Four-loop Feynman diagrams in three dimensions

Pro gradu -tutkielma Teoreettinen fysiikka

Four-loop Feynman diagrams in three dimensions

Aleksi Vuorinen

2001

Ohjaaja: Prof. Keijo Kajantie Tarkastajat: Prof. Keijo Kajantie, Prof. Claus Montonen

HELSINGIN YLIOPISTO FYSIKAALISTEN TIETEIDEN LAITOS

PL 64 00014 Helsingin yliopisto

Acknowledgements I would like to thank Keijo Kajantie for being my supervisor and for giving me invaluable advice not only during the writing of this thesis but in the whole course of my university studies. In addition I wish to express my gratitude to York Schroder, who has helped me a great deal in the diagrammatic calculations and on whose work much of the material of chapters 4-6 is based. During the past year my student friends Antti Gynther, Tuomas Lappi and Mikko Vepsalainen have taught me a lot of physics and showed what being a physicist is all about. They certainly must be acknowledged. There are also other students, such as Janne Hogdahl, Jonna Koponen, Vesa Muhonen and Teemu Ojanen, with whom I have often discussed matters at least remotely related to physics and who I want to thank. A special thanks naturally belongs to my parents for all their support.

Contents 1 Introduction 2 The free energy of hot QCD

2.1 2.2 2.3 2.4

QCD at nite temperatures . . . . . . . . Dimensional reduction and eective QCD The three-dimensional free energy . . . . The skeleton expansion . . . . . . . . . .

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4

4 7 12 17

3 Evaluation of scalar diagrams in three dimensions

20

4 Diagrams evaluated in momentum space

22

5 Diagrams evaluated in coordinate space

25

6 The remaining diagrams

37

7 Conclusions

43

4.1 Diagram a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Diagram b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Diagram c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Diagram d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Diagram e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Partial dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22 24 24

25 26 28 37 41

Chapter 1

Introduction Richard P. Feynman is quoted saying "the purpose of physics is to produce numbers". One of the most interesting numbers concerning QCD, the theory of strong interactions, is the value of its free energy as a function of temperature. The purpose of this master's thesis is to explore the process of improving the perturbative expansion of this quantity. By providing results for previously unevaluated four-loop scalar Feynman diagrams I also hope to be able to make a small contribution to this work. Soon after the rapid development of the theoretical structure of QCD in early 1970's [1] it was proposed that when a hadronic system is heated or compressed enough, one will at some point observe a dramatic change in its physical properties [2]. This would happen when the wave functions of the hadrons start overlapping and would lead to a transition of the system from a low-temperature phase of baryons and mesons to a high-temperature phase described by a gas of free quarks and gluons. The setup has since then been studied intensively and there now exists strong evidence suggesting that at a critical temperature of Tc  170 MeV QCD matter indeed undergoes a decon ning phase transition into quark-gluon plasma [3]. The reason for the importance of having an accurate theoretical prediction for the free energy at T > Tc is that it aects the probes used in studying the phase transition e.g. in heavy-ion collisions. This is a very hot topic in the current research of high energy physics, since summer 2000 witnessed the start of a new and powerful heavy-ion collider RHIC at Brookhaven, New York. Another collider designed for colliding massive hadronic particles, LHC, is currently being built at CERN and will hopefully start producing data in 2006. The theoretical study of nite temperature QCD matter has traditionally employed two very distinctive methods: high T perturbation theory [4, 5] and lattice simulations [6, 7]. The latter approach is non-perturbative but is very diÆcult to apply if dynamical quarks are present or, especially, if the baryon chemical potential is nonzero [8]. It has however been successfully used in evaluating the B = 0; Nf = 0 free energy at a temperature scale ranging from below the critical temperature up to a few times Tc [9, 10]. Perturbative QCD is, on the other hand, a method based on expanding dierent physical quantities with respect to the gauge coupling g, and is most easily applied in the region T  Tc , where g is small. This is a serious limitation since one is in general not only interested in the limit of asymptotically high temperatures, where the plasma behaves almost like an ideal gas, but 1

also in temperatures near Tc. Another setback comes from the fact that e.g. the free energy gets contributions from dierent momentum scales ranging from T down to g2 T so that the problem is a multiscale one: even if g were small at energies of order T its value may well be considerably larger at other relevant scales. In order to be able to extend the perturbative regime of the free energy down to the interesting region of temperatures of only a few times Tc, one is forced to tackle the technically very demanding calculation of high-order terms in the corresponding perturbation expansion. A straightforward evaluation of these terms in full thermal QCD is, however, not suÆcient all the way, since the loop expansion of F breaks down at order g6 [11]. An eÆcient approach for solving the problem is oered by the construction of eective three-dimensional eld theories [12, 13], which can be used to extract the contributions of small momenta. These theories can in turn be studied by a combination of analytic calculations and lattice simulations [14]. At high perturbative orders it becomes necessary to evaluate large numbers of multi-loop Feynman diagrams in the eective theory. Reaching beyond order g5 in the free energy expansion one is faced among other things with the challenge of calculating numerous fourloop vacuum diagrams in a three-dimensional 'SU(3) + adjoint Higgs' theory without fermions [15]. A software capable of simplifying complicated tensor products, such as FORM [16], can be used to reduce these into a set of four-loop scalar diagrams containing powers and scalar products of loop momenta in the numerators of the corresponding integrands. The evaluation of a subset of these is the main goal of my thesis. The calculation of multi-loop scalar diagrams constitutes an interesting area of research as such and has been approached by many authors both in three and four dimensions: [13, 17, 18, 19, 20] to name a few. The diagrams have applications in various eld theories their use ranging from nite temperature QCD to condensed matter physics. They become necessary whenever a perturbative expansion is performed with respect to a parameter of nite magnitude and numerically accurate predictions are sought after. In this thesis my objective is to evaluate all non-trivial four-loop 2PI (two-particle irreducible) vacuum diagrams of a scalar theory containing couplings of any number of particles. There are altogether ten such diagrams of dierent topology and I shall calculate them analytically and with arbitrary masses of propagators whenever possible. As a regularization scheme for ultraviolet divergencies I am going to use dimensional regularization, i.e. work in 3 2 dimensions, which will in the case of divergent diagrams result in Laurent expansions with respect to . The thesis is organized as follows. Chapter 2 presents an overview of the properties of the QCD free energy and explores its evaluation in more detail. After brie y reviewing the formalism used in nite temperature quantum chromodynamics I concentrate in section 2.1 on the history and present status of the research aimed at nding F . In 2.2 I then explain the construction of the three-dimensional eective theories through dimensional reduction following to some extent the treatment of [21]. After that I concentrate on the recent work of Kajantie et al. [14, 22] and introduce in section 2.3 their setup for obtaining the threedimensional free energy. In 2.4 I further review the skeleton method used in constructing loop expansions for this quantity and then conclude by explaining how the scalar diagrams evaluated in this thesis enter the calculations. Chapters 3 to 6 are entirely devoted to the evaluation of the diagrams mentioned above. In chapter 3 I give a general introduction to the diagrammatic calculations and explain 2

the dierent notations and conventions. The integrals are then divided into three chapters based on the calculational method used: chapter 4 contains three diagrams evaluated directly in momentum space, chapter 5 two diagrams, results for which are obtained after Fouriertransformation into coordinate space and chapter 6 nally ve diagrams with whom other methods have to be applied. In chapters 4 and 5 arbitrary masses are used on each propagator line and results are obtained either in the form of an analytic expression or as 1- or 2-dimensional integrals. For the diagrams of chapter 6 integral representations of various dimensions are constructed in the case where the masses are identical, and they are then evaluated numerically. In addition to this, rst order partial dierential equations with respect to the dierent masses are obtained for these diagrams with the exception of one graph. The topologies of the diagrams evaluated in this thesis are printed below.

;

Diagrams of chapter 4 :

;

:

;

Diagrams of chapter 5 :

;

Diagrams of chapter 6 :

: ;

;

; :

(1.1)

To conclude this introductory section I now mention a few general conventions that will be applied in the following. In all considerations of chapter 2 the masses of the relevant quarks as well as the net baryon density and the baryon chemical potential are assumed to be zero. The rst condition is a valid approximation based on the small masses of the up and down quarks in comparison with Tc . The second constraint is, on the other hand, more severe since e.g. in heavy-ion collisions the net baryon density certainly does not vanish. The generalization of the perturbative calculations to the B 6= 0 -region is, however, in principle straightforward [23] and is not expected to alter the qualitative results signi cantly at least for small values of the chemical potential. As is customary in the literature I will hereafter make a choice of units such that h = c = kB = 1. This is merely a matter of convenience and can at any point be undone.

3

Chapter 2

The free energy of hot QCD 2.1 QCD at nite temperatures Quantum chromodynamics is the gauge eld theory that describes strong interactions. Its symmetry group is SU(3) and it comprises of Nf families of quarks fi in the fundamental representation and the gluons Ai in the adjoint one. Here i and a stand for the color degrees of freedom and run from 1 to 3 for the spin-1/2 quarks and from 1 to 8 for the spin-1 gauge boson, gluon. Nf is not xed but denotes the number of quark avors relevant in each case. In Minkowskian metric the Lagrangian of QCD is X LQCD = 41 Fa F a + f [i  D mf ] f ; (2.1) f where the quarks are represented by 4-component spinors in coordinate space and by 3-vectors in color space, and a = @ Aa @ Aa abc b c F     gf A A ; D = @ + igAa T a :

(2.2)

Here the 3  3 matrices T a ; a 2 f1; :::; 8g are the generators of SU(3) (proportional to the Gell-Mann matrices). The above Lagrangian is invariant under local SU(3) gauge transformations. In their in nitesimal form the transformations read

Aa

! Aa + gf abcAb c @ a; ! (1 + iga T a) ;

(2.3)

where a; a 2 f1; :::; 8g are arbitrary in nitesimal functions of space-time. This invariance leads to problems when constructing a covariant zero-temperature perturbation theory for QCD, since one then needs some method of xing the gauge in the functional integral corresponding to the generating functional of Green's functions. The gauge xing is required in order to avoid an overcounting of physical degrees of freedom and is achieved by introducing unphysical anticommuting scalar particles, Faddeev-Popov ghosts C a , a 2 f1; :::; 8g, in the 4

theory. In nite temperature QCD with zero baryon chemical potential one is eventually lead to a representation for the partition function in the form

Z  Tr e

H

=

Z

D

Aa

D D

DC aDC a exp

"Z



0

d

Z

d3 x

#

Leff ;

(2.4)

where

Leff = LQCD + @ C a@  C a + gf abcC a @Ab C c 21 (@ Aa )2 :

(2.5)

Here  1=T , time has here been rotated into an imaginary one  = {t and 21 (@ Aa )2 is the usual covariant gauge xing term. As always in path integral representations of the partition function, the only bosonic elds that contribute are the ones periodic on the interval 0    and correspondingly the only contributing fermionic elds are the antiperiodic ones. The functional integral in (2.4) cannot be evaluated analytically. It can, however, be used in calculating the dierent thermodynamic observables of QCD to dierent orders in nite temperature perturbation theory, where the coupling constant g is treated as a small expansion parameter (g  1). This seems to be justi ed at high temperatures, since the lowest order solution to the renormalization group equation for g shows that the coupling decreases logarithmically as the energy scale  is increased:

 () 

6 g2 () = : 4 (33 2Nf ) ln (=QCD )

(2.6)

Here QCD is an integration constant originating from the RGE and signaling the divergence of (2.6) at low momenta. Despite the above result, one must be careful in applying perturbation theory even at high T since dierent physical quantities receive contributions also from momentum scales considerably lower than T . Even if g were small at scale T thus allowing a perturbative treatment, the situation may well be dierent at the other relevant scales. The simplest and at the same time the most fundamental observable of a high temperature system is its free energy density f = VF = VT ln Z , which determines various static thermodynamic quantities through standard equations of statistical mechanics. Since in the case of QCD f essentially equals the functional integral of (2.4), the whole nite temperature theory can be seen to be de ned by its value as a function of T and the other parameters, such as coupling constants. Assuming the temperature to be high enough so that the masses of relevant quarks can be neglected one may write the perturbative expansion of f in the form [21] ( being the renormalization scale) "

  82 4  ()  () 3=2  () 2  () f = T F0 + F2 + F3 + F4 ln 45     #  2  5=2    ()  () +F5 + F6 + O 3 () ln  () ; 3 () ;  

5

(2.7)

where the coeÆcients are 21 F0 = 1 + Nf 32  5 15 1 + Nf F2 = 4 12  1 3=2 F3 = 30 1 + Nf 6   135 1 F4 = 1 + Nf 2 6     1 1 135 2 1 + Nf ln 1 + Nf F5 = 237:2 + 15:97Nf 0:413Nf + 2 6 6   5 5  1 + Nf (33 2Nf ) ln 8 12 2T 1=2  1 799:2 21:96Nf 1:926Nf2 F6 = 1 + Nf 6 !  15  1  + 1 + Nf (33 2Nf ) ln : 2 6 2T As can be easily veri ed using (2.6), the -dependence of (2.7) cancels between the dierent terms exactly in such a way that d(lnd ) F = 0 to the order shown. This is of course essential, since  is not a physical parameter. The absence of an  ()5=2 ln  () term from (2.7) is not immediately obvious but is due to remarkable cancellations that occur between various diagrams [24]. To zeroth order in the coupling a strongly interacting system consists of merely noninteracting quarks and gluons. Therefore the leading order term in the above expansion, F0 , corresponds to the well-known result of the free energy of an ideal gas of 8 massless gauge bosons and 3Nf massless fermions (see e.g. chapter 17 of [25]). It depends on the number of colors but is otherwise independent of the structure of QCD. The rst non-trivial coeÆcient, F2 , is on the other hand already clearly characteristic to QCD and was evaluated by Shuryak in 1978 [26]. This required the calculation of a set of four 2-loop diagrams in thermal QCD: 1 1 1 1 F2  + + : (2.8) 2 2 12 8 The coeÆcients F3 and F4 of the rst non-analytic terms in the expansion were obtained, respectively, in 1979 by Kapusta [27] and in 1983 by Toimela [28]. The authors evaluated to leading order the Debye length for color charge, which describes the screening of the chromoelectric force. This involved calculating an in nite set of ring diagrams of the form: " #  1 1 1 +: : : ; (2.9)   2 2 3  where the one-loop gluon self-energy reads 1 1  + + : (2.10) = 2 2 6

In addition there were three 3-loop diagrams that had to be obtained to get all contributions to F3 and F4 . F5 was evaluated by Arnold and Zhai in 1994 [29] and F6 by Kastening and Zhai in 1995 [30]. At this order the running of the coupling constant g must already be taken into account resulting in a renormalization scale dependence in the formulas for F5 and F6 . The authors had to perform a resummation for the gluon propagator just as in the two previous orders and were in addition required to evaluate a large set of 2- and 3-loop diagrams. Although these calculations were originally performed in four-dimensional thermal QCD, the same results were soon derived also using another approach, namely a three-dimensional eective theory [21]. The convergence of the above free energy expansion is fairly weak at temperatures close to Tc . It is therefore essential to attempt to extend it further to the next order, where the use of an eective theory approach already becomes vital. Based on calculations in the full theory it was actually believed for a long time that beyond the g5 term the expansion (2.7) is completely meaningless due to non-perturbative eects that enter at order g6 [11]. The loop expansion for the free energy indeed breaks down at this order because of the chromomagnetic force, which is unscreened at scale gT and renders inapplicable the resummation needed to take into account the chromoelectric screening. For a magnetic screening mass of order g2 T it can be easily seen that diagrams of all loop orders higher than three contribute to F at order g6 (see e.g. [5]). The problem was, however, at least partially solved in the mid 1990's by Braaten and Nieto [12, 13, 21], who used the idea of dimensional reduction to develop two eective three-dimensional theories that agree with QCD at distances larger than 1=(gT ) and 1=(g2 T ), respectively. These theories can be used to extract the contribution of the soft momentum scales gT and g2 T to the free
energy and they thereby provide a practical method for evaluating the next term (O g6 ln g ) in (2.7). The construction of the eective theories is the subject of the next section of this thesis.

2.2 Dimensional reduction and eective QCD The method of dimensional reduction in 3+1 -dimensional eld theories is based on the observation that at suÆciently high temperatures the large distance static correlation functions can be obtained via an eective theory in three dimensions. In the following I will explain the reasoning that leads to this result. Let us consider a generic eld theory, where the masses of the dierent elds are of negligible magnitude in comparison with the temperature. Just as in QCD the partition function may be represented by a functional integral over the dierent elds, where the integrand is an exponential of the eective action with the ordinary time replaced by an imaginary one. The bosonic elds have again periodic and the fermionic elds antiperiodic boundary conditions on the interval [0; ]:

 (x;  = ) =  (x;  = 0) ;

7

(2.11)

which leads to a Fourier representation for them in the form 1 X  (x;  ) = T ei!n  n (x) : (2.12) n= 1 Here the !n's are the famous Matsubara frequencies, which for bosons equal !n = 2nT and for fermions !n = (2n + 1) T . The propagators of the eld modes are of the form !n2 +1 p2 showing that the nonstatic modes have eective masses proportional to T that grow very large, when the temperature approaches in nity. There exists a decoupling theorem due to Appelquist and Carazzone [31], which states that for a renormalizable zero-temperature eld theory containing elds with masses m1  m2 , the Green's functions with typical momentum scales p  m2 (and distance scales x  1=m2 ) m1 , p be calculated using a Lagrangian with the heavy may up to corrections of order m 2 m2 elds removed. In the new Lagrangian the coupling constants have modi ed values and one furthermore needs to add new, possibly non-renormalizable interaction terms in order to reproduce the results of the old theory to a good accuracy. In the nite temperature case it is clearly the bosonic zero modes that play the role of the light elds and become the only degrees of freedom contributing to large distance correlation functions as T increases. The non-static modes are correspondingly the heavy elds and can be integrated out from the theory at high enough temperature. What happens in this process is that the  -dependence vanishes from the Lagrangian making the theory a three-dimensional one. The new theory generally has a simpler structure than the original one due to the lack of fermions and the disappearance of tedious frequency sums from diagrammatic calculations. In addition e.g. 3d lattice simulations are considerably easier and less CPU time consuming than fourdimensional ones. In the case of dimensionally reduced QCD the quarks have been integrated out leaving as the real degrees of freedom the n = 0 -term of the expansion of the gluon eld 1 X Aa (x;  ) = T ei!n  Aa;n (x) ; (2.13) n= 1 where !n = 2nT . The components of the zero-frequency mode are essentially identi ed with the electrostatic scalar eld Aa0 (x) and the magnetostatic gauge eld Aai (x) of a threedimensional eective theory called electrostatic QCD (EQCD), which was constructed in [12] to produce the contribution of momenta of order gT to the full free energy. Although one in principle could construct eective QCD by explicitly integrating out all eld modes with nonzero Matsubara frequency, a more eective way is to simply write down the most general Lagrangian that involves the elds Aa0 (x) and Aai (x) and respects the symmetries of the theory [21]. The Lagrangian constructed in this manner contains free parameters that have to be chosen in such a way that the static correlation functions of the eective theory agree with the ones of the full theory at large distances. In the case of EQCD this distance scale is 1= (gT ) and the Lagrangian is the super-renormalizable one of a three-dimensional 'SU(3) + adjoint Higgs' theory: (2.14) LEQCD = 41 Fija Fija + 12 Di Aa0 DiAa0 + 12 m2E Aa0 Aa0 + 18 E (Aa0 Aa0 )2 ; 8

where

Fija = @i Aaj

@j Aai + gE f abc Abi Acj :

(2.15)

A term proportional to the unit operator appearing in the EQCD Lagrangian has not been included in the de nition of LEQCD and there are in addition higher order non-renormalizable operators involving Aa0 (x) and Aai (x) that have been suppressed here. It has been shown that at large temperatures the contribution of the higher order terms can be neglected to a good accuracy. Using (2.14) the partition function of QCD can be expressed in the form

Z = e = e

V fE (E ) Z

EQCD

V fE (E )

Z E

D D exp Aa

b 0 Ai



Z

d3 x



LEQCD ;

(2.16)

where fE (E ) is the coeÆcient of the unit operator mentioned above and an ultraviolet cut-o E has been introduced in the functional integral corresponding to the partition function of the eective theory. The reason for the appearance of E is that EQCD has been constructed to reproduce the results of the full theory only at large distances and clearly cannot be expected to be correct at arbitrarily high momenta. E plays the role of a factorization scale separating the EQCD regime from the region of high momenta that is described correctly only by full QCD. The coeÆcient of the unit operator, fE , clearly represents the explicit contribution of hard momenta of order T to the QCD free energy and can be evaluated by constructing a strict perturbation expansion for f in the original theory. The computation is performed in [21] and produces the result "





!2

82 3 1 21 g2 () g2 () 5 fE (E ) = T + Nf N + 3 + f 9 5 160 4 (4)2 (4)2   22  116 148  0 ( 1) 38  0 ( 3) E  9 48 ln 4T 3 ln 4T + 5 + 4 + 3  ( 1) 3  ( 3) 3  E 47  367 271 74  0 ( 1) 1  0 ( 3)  + Nf 48 ln ln + ln 2 + 8 + 2 4T 3 4T 20 15 3  ( 1) 3  ( 3)   !# 0  1 88 16  ( 1) 8  0 ( 3) 1 2 20 ln + ln 2 + 4 + + Nf 4 3 4T 3 5 3  ( 1) 3  ( 3)   +O g6 ; (2.17) where the order g6 term is also in principle calculable through four-loop perturbation theory. The Euler constant is numerically  0:577216 and the values0 of the ratio of the Riemann 0 zeta function to its derivative required here are approximately  (( 1)1)  1:98505 and  (( 3)3)  0:645429. The parameter  may be considered arbitrary, since just as in the case of (2.7), fE can be shown to be independent of it. The expression (2.17), however, depends on the 9

factorization scale E explicitly and fE therefore has a non-trivial renormalization group equation with respect to this parameter. The RGE has the solution

fE (E ) = fE 0E




3 2 2 E g m ln : 22 E E 0E

(2.18)

The contribution of lower momentum scales to dierent observables of thermal QCD can be calculated in the eective theory, where one needs expressions for the coupling constants gE , m2E and E . The matching of static correlation functions of EQCD to those of original QCD at distances of order 1= (gT ) and larger leads to expansions for these parameters as functions of T , E and g and implies that besides through fE , the contribution of the energy scale T to the original free energy enters through them. In [21] the quantities needed in the matching process are evaluated as strict perturbation expansions in g, and the same infrared cut-o is applied both to EQCD and to full QCD. This is not the physically correct way of dealing with the infrared divergencies of the theory since one is then totally neglecting the screening eects. It however leads to correct values for the EQCD parameters, which is due to their leading contributions coming from the hard momentum scales. In the following I will brie y review the calculation of the parameters performed in [21] and [22], where relative errors are of order g4 . The three-dimensional gauge coupling gE may at lowest order be directly read o from the QCD Lagrangian, where the gluon elds have been expressed in terms of the EQCD elds using (2.13). Higher order corrections can be obtained either by matching scattering amplitudes in EQCD and full QCD or by calculating the contribution of the neglected eld modes to the correlation function of Aai and Abj . One obtains [22]

g2

E

= g2 () T

"

g2 ()  2  1+ (33 2Nf ) ln 2 T (4) 3

8 N ln 2 + 1 3 f

#

;

(2.19)

where T = 4e T . As the other EQCD parameters, gE is scale independent to order g6 : 2
 @g@E = O g6 . Using (2.6) one may therefore express (2.19) in terms of any scale and  may be regarded as an arbitrary mass parameter. The coupling is independent of the factorization scale E . The mass parameter mE , which is the contribution of scale T to the chromoelectric screening mass mel , can at two-loop level be obtained by matching EQCD and QCD results for the strict perturbation expansion of mel [21]. Just as the works of Kapusta and Toimela [27, 28], also this calculation involves the evaluation of the 00-component of the gluon selfenergy at p0 = 0 in full thermal QCD. Following the gE2 -calculation one may express m2E in terms of an unspeci ed scale  instead of the scale of dimensional regularization. This leads to:  1  m2E = 1 + Nf g2 () T 2 6 " !# 90 + 2Nf2 + 3Nf g2 () 2  8  1 + 2 3 (33 2Nf ) ln  3 Nf ln 2 + 18 + 3N : (2.20) (4) T f 10

The dependence of (2.20) on E is only weak:


@m2 2
8
3gE2 E E E E = @ E (4)2

(2.21)

In order to evaluate E one needs the two-loop eective potential of the full four-dimensional theory. It is calculated in [32] and leads to the result [22]:

E =



1  g4 () T Nf 3 82 " 2  1 + 2 g (2) 23 (33 2Nf ) ln  (4) T

(2.22)

3



8 ln 2 3



7=2 23Nf =18 2 Nf + 3 1 Nf =9

!#

;

where  is arbitrary as before. Clearly E is independent of the factorization scale. Using the above expressions for the dierent parameters of the EQCD Lagrangian one is now able to e.g. evaluate the free energy of the full theory to order g5 [21]. Since the ultimate purpose is, however, to calculate f to an even higher order, one needs to construct another eective theory, magnetostatic QCD (MQCD) [12], in order to further separate the contributions of the scales gT and g2 T . This is achieved by integrating out the electrostatic eld Aa0 (x) and leads to a Lagrangian for MQCD that corresponds to a three-dimensional pure gauge theory:

LMQCD = 41 Fija Fija :

(2.23)

Here Fija is as in (2.15) but with gE replaced by gM , the MQCD gauge coupling constant. Just as in (2.14) a term proportional to the unit operator has been left out from the de nition of LMQCD and higher order operators have been suppressed. The partition function of MQCD is de ned in close analogy with the construction of ZEQCD in (2.16):

ZEQCD = e = e

V fM (E ;M ) Z

MQCD

V fM (E ;M )

Z M

D

Aai

exp



Z



LMQCD :

d3 x

(2.24)

Here fM (E ; M ) is now the coeÆcient of the unit operator of the MQCD Lagrangian and the ultraviolet cut-o M has been introduced to explicitly separate the scales gT and g2 T . The coeÆcient fM as well as the coupling gM can be obtained in terms of the EQCD parameters by matching MQCD and EQCD calculations. The rst one has been evaluated to three loops in [21] with the result being "

fM (E ) =

2 3 m 1 3 E 3gE2 + 4mE

!2

11

3gE2 4mE 89 8



9 E  + 3 ln 4 2mE

!#

2 11 + ln 2 2 2

:

(2.25)

Using (2.16) and (2.24) one may now write down the free energy of full QCD in terms of the eective theory quantities fE , fM and ZMQCD [21]:     1 f = T fE (T; g; E ) + fM m2E ; gE ; E ; E ; M ln ZMQCD (gM ; M ) : (2.26) V Here the eects of the dierent momentum scales are clearly visible. The coeÆcient fE contains the contribution of large momenta of order T to the free energy and can be calculated in full thermal QCD as a power series expansion in g2 (2.17) starting at order g0 . The second part, fM , represents the scale gT in the calculation and is obtained as a perturbative expansion in EQCD (2.25) with the leading term being proportional to m3E . It therefore contributes to the full free energy starting at order g3 . The last term of (2.26) contains the logarithm of ZMQCD and cannot be obtained perturbatively. Lattice simulations are anyway expected to lead to a result for it, where the leading contribution is of order g6 ln g. The role of the dierent momentum cut-os used in the above is now obvious, too. There are altogether four scales that have appeared: the renormalization scale, the factorization scales E and M and in addition the infrared scale QCD . As explained before, the last one is an integration constant coming from the RGE of the gauge coupling g and corresponds to the logarithmic divergence of (2.6) at small momenta. It can be viewed as the natural infrared cut-o of full QCD and it provides a scale for all considerations here. M , on the other hand, merely separates the MQCD and EQCD regions being at the same time both the ultraviolet -cut o of MQCD and the IR -cut o of EQCD. The role of E is similar to that as it acts as the scale separating EQCD from the full theory. Finally, the momentum parameter corresponding to the renormalization of the original theory is an ultraviolet one of order 1/(fm) but has no actual physical meaning.

2.3 The three-dimensional free energy Ever since the g5 term of the free energy expansion (2.7) was rst obtained, the both theoretically and technically very challenging task of further improving the validity of the series has been tackled by various people (see e.g. [6, 14, 33]). The evaluation of the next order term was rst outlined in [21] by Braaten and Nieto, who took advantage of the separation of scales T , gT and g2 T in (2.26) provided by their eective theories EQCD and MQCD. The contribution to the order g6 term from momenta of order T was shown to be available through a four-loop calculation of fE in the full four-dimensional theory and an O() evaluation of the constants gE2 and E . In order to obtain the gT part one must, on the other hand, evaluate fM as a perturbative expansion in EQCD up to four-loop order and there use the order g4 result (2.19) for the parameter gE2 . Finally the authors argued that the contribution from the magnetic sector can be calculated by determining the coeÆcients a and b in the expansion 6 . According to them a is non-perturbative and may only be ln ZEQCD =V = (a + b ln g2M )gM M obtained by means of lattice simulations in a three-dimensional pure gauge theory, whereas b can be calculated analytically by simply evaluating four-loop MQCD vacuum diagrams. It has, however, been shown in [22] that such a straightforward picture has an inconsistency in the construction of MQCD, since it is only at unreasonably high temperatures that the 12

magnetic sector contributions to the free energy of the three-dimensional theory can be seen to be small in comparison with the electric ones. Kajantie, Laine, Rummukainen and Schroder have recently applied methods similar to the ones described above to combine analytic calculations with lattice Monte Carlo simulations in order to extract the contributions of the dierent energy scales to f . In particular they have built in [14] a new method for resumming the long-distance contributions to the free energy of a three-dimensional eective QCD. This work is based on the papers [22], where a 3d 'SU(N) + adjoint Higgs' theory with a Lagrangian corresponding to (2.14) is studied at high temperatures.
In [22] the authors de ne dimensionless parameters x = E =gE2 and y = m2E 3 = gE2 =gE4 (3 being the MS dimensional regularization scale in three dimensions) and show that choosing such renormalization scales for the eective theory parameters that their next-to-leadingorder corrections vanish leads to: 242 1
T; 33 2Nf ln g T =MS 9 Nf 1
and x = 33 2Nf ln x T =MS (9 Nf ) (6 + Nf ) 486 33Nf 11Nf2 2Nf3 y = + + O (x) ; 1442 x 96 (9 Nf ) 2

gE2 =

(2.27)

where N has been set equal to three, !

i = exp

54 22Nf 3ci + 4Nf ln 4 4 ; cg = 1; cx = Nf + ; MS  TC 66 4Nf 3 9 Nf

(2.28)

and T is as in (2.19). With help of equations (2.19)-(2.22) it can be observed that to leading order x  g2 and y  1=g2 . Using the parameters x and y Kajantie et al. write in [14] the pressure (= f ) of full QCD in the form "

p (T ) = p0 (T ) 1 45 + 2 8

gE2 T

5 x 2

!3 

 3  3d y ln +Æ 22 T



FMS (x; y) + O



g6



#

;

(2.29)

where 3d may be identi ed with E and Æ is an analytically known constant of negligible  2 4  magnitude that can be obtained from (2.17). The factor p0 (T ) = 845T 1 + 21 32 Nf represents the ideal gas result discussed above (the term F0 in (2.7)), and when written in terms of the full QCD gauge coupling g, the rst three terms of (2.29) have perturbative expansions starting respectively at orders g0 , g2 and g4 . The term FMS (x; y), which essentially equals the previous gE6 fM , is the dimensionless free energy of three-dimensional QCD and represents 13




the contribution of momenta of order gT and smaller to the original f . The O g6 error in (2.29) originates from higher momentum scales and is therefore not connected to FMS . In the new notation FMS (x; y) can be written as [14]

FMS (x; y) 8

"

#

1 1 1 9 3  5 = + p
2 ln 4y + 3 ln 32d + x 4py 3 4 2 2 g 4 y E "   1 267 32 33 1 + p
3 + ln 2 15 ln 4y x 8 2 2 2 4 y  #! 
F (x; y) 1 ln 16y x2 + MS ; +5 2 8

y2

(2.30)

where the expansion parameters are now x and 1= 4py . This expression is equivalent to (2.25) except that here the self-coupling of the Aa0 elds has been taken into account (x has not been set equal to zero). Apart from the other x-dependent terms it is clearly
FMS (x; y) that represents the previously undetermined part of f . The approach of [14] is to dierentiate
FMS (x; y) with respect to x and y, calculate the derivatives by means of lattice studies and then nally numerically integrate the result back in the form



FMS (x; y) =
FMS (x0 ; y0 ) ! Z y @
FMS (x (y) ; y) dx (y) @
FMS (x (y) ; y) + dy + : (2.31) @y dy @x y0 The partial derivatives can easily be related to expectation values of dierent operators using the path integral representation of ZEQCD (2.16). In fact the two terms in the integrand of (2.31) are respectively the dierences of the quadratic and quartic A0 condensates and their perturbative results in MS scheme, and the nonperturbative values can be obtained from lattice ones through multi-loop calculations both in ordinary and lattice perturbation theory. For the y-derivative the equation relating it to the condensates is

@
FMS Aa Aa = 0 20 @y 2gE MS *

+

*

+

Aa0 Aa0 ; 2gE2 MS; pert.

(2.32)

where the perturbative result can be directly read o from (2.30). The dierence as measured with dierent lattice spacings is shown in gure 2.1. The integration constant in equation (2.31),
FMS (x0 ; y0 ), is on the other hand expected to have the form 8
33 x 3
FMS (x0 ; y0 ) = e0 1 + O 0 ; 1=2 4 3 (4) 4y0

!!

;

(2.33)

where e0 is an unknown parameter. If
FMS (x0 ; y0 ) is xed at a temperature T0 = 1011 MS , where x0 = 0:01 and y0 = 3:86, the terms not explicitly shown in (2.33) can be neglected 14

_

log10(T/ΛMS)

0.0

0.4

2.0

4.0

6.0

8.0

10.0 3

3

3

3

3

3

3

3

βG=12, vol=12 −32 βG=16, vol=16 −32

− pert

0.3

βG=24, vol=24 −32 βG=32, vol=32 −64 continuum limit

0.2

0.1

0.0 0.0

1.0

2.0

3.0

4.0

y

Figure 2.1: Equation (2.32) as
measured using Monte Carlo simulations with dierent lattice spacings a. Here G = 6= gE2 a and the continuum limit naturally corresponds to G ! 1. The gure is taken from [14].

1.5 e0=0

e0=9

1.0

e0=10

p/p0

e0=11 e0=14

0.5

4d lattice 0.0

1

10

_

100

1000

T/ΛMS

Figure 2.2: The behaviour of the QCD pressure as a function of e0 [14]. For e0 = 10 statistical errors are indicated. The lattice data is from [9]. 15

and knowing the value of e0 suÆces. As can be seen from gure 2.2, at temperatures below (20 50)Tc the behaviour of the perturbative pressure depends strongly on e0 , which is presently being treated as a free parameter. The gure is taken from [14], where the authors note that the expansions computed by them are reliable only when T  5Tc but that by taking into account also the quartic condensate neglected so far, one will in principle be able to reduce the lower limit down to 2Tc . The next logical step in the perturbative determination of the free energy at temperatures near Tc would seem to be the explicit evaluation of e0 through extensive lattice studies. This is, however, a very complicated task requiring e.g. four-loop calculations in ordinary and lattice perturbation theory and is not presently being pursued. The complexity of the problem is not surprising at all, since one easily sees from (2.29)-(2.33) that e0 represents the contribution of the infamous g6 term to the original free energy expansion. As mentioned earlier, it was shown by Linde already in 1980 [11] that this part is entirely nonperturbative, since it gets contributions from full QCD diagrams of all loop orders greater than three. Regarding e0 still as a free parameter it is however notable that for values e0 = 10:0  2:0, as Kajantie et al. estimate, the above calculations produce a convincing agreement between the perturbative results and four-dimensional lattice data ( gure 2.2). The setup of [14] has therefore produced a theoretical framework, which has the capacity of explaining the behaviour of the free energy on a temperature range from a few times Tc all the way to in nity, while taking the long-distance contributions systematically into account. Using this approach one is in particular able to see, why the long-distance eects do not necessarily cause a large deviation between the measured pressure and its g5 -result ( gure 2.3). In comparison with previous results and other approaches, e.g. [6, 33], this is a signi cant improvement. 1.5

p/p0

1.0

F=0 3/2 F up to y F up to y

0.5

1/2

F up to y 4d lattice 0.0

1

10

T/Λ_

100

1000

MS

Figure 2.3: The perturbative pressure of (2.29) as taken from [14], where dierent orders of the expansion (2.30) have been taken into account. 16

2.4 The skeleton expansion Determining the loop expansion of the free energy in a three-dimensional 'SU(3) + adjoint scalar' theory clearly plays an essential role in the computation of the perturbative free energy of full nite temperature QCD. At four-loop level the number of dierent diagrams contributing to FMS is already considerable and some method is needed for organizing and grouping them. This is provided by the so called skeleton expansion [34, 35], which was used in [15] for writing down the loop expansion of the QCD free energy in terms of two-particle irreducible diagrams (skeletons) and ring diagrams containing self energy insertions. It was further shown in [15] that the self energies needed in the computation can be straightforwardly obtained from lower order skeletons. Below I shall brie y introduce this method and explicitly list all skeleton diagrams up to four-loop order that are needed in the calculation of the free energy of the eective theory and that eventually lead to the scalar integrals of the later chapters. Let us rst study the free energy of a generic eld theory along the lines of [15] and denote the full propagators of the dierent elds by Di , the free ones by
i and the proper selfenergies by i . The starting point of the skeleton approach is a formula derived in [34, 35], which gives the free energy in terms of a function  [D] containing all vacuum skeletons:

F [D] =

X

i



ci Tr ln Di 1 + Tr i [D]Di



[D] :

(2.34)

Here the sum goes over bosons and fermions with ci being 1=2 for bosons and 1 for fermions. The equation can be modi ed by writing the right hand side in a form containing only free propagators, which is an essential simpli cation from the point of view of practical calculations. Denoting the terms in the loop expansion of  by n and expanding the Di 's in terms of self-energies, the authors of [15] obtain:

F =

F0 + 2 [
] + 3 [
] + + 4 [
] + + 5 [
] + + 12 2

X

i X

i X

i

2 + 12 2

ci

ci

ci







1 2 1

1 31

1 1

!

1 + 1

2 + 21 1

2

!

2 2 1 1 1 + 1 + 4 1 1 1 1 21 1

2 + 1

3 + 12 1

3 + 13 1

3



;

(2.35)

were the circles and squares are respectively irreducible and reducible self-energies and the loop order is indicated by the number inside. As mentioned above, all self-energies are available through the n 's. Since the ring diagrams are generally considerably easier to evaluate than the skeleton ones it can be seen from (2.35) that what is most urgently needed in order to obtain the free energy to dierent orders are the terms n that originate from the loop expansion of . In 17

the case of the three-dimensional QCD containing as particles Aai -gauge bosons -scalars and ghosts obtains [15]: + 121

1 2

2 =

1 8

3 =

1 24

1 3

1 4

+ 21

+ 14

+ 81

4 =

1 72

;

, and as vertices

1 3

1

+ 14 + 481

1 2

+ 61

+ 61

+ 18

1 2

+ 18

+ 18

+ 121

1 3

+ 16

+ 121

+ 21

+ 14

1 2

+ 14

+ 81

+ 12

+ 21

+ 18

+ 21

+ 12

+ 81

+ 14

+ 21 + 41

+ 161

+ 16

+ 14

+ 18

+ 161

+ 18

+ 161

+ 81

1 4

+ 14 + 21

+ 21 + 81 +1

+ 41

+ 161

+ 12 + 41

+ 481

+ 41 +1

+ 81

+ 14

(2.37)

+ 481

+ 41

+ 41

(2.36)

1 2

+ 161 + 41

;

;

+ 121

+ 14

+ 18

+ 481

1 6

; one

and

+ 61

+ 18

+ 18

;

+ 41

+ 18

1 4

;

, Aa0

+ 161 + 21

:

(2.38)

The evaluation of the loop expansion of FMS to dierent orders has here been reduced to the laborious task of calculating well-de ned but large sets of Feynman diagrams in three dimensions. The diagrams contain propagators and vertices of gauge bosons, ghosts and massive scalars making their evaluation technically very diÆcult. A software designed for the symbolic manipulation of mathematical expressions such as FORM [16] can be used as a powerful tool in further reducing (2.36)-(2.38) into a set of scalar diagrams containing powers and scalar products of loop momenta in the numerators of their integrands. The evaluation of these scalar integrals remains, however, still to be done. The remaining of this thesis is devoted to the calculation of a subset of the four-loop scalar diagrams mentioned above, namely all scalar graphs originating from 4 but having no contributions in the numerators of the corresponding integrands. The dierent topologies 18

found from (2.38) constitute the diagrams of chapters 4 and 6, but in addition to this I will in chapter 5 evaluate two four-loop diagrams containing ve-vertices. They are needed in lattice perturbation theory, where the n 's get additional contributions with the following topologies [15] (these diagrams are for a generic eld theory):

3 lat =

1 12

4 lat =

1 8



+ 481

+ 481 + 121

;

(2.39)

1 + 240

+ 721

+ 121 + 481

+ 81

+ 161

+ 481

1 + 384

(2.40) :

As scalar diagrams the non-trivial ones in the above expression for 4 lat are clearly the second and third diagrams, since the rst is calculationally equivalent to the N topology (see chapter 3) appearing already in (2.38) and the others factorize into products of lower-order integrals.

19

Chapter 3

Evaluation of scalar diagrams in three dimensions The treatment of the four-loop diagrams of this thesis will be somewhat analogous to the three-loop considerations of Rajantie in [20]. Due to the nature of the eective threedimensional QCD, the Feynman rules applied can be chosen to be extremely simple.
Each solid line with momentum p will be represented by the propagator
(p)  1= p2 + m2a and in all vertices momentum is conserved but no additional vertex function or coupling constant is inserted. The metric is Euclidean and the integrations will be performed in 3 2 -dimensions in order to regularize ultraviolet divergencies; no IR-singularities will arise, since the propagators are massive. The integration measure chosen here is the MS one: Z 4m1+2 Z d3 2 p = ; (3.1) (e ) (2)3 2 p where m is an arbitrary mass parameter. In the calculations I will use arbitrary propagator masses whenever possible, which is essential not only from point of view of generality but also since in certain applications of the eective QCD dierent kinds of mass con gurations may appear. When a scalar integral has been evaluated with arbitrary masses, one is in addition able to instantly obtain results for diagrams with the same topology but with propagators raised to arbitrary powers through simple dierentiation of the initial result. There furthermore exist analytic relations such as the equation 1 of [20], which relates n-1 -loop 2-point functions to n-loop vacuum diagrams, that may be applied to the results of this thesis and that naturally are considerably more useful when arbitrary propagator masses are used. To illustrate the above conventions let us now consider the simplest possible vacuum scalar diagram in three dimensions, which is nothing but the single bubble: a Z  I (ma )  p2 +1 m2 : (3.2) p a Performing the integration using standard formulae results in ! 4m1+2 23=2  Z 1 dp m2a 2  I (ma ) = p 1 2 (e ) (3=2 ) 0 (2)3 2 p + m2a 20

m1+2 m1a 2 e  Z 1 x 1=2  p dx 21 2  (3=2 ) 0 x+1 1+2  1 2 

 m p ma e (1=2 + ) (1=2 ) 1 2  2  (3=2 ) (1) !

2 1 (1=2 + ) 4e m mma 1 2 (1=2) m2a

= = =





m (1 + 2) (1  ( + 2 ln 2)) 1 +  2 ln 2 + + 2 ln ma     m  2 mma 1 + 2 1 + ln +O  ; ma

= =



mma (3.3)

where the vanishing of dimensionless integrals under dimensional regularization has been used 0 along with the known value of the function (z )  ((zz)) at z = 1=2: (1=2) = 2 ln 2. If one were interested in the value of (3.2) with the propagator squared, a simple dierentiation with respect to m2a would produce the correct result. As will be seen later, calculations to higher loop orders are in many ways analogous, but they often require the use of more sophisticated integration methods. In the case of convergent diagrams I will hereafter set  = 0 already in the beginning of the calculation and will actually not evaluate the O () part of any diagram. Each result is therefore to be viewed as being of order 0 . The set of diagrams considered in this thesis is printed in gure 3.1. In the following I shall denote the value of each diagram as a function of its propagator masses as Ii (mi1 ; :::; min ), where the ordering of the masses can in each case be found upon comparison with the diagrams below, which correspond to Ii (m1 ; :::; mn ). This canonical expression will from now on be abbreviated simply by Ii . For a few diagrams I will also use speci c names given below, the origin of which is obvious from the gure. It should, however, be kept in mind that the illustrations of the dierent topologies in g 3.1 are by far not unique: e.g. the 'wheel' diagram can easily be seen to be equivalent to the letter 'A' inside a circle.

a)

1

26 4 3

5

b)

1 3

2 4 6

6

7

c) 1

5

f)

3

5

0 triangle0 6 4 5 7 8 1 2 3

27 4

1

0N 0

g) 6

4 15 2 3

7

h)

1 2 4 6 3 7 5 8

0 wheel0

d) 1

23 4

e) 2

5

0 basketball0

i) 4

1 2 7 3 6 8 9

5

j)

0H 0

Figure 3.1: Diagrams evaluated in this thesis. 21

3 4 56

0 twisted H 0

Chapter 4

Diagrams evaluated in momentum space 4.1 Diagram a The triangle diagram has been studied both in three and four dimensions [13, 36] with the result that its divergent part has been found. As will be seen here, the nite term of the three-dimensional case is also available through straightforward calculations. Using the Feynman rules stated above the value of the diagram can be seen to correspond to the four-fold momentum integral Z Z Z Z 1 1 Ia  2 2 2 p q r s p + m1 (p s) + m22 1 1 1  q2 +1 m2 (4.1) 2 2 2 2 2 2: 3 (q s) + m4 r + m5 (r s) + m6 Counting the powers of momenta in the numerator and denominator one easily observes that this expression indeed diverges in three dimensions necessitating the use of d = 3 2. The diagram contains three identical one-loop subdiagrams in the form of a two-point function having the representation (see e.g. [37]): p

1 2

Z

1 1 2 2 2 2 q + m q 1 (q p) + m2 Z e  m1+2 (1=2 + ) 1 1 = 1 2 p 2 1=2+ dx

2 0 2  (p ) x(1 x) + xm1 + (1 x)m22 =p2 1=2+

 1+2 (4.2)  e 1m2p (12=21=+2+) B (p; m1; m2 ; ) ; 2  (p )



where the function B possesses the properties

p

22  (1=2 ) lim B ( y;  ;  ;  ) = 1 2 y!1 (1 ) 22

(4.3)

y  : (4.4) B (y; 1 ; 2 ; 0) = 2 arctan 1 + 2 The use of this result simpli es the evaluation of the original integral signi cantly. The angular part of (4.1) becomes trivial due to (4.2) and one is lead to a one-dimensional integral representation for the triangle diagram: (1=2 + )3 Ia = 2 3  2 m4 (4e )4 (4.5) (3=2 )       Z 1  dy y 1 8B y; mm1 ; mm2 ;  B y; mm3 ; mm4 ;  B y; mm5 ; mm6 ;  : 0 The divergence of the initial diagram is here still contained in the integral, whose expression has a pole in . That can, however, be conveniently separated by performing a partial integration. Due to (4.3) the boundary terms vanish and one is left with the expression (1=2 + )3 1 Ia = 2 3  2 m4 (4e )4 (4.6) (3=2 ) 8        Z 1  dy y 8 dyd B y; mm1 ; mm2 ;  B y; mm3 ; mm4 ;  B y; mm5 ; mm6 ;  ; 0 where the remaining integral is completely nite. The factor y 8 can now be expanded in  leading to (1=2 + )3 Ia = 2 3  2 m4 (4e )4 (3=2 )       m1 m2 m3 m4 m5 m6  81 1 B y; ; ;  B y; ; ;  B y; ; ;  0 m m m m m m !     Z 1 d m1 m2 m3 m4   m5 m6  dy ln y B y; ; ;  B y; ; ;  B y; ; ;  dy m m m m m m 0 

p

!3

(1=2 + )3 1 22  (1=2 ) = 2 (4.7) (3=2 ) 8 (1 ) !    Z 1 d  my  my  my  8 dy ln y arctan arctan arctan : dy m1 + m2 m3 + m4 m5 + m6 0 3  2 m4 (4e )4

Here I have used the result (4.4) and in addition suppressed O () terms. Using the value (1) = one may expand also the rst term in (4.7) nally obtaining:  1 5 2 4 Ia = 2  m + 12 ln 2 + 2 (4.8)         64 Z 1 d my my my dy ln y arctan arctan arctan : 3  0 dy m1 + m2 m3 + m4 m5 + m6 It seems that the remaining integral can be performed analytically only in some special cases. One such case is when m1 + m2 = m3 + m4 = m5 + m6  M . One then has:    M 21 1 5 2 4 + 12 ln 2 + 2 4 2 ln + 2  (3) ; (4.9) Ia = 2  m  m  23

which in the case of identical propagator masses (mi = m 8i) gives numerically:   1 Ia (m; :::; m)  0:3084251375 5:4580928494  m4 

(4.10)

4.2 Diagram b If an extra propagator with mass m7 is added to the triangle diagram, it becomes nite in three dimensions. Using the above calculations one may then instantly write it in the form analogous to (4.5): 2m2 = 

Z

1





1 my  arctan dy  Ib 2 m m1 + m2 0 y y2 + m72 !   my  my   arctan m + m arctan m + m : (4.11) 3 4 5 6 This integral can easily be evaluated numerically with each mass con guration. When all masses are identical one obtains:

Ib (m; :::; m) = 0:1291074598  m2 :

(4.12)

4.3 Diagram c The N diagram is also clearly nite in three dimensions so one may at once set  = 0. Taking advantage of the familiar subdiagrams one is able to perform two momentum integrations, which eventually results in the formula

Ic

m4 = 4 4

Z

d3 p

Z

1 d3 q p 2 2 pq

arctan



jpj

m1 + m2

 p2 +1 m2 q2 +1 m2 (p q)12 + m2 : 5 6 7



arctan



jqj



m3 + m4 (4.13)

The angular integrals can be calculated in a straightforward way leading to a twodimensional integral representation for the N diagram:     mx my m2 Z 1 Z 1 dx dy arctan arctan Ic = 2  0 m1 + m2 1 m3 + m4 0 0 m27 2  2 1 m25 2 1 m26 ln @ (x + y)2 + mm227 A : (4.14) x + m2 y + m2 (x y) + m2 Again it seems that the remaining integrals can only be performed numerically with the possible exception of some special mass con gurations. In the case mi = m 8 i one obtains Ic (m; :::; m) = 0:1077181618  m2 : (4.15) 24

Chapter 5

Diagrams evaluated in coordinate space 5.1 Preliminaries When the number of vertices in a diagram is small, one possibly fruitful approach is to convert the momentum space integrations into coordinate space ones by taking Fourier transforms of the propagators. This is based on the fact that the number of coordinate space integrals needed in each case is one less than the number of vertices. The coordinate representation of a propagator is, however, considerably more complicated than the momentum one and in addition the angular integrals often prove to be very diÆcult in coordinate space. This method has been used extensively e.g. by Braaten and Nieto [13] and by Groote, Korner and Pivovarov [19], whose results in angular integrals I will quote in the following. The fundamental formula in coordinate space calculations is the 3 2 -dimensional Fourier transform of a massive propagator:  m2 Z 3 2 e {p
x ma 1=2  m2 D (x; ma )  d p = K1=2  (ma x) ; (5.1) p2 + m2a (2)3=2  x (2)3 2 where K1=2  denotes a modi ed Bessel function. When expanded with respect to x at small values of the variable the propagator gives !

  (ma x)2 m2 (1=2 ) 1+2 4 D (x; ma ) = x 1 + + O ( m x ) a 41  (1=2) 2 (1 + 2)   m2 ( 1=2 + ) 1 2  2 m 1 + O ( m x ) a a (4)1  ( 1=2) " !   (ma x)2 1+2  4 1 2  + O (ma x)  ma  (max) 1+ 2 (1 + 2) 



1 + O (ma

x)2



25

#

;

(5.2)

where the coeÆcients  and have the following -expansions: h  i m2 (1=2 ) 1 2 = 1 + 

+ ln 4 m 41  (1=2) 4 !     2    2 2 2 2 2 3 +  + 2 + 4 ln 4m + 2 4m +O  ; 4 h  i   m2 ( 1=2 + ) 1  2 + O 2 : = = 1 +  2

+ ln m (4)1  ( 1=2) 4

 =

(5.3)

Correspondingly, the -expansion of D itself is D (x; ma ) =

e

!

!

!

  2m2 x + e2ma x Ei ( 2ma x) + O 2 ; 1 +  ln ma

ma x

4x

(5.4)

where Ei (x) denotes the exponential integral and the relations (equations 8.469 and 8.486(1) of [38]) r

K1=2 (x) = 

@ K (x)  = @  =1=2

 x e 2x   1=2 x e Ei ( 2x) 2x

(5.5) (5.6)

have been used.

5.2 Diagram d The basketball-type diagrams containing only two vertices are especially well suited for integration in coordinate space due to their trivial angular structure there. After writing the propagators in terms of their Fourier transforms one obtains in the four-loop case:

Id = m

10

4m1+2 (e )

=

4m1 =2 (e )



4m1 =2 (e )

!4 !4

!4 Z

d3

23=2  (3=2 )

5 2 x Y D (x; m i=1

1

Z

0

23=2  I: (3=2 )

dx x2

i)

5 2 Y D (x; m i=1

i)

(5.7)

It can be easily seen that the integral I diverges near the origin. One may then simplify the calculation by introducing a small parameter r and dividing the radial integration line into two subintervals [0; r] and [r; 1[ (I  I1 + I2 ). On the rst interval the small-x expansion of the propagators can be used, whereas on the latter one may simply set  = 0. The parameter r is considered to be arbitrarily small but nonzero so that all contributions to the integrals 26

proportional to its positive powers can immediately be neglected. If the calculations are performed correctly, all r-dependence will in any case naturally vanish from the nal result. Using (5.2) one obtains on the rst interval:

I1 =

Z r

0



dx x2

5 Y i=1

2

(5.8)

m1 2

"

1+2

 (mi x)

i

!

  (mi x)2 + O (mi x)4 1+ 2 (1 + 2)





1 + O (mi

x)2

#!



which reads after neglecting the terms that vanish in the limit r ! 0:

I1 =

Z r

0

dx x2

2



5 x

5+10

4

5 X i=1

m1i

2

x

4+8

 3 2 X (mi mj )1 2 x 3+6 : (5.9) 2 (1 + 2) i=1 2 i=6 j If  is assumed to be large enough for the lower limits to vanish (later one may analytically continue the result to the neighborhood of  = 0) and irrelevant (higher order in ) terms are dropped, the integrations can be straightforwardly performed with the result:

5 X

5

+

m2i x

3+10 +

M

5 r 2

z }| { 5 X

5 X

3 2 r4 X (mi mj )1 2(5.10) : 8  i=1 i=1 i= 6 j On the second interval the calculation is considerably easier, since with  = 0 one only needs to evaluate the integral

I1 =

2 + 4

mi r

1+

5 r8 16 (1 + 2)

m2i +

1 Z 1 e Mx dx 3 : (5.11) x (4)5 r Performing several partial integrations and at each step dropping terms that vanish when r ! 0 one eventually obtains: ! 1 e Mr Me Mr M 2 e Mr ln (Mr) M 2 Z 1 x I2 = + dx ln x e 2r 2 2 0 (4)5 2r2 !  2 ln (Mr ) 1 M 1 3 M 1 = +

M2 : (5.12) r 2 2 2 (4)5 2r2

I2 =

Adding up I1 and I2 , expanding to 0th order in  and simplifying the result in a straightforward manner one may now write down the result for the whole diagram:

Id = m4





h

1 +  2 + 3

1 16

2 4

5 X i=1

m2i + 2



ln 45 m2 X

i6=j

i

3

mi mj 5 +

1 16 27





2 + 5 + 5 ln 4m2

5  X i=1

m2i +

;

  X 1 4 + + ln 26 5 m10 mi mj 8 i6=j

M2 3 + 2 2

2



M 2 ln M 2

!

1X m m ln mi 2 i=6 j i j

!

(5.13) 3

!

  X M 2 m4 1 4 X mi mj 5 3 2m mi mj   m  3  = 1+ + + ln + ln + : 2 8 M2 2 M M2 mi 2 i6=j i6=j

When all masses are equal one obtains:     40 2 1 45m6 1 + 12 + ln  2:8125  + 7:9275967472  m6 (5.14) Id (m; :::; m) = 16  9 5

5.3 Diagram e This divergent diagram has a rather complicated structure and its analytic evaluation in momentum space might well turn out to be impossible. First Fourier-transforming the propagators and then performing the momentum integrals with help of delta-functions one, however, ends up with the more manageable formula:

Ie

!4

4m1  Z 3 2 Z 3 2 = d x d y D (j x y j; m1 ) (e ) D (x; m2 ) D (x; m3 ) D (y; m4 ) D (y; m5 ) D (y; m6 ) ;

which is illustrated by the picture

(5.15)

x1 y 2

3 4 56

:

(5.16)

0 The angular integrations of (5.15) can be performed using the identity Z K (j r  j) I () K (r) d   = (2)+1     (5.17)  jr j  r derived by Groote et al. [19] and which holds when r > . This leads to the expression:   3 6 4m 4 Z 1 Z 1 2 2 y 2 2 Y D (x; m ) Y D (y; m ) Ie = d x d y x i i (e ) 0 0 i=2 i=4  (D (x; m1 ) G (m1y)  (x y) + D (y; m1) G (m1 x)  (y x))   4m 4  (e ) I ; (5.18) where 25=2  3 2 I1=2  (ma x) G (ma x) = : (5.19) (3=2 ) m4 (ma x)1=2  28

The function G has the small-x -expansion    43 2 2 G (ma x) = 1 + O ( m x ) a (3=2 )2 m4      1 + O (max)2 ;

(5.20)

where the coeÆcient  further has the -expansion h  i 43 2 2 1 + 2 2 ln 4m2 = (4  )  = (3=2 )2 m4 " # !    2   2 2 2 2 2 3 8 + 2 + 4 ( 2) ln 4m + 2 4m + O  :(5.21) + 12 2 With  set to zero G reads sinh (ma x) G (ma x) = (4)2 : (5.22) ma x Since it is obvious that the divergence of I comes from the UV area, one may again simplify the calculation by dividing the integration plane into dierent regions: A : 0  y  x  r B : 0  x  y  r C : 0  y  r  x < 1 D : 0  x  r  y < 1 E : r  y  x < 1 F : r  x  y < 1; (5.23) which is depicted in the diagram below. The parameter r can as before be thought to be arbitrarily small but nonzero, and hence I will at each stage of the calculations discard terms that vanish as r ! 0.

y F

D

(5.24)

E r

B

A

C r

x

From now on I will use a new notation for the sums and products of masses: mi + mj + mk + :::  Mijk::: mi  mj  mk  :::  M ijk:::: 29

(5.25)

Region A In this integration region the variables x and y are arbitrarily small so one may certainly use the small-x expansions. Denoting y = tx (Jacobian = x) and neglecting terms that vanish in the limit r ! 0 one obtains Z r

IA =

0

6

=

Z x

dx

Z r

0

0

dy x2

dx

Z 1

0

2 y 2 2 G (m

1 y)

1+8 t 1+4

dt x

3 Y i=1

D (x; mi )

= 6



r8 322

6 Y i=4

D (y; mi )



  1 4 1 2 r2 + 1 +

+ ln 4 m = 32 (4)4 2  !      2  2 +4 3 + + 4 + 2 2 + 4 (1 + ) ln 4m2 r2 + 2 ln 4m2 r2 : (5.26) 4

Region B In region B the calculation proceeds exactly in the same fashion as in the previous one and one is straightforwardly lead to the expression

IB = =

Z r

0

dy

6

Z y

0

dx x2

3 6 2 y 2 2 G (m x) D (y; m ) Y D (x; m ) Y D (y; m ) 1 1 i i i=2 i=4

r8 1 = 8 (1 + 2) 8 (4)4





h  i 1 + 2 1 + 2 + 2 ln 4m2 r2 : 

(5.27)

Region C In the evaluation of the diagram e it is the region C that clearly produces the most diÆculties. This is due to the fact that while the y-integral contains here a divergent contribution from the region near the origin necessitating the use of a nonzero , the x-integration is being performed from r to 1 making the application of the small-x -expansion for D (x; mi ) impossible. Proceeding as before one obtains:

IC = =

1

Z

r

3

= 3

dx

Z r

1

Z

r

0

dx 1

r4 Z 4

dy

r

x2 2 y2 2 G (m

1 y)

3 Y 2 2  x D (x; m i=1

dx x2

2

i)

3 Y i=1

3 Y

i=1 Z r

0

dy y

D (x; mi ) :

30

D (x; mi )

6 Y i=4

D (y; mi )

1+4

(5.28)

In order to obtain also the nite part of (5.4) in the x-integral resulting in 1

Z

3 2 Y D (x; m

dx x2

r

1 = (4)3

IC one must now apply the -expansion of D

1

Z

r

i=1

dx

i)

e

M123 x

x

(5.29)

"

1 +  ln x + 3 ln



2m2



3 X ln M 123 + e2mi x Ei ( i=1

!#

2mi x)

:

The evaluation of this integral requires the calculation of several elementary one-dimensional integrals on the interval [r; 1[. Performing partial integrations, using standard formulae and neglecting terms that vanish in the limit r ! 0 one obtains: Z 1 e kx  Ei ( kr) dx x r Z 1 Z kr 1 = kr ln x e x + dx ln x e x dx ln x e x 1

Z

r

dx

e

kx

x

= =

ln x = = =

Z

r

1

e

dx

kx

x

 Ei ( 2mx) =

=

=

0

0

ln (kr) e kr

ln (kr) (5.30) Z 1 x e dx (ln x ln k) x kr Z 1 Z 1 1 1 e x 2 x+1 2 e x ln k (ln x ) d x (ln x) e 2 kr 2 0 x kr ! 2 1  1 (ln (kr))2 + + 2 ln k ( + ln (kr)) 2 2 6 f (k; r) (5.31) 2 m   Z 1 1 2m e (1+ k )x x x d x ln x ln x e Ei kr k  x kr Z 1 2m + dx ln x e x Ei x k  0   2m k  kr ln (kr) e Ei ( 2mr) f + 1; kr Li2 k 2m      ! 1 k 2m 2m + ln 1 + 2 + ln 1+ (5.32) 2 2m k k !  k  1 1 2 2 + Li2

ln (2mr) (ln (2mr))2 : 2 6 2m 2

In the nal stage of the last integral the results of the two previous ones have been used. Returning to (5.29) one may now write Z

r

1

dx x2

3 2 Y D (x; m i=1

i)

31

=

1 (4)3 "

ln (M123 r)





3  X

1 2

+ 3 ln 2m2 +

i=1



ln M 123

2 + 2 6



!

Li2





ln (M123 r) + f (M123 ; r)

M123 2mi 

ln (2mi r) 2mi

#!

 1 (ln (2mi r))2 2

  h    i 1 2 123

+ ln ( M r ) 1 +  3 ln 2 m ln M M 123 123 (4)3 h 2 1 +

2 (ln (M123 r))2 6 2 !  3 1 X M123  i 2 (ln (2mi r)) + ln (2mi r) + Li2 1 : 2mi i=1 2

=

(5.33)

Using this result in (5.28) and simplifying the result the contribution of region C to I is now nally available:

IC





  1     1 2r 123

+ ln ( M r ) + 4 +

+ ln 2 + 4 ln 2 m ln M M = 123 123  4 (4)4 2 1 + + 2 + (ln (M123 r))2 6 2 !  3  X M123   1 2 (ln (2mi r)) + ln (2mi r) + Li2 1 : (5.34) + 2mi i=1 2

Region D This region produces no divergences so one may set  = 0 in the beginning. Then the integrals reduce to familiar types and going once again to the limit r ! 0 one obtains

ID =

Z r

0

dx

Z

r

1

dy x2 y2 G (m1 x) D (y; m1 )

3 Y i=2

D (x; mi )

6 Y

i=4 M1456 y

D (y; mi )

Z r Z 1 1 e M23 x e = dx sinh (m1 x) dy 4 x y2 (4) m1 0 r ! 1 e M1456 r = m1 r + M1456 Ei ( M1456 r) r (4)4 m1  M1456 r  1 1 = 1 + + ln (M1456 r) = : 4 M1456 r (4) (4)4

32

(5.35)

Region E Since the region E contains no points near the origin, one easily observes that its contribution to the integral I is also nite. Proceeding as before one may write Z 1 Z 1 3 6 Y Y IE = dy dx x2 y2 G (m1 y) D (x; mi ) D (y; mi ) r

y

i=1

i=4

Z 1 Z 1 1 e M456 y e M123 x = d y sinh ( m y ) d x 1 y2 x (4)4 m1 r y Z 1 M y e 456 1 dy sinh (m1 y) Ei ( M123 y) = 4 y2 (4) m1 r  M m1 +M456 y  1456 y M123 Z 1 2 M = dy y Ei ( y) e 123 e M123 : 2 (4)4 m1 M123 r R1

(5.36)

It is obvious that one now has to evaluate an integral of the form r dy y 2 Ei ( y) e ky . In the limit r ! 0 one obtains Z 1 Z 1 Z 1 1 e ky e ky e (k+1)y e ky Ei ( y) k dy Ei ( y) + dy dy 2 Ei ( y) = r y y y y2 r r r Z 1 1 e kr Ei ( r) k r ln y e ky Ei ( y) k2 dy ln y e ky Ei ( y) = r 0 Z 1 Z 1 1 e (k+1)y e (k+1)y e (k+1)y +k dy ln y r (k + 1) dy y y y r r kr e = Ei ( r) + k ln r e kr Ei ( r) r  1 2 k ln (k + 1) + (ln (1 + k)) Li2 ( k) 2 e (k+1)r +kf (k + 1; r) + + (k + 1) Ei ( (k + 1) r) r ln r 1 + k = + + (ln r)2 + (1 + k) ln r k 1 + + r r 2 ! k 2 2 (k + 1) ln (k + 1) + kLi2 ( k) + + 2 6 ln r 1 + (ln r)2 = + + ln r 1 + + k + ln r 1 + r r 2 !! 1 2 2 + + (k + 1) ln (k + 1) : (5.37) Li2 ( k) + 2 6 Using (5.37) IE can be straightforwardly written down:

IE

1 = 2 (4)4

M1123456  M1123456  2 2 ln 2+ + + 6 m1 M123 33

M23456  M23456  ln m1 M123

M1456  M1456   Li + 1 + m1 2 M123

M456   Li2 m ! 1

m1 + M456  M123

+2 ln (M123 r) + (ln (M123 r))2 :

(5.38)

Region F The calculation in region F follows very closely that of the preceding section. Setting  = 0 one gets Z 1 Z 1 3 6 Y Y IF = dx dy x2 y2 G (m1 x) D (y; m1 ) D (x; mi ) D (y; mi ) r

x

i=2

i=4

Z 1 Z 1 1 e M23 x e M1456 y = d x sinh ( m x ) d y 1 x y2 (4)4 m1 r x ! Z 1 e M23 x 1 e M1456 x dx = sinh (m1 x) + M1456 Ei ( M1456 x) x x (4)4 m1 r  1  M23456 r 1 M1123456 r e e = (4)4 m1 2r M M1123456 + 23456 Ei ( M23456 r) Ei ( M1123456 r) 2 2 !   M1456 Z 1 1 ( m + M ) x M x 1 23 123 dx x Ei ( M1456 x) e e : (5.39) + 2 r

Going once again to the limit r ! 0 and using a previous result (5.32) one obtains

IF =

1 M M1123456 2 (1 ) + 23456 ln (M23456 r) ln (M1123456 r) 4 m1 m1 2 (4)      ! M1456 M123 m1 + M23 + Li2 Li2 : (5.40) m1 M1456 M1456

Final result Adding up the contributions from the dierent regions one may now nally write down the result for the whole diagram. After some straightforward simpli cations it reads "

Ie

#

 m4 1 8 2m  = + 1 + ln 32 2  M123 "    72 2m  2m 2 +4 13 + 3 (ln 2)2 + 16 ln + 8 ln 12 M123 M123 ! !
M 123 2 (M123 )3 + ln M ln +2 ln 2 ln 123 M 123 (M123 )3

34

M1456  M1456   M456   m1 + M456  Li +4 1 Li2 +4 m1  2  M123  m1 M123   M M123 m1 + M23 +4 1456 Li2 Li2 m1 M1456 M1456  #!   3 X M123 : (ln mi )2 + 2Li2 1 2mi i=1

(5.41)

All r-dependence has cancelled just as it should have so one may feel quite con dent about the correctness of the result. It is interesting to note that the divergent term only depends on the rst three masses through the combination M123 = m1 + m2 + m3 . The reason for its independence of the three other masses becomes obvious when the diagram is written in the form of an integral in momentum space and could have been anticipated already previously on the basis of the structure of the diagram in coordinate space (5.16).

Special mass con gurations An important result needed in applications of the eective QCD is obtained when the propagator masses have the values m3 = 0; m1 = m2 = m4 = m5 = m6 = m. Using the representation of the dilogarithm function Z 1 2 1 ln (1 + t) Li2 ( x) = (ln (1 + x))2 + dt (5.42) 6 2 t (1 + t) x and the relation Li2 ( x) + Li2





1 1 = (ln ( x))2 x 2

one has:

"

2 6

(5.43)

 m4 1 8 7 2 m 2 Ie (m; m; 0; m; m; m) = + + 4 13 + + 3 (ln 2) + 2 ln 2 ln 32 2  12 m3 ! mm23 + 16Li2 ( 2) 8Li2 ( 1) + ln (2m) ln 8    1 +16 Li2 Li2 (0) 2 (ln m)2 (ln m3 )2 2 ! #! 2 1   m 2 4Li2 (0) + 2 + ln 6 2 m3  ! 132 m4 1 8 2 + + 4 13 8 (ln 2) : (5.44) = 32 2  12

With identical masses the result is correspondingly:

m4 1 8  2 + 1 + ln Ie (m; :::; m) = 32   3 35

"













2 2 2 3 72 + 5 ln + 16 ln + 16 Li2 + Li2 12 3 3 4      #!  1 1 2 6Li2 16Li2 8Li2 3 2 4   1 1 0:03125 2 + 4:7562791351 11:1450820899  m4  

+4 13 +

36



4 3



(5.45)

Chapter 6

The remaining diagrams There exists a class of four-loop scalar vacuum skeletons (the diagrams f - j of gure 3.1) with such complicated topologies that neither a straightforward momentum space calculation nor a Fourier transformation and coordinate space evaluation is enough to solve them. They do not contain any simple subdiagrams and the number of vertices is too high for a successful coordinate space integration - at least using the standard formulae of Gradshteyn and Ryzhik [38] and other such tables. There remain, however, a few calculational methods to consider. Restricting oneself to previously chosen values of the propagator masses (in this chapter I will only consider the case mi = m 8i) one may perform numerical integrations exploiting the fact that all the diagrams in this class are UV- nite. Another possibility is to resort to the method of partial integration used succesfully by Rajantie in [20] and derive partial dierential equations that the diagrams must satisfy. Here the resulting equations will, however, be much harder to solve than the ones of [20], since the diagrams considered in this thesis do not exhibit such beautiful symmetries as the famous Mercedes diagram. I will therefore restrict myself to merely constructing the equations and will not address the question of actually solving them.

6.1 Numerical calculations In this section I will evaluate the diagrams f - j numerically assuming the masses of the dierent propagators to equal the m appearing in the MS measure (3.1). This procedure may seem very restrictive but it is useful to note that the methods used below can easily be generalized to any previously chosen mass con guration. The actual integrations will be performed using Mathematica and a very limited amount of CPU-time. It is therefore certainly possible to improve the accuracy of the results obtained here.

Diagram f The structure of this diagram is clearly very complicated, since it contains as a subdiagram the ' sh' two-point function, for which no analytic expression has been found. The sh part has been considered e.g. by Rajantie in [20], where he was able to construct an onedimensional integral representation for it in exactly three dimensions. Due to the f diagram 37

being UV- nite one may also here set  = 0 enabling the use the result, which for identical propagator masses reads: Z m 1 m = 2p 2 dx p 2 (6.1) 2 p p + 3m 0 p + 4m2 x2 " !#    2p p  p  p2 + (2m + x)2  2m + x arctan 2m + x arctan 2m + ln : (2m + x)2

p

Using the above relation together with (4.2) and performing a trivial angular integration one now obtains
arctan y2 2Z1 Z1 p p (6.2) If (m; :::; m) = dy dt  0 y (1 + y2 ) y2 + 3 y2 t2 + 4 0 " !#     2y y  y y2 + (2 + t)2  2 + t arctan 2 + t arctan 2 + ln (2 + t)2 : This can be easily calculated numerically and leads to

If (m; :::; m)  0:007861658675:

(6.3)

Diagram g With identical masses on the propagator lines the evaluation of this diagram proceeds in exact analogy with the previous one resulting in y
2m2 Z 1 Z 1 p arctan 2 p dy dt Ig (m; :::; m) = 2 + 3 y 2 t2 + 4  0 y y 0 !# "      y y y2 + (2 + t)2 2y  2 + t arctan 2 + t arctan 2 + ln (2 + t)2  0:07753725612  m2: (6.4)

Diagram h Denoting Z

d3 s

1

s2 + m2

1 1 2 2 (s p) + m (s q)2 + m2

 g (jpj; jqj; p;q )

(6.5)

one easily sees that the wheel diagram may be written in the form: 

m 4 Z 3 Z 3 1 1 dp dq 2 g (jpj; jqj; )2 2 2 2 2 p + m q + m2 1 Z1 Z1 Z x2 y2 = d x d y d  sin  g (x; y; ; m = 1)2 : (6.6) 6 2 2 2 0 x +1 y +1 0 0

Ih (m; :::; m) =

38

The function g can be obtained most easily by introducing two Feynman parameters before evaluating the momentum integral: g (x; y; ; m = 1) Z

Z 1

Z 1

2z 0 0 z ) y2 2s
(wz x + (1 z ) y))3 Z Z 1 2 1 z = dw dz 2 0 0 (1 + wz (1 wz ) x2 + z (1 z ) y2 2wz (1 z ) xy cos )3=2 p Z 1 2 + y2 + w x2 2 1 + w (1 w) x2 2w xy cos  2 p = dw 1 + w (1 w) x2 0 =

d3 s

dw

dz

(s2 + 1 + wz x2 + (1







 w2 x2 4 4xy cos  + x2 + y2 4 4wxy cos  + y2 +2wx2 y2



1 + 2w (cos )2



8wxy cos 

! 1

:



(6.7)

The remaining w-integral can, too, be calculated and yields merely elementary functions, but the result is far too long to be printed here. Using (6.7) one may now anyway perform the numerical integral of (6.6), which eventually leads to the result for the wheel diagram

Ih (m; :::; m)  0:005:

(6.8)

Diagram i The numerical evaluation of the H diagram is very similar to the preceding case. Setting all masses the same one is able to write the integral in the form 



1 1 1 m 4Z 3 Z 3 dp dq 2 g (jpj; jqj; )2 (6.9) Ii (m; :::; m) = 22 p + m2 q2 + m2 (q p)2 + m2 1 Z1 Z1 Z x2 y2 g (x; y; ; m = 1)2 = d x d y d  sin  : 26 m2 0 x2 + 1 y2 + 1 x2 + y2 2xy cos  + 1 0 0 The nal numerical integration is easily performed using the expression for g given in (6.7) and yields

Ii (m; :::; m)  0:0009  m 2 :

(6.10)

Diagram j The 'twisted H' diagram is especially diÆcult to evaluate, and one is even unable to take advantage of the known form of the g-function constructed above. The most straightforward way to proceed is to rst combine the dierent propagators by introducing multiple Feynman parameters and then to use standard formulae to perform the momentum integrals. In the 39

end one is, however, left with several one-dimensional integrals that have to be evaluated numerically. Using the standard parametrization formula (see e.g. [37]) 1 (a1 + a2 + ::: + ak ) = (D1 )a1 ::: (Dk )ak (a1 ) (a2 ) ::: (ak ) one obtains:



Ij (m; :::; m)

Z

1

Æ (1 x1 ::: xk ) xa11 1 :::xakk 1 dxi (6.11) (D1 x1 + ::: + Dk ak )a1 +:::+ak i=1

Z 1Y k

0

1

1

1

p;q;r;s p2 + m2 q 2 + m2 r 2 + m2 s2 + m2

1 (p r)2 + m2

 (p s)12 + m2 (q r)12 + m2 (q s)12 + m2 (p + q r 1 s)2 + m2 =

(9)

Z 1Y 9

Z

z

p;q;r;s 0 i=1 y2

+q2 (x

9 X

dxi Æ 1

}|

2 + x7 + x8 + x9

i=1

{

z

xi

!

z

y3

}|

) +r2 (x

y1

}|

m2 + p2 (x

3 + x5 + x7 + x9

{

z

) +s2 (x

2q
s (x8 + x9 ) + 2p
q x9 + 2r
s x9

y4

}|

{

4 + x6 + x8 + x9 )

2p
r (x5 + x9 ) 2p
s (x6 + x9 ) 2q
r (x7 + x9 )  9

{

1 + x5 + x6 + x9 )

:

(6.12)

The momentum integrations can now be performed using the simple relation Z

d2! p

 ! (m ! ) 1 1 = m 2 2 2 (p + m + 2p
q) (m) (m q2 )m

which leads to:

Ij (m; :::; m) =

1

82 m2

Z 1Y 9

0 i=1



dxi

Æ 1



(AC

P9

;

!

(6.13)



9=2 3=2 i=1 xi y1 A

E 2 ) (AB D2 ) (AF

DE )2

3=2

:

(6.14)

The functions A, ..., F in the nal form are

A B C D E F

= = = = = =

y1 y2 x29 y1 y3 (x5 + x9 )2 y1 y4 (x6 + x9 )2 y1 (x7 + x9 ) x9 (x5 + x9 ) y1 (x8 + x9 ) x9 (x6 + x9 ) y1 x9 (x5 + x9 ) (x6 + x9 ) :

Evaluating the parametric integrals numerically one nally gets

Ij (m; :::; m)  0:0004  m 2 : 40

(6.15)

6.2 Partial dierential equations As another method of tackling the evaluation of the integrals of this chapter let us now derive partial dierential equations with respect to the propagator masses that the diagrams f to i must satisfy. This will be done in analogy with the treatment of the Mercedes diagram in [20] and will employ the idea developed by Kotikov [39]. However, as already mentioned, nding solutions to the equations obtained here seems to be a laborious task - probably one requiring the use of numerical methods - and that question will not be addressed in the thesis. In the following let us use a compact notation for the propagators and diagrams adopted from [20]. A propagator with momentum p and mass ma will be denoted by
ap , a diagram I with one propagator (mass ma ) squared by Ka and the same diagram with one propagator (mass ma ) squared and one propagator (mass mb ) removed by Jab . Consider then a convergent ( = 0) planar four-loop diagram with the structure

I

Z

p;q;r;s


1p
2p q
3p r
4q
5r f (q; r; s) ;

(6.16)

where f is an in principle arbitrary function of three variables. This is illustrated by the gure below and appears in all integrals considered here. (6.17) 2 3 1 4

5

Performing a partial integration with respect to p and neglecting the vanishing boundary terms one obtains: i 1Z @ h 1 2 3 4 5 I = pi
p
p q
p r
q
r f (q; r; s) 3 p;q;r;s @pi   2Z =
1p
2p q
3p r
4q
5r f (q; r; s) p2
1p + p
(p q)
2p q + p
(p r)
3p r 3 p;q;r;s 2Z
1
2
3
4
5 f (q; r; s) = 3 p;q;r;s p p q p r q r 

 1 m21
1p m21
2p q +
1p

I

=



1 1 + m22 2 1  2 + 1 + m3 2

+

()








1p



 1 1


p

2m21 K1 + m21 + m22

 1

 1



+ m21


4q



+ m21




2p

 1

 5 1


r





m21
3p r +
1p

q



+ m24
2p

m24 K2 + m21 + m23




3p

r



q



+ m25
3p

 1

!

r

m25 K3 J21

(6.18)

J31 + J24 + J35 ; (6.19)

which is equivalent to eq. 29 of [20] (up to a few misprints in [20]). Using the above result one is able to write down partial dierential equations for all diagrams of this chapter with the exception of the non-planar 'twisted H' -case. The equations are of rst order and can in principle be solved by standard methods starting from the g and 41

f cases. For these two diagrams the equations contain as parameters values of only such diagrams that have already been evaluated in this thesis with arbitrary masses. Below I shall merely list the equations.

Diagram f 

@ 1 + 2m21 2 @m1

=

+ (m21 + m22

@ m24 ) 2 @m2

+ (m21 + m23

@ m25 ) 2 @m3



If

(6.20)

@ @ Ib (m2 ; m3 ; m4 ; m5 ; m7 ; m8 ; m6 ) + 2 Ib (m2 ; m3 ; m4 ; m5 ; m7 ; m8 ; m6 ) 2 @m2 @m3 @ @ I (m ; m ; m ; m ; m ; m ; m ) I (m ; m ; m ; m ; m ; m ; m ) @m22 c 1 5 7 8 3 6 2 @m23 c 1 4 7 8 2 6 3

Diagram g 

 @ 2 + m2 m2 ) @ + (m2 + m2 m2 ) @ + ( m (6.21) 1 2 4 @m2 1 3 5 @m2 Ig @m21 2 3 @ @ = I (m ; m ; m ; m ; m ; m ) + I (m ; m ; m ; m ; m ; m ) @m22 a 2 3 4 5 6 7 @m23 a 2 3 4 5 6 7 @ @ Ie (m3 ; m1 ; m5 ; m8 ; m2 ; m6 ; m7 ) I (m ; m ; m ; m ; m ; m ; m ) 2 @m2 @m23 e 2 1 4 8 3 6 7

1 + 2m21

Diagram h 

=

@ 1 + 2m21 2 @m1

+ (m21 + m22

@ m24 ) 2 @m2

+ (m21 + m23

@ m25 ) 2 @m3



Ih

(6.22)

@ @ Ig (m8 ; m4 ; m5 ; m6 ; m7 ; m2 ; m3 ) + 2 Ig (m8 ; m4 ; m5 ; m6 ; m7 ; m2 ; m3 ) 2 @m2 @m3 @ @ I (m ; m ; m ; m ; m ; m ; m ) I (m ; m ; m ; m ; m ; m ; m ) @m22 g 5 1 8 3 7 2 6 @m23 g 4 1 8 2 6 3 7

Diagram i 

 @ 2 + m2 m2 ) @ + ( m (6.23) 1 3 5 @m2 Ii @m22 3 @ @ Ih (m6 ; m2 ; m4 ; m7 ; m9 ; m3 ; m5 ; m8 ) + 2 Ih (m6 ; m2 ; m4 ; m7 ; m9 ; m3 ; m5 ; m8 ) = 2 @m2 @m3 @ @ If (m5 ; m1 ; m9 ; m3 ; m8 ; m7 ; m2 ; m6 ) I (m ; m ; m ; m ; m ; m ; m ; m ) 2 @m2 @m23 f 4 1 9 2 6 7 3 8

1 + 2m21

@ + (m21 + m22 @m21

m24 )

42

Chapter 7

Conclusions Based on the complexity of already existing three-loop computations ([19, 20], etc.) and the fact that even at this order one has been forced to resort to numerical methods, the evaluation of four-loop scalar diagrams could already in advance be anticipated to be a highly non-trivial task. The two-particle irreducible diagrams, skeletons, represent furthermore an especially tedious subset of these, since no immediate way is available for reducing the integrals into lower-order calculations. In this thesis it has been observed that these fears are indeed well-motivated. One has, however, also seen that even analytic results can be obtained for diagrams which at rst sight appear to be extremely complicated. Of the ten diagrams considered here I have been able to evaluate completely analytically two while keeping the masses of the propagators arbitrary. For three other ones simple oneor two-dimensional integral representations have been constructed with arbitrary masses, and when setting the masses equal one of these integrals has been shown to be analytically calculable. With the exception of one special case the evaluation of the remaining diagrams with equal propagator masses has too been reduced to the calculation of two- and threedimensional integrals that can easily be performed numerically. In the identical mass -case explicit numerical results have been given for all diagrams of this thesis making a direct comparison of their magnitudes possible. It is in addition notable that all divergent parts of the diagrams have been determined analytically. The fact that I have failed to obtain simple results for the nite parts of some integrals certainly does not imply that this task would be insurmountable. Analytic expressions for the diagrams surely exist and to obtain them one merely needs to do more work: there are a great number of dierent calculational techniques and tricks left to try. A lot of eort is also required in evaluating all the additional diagrams that perturbative calculations in three-dimensional eective QCD will produce. It is evident that much remains to be done in the eld of evaluating multi-loop Feynman diagrams. A need for such calculations is present in many areas of modern day physics and new techniques for the analytic treatment of diagrams are constantly emerging (see e.g. [17, 41, 40]). Research in the eld will certainly remain active for years to come.

43

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46