massless line. In x-space, using eq.(4.8) and (4.9) we get: = ( Γ( d d d x ) ) 3 ( K d (x) ) (4.35) 4π d x d (π) d x d Performing the trivial angular integration this simplifies to: = Γ( d )3 π d 4 3 Γ( d )π 3d (π)d d d xx 7 3d K d (x) (4.36) Now we can use the equation (4.3), which in this case simplifies to 0 dxx α K µ(x) = α 3 Γ(α) + µ Γ(α )Γ ( α and setting α = 8 3d and µ = d we get: µ )Γ(α ) (4.37) = Γ(5 d)γ (4 3d )Γ(3 d)γ3 ( d ) (4π) d Γ(8 3d)Γ( d ) (4.38) Notice that this result looks different then the one in for example appendix of [3]. ( To match it with their result one has to multiply our result by ) (4π) 4 e 4ɛ, γ π which is due to a different normalization of the integral measure. An example for a three massive lines graph would be: = 0 ( x d Γ( d ) ) x d K d (x) 4π d (π) d 3 (4.39)

35 4.4. MELLIN-BARNES METHOD 35 Since we set all three masses to be m = m = m 3 =, the F 4 -function from equation (4.30) simplifies to 4 F 3 - and 3 F -hypergeometric functions. The (somewhat long) result in terms of these functions is given in the appendix. It is worth noticing, that not only the sunrise-type graphs can be computed with this method, also other topologies, whose massless lines can be combined in a way which gives effectively sunrise topology, can be computed. Take, for example, the graph T = as : where G is the massless scalar one loop integral:. The massless lines can be written = [G(, )] (4.40) G(α, β) (k ) α+β d d d p π d (p ) α ((p k) ) β = Γ(α + β d )Γ( d α)γ( d β) π d Γ(d α β)γ(α)γ(β) (4.4) Then one proceeds like in the example above, since then one has the same topology, with the massless propagator being of course different. The result is: 8 d 3 Γ 3 ( J 4 T = )Γ(6 d)γ3 ( d ) sin( 3πd 3d )Γ( )Γ ( d )Γ (d ) which coincides with the result given in [3] and [36]. (4.4) 4.4 Mellin-Barnes Method In this section we present the so called Mellin-Barnes (MB) method for solving massive Feynman diagrams as was done in [46]. The method consists of the substitution for a massive denominator: ( ) (k m ) β = i m s (k ) β ds Γ(β)πi i k Γ( s)γ(β + s) (4.43) In other words one is reducing (or rewriting) massive lines in a Feynman diagram to massless ones at a price of an additional integration of Mellin- Barnes type. This integration can sometimes be performed using first Barnes

36 36 CHAPTER 4. METHODS OF COMPUTATION lemma, i dsγ(a + s)γ(b + s)γ(c s)γ(d s) πi i Γ(a + c)γ(a + d)γ(b + c)γ(b + d) = Γ(a + b + c + d) (4.44) the second Barnes lemma, i Γ(a + s)γ(b + s)γ(c + s)γ(d s)γ(e s) ds πi i Γ(f + s) Γ(a + e)γ(a + d)γ(d + c)γ(b + d) = Γ(a + b + d + e)γ(a + c + d + e) Γ(b + e)γ(c + e) Γ(b + c + d + e) (4.45) where f = a + b + c + d + e and/or by closing the integration contours in the complex plane and summing up the corresponding series. Simple one-loop example would be: I(a, b; m) Using equation (4.43) we get: Γ(b) πi i i d d p (p ) a ((k p) m ) b (4.46) ds( m ) s Γ( s)γ(β + s) d d p (p ) a ((k p) ) b (4.47) Now, the last integral is a massless one-loop integral which is known analytically in terms of Gamma functions to be d d p (p ) a ((k p) ) b = π d (k ) d a b Γ( d a)γ( d b)γ(a + b d ), (4.48) Γ(a)Γ(b)Γ(d a b) so using this result and linearly shifting the variable of integration s = a b s we get: d I(a, b; m) = π d ( m ) d a b Γ( d a) Γ(a)Γ(b) ( ) i s ds k Γ( s)γ(a + s)γ(a + b d + s) πi i m Γ( d + s) (4.49)

37 4.4. MELLIN-BARNES METHOD 37 Closing the contour on the right, we obtain: I(a, b; m) = π d ( m ) d a b Γ( d a) Γ(a)Γ(b) ( ) k j Γ(a + j)γ(a + b d + j) j! m Γ( d + j) (4.50) j=0 where we used the basic formula for the residuum of Gamma functions res(γ( x + a), x = a + j) = ( )j j!. If we look at the sum in the above equation and compare it with our definition of hypergeometric function in eq. (3.8) we see that we can write our result as: I(a, b; m) = π d ( m ) d a b Γ( d a)γ(a + b d ) Γ( d )Γ(b) F (a, a + b d ; d ; k m ) (4.5) These hypergeometric functions can then be expanded in ɛ with the program in the appendix, as long as the coefficients are integer valued. Another, more complex example is, which we cannot solve with methods from previous sections. With calculation analog to the one we just presented, can be also written in terms of one hypergeometric function (we look at the case where all three masses are equal, for the general case of different masses see [45]): { 3(d ) J = F ( d 4(d 3), ; 5 d ; 3 4 ) 3 d 5 πγ(5 d) Γ( d )Γ(3 d ) } (4.5) In d = 4 ɛ dimensions this graph is needed in Standard Model calculations, however the hypergeometric function has rational number coefficients and cannot yet be expanded in ɛ in four or three dimensions. We have found a relation in the literature which transforms this hypergeometric function appearing in in four dimensions into a hypergeometric function which has only integer valued coefficients, namely [4]: F (ɛ, ; + ɛ; z) = ( z + z) ɛ z F (ɛ, ɛ; ɛ; z z + z) (4.53)

38 38 CHAPTER 4. METHODS OF COMPUTATION where z = 3 4. Finally, we can write the result in d = 4 ɛ as: { 3( ɛ) J = F (ɛ, ɛ; ɛ; z 4( ɛ) z + z) } 3 ɛ πγ( + ɛ) Γ(ɛ)Γ( + ɛ) (4.54) In the appendix we have a sample code from [3] which computes the expansion of this hypergeometric function to the second order and we also printed the result of to the same order there.

39 Chapter 5 Applications We would like to apply the methods introduced in the previous chapter to the following set of vacuum integrals:,,,,,,,,, (5.) They are all normalized to massive tadpole J() to the power of the number of loops of the diagram, that is, a n-loop diagram will be normalized (divided by) J n (). Since all diagrams have only one scale, we will set m =. The mass can in the end be reconstructed by dimensional analysis. 5. and Loop The master integrals for the SU(N c ) + Adjoint Higgs Theory are given in equation (5.). There are no graphs at two loop level and at the one loop level there is only one graph which we solved in section 4. equation (4.) Loop At this level there are two integrals which have to be solved, V3 and V4. The V3 is most conveniently solved by using the difference equations method. All difference equations in this thesis are taken from [38]. The difference equation for V3 is: V 3(x) J 3 39

40 40 CHAPTER 5. APPLICATIONS PINCH Figure 5.: Pinching one line 0 = (x + 6 d)v 3(x + ) + (x + 4 3d )V 3(x) + + Γ(x + d )( d) (x + 3 d)γ(x + )Γ( d ) (5.) This is a first order difference equation, which has a formal solution (see equation (4.0)). What we still need is an initial condition, say at x = 0. Setting x to zero means that we pinch the line to which the dot is attached, that is, the neighboring vertices are pulled to one vertex (see fig. 5.). So we have to calculate: V 3(0) = B() = J 3 (5.3) Since this is a sunrise type graph and it has only two massive lines, it can be computed simply in coordinate space with the method described in section 4.3. Straightforward calculation leads to a result: B() = J 3 = d 3 Γ(4 3d 3 d )Γ( )Γ( d ) Γ( 7 d)γ( d ) (5.4) which coincides with [3], [36]. Now setting x = 0 in equation (5.), we already get what we want, that is V 3(): V 3() = = V 3() = (d ) 4(d 3) + 3d 8 4(d 3) V 3(0) (d ) 4(d 3) + d 5 (3d 8)Γ(4 3d 3 d )Γ( )Γ( d ) (d 3)Γ( 7 d)γ( d ) (5.5) This gives in d = 4 ɛ and d = 3 3ɛ: V 3() d=4 ɛ = ɛ + 5ɛ + ( ζ 3 3 ) ɛ 3 + O(ep 4 )

41 5.. 3 LOOP 4 V 3() ( d=3 ɛ = π π ζ ) ( ) 3 ɛ + π 3 9π4 40 0ζ 3 ɛ + +O(ɛ 3 ) (5.6) Actually, by choosing a different basis of three loop master integrals, one would get from IBP relations [3]: = J ( 3 (d ) ) ( ) 4(d 3) + (5.7) (d 3)(3d 8) (3d 8) ( ) ( ) = J 3 (d ) 4(d 4) + (5.8) (d 3)(3d 8) (3d 8) So equivalently, one could at three loop level also try to solve B and B4. We will use the relation from equation (5.8) and solve that, and from. The difference equation doesn t bring much help, since it is a second order inhomogeneous difference equations, which we couldn t manage to solve [38]: B4(x) = J 3 0 = 6(x + 3 d)(x + )B4(x + ) + (7x 0xd + 7x + 3d with boundary conditions: 7d + 4)B4(x + ) + (x + d)(x + 6 3d)B4(x) Γ(x + d )3(d ) Γ(x + )Γ( d ) (5.9) B4(0) =, and B4(x ) = Γ(x d ) Γ(x)Γ( d ) x d Γ( d ) (5.0) Here one can see that we have only one initial condition, where the other one is a asymptotic condition. The Laplace transform of the homogeneous part of this difference equation leads to the following differential equation: 0 = {t [ 6t + 4t + ] t t[6( d)t + ( 0d)t + 8 5d] t + [(48d + 6)t + (6d 4d + 8)t + 3d 6d + ]}v(t) (5.)

42 4 CHAPTER 5. APPLICATIONS where v(t) gives B4(x) according to equation (4.4). This equation cannot be solved by means we have. Here we sketch a differential equation method and a result of B4() from [37] which manages to solve B4() with harmonic polylogarithms. First we define: Φ(d, x) M p p +M p...p 3 p + x M p + x M p 3 + M (p + p + p 3 ) + M (5.) which due to the symmetry of the the underlying Feynman diagram has to fulfill: Φ(d, x) = x 3d 8 Φ(d, x ) (5.3) We can recover B4 if set x =. By setting x = 0, we get the initial condition: Φ(d, 0) = B() = d 3 Γ(4 3d 3 d )Γ( )Γ( d ) Γ( 7 d)γ( d ) (5.4) Φ(d, x) satisfies following differential equation taken from [37]: (x( x ) x ( x )(d 3) x x(d 3)(3d 8))Φ(d, x) Now, what one does is the Ansatz (in 4-dim): = (d ) x d 3 (x d ) (5.5) Φ(d, x) = ɛ n Φ n (x) (5.6) n=0 and solving the differential equation order by order in ɛ, in terms of harmonic polylogarithms. Plugging this expression in equation (5.5) we get: (x( x ) x ( x )(d 3) x 4x)Φ n (x) = K n (x) (5.7) where the inhomogeneous term K n (x) is given by: K n (x) = 4x 3 ( 4) (H n 0 n (x) + H 0 n (x) + ) 6 H 0 n (x) +4x( ) (H n 0 n (x) + H 0 n (x) + ) 4 H 0 n (x) [( x ) x + 7x]Φ n (x) + xφ n (x) (5.8)

43 LOOP 43 where H 0 0 =, H 0 n<0 = Φ i<0 (x) = 0 and 0 n is a vector with n zeros. The two linearly independent solutions of the homogeneous equation associated with equation (5.7) are: Φ (x) = ( x ) = (( x)( + x)) Φ (x) = x( + x ) ( x ) [H (x) + H (x)] = x( + x) 4x (( x)( + x)) [H (x) + H (x)] (5.9) With these two solutions one can construct the general solution of the inhomogeneous equation (5.7) by variation of constants: ( ) Φ (x)k n (x) Φ n (x) = Φ (x) c,n + dx 6x 3 (( x)( + x)) 4 ( ) Φ (x)k n (x) + Φ (x) c,n + dx 6x 3 (( x)( + x)) 4 (5.0) where c i,n are integration constants, which have to be fixed by relations (5.3) and (5.4) order by order. Since Φ, (x) and K n (x) are all functions of x, x, x, +x, it follows that the integrand contains harmonic polylogarithms, according to the definition in section 3.3. It also follows that the integration from 0 to x as in equation (5.0) of HPL s gives again HPL s, due to the recursive definition, so that to all orders in ɛ the solution of Φ(d, x) will be in terms of HPL s. Even more, since we are interested in B4(), which is Φ(d, x = ), it follows that, to all orders in ɛ, the B4() is given in terms of HPL s with argument, which are harmonic/euler-zagier sums at infinity, which are multiple zeta values. We quote here only the first few coefficients from [37]: B4() B4() d=4 ɛ = 5 3 ɛ ɛ + O(ɛ 3 ) d=3 ɛ = ɛ + 4 ln + (6 8 ln + 4 ln + 4ζ )ɛ + O(ɛ ) (5.) This result of B4 gives at the same time the result of (5.8), since J() is known. via the equation Loop At four loop level there are eight master integrals which have to be solved. First let us quote the first few orders (in 4-dim.) of those integrals (T 6, T 4,

44 44 CHAPTER 5. APPLICATIONS BB4) which we couldn t solve with methods presented in this thesis, but which have been solved by method analog to the one used in case of B4() (from [37]): J 4 T 4 d=4 ɛ = ɛ + 3 ɛ + O(ɛ 3 ) J 4 T 6 d=4 ɛ = ɛ + 9 ɛ + O(ɛ 3 ) J 4 BB4 d=4 ɛ = ɛ ɛ + O(ɛ 3 ) (5.) Now there are five integrals to solve, namely T 3, T, BB, V Ba, V Bb. Two of them, T, BB can be solved easily. T has already been solved in section 4.3, where it was shown that one can rewrite the massless lines into one massless line with different power, so that one can use the x-space evaluation method, which then gave: 8 d 3 Γ 3 ( J 4 T = )Γ(6 d)γ3 ( d ) sin( 3πd 3d )Γ( )Γ ( d )Γ (d ) (5.3) We have solved BB using the same method in that chapter, but here we want to illustrate another way of obtaining the result, which we will also use on T 3. First let us look at the difference equation for BB [38]: BB(x) J 4, (d 4 x)(3d 6 x)bb(x) = x(7 3d + x)bb(x + ) (5.4) This is a first order homogeneous difference equation, whose solution is: x (d 4 i)(3d 6 i) BB(x) = BB(x 0 ) (5.5) i(7 3d + i) i=x 0 Now, a closer look at the difference equation (5.4) shows that it is only valid for x > 0. That means that we have to choose our x 0 in the solution (5.5) to be at least one: BB(x) = [ x i= ] (d 4 i)(3d 6 i) BB() (5.6) i(7 3d + i)