Automated Generation and Computation of Feynman ... - CiteSeerX

GEFICOM: Automated Generation and Computation of Feynman Diagrams in Quantum Chromodynamics up to Three Loops K.G. Chetyrkin, B.A. Kniehl and M. Steinhauser Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, München Abstract The interpretation of electroweak precision data in particle physics makes it necessary to compute higher-order quantum corrections in perturbation theory. The traditional manual treatment of the corresponding Feynman diagrams is limited by the large amount of their occurrence and the enormous complexity of the mathematical expressions. In the framework of our research project in quantum chromodynamics, we have succeeded in automating the generation and computation of Feynman diagrams up to three loops. This is insofar a methodical breakthrough as with conventional methods only a limited class of two-loop diagrams could be computed. Our method makes a lot of new and physically interesting applications accessible.

1 Introduction The standard model (SM) of elementary particle physics is a renormalizable quantum field theory. This means that we may calculate physical observables like scattering cross sections, decay widths etc. with arbitrary precision in perturbation theory provided the coupling constants which we use as expansion parameters are sufficiently small. In the context of quantum chromo127

dynamics (QCD), i.e. the sector of the SM which characterizes the interactions of quarks and gluons, this implies that we should consider small distances, which according to Heisenberg’s uncertainty principle is equivalent to large energy scales  of the interactions, where the strong coupling constant satisfies s ()  1. This follows from the fact that QCD is asymptotically free, i.e. s () ! 0 as  ! 1. At the energies available in contemporary collider experiments, s () is not actually very small. For example, at energies which correspond to the Z -boson mass MZ  100  mproton , one has s (MZ )  0:118. Over the years, the experiments at the European Laboratory for Particle Physics (CERN) and other laboratories throughout the world have accumulated vast data samples, so that the experimental uncertainties in the relevant QCD observables are challenging the state of the art of the corresponding theoretical calculations. It is therefore of crucial importance to progress to higher orders of the perturbative expansion in s (). Unfortunately, such perturbative calculations are very cumbersome. For one thing, the number of contributing Feynman diagrams, which visualize the various possibilities for the considered process to take place at the respective order, is rapidly increasing as one goes to higher orders. On the other hand, every additional order increases the number of loops that may occur in these Feynman diagrams by one. By the same token, the resulting mathematical expressions are getting more and more complex, since every loop involves one integration over the four-momentum of the particle circulating in that loop. The traditional way to evaluate Feynman diagrams is by manual calculations. These have reached a grinding halt at two loops, where only the simplest topologies of diagrams, i.e. those with two external legs or less, have been tackled. Not rarely, two different groups found results which were in disagreement with each other, and it took a third group to settle the controversy. Typically, a few diagrams were missed out or evaluated wrongly. In this situation, it suggests itself to try and automate the generation and evaluation of Feynman diagrams so as to exclude as far as possible human error sources. In the following, we shall present a successful example of such an endeavour, exploiting modern technology in algebraic manipulation. In Section 2, we shall give a brief introduction to our program package GEFICOM, which manages to automatically generate and analytically evaluate all Feynman diagrams contributing to a given process within QCD, up to the three-loop order. In Section 3, we shall describe in some detail a recent application, namely the calculation of the hadronic decay width of the Higgs boson to order 4s [1]. In Section 4, we shall review more examples of such three-loop calculations and present an outlook to future work. 128

2 Description of the program The program package GEFICOM provides the possibility to compute Feynman diagrams in QCD up to three loops. Of course, the computation of general three-loop integrals with several different external momenta and masses is out of range. However, for the important case of two-point functions it is possible to solve the integrals with either one mass or one external momentum different from zero in analytic form. With these classes of diagrams, already a wide range of very interesting physical problems may be covered. In addition, GEFICOM provides the opportunity to expand in small masses and/or momenta, which leads to a great flexibility in the applications. In physical processes, it often happens that a hierarchy between different mass scales exist. Typical examples are processes in which the top quark only appears virtually in the loops. Then, because of the large top-quark mass Mt , it makes sense to expand in 1=Mt and finally end up with expressions which have one mass scale less than the original one. A typical example is the decay of the Higgs boson into gluons. This process is mediated by a top quark loop. For illustration, some example diagrams are depicted in Fig. 1. Below, we shall demonstrate how these diagrams can be generated and computed with the help of GEFICOM.

O

!

Fig. 1: Typical diagrams generating (2s ) corrections to ,(H gg). Looped, thin, boldfaced, and dashed lines represent gluons, light quarks, top quarks, and Higgs bosons, respectively.

GEFICOM is a combination of four different programming languages, which exploits the advantages of each one. In the following, we shall briefly describe the structure of GEFICOM and also mention which language is used at which point. An overview of the package and the possible interactions with the user is presented in Fig. 2. The package splits into three parts: the generation of the diagrams; the recognition of the topologies; and the very computation of the produced am129

GEFICOM

automatic GEneration, FInding topologies and COMputataion of Feynman diagrams

HUMAN

Problem Specification

COMPUTER QGRAF

Problem dependent Input Files Command "doq2f -p - l "

FindTop

Command "do - l "

Database of all diagrams with all administrative (make -) les MINCER

MATAD

Summation of all diagrams with FORM program

Renormalization

File with results (in unrenormalised form)

Writing the article

Fig. 2: The structure of GEFICOM.

plitudes. For the very generation, a public-domain Fortran program, QGRAF [2], has been used. However, in order to guarantee a convenient handling, we had to write an interface written in AWK in order to get enough flexibility concerning the input for QGRAF. The output is immediately transformed into a format which can be read by MATHEMATICA. The heart of the second part is written in MATHEMATICA. Some auxiliary files are again written in AWK. The file containing the diagrams is read. Then, the topology of each diagram is determined, and finally it is trans130

formed into a format suitable for the third part. For each diagram, also a corresponding colour diagram in generated. At first sight, the recognition of the topology seems to be trivial. For a human being, this is definitely true. However, the implementation of an algorithm in a computer is highly nontrivial. The MATHEMATICA program has also to provide the administrative files, which are necessary to handle a large number of diagrams. This includes the generation of make-files which guarantee that all files containing the results are up-to-date. Furthermore, special files are necessary which multiply all diagrams with the correct colour factors and take the sum at the end. For the third part of GEFICOM, only the algebraic manipulation program FORM is used. This part, which is responsible for the evaluation of the integrals, could also be used independently. In the latter case, the files which are generated automatically by GEFICOM have to be created by hand. This is at most tractable for a small number of diagrams. Specifically, the third part of GEFICOM constist of two independent packages, MATAD [3] and MINCER [4]. MINCER computes the massless integrals and is publicly available. MATAD has been written for the purpose of treating the massive diagrams. The structures of these two packages are very similar. In a first step, the fermion traces over the matrices are performed. Then, if necessary, an expansion in small momenta or masses is done and appropriate projectors are applied in order to end up with scalar integrals. After that, it is possible to decompose the numerators in terms of the denominators. At this step, it may already happen that millions of different integrals have to be treated. Of course, this is impossible to do. For this reason, in a next step, recurrence relations are used in order to end up with just a small number of integrals which actually have to be computed. In the case of MATAD, it is possible to reduce the number to three. At the end, the result for an individual diagram in rather compact. In intermediate steps, when the recurrence relations are applied, it may, however, happen that the expression blows up and an intermediate disk space up to several Gigabytes is needed. The colour diagrams are treated in a similar way. However, their computation is very fast and takes in general only a few milliseconds. Each diagram is treated in the described way, and the results are saved to hard disk. Finally all diagrams are summed up. In the following, we want to give a small description on how to use GEFICOM. This will be illustrated with the example already mentioned above, the computation of the vertex diagrams for the Higgs-boson decay into gluons. The user has to generate two files or more. One of them has to contain specifications like the process under consideration and the masses of the different particles. Also some optional parameters are allowed which restrict the number of diagrams according to specified rules. For instance, it is possible to select only those diagrams which contain at least one line of a certain quark species. In our example the corresponding file would look as follows: 131

*** MATAD * power 2 * gauge 1 * dala12 list = symbolic ; lagfile = ’q.lag’ ; in = h[q1p2]; out = g[q1], g[q2]; nloop = ; options = ; true = bridge[ g,q,c, 0,0 ]; false = iprop[ t ,0,0 ]; The first four lines specify parameters which tell the program that MATAD has to be used and an expansion in the momenta up to second order has to be performed. Furthermore specifications concerning the gauge in which the calculation has to be done, and the kind of derivatives w.r.t. the external momenta are given. The remaining part is necessary for the generation of the diagrams. For example, “in” and “out” specify the process under consideration, and the last line is to make sure that only those diagrams which contain at least one top-quark line are included. The second file contains the propagators and vertices contained in the model. In our example, it reads: *

propagators [q,Q;2-] [t,T;2-] [c,C;1-] [g,g;3+t] [h,h;1+p] [sigma,sigma;1+t] * vertices [Q,q,g] [T,t,g] [C,c,g] [T,t,h] [g,g,g] [g,g,sigma] In addition, the user can create some auxiliary files specifying projectors or additional commands which are to be executed during the computation of the diagrams. For the process we are interested in, the coefficient of the transversal structure formed by the two external momenta has to be projected out. This is done by the following commands: 132

drop dia; G diah1 = dia; .store G diahc1 = d_(mu2,mu4)*diah1; .store G diahc2 = q2(mu2)*q1(mu4)*diah1; .store G diahc3 = q2(mu4)*q1(mu2)*diah1; .store G dia = deno(2,-2)*(q1.q2*diahc1-diahc3-diahc2); .sort Also a projector in colour space is needed. This is achieved by a program containing the simple command multiply, prop(a(2),a(4)); This corresponds to the multiplication with a delta function in the colour indices of the two gluons. Once these simple files have been generated, GEFICOM can be called by executing the shell-script command doq2f: > doq2f -p -l After the option “-p” the name of the problem has to be given and “-l” specifies the number of loops. In our case, altogether 657 diagrams are generated, which takes roughly 15 minutes of CPU time. The computation of the diagrams is not started automatically. Instead, GEFICOM has generated a shellscript program which allows the user to compute, e.g., only a subset or only a single diagram. With the command > do -l all diagrams are computed. For our example, this takes approximately one week of CPU time.

3 Example of a physics application The Higgs boson, H , is the missing link of the standard model (SM) of elementary particle physics. Its experimental discovery would eventually solve the longstanding puzzle as to whether nature makes use of the Higgs mechanism of spontaneous symmetry breaking to generate the particle masses. So far, direct searches at the CERN Large Electron Positron Collider (LEP) have only been able to rule out the mass range MH  65:6 GeV at the 95% confidence level (CL) [5]. On the other hand, exploiting the sensitivity to the 133

Higgs boson via quantum loops, a global fit to the latest electroweak precision data predicts MH = 149 +148 ,82 GeV together with a 95% CL upper bound at 550 GeV [6]. The coupling of the Higgs boson to a pair of gluons, which is mediated at one loop by virtual quarks [7], plays a crucial rôle in Higgs phenomenology. The Yukawa couplings of the Higgs boson to the quark lines being proportional to the respective quark masses, the ggH coupling of the SM is essentially generated by the top quark alone. The ggH coupling strength becomes independent of the top-quark mass Mt in the limit MH  2Mt . In fact, in extensions of the SM by new fermion generations, this property may be exploited by using the ggH coupling as a device to count the number of high-mass quarks [7]. In contrast to the electroweak  parameter [8], the ggH coupling is also sensitive to quark isodoublets if they are mass-degenerate. At this point, we also wish to remind the reader that, by the Landau-Yang theorem [9], spin-one particles such as the photon or the Z boson cannot couple to two real gluons, while spin-zero particles such as the Higgs boson do. The prospects for the Higgs-boson discovery at the CERN Large Hadron Collider (LHC) vastly rely on the gluon-fusion subprocess, gg ! H , which will be the very dominant production mechanism over the full MH range allowed [10]. The cross section of inclusive Higgs-boson production in protonproton collisions, pp ! H + X , is significantly increased, by approximately 70% under LHC conditions, by including its leading-order (two-loop) QCD corrections [11, 12], which are intimately related to the ggH coupling. Under such circumstances, the theoretical prediction for this extremely relevant observable can by no means be considered to be well under control, and it is an urgent matter to compute the next-to-leading-order QCD corrections at three loops, since there is no reason to expect them to be negligible. Recently, a first step in this direction has been taken by considering the resummation of soft-gluon radiation in pp ! H + X [13]. An important ingredient in this complex research programme is the O(2s ) three-loop correction to the ggH coupling. Typical Feynman diagrams that contribute in this order are those obtained by attaching two virtual gluons to the primary top-quark triangle. There are also other classes of diagrams, and they all come in large numbers. The ggH coupling also appears as a building block in the theoretical description of the crossed process, H ! gg , which contributes to the hadronic decay width of the Higgs boson. In the low to intermediate mass range, MH <  150 GeV, this decay mode has a branching fraction of up to 7% [14, 15]. Observing that a Higgs boson in this mass range almost exclusively decays to bb pairs, this number may be quickly understood by taking the ratio of the H ! gg and H ! bb partial widths in the Born approximation, which gives (s MH =mb )2 =27.

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The O(s ) correction to the H ! gg decay width was originally derived [16] in the limit MH  2Mt by constructing a heavy-top-quark effective Lagrangian and subsequently confirmed by a diagrammatic calculation [12] and via a low-energy theorem (LET) [17] in Refs. [11, 12]. This correction consists of two-loop contributions connected with gg production and oneloop contributions due to ggg and gq q final states, where q stands for the first five quark flavours. In contrast to the H ! q q decay with subsequent gluon radiation, in the H ! gq q diagrams of interest here, the q q pair is created through the branching of a virtual gluon, so that these contributions survive in the limit of vanishing q -quark mass. In fact, if all quark masses, except for Mt , are nullified, the hadronic decay width of the Higgs boson is entirely due to H ! gg and the associated higher-order processes under consideration here. Depending on the experimental setup, the heavier quarks Q = c; b may be detectable with certain efficiencies. The secondary Q quarks from H ! gg ! gQQ will typically be much softer than the primary ones from H ! QQ ! gQQ , which may serve as a criterion to distinguish between these two production mechanisms. Alternatively, one may attempt to subtract  contributions from the QCD-corrected H ! gg decay width [15]. the gQQ For simplicity, following Refs. [12, 16], we shall not consider such a subtraction for the time being. Futhermore, as in Refs. [12, 16], we shall concentrate on the limit MH  2Mt , which is most relevant phenomenologically. Although the LEP1 lower bound on MH [5] then implies that nl = 5 light quark flavours contribute at the renormalization scale  = MH , we shall keep nl arbitrary. Thus, the Born result reads (n ) 3 ,Born (H ! gg) = GF MpH s  () 36 2 l

where GF is Fermi’s constant. The multiplying Eq. (1) with [12, 16]

!2

;

(1)

O(s ) correction may be included by

 95 7  11 1  2  (n )  (  ) s K =1+  (2) 4 , 6 nl + 2 , 3 nl ln MH2 : For  = MH = 100 GeV, this amounts to an increase by about 66%. Since l

such a sizeable correction is unlikely to provide a useful approximation, it is indispensable to go to higher orders. The purpose of this project is to take the next step by extending Eq. (2) to O(2s ). To this end, we need to calculate three-loop three-point, two-loop four-point, and one-loop five-point amplitudes. The contributing final states are gggg , ggq q, q qq 0 q0 , ggg , gq q, gg , and q q. Typical diagrams are depicted in Fig. 1. 135

Our procedure is similar to that of Ref. [16]. We construct an effective Lagrangian, Le , by integrating out the top quark. This Lagrangian is a linear combination of certain dimension-four operators acting in QCD with five quark flavours, while all Mt dependence is contained in the coefficient functions. We then renormalize this Lagrangian and compute with it the H ! gg decay width through O(4s ). For brevity, we do not list here all operators that enter our analysis in intermediate steps. Instead, we immediately proceed to the final version of Le ,

Le = ,21=4 G1F=2 HC1 [O10 ] :

(3)

0 G0a0 , [O10 ] is the renormalized counterpart of the bare operator O10 = G0a where Ga is the colour field strength, the superscript 0 denotes bare fields, and primed objects refer to the five-flavour effective theory. C1 is the corresponding renormalized coefficient function, which carries all Mt dependence. Note that C1 and [O10 ] are not separately renormalization-group (RG) invariant through the order considered, while their product is. From Eq. (3) we may derive a general expression for the H ! gg decay width,

p

,(H ! gg) = M2GF C12 Im h[O10 ] [O10 ]i; H

(4)

where h[O10 ] [O10 ]i is the vacuum polarization of the Higgs field induced by 2 , with q being the external four-momentum. the gluon operator at q 2 = MH In order to cope with the enormous complexity of the problem at hand, we make successive use of our symbolic manipulation program GEFICOM. Specifically, we generate the contributing diagrams with the package QGRAF [2] and convert the output to a form that can be used as input for the packages MINCER [4] and MATAD [3], which solve massless and massive three-loop integrals, respectively. The cancellation of the ultraviolet singularities, the gauge-parameter independence, and the RG invariance serve as strong checks for our calculation. We adopt two independent methods to calculate C1 . One is based on the LET [17] and naturally extends the analysis of Ref. [16] by one order in s . This leads us to consider the top-quark contributions to the gluon and ghost propagators as well as the gluon-ghost coupling through O(4s ) with all external four-momenta put to zero. Specifically, we need to compute 189, 25, and 228 three-loop diagrams, respectively. The external Higgs line is then attached through differentiation with respect to the top-quark mass according to the LET. From the resulting three expressions, C1 is then obtained by solving a linear set of equations [16]. The second method is the brute-force calculation of the 657 three-loop three-point diagrams which contribute to C1 . Both 136

methods lead to the same result, which upon renormalization reads

C1

(



(6) 1 (6) s () 1 + s () 11 , 1 ln 2 = , 12   4 6 Mt2

+

(6) s ()  

!2 



2693 , 25 ln 2 + 1 ln2 2 288 48 Mt2 36 Mt2

67 + 1 ln 2 + nl , 96 3 Mt2



;

(5)

where s is defined in the MS scheme and Mt is the top-quark pole mass. Since C1 appears as an overall factor in Le , it also enters the calculation of the gg ! H parton-level cross section at next-to-leading order [13]. We should mention that Eq. (5) disagrees with the corresponding result recently found in Ref. [13], although the numerical difference is relatively small. We now turn to the second unknown ingredient in Eq. (4), Im h[O10 ] [O10 ]i. In fact, it is convenient to calculate h[O10 ] [O10 ]i first and then to take the absorptive part of it. There is a total of 403 three-loop diagrams to be evaluated. After renormalization, the result is

(



(n ) 2 Im h[O10 ] [O10 ]i = (q2 )2 2 1 + s () 73 + 11 ln 2





l

 2 

4

2

q !2 

(n ) 37631 + s  () 96 495  (3) + 2817 ln 2 + 363 ln2 2  (2) , , 363 8 8 16 q2 16 q2 11  (2) + 5  (3) + + nl , 7189 144 2 4

, nl 76 + 31 ln q2

l

2 , 11 ln2 2  , 263 ln 12 q2 4 q2  127 1  7 ln 2 + 1 ln2 2 + n2l 108 , 6  (2) + 12 (; 6) q2 12 q2 where  is Riemann’s zeta function, with values  (2) =  2 =6 and  (3)  1:202. We are now in a position to find the O(2s ) term of the K factor in Eq. (2). 2 into the master forTo this end, we insert Eqs. (5) and (6) with q 2 = MH mula (4) and factor out the Born result of Eq. (1). In order to get a compact (6) (n ) expression, we also eliminate s () in favour of s l () [18] and choose 137

 = MH . We thus obtain

  (n ) 7n , K = 1 + s (MH ) 95 4 6 l ! 2 (n )  ( M ) 149533 , 363  (2) , 495  (3) s H +  288 8 8 Mt2 , 198 ln M 2  H 2 4157 11 5 2 M t + nl , 72 + 2  (2) + 4  (3) , 3 ln M 2 H  127  1 + n2l 108 , 6  (2) l

l

(5)  1 + 17:917 s (MH )

+

(5) s (MH ) 

!2 

2 M t 156:808 , 5:708 ln M 2 ; H

(7)

where we have substituted nl = 5 in the last step. If we also use the measured (5) values Mt = 175 GeV and s (MZ ) = 0:118, and assume MH = 100 GeV, we have (5) (5) K  1 + 17:917 s (MH ) + 150:419 s (MH )  1 + 0:66 + 0:21:

!2

(8)

We observe that the new O(2s ) term further increases the well-known O(s ) enhancement by about one third. If we assume that this trend continues to O(3s ) and beyond, then Eq. (7) may already be regarded as a useful approximation to the full result. Inclusion of the new O(2s ) correction leads to an increase of the Higgs-boson hadronic width by an amount of order 1%. Equation (7) may be RG-improved by resumming the terms proportional 2 ) as described in Ref. [16]. This leads to to ln(Mt2 =MH (5) (6) K  1 + 14:938 s (MH ) + 2:978 s (Mt ) !2 (5) (6)  ( M ) (5) s s (MH ) s (Mt ) H + 104:499 + 44 : 491   

138

(6) + 7:818 s (Mt )

!2

:

(9)

For the MH values of interest here (65.6 GeV < MH  2Mt ), this amounts to an insignificant reduction of the absolute value of K , by at most 0.6%, for MH = 65:6 GeV. In particular, the second line of Eq. (8) remains valid within its accuracy. Finally, we wish to mention that the K factor of Eq. (7) also applies to the neutral CP-even Higgs bosons of two-Higgs-doublet models such as the minimal supersymmetric extension of the standard model, as long as their couplings to gluon pairs are dominantly generated via top-quark loops.

4 More examples and outlook In this section, we shall mention more examples where GEFICOM has been applied successfully, while a computation by hand would not have been feasible. In renormalization schemes like the widely used modified minimal-subtraction (MS) scheme, the decoupling of heavy particles with masses much larger than the considered energy scale does not happen automatically. Instead, one has to use the language of effective field theory. In QCD with nl light quarks and one heavy quark, with mass mh , this leads to a non-trivial matching condition for the strong coupling constants of the effective and full (n ) (n +1) theories, s l () and s l (), respectively. Whereas for  = mh both are equal in leading and next-to-leading order, this is no longer true at the two-loop level. By the way, for a long time, there was a disagreement between two different two-loop results in the literature [19, 18]. We were able to fix this controversy and add one more loop so as to arrive at the three-loop decoupling relation for s () [20]. Typical diagrams are depicted in Fig. 3.

Fig. 3: Typical three-loop diagrams which contribute to the decoupling relation for s . Looped, dashed, and solid lines represent gluons, Faddeev-Popov ghosts, and heavy quarks, respectively.

139

Fig. 4: Typical three-loop diagrams which contribute to the decoupling relation for the light quark masses. Solid, bold-faced, and looped lines represent massless quarks, heavy quarks, and gluons, respectively.

Fig. 5: Typical Feynman diagrams contributing to the decay rate of the Higgs boson into quarks.

In Ref. [21], the decoupling relations for the light-quark masses mq () were considered. This requires the evaluation of three-loop corrections to the quark propagator. Typical specimens are shown in Fig. 4. The dominant decay channel for an intermediate-mass Higgs boson is the one into bottom quarks. With the help of GEFICOM, it was possible to evaluate the imaginary parts of the four-loop diagrams [22] which give the O(3s ) corrections to the H ! q q decay width. We were also able to evaluate the class of diagrams where the top quark appears as a virtual particle. After going to the effective theory, one ends up with three-loop diagrams of the type depicted in Fig. 5 [23]. The and m functions, two fundamental quantities of QCD, govern the running of the strong coupling constant s () and the light-quark masses, respectively. Over the years, much effort has been invested into their evaluation to higher order. Only recently, it has become possible, with the help of GEFICOM, to deal with the approximately 2000 four-loop diagrams to compute the O(4s ) coefficient of the mass anomalous dimension m [24]. It is not always possible to naively Taylor-expand a Feynman diagram in its small quantities. Doing so, new divergences may appear which were not 140

present in the original diagram. Consistent methods for the expansion of diagrams with high internal masses or large external momenta are provided by the hard-mass (HMP) and large-momentum procedures (LMP), respectively. The rules for the application of these procedures are well defined and are well suited for the implementation on a computer. Prototypes of programs realizing the HMP and LMP already exist. In the future, we shall include them into GEFICOM and thus greatly extend the applicability of the package. Furthermore, we intend to use GEFICOM to compute other important physical quantities in the SM and beyond. For instance, complete four-loop renormalization-group analyses of the SM and its supersymmetric generalizations are amenable and will be undertaken.

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