Abusing qgraf

Abusing qgraf

ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 559 (2006) 220–223 www.elsevier.com/locate/nima Abusing QGRAF P. Nogueira C...

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ARTICLE IN PRESS

Nuclear Instruments and Methods in Physics Research A 559 (2006) 220–223 www.elsevier.com/locate/nima

Abusing

QGRAF

P. Nogueira Centro de Fı´sica das Interacc- o˜es Fundamentais (CFIF), Edifı´cio Cieˆncia, Instituto Superior Te´cnico, 1049–001 Lisboa, Portugal Available online 13 December 2005

Abstract A few chosen examples of Feynman diagram generation with special constraints—and which may be used to speed up some multiloop calculations—are presented in this paper. It will be shown that with a little help QGRAF can be used to solve some types of problems that may seem to be out of its current reach. Those examples also serve to expose some weaknesses of the package, which will hopefully be lessened in future releases. r 2005 Elsevier B.V. All rights reserved. PACS: 12.38.Bx; 11.15.Bt Keywords: Feynman diagram generation; Perturbative massless QCD

1. Introduction The complexity of multiloop calculations of Feynman diagrams frequently demands high computational power. For example, the calculation [1] of the first few moments of the flavour singlet QCD structure functions (which followed the non-singlet counterpart [2]) took the equivalent of a number of years in the typical computers available at the time (ca 1995). Some extra moments were computed more recently [3,4], using much more powerful computers (note that the 3-loop expressions for the structure functions are now known [5]). However, it seems all too likely that in addressing more complex problems—for instance, pushing calculations one loop higher—the computational power problem will be back. Even if that problem may be lessened by using some sort of parallel computation, it should be a good idea to try to reduce the CPU time by other means, whenever possible. In this paper we present some examples of Feynman diagram generation that can be used in particular with nonabelian gauge field theories, especially massless QCD. Although those examples cannot be dealt straightforwardly with QGRAF [6], it will be shown that a little work from the user is enough to address them satisfactorily. A basic point to stress is that whenever a diagram generator fails to E-mail address: [email protected]. 0168-9002/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2005.11.151

provide a certain answer (at least in a direct and simple way) then one should try to find whether some less obvious approach will do. That includes things like constructing a new model, providing as input a new process and postprocessing of diagrams. Before proceeding, let us stress that the examples presented below rely on some assumptions about the model. They are applicable to massless QCD and some other non-abelian gauge theories, for example, but there are certainly many models for which they do not apply. 2. Example 1 Many n-loop diagrams (n41) can be obtained from those of a lower order (say, k-loop) by replacing every treelevel propagator from the k-loop diagrams with the propagator expansion up to order n–k (and then discarding the diagrams whose loop order does not match). Hence, provided one knows how to process such composite diagrams efficiently, one may think of using propagator ‘blobs’ (see Fig. 1) representing 2-point functions at fixed orders of perturbation theory to decrease the number of diagrams to evaluate and thus compute higher-order results faster. Let us denote by G ðkÞ ðfi fj Þ the k-loop (diagrammatic) contribution to the connected 2-point function defined by the product fi fj . Herein we will assume—and this is a

ARTICLE IN PRESS P. Nogueira / Nuclear Instruments and Methods in Physics Research A 559 (2006) 220–223

D0

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extended by increasing the number of propagators and vertices). In the notation of QGRAF, the propagator section of the pseudo-model file would include something like

1

[quark0, QUARK0, - ; w = 0] [quark1, QUARK1, - ; w = 1] D1

D2

D3

Fig. 1. A diagram that includes a 1-loop propagator blob (top) and some of the diagrams it represents (diagrams D1 , D2 , and D3 ).

critical condition—that the mixed propagators are identically zero. That is certainly the case in QCD, where the three basic fields—the gauge field Aam , the fermionic field C, and the ghost field cb —simply do not mix. Another important assumption is that the diagrams under consideration have at least two legs. Obtaining the above-mentioned composite diagrams— instead of the usual ones—can be accomplished rather simply with the help of an auxiliary model (that we will call pseudo-model). Step 1: Let us assume we want to deal with (at most) nloop amplitudes. Then, for each original field fi from the real model there will be n þ 1 fields in the pseudo-model, say Fi;0 , Fi;1 , y, Fi;n . Step 2: For each original propagator in the real model there will be a set of n þ 1 propagators in the pseudo-model; more precisely, if hfi fj i is a nonzero propagator from the real model, then hFi;k Fj;k i ðk ¼ 0; 1; . . . ; nÞ are propagators from the pseudo-model. Step 3: Define a propagator function w such that wðhFi;k Fj;k iÞ ¼ k, thereby associating the propagator hFi;k Fj;k i with the k-loop blob G ðkÞ ðfi fj Þ. Step 4: The vertices of the pseudo-model can be obtained in a simple way: for each vertex fi1 fi2 . . . fim from the real model define all the possible vertices Fi1 ;k1 Fi2 ;k2 . . . Fim ;km in the pseudo-model (excluding duplicates related by permutation of the fields). For the sake of efficiency one should also discard all vertices such that k1 þ k2 þ . . . þ km 4n. Step 5: Perform n þ 1 runs of QGRAF (for nloop equal to 0; 1; . . . ; n) requiring that (1) there are no 2-point function insertions (i.e., use option nosigma), and P (2) the effective loop order is correct i.e., p wðpÞ ¼ n  nloop, where the summation extends over the propagators p of the diagram (this is a propagator weight constraint that can be implemented very simply in version 3.1). There is actually an exception to Step 5 when dealing with propagator diagrams. In that case one should exclude the run for nloop ¼ 0 (also, this process does not apply if n ¼ 1). In order to provide an explicit example, let us assume we are dealing with QCD at 2-loops (that bound can be easily

[quark2, QUARK2, - ; w = 2] instead of a single quark propagator declaration. The gluon and ghost propagators would have to be dealt with similarly. On the vertex side, the list generated e.g. from the quark–gluon vertex would be as follows: ½quark0; QUARK0; gluon0 ½quark0; QUARK0; gluon1 ½quark0; QUARK1; gluon0 ½quark1; QUARK0; gluon0 ½quark0; QUARK1; gluon1 ½quark1; QUARK0; gluon1 ½quark1; QUARK1; gluon0 ½quark0; QUARK0; gluon2 ½quark0; QUARK2; gluon0 ½quark2; QUARK0; gluon0 Given the way the field names of the pseudo-model have been chosen in this example it is easy, if desired, to write the name of the original field in the output file: just use the construction hfieldihbacki in the style file. 3. Example 2 A well-known inconvenience that appears in non-abelian gauge theory calculations is that—due to the 4-gluon vertex expression, which contains three different pieces— the so-called colour factor of a Feynman diagram is not necessarily a global colour factor. It means that in general it is not possible to factor out the colour contribution and simply deal with the Lorentz part of the amplitudes separately. If nothing is done about it, then in general one will have to deal with longer expressions because (i) when a global colour factor exists and such factor has k terms ðk41Þ, we miss the opportunity to deal with only a fraction 1=k of the original number of terms, and (ii) the (usually symbolic) colour contribution is still present, and so it still contributes to the size of every term. A possible attempt to improve the situation would be to rewrite the amplitudes as a sum of products of a colour factor and a colour independent part, where one would ideally minimize the number of terms in that sum. Then each term would be dealt with separately, as if it were a separate amplitude. Herein we highlight a different possibility: to rewrite the Lagrangean density of the non-abelian gauge theories such that there is no 4-gluon vertex (nor any other undesirable vertex). That turns out to be a real possibility, and the resulting Lagrangean density has cubic vertices only. The

ARTICLE IN PRESS P. Nogueira / Nuclear Instruments and Methods in Physics Research A 559 (2006) 220–223

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a,λ

k,σ,τ

(g/sqrt(2)) fkab Ωλµστ

~

b,µ k,σ,τ

l,α,β

~ δkl gσα gτβ

a,λ

c,ν ~

b,µ

g2 fkabfkcd Ωλµνρ

d,ρ

Fig. 2. The new cubic vertex (top), the new propagator (centre), and how the quartic vertex is effectively recreated. Factors like þi or i have been neglected.

fs = 1/4

fs = 1/2

fs = 1/4

Fig. 3. A diagram from QCD (top) together with the corresponding diagrams in the cubic gauge (bottom).

cost of this transformation is the introduction of a new ficticious field X a;b . That field is non-propagating i.e., the propagator is constant. As illustrated in Fig. 2, a single additional vertex ggX (top) will lead to an effective 4-gluon vertex (bottom). The quantity O is the tensor: Ol;m;n;r ¼ gl;n gm;r  gl;r gm;n

(1)

which satisfies the relation Ol;m;s;t On;r;s;t ¼ 2Ol;m;n;r .

(2)

Apparently this effective vertex reproduces only one piece of the real 4-gluon vertex, but it should be noted that there are usually more diagrams in the cubic gauge (see Fig. 3). The combinatorics will work out automatically. It should be noted that this approach seems better than the simple expansion of the 4-gluon vertex into 3 terms (which generates 3k terms for a diagram with k such vertices). That is because some diagram symmetries are naturally taken into account in the cubic gauge—see Fig. 3, where a diagram with a quartic vertex is replaced by two diagrams (not three). This approach will lead to an increase in the number of diagrams but, based on the above evidence, we expect it will also lead to somewhat faster calculations (nothing dramatic, but visible).

the slightly more general setting of non-abelian gauge theories. One simply has to address the extra complication coming from the so-called colour factor. The basic idea is to choose a single representative from each set of diagrams that differ only in the orientation of the fermion loops. The result obtained in the calculation of the amplitude of that representative may then be used to obtain the amplitudes of the remaining diagrams in the set—one simply has to compute the relative sign and the colour factor for those diagrams. How can one obtain the representative diagrams only, using a Feynman diagram generator? There are at least two possibilities: (i) implementing such a feature in the diagram generator, which can be done by allowing a single orientation for the fermion loops, and (ii) temporarily transforming the (Dirac) fermion into a scalar particle. The former possibility will hopefully be integrated into QGRAF, but it may be instructive to look at the latter possibility. Let us consider the 3-loop quark propagator to present a specific example. In this case we will transform the open line into a closed one because we want that diagrams like D1 and D2 (from Fig. 4) not to be both generated (assuming the respective contributions to be equal). So we will do the following: first we build a pseudo-model where the quark is now a real scalar field H, but keep the gluon and the ghost as they are; now we add an external field Y as well as a vertex YH H; then we run QGRAF for the tadpole of Y at 4-loops (one more loop than the desired order). Some of the resulting diagrams are presented in Fig. 5. At this point those diagrams are processed as follows. Firstly, we revert to (oriented) fermions, by selecting an orientation for the respective lines, and then compute the standard diagram sign. Then to recover the contribution from the omitted diagrams we need two pieces of

D1

D2

D3 Fig. 4. Two diagrams (D1 and D2 ) that are dual under fermion flow inversion, and a diagram (D3 ) that is invariant under that operation.

D4

D5

4. Example 3 A well-known feature of QED is that any two Feynman diagrams that differ only in the orientation of the fermionic loops have very similar amplitudes (i.e., either identical or opposite in sign). It is possible to make use of that result in

Y

Y

Fig. 5. Two diagrams related to the diagrams of Fig. 4.

ARTICLE IN PRESS P. Nogueira / Nuclear Instruments and Methods in Physics Research A 559 (2006) 220–223

information for all the quark loops (let us assume there are no open lines). One is the relative sign generated by the inversion of the fermion number flow. The other is the string of group generators T a1 T a2 . . . T ak that results from the Feynman rule for the quark–gluon vertex and that will lead to the colour factor. It is not difficult to see that the desired result may be obtained replacing each such string with T a1 T a2 . . . T ak þ ð1Þk T ak . . . T a2 T a1 .

(3)

Finally we delete the YH H vertex to revert to propagator type diagrams (and change the diagram sign if needed). Although it may not be obvious, the combinatorics also works automatically in this case. For example, let us analyse again Figs. 4 and 5. Diagrams D1 and D2 , which have f s ¼ 1, are represented by D5 which has f s ¼ 1=2; however, the loop of length 2 provides (through Eq. (3)) a factor of 2, while the longer loop provides the contributions for two diagrams. Let us check diagram D3 too, which has f s ¼ 1, and no dual diagrams (because there is a certain symmetry for fermion flow inversion); hence, diagram D4 is symmetric (f s ¼ 1=2) but Eq. (3) gives a factor 2 that cancels the symmetry factor. A short mention about the way the diagrams for the 3loop 4-point functions were obtained in Ref. [1] is also due. Because one was looking at even moments of the structure functions one could pair the legs of those functions and apply a similar trick to take symmetries into account. Hence the diagrams for those functions were obtained from 5-loop propagators. 5. Recent developments Two new versions of QGRAF have been recently released [7], namely version 3.0 (a major version) in May 2004 and version 3.1 in May 2005. Version 3.0 presented at least one main benefit: it allows the definition of parameters in the model file, and then, by an extension of the style file syntax, it also allows them to be written in the output file. This is an important step towards the goal of allowing the model file to contain all the relevant information about the model, and also towards QGRAF being able to write general symbolic amplitudes. With version 3.1 users may now impose weight-like constraints based on the numerical parameters defined in the model file. This feature allows, e.g. for a straightforward selection of gauge invariant subsets of diagrams—

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subsets defined by the powers of the gauge coupling constants—in models with two or more independent gauge couplings. Another possible use has been illustrated in Section 2. 6. Final remarks Examples 1–3 were actually implemented in 1994 during the early stages of the work leading to the above-mentioned calculation of the moments of structure functions [1]. The present paper may be seen as a detailed description of the brief comment made in that paper (see page 350) about the diagram counting being non-standard because a number of tricks had been used (actually, the bit about the 4-gluon vertex seems now misleading—the exact method used at the time is the one described herein). Acknowledgements I would like to thank my then collaborators, Timo van Ritbergen, Sergei Larin, and especially Jos Vermaseren, for various discussions. I gratefully acknowledge that my stay at NIKHEF was partially supported by Fundac- a˜o para a Ci^encia e a Tecnologia (FCT), Lisbon. Then I would also like to thank Mikhail Tentyukov and Andre´ van Hameren for pointing out (during the presentation of my contribution at ACAT05, Zeuthen) parallel/ independent work related to Example 2. The former author claimed that the replacement of the quartic vertex by a cubic one is also used by some authors involved in multiloop QCD calculations. The latter author brought to my attention a publication [8] in the subject of very high energy multijet production where the use of (only) cubic vertices provided a strong reduction of the CPU time. References [1] S.A. Larin, P. Nogueira, T. van Ritbergen, J.A.M. Vermaseren, Nucl. Phys. B 492 (1997) 338. [2] S.A. Larin, T. van Ritbergen, J.A.M. Vermaseren, Nucl. Phys. B 427 (1994) 41. [3] A. Retey, J.A.M. Vermaseren, Nucl. Phys. B 604 (2001) 281. [4] J. Blu¨mlein, J.A.M. Vermaseren, Phys. Lett. B 606 (2005) 130. [5] S. Moch, J.A.M. Vermaseren, A. Vogt, Phys. Lett. B 606 (2005) 123. [6] P. Nogueira, J. Comput. Phys. 105 (1993) 279. [7] http://cfif.ist.utl.pt/paulo/qgraf.html. [8] P. Draggiotis, R.H.P. Kleiss, C.G. Papadopoulos, Phys. Lett. B 439 (1998) 157.