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NAVIGATING AROUND THE ALGEBRAIC JUNGLE OF QCD:

arXiv:hep-ph/9207266v1 28 Jul 1992McGill/92-32hep-ph/9207266(July, 1992)NAVIGATING AROUND THE ALGEBRAIC JUNGLE OF QCD:EFFICIENT EVALUATION OF LOOP HELICITY AMPLITUDESC.S. Lam∗Department of Physics, McGill University, 3600 University St.Montreal, P.Q., Canada H3A 2T8AbstractA method is developed whereby spinor helicity techniques can be used to simplifythe calculation of loop amplitudes.This is achieved by using the Feynman-parameterrepresentation where the offending off-shell loop momenta do not appear.

BackgroundFeynman gauge also helps to simplify the calculations. This method is applicable to anyFeynman diagram with any number of loops as long as the external masses can be ignored,and it is at least as efficient as the string technique in the special circumstances whenthe latter can be used.

In order to minimize the very considerable algebra encountered innon-abelian gauge theories, graphical methods are developed for most of the calculations.This enables the large number of terms encountered to be organized visually in theFeynman diagram without the necessity of having to write down any of them algebraically.A one-loop four-gluon amplitude in a particular helicity configuration is computed explicityto illustrate the method. * email address: Lam@physics.mcgill.ca0

1. IntroductionThe number of diagrams and the number of terms in a QCD calculation increasedramatically with the multiplicity of the external particles as well as the number of loops.Even in the absence of quarks, a pure QCD tree process describing the production of sixgluon jets from a glue-glue collision is given by the sum of some 34,000 diagrams, androughly half a billion terms.

A one-loop pure QCD glue-glue elastic scattering amplitudehas 39 diagrams and some ten thousand terms. The large number of diagrams is due tothe large number of ways triple and quadruple gluon (and ghost) vertices can be assembledtogether, and the large number of terms is due to the presence of six terms at each vertex.The necessity of having to sum over intermediate color indices makes it more complicated;if loops are present loop integrations must be done and the problem gets worse.

Similarcomplexity occurs in electroweak computations. These difficulties are not something thatone can ignore in practice because a large number of jets is present at high energies, andbecause loop calculations are increasingly demanded for precision comparisions with theStandard Model.

To make progress one must find a way around this algebraic jungle.Unnecessary algebraic complications are already present in QED bremstrahlung cal-culations as is evidenced by the fact that simple results emerge from complicated covarianttechnique calculations [1]. It was later discovered that the use of the spinor helicity tech-nique enables one to obtain the final result directly in a much simpler way [1,2].

Over thelast ten years or so this technique has been further developed and applied to various QCDand electroweak processes in the tree approximation [1–18]. It leads to a tremendous sim-plification in the calculations, reducing impossibly large number of terms into manageablesizes.

As a result of these techniques, many tree amplitudes which are too complicated tocalculate by ordinary means have been successfully computed. See Ref.

[16] for an excellentreview of the techniques and the results.The basic idea of this technique is that quark masses are negligible at high energies.If they are neglected, chirality is conserved, and this conservation can be exploited to sim-plify the calculations. For example, the trace tr(γp1 · · ·γp2n) which according to the usualformula is given by the sum of (2n)!/2n(n)!

terms, can be written using this technique asa sum of merely two terms if all p2i = 0 (see eq. (39) below).

This simplification is equallyapplicable to processes involving gluons and photons if their wavefunctions are written ina multispinor basis. Moreover, gauge freedom allows one to choose these wavefunctions tobe orthogonal to any massless ‘reference momentum’, thereby further simplifying the cal-culations by rendering many terms zero.

On top of that, recursion [4] and supersymmetry[17,12] relations may be exploited to further simplify the calculations.Unfortunately it is difficult to apply this beautiful technique to the computation ofloop diagrams.The internal loop momenta are offshell; chirality is not conserved andmassless spinor methods are not useful for these momenta.In an era when precisionexperiments are increasingly called for this is a serious handicap and it is important tofind a way around this obstacle. This turns out to be possible and this is the subjectmatter of the present paper.The idea is very simple, though the detailed implementation of the idea is far frombeing trivial.If a representation of the scattering amplitude can be found where theinternal loop momenta do not appear, then every momentum in the problem is a linear1

combination of the external massless momenta and the spinor helicity technique can beused.This happens to be true for certain one-loop amplitudes which have a string extension.The point is that a one-loop string scattering amplitude can be written as an integral overthe Koba-Nielsen variables, without the explicit presence of any internal momenta. Bytaking the string tension to infinity, one obtains a formula for the scattering of the masslessparticles in the string in which no internal momentum is present and the spinor helicitytechnique can be used.

The simplification thus achieved is very considerable [19–21].This ingenious method has its limitations. It is difficult to find a simple string formulabeyond one loop, and in any case the technique is not applicable if the corresponding fieldtheory has no string extension, or that the one-loop formula for the desired external particleis difficult to write down, as is the case for external fermions.

Moreover, one cannot helpbut feel that there must be a purely field-theoretical way of calculating a field-theoreticalscattering amplitude, without having to resort to the artificial, though ingenious, means ofcreating an intermediate string and then destroying it again by taking the infinte tensionlimit.There is indeed a well-known and purely field-theoretical way of getting rid of theinternal loop momenta: by introducing the Feynman parameters to combine the propaga-tors, the loop-momentum integrations can be explicitly carried out. The result is again aformula in which the only momenta present are the external massless momenta, and thusthe spinor helicity technique can again be used.

It is this route that we would like toexplore here. Unlike the string technique, Feynman parameter representations are avail-able for any Feynman diagram with any number of loops, so this method can be used forall processes subject only to the validity of ignoring the external masses.

As a matter offact, in the known cases [19–21], the Koba-Nielsen parameters reduce themselves to theFeynman parameters in the infinite tension limit of the string, thus suggesting the closeconnection of the two methods.This simple idea must be supplemented by a number of other developments to makeit useful, for otherwise the amount of algebra necessary to carry out the calculations usingelementary means is too unmanageable. These means are available; they will be mentionedbelow and discussed in much more detail in the next two sections.First of all, one needs a set of rules analogous to the usual momentum-space Feynmanrules to write down the Feynman parameter representation directly from the Feynmandiagram.

Otherwise if we had to do the internal loop integrations explicitly every timethen the task would be too complicated. These Feyman-parameter rules are available [22]and will be reviewed in Sec.

2.The number of terms in the Feynman-parmeter representation is even larger thanthe number of terms in the conventional momentum representation. This would not haverepresented progress towards simplification except for the fact that the spinor helicitytechnique is now available to render many terms zero.

But to be able to do that one mustfirst find a way to organize the large number of terms in a simple and systematic way sothat one can recognize beforehand which terms to discard in the calculation. The way todo that is to reorganize these terms into gauge-invariant color subamplitudes.

It is knownhow this can be done algebraically in tree and one-loop processes [16,19]. We shall discuss2

in Sec. 3 how this can be one for any number of loops graphically by introducing color-oriented Feynman diagrams.

In this graphical language the different terms in the scatteringamplitude correspond to different covering paths. The use of this graphical language doesnot in any way reduce the number of terms, but it gives a way to organize them in avisual way without the necessity of writing down anything algebraically.

This graphicalorganization can be used in the usual momentum-space representation of a scatteringamplitude, as well as the Feynman-parameter space representation discussed in Sec. 2.This graphical technique is particularly useful when it is combined with the spinorhelicity technique, which as a result of chirality conservation renders many terms zero.

Ingraphical language this means that paths of certain topologies lead to vanishing resultsand do not have to be included. We shall also find that the use of Feynman gauge in thebackground field method reduces further the amount of labour of calculation by renderingmore terms zero.

A review of the spinor helicity technique and how this can be implementedgraphically will be discussed in Sec. 4.We choose to illustrate the present method in Sec.

5 by computing the one-loop gluon-gluon elastic scattering amplitude in a particular helicity configuaration. This amplitudein the absence of quark loops has already been computed in the string method [19–21];we choose it to illustrate our method so that the efficiency of the two techniques can becompared.

We shall find that within the present framework there are two ways to computethis amplitude. The direct way yields a result as efficient as the string method; the indirectway making use of supersymmetry is even simpler and the result can be obtained in only afew lines.

This is to be compared with the ordinary Feynman diagram calculations wheresome ten thousand terms appear.It should be emphasized that chirality conservation affects only the spin flows, viz.,the derivative couplings and the numerators of the propagators in the usual momentum-space representation. The denominators of the propagators may remain massive without inany way affecting the effectiveness of these techniques.

This means that while the externalparticles must remain massless, exchange and internal particles may often be massive, as isthe case for the Z and W bosons. Thus the present technique can be used to compute heavyparticle productions if their subsequent decays into light particles are also incorporatedinto the diagrams.2.

Feynman-parameter RulesConsider a Feynman diagram in d-dimensional spacetime, with N internal lines andℓloops. Let pA be the external outgoing momenta, qr (r = 1, · · ·, N) be the momentumof the rth propagagor, and mr be the mass of the particle being propagated.

Every qr isgiven by a linear combination of pA and the ℓloop-momenta ka, the specific combinationdepends on the topology of the diagram.The scattering amplitude corresponding to a Feynman diagram expressed in momentum-space representation is of the formM = −i(2π)4ℓZℓYa=1(ddka)S(q, p)QNr=1(−q2r + m2r −iǫ),(1)3

where S(q, p) receives its contributions from the vertices and the numerators of propaga-tors. It also contains the symmetry factor and the minus sign for each fermion loop.

Byintroducing the Feynman parameters αr and carrying out the integrations over ka, it canbe shown [22] thatM =πd/2−416ℓXk=0ΓN −dℓ2 −k ZDNα∆(α)d/2Sk(J, p)D(α, p)N−dℓ/2−k ≡Xk=0Mk,(2)whereDNα =(NYr=1dαr)δ(NXr=1αr −1),(3)∆(α) =XT1(ℓYα),(4)D(α, p) =NXr=1αrm2r −P(α, p),(5)P(α, p) =∆(α)−1 XT2(ℓ+1Yα)(ℓXp)2,(6)Jr =∆(α)−1 XT2(r)α−1r (ℓ+1Yα)(ℓXp),(7)S0(J, p) =S(J, p). (8)The quantities appeared in (2) have a very simple physical interpretation.If weconsider the Feynman diagram as an electrical circuit, with the external momenta pA asthe external currents, and the Feynman parameters αr as the resistance of the rth line,then Jr is simply the current flowing through the rth line, and P(α, p), which can beproven to be equal to PNr=1 J2r αr, is just the power dissipated in the circuit.

The crypticformulas (3)–(8) offers a simple and practical way to compute these currents and the powerdirectly from the Feynman diagram.We shall now elaborate on these cryptic formulas. A connected diagram with ℓloopscan be made into a connected tree diagram if ℓinternal lines are cut.

There are manyways to do this, each resulting in a different (one-)tree T1. The sum in (4) is taken overall such one-trees T1, with each term in the sum equal to the product of all the Feynmanparameters α of the cut lines.

As a result, ∆(α) is of a homogeneous degree ℓin the α’s.Similarly, ℓ+ 1 cuts can bring the diagram into two connected trees that are disjoint,or a ‘two-tree’ T2. The sum in (6) is over the set of all such two-trees T2.

This time eachterm consists of the product of the ℓ+ 1 Feynman parameters of the cut lines, times thesquare of the sum of all the external momentum pA attached to one of the two trees. It doesnot matter which tree we choose to compute the momentum sum because of conservationof momentum.4

Now we come to the numerator Sk(J, p) in (2), for k = 0, 1, 2, · · ·. The first termS0(J, p) is just the numerator factor S(q, p) in (1) with each qr replaced by Jr.

The rulefor computing the current Jr is given in (7), where the sum is over the set of all two-treesT2(r) obtained by having the line r always cut. The summand consists of the product ofthe α’s of the cut lines except the rth (so it is of homogeneous degree ℓin α), times amomentum factor given by the sum of all the external momenta attached to one of the tworesulting trees.

If the momentum qr flows from tree 1 to tree 2, then it is the sum of pAattached to tree 2, or minus the sum of pA attached to tree 1, that should be used in thesum. This convention presumes that all the external momenta pA are outgoing.

In otherwords, the sign is such that the direction of the flow of the current Jr must match that ofthe external currents pA.We are now in a position to describe Sk(J, p) for k > 0. It is obtained from S0(J, p)by contracting k pairs of J’s in all possible ways, and summing over all such contractions.If no contractions are possible then Sk = 0.

For each pair Jr, Js in the contraction, onemakes the replacementJµr Jνs →−12gµνHrs,(9)and the factor Hrs is given byHrr = −∆(α)−1∂∆(α)/∂αr,(10)Hrs = ± ∆(α)−1 XT2(rs)(αrαs)−1(ℓ+1Yα),(r ̸= s). (11)This time the sum in (11) is over the set of all two-trees T2(rs) in which lines r and s musthave been cut, and each term in the sum is a product of the α’s of the cut lines except therth and the sth.

The sign in front is +1 if both qr and qs flow from tree 1 to tree 2, and−1 otherwise.This concludes the description of the quantities in (2). We shall now illustrate theserules with one-loop diagrams.

See Ref. [22] for an illustration of these rules for a two-loopdiagram.A tree is obtained from a one-loop diagram by cutting any of its N internal lines.Thus for any one-loop diagram, (4) with (3) yields∆(α) =NXr=1αi = 1(12)and (10) givesHrr = −1.

(13)Now specialize to a box diagram (Fig. 1(a)) and a vertex diagram (Fig.

1(b)). Using (6),(7), (11) and (12), one gets for the box diagramP(α, p) = α1α3(p1 + p2)2 + α2α4(p1 + p4)2 + α1α2p21 + α2α3p22 + α3α4p23 + α4α1p24,J1 = α2p1 + α3(p1 + p2) + α4(p1 + p2 + p3),5

J2 = α3p2 + α4(p2 + p3) + α1(p2 + p3 + p4),J3 = α4p3 + α1(p3 + p4) + α2(p3 + p4 + p1),J4 = α1p4 + α2(p4 + p1) + α3(p4 + p1 + p2),Hrs = −1(r ̸= s),(14)and for the vertex diagramP(α, p) = α1α2(p2 + p3)2 + α2α3p22 + α1α3p23,J1 = −α3p3 −α2(p2 + p3),J2 = α3p2 + α1(p2 + p3),J3 = α1p3 −α2p2,Hrs = −1(r ̸= s). (15)Before leaving this section let us say a word about renormalization.

For theories thatare no more than logarithmically divergent, primitive ultraviolet divergence, if any, comesfrom the term with the highest power of q in the numerator of (1), and this power is2(kmax) = 2(N −2ℓ). In the language of (2), this occurs only in the term Skmax(J, p)with the maximum number of contractions.

If we let d = 4 + 2ǫ, then this term isMkmax =π2+ǫ16ℓΓ(ℓǫ)ZDNα∆(α)2+ǫSkmax(J, p)D(α, p)ℓǫ . (16)Renormalization in the MS or MS scheme is therefore easy to carry out.3.

Color Decomposition and Spin FlowWe specialize now to QED and SU(N) QCD. With simple modifications this canalso be applied to the electroweak processes.

The quark (q) belongs to the fundamentalrepresentation and both the gluon (g) and the Fedeev-Popov ghost (G) belong to theadjoint representation. For the purpose of discussing color decomposition, there is no needto distinguish ‘g’ and ‘G’ so we shall collectively refer to them as ‘G’.The factor S(q, p) in (1) is composed of vertex contributions and the numerators ofpropagators.

As such it contains information on both spin and color. It is this factor thatcontains the large number of terms mentioned in the Introduction.

The purpose of thissection is to discuss how this quantity (and the corresponding scattering amplitude M) canbe reorganized to simplify calculations. Specifically, the factor S(q, p) (the amplitude M )is to be decomposed into the form Pi Cimi, where Ci is the color factor, which consists ofproducts of the (S)U(N) generators and their traces, and mi carries spin and momentumbut no color information.We shall refer to mi in the decomposition of S(q, p) as thespin factor (other than some trivial factors of the coupling constant g to be discussedlater), and mi in the decomposition of M as the color subamplitude.

There are at leastthree advantages for such a decomposition. First of all, many Ci’s differ from one another6

only by permutations of the color indices. Consequently only one of these mi’s has to becomputed explicitly; the rest of them can be obtained by similar permutations.

Secondly,the color factors Ci are independent. As a result each spin factor and color subamplitude areinvariant under an arbitrary gauge change of an external polarization vector.

This aspectof it will be particularly useful in the spinor helicity technique because we can choose thereference momenta independently for each mi. Thirdly, the mi’s satisfy other importantidentities like the ‘dual Ward’s identity’ which can be utilised in practical computations.Color decomposition has been carried out algebraically for tree and one-loop diagrams[16,19]; we shall do it to all loops and do it graphically in order to minimize the algebra.For that purpose we will introduce color-oriented Feynman diagrams, from which colorfactors as well as spin factors can be read offdirectly.The discussion in this section is independent of the last section.

Hence the resultsare equally applicable to momentum-space representations as well as Feynman-parameterrepresentations.It is convenient to extend the gauge theory by an extra U(1) factor to complete itto an U(N) = U(1) × SU(N) gauge theory. This simplifies the algebraic manipulationwithout losing any information, for as we shall see QCD (SU(N)) expressions can be readofffrom the simpler results of an U(N) gauge theory.Let T A (A = 0, a; a = 1, · · ·, N 2 −1) be the U(N) = U(1) × SU(N) generators in thefundamental representation.

T 0 = (1/√N)1 is the U(1) generator and T a are the SU(N)generators. The normalization of the U(1) factor is chosen to satisfy the normalizationTr(T AT B) = δAB.

(17)The structure constant f ABC can be obtained from the commutation relation [T A, T B] =if ABCT C by the formulaf ABC = −i{Tr(T AT BT C) −Tr(T AT CT B)},(18)and is seen to be antisymmetric in its U(N) indices. Note from this that f 0BC = 0, whichreflects the physics that the SU(N) and the U(1) gauge bosons do not interact directlywith each other.

This is an important fact which will allow us to project out the U(1)bosons to regain QCD.The completeness relation dual to (17) is(T A)ij(T A)kl = δjkδil,(19)where summation over the N 2 repeated indices A is understood. From this one obtainsf ABEf ECD = (−i)2{Tr(ABCD) −Tr(BACD) −Tr(ABDC) + Tr(BADC)}.

(20)For brevity, we have chosen to write Tr(T AT BT CT D) simply as Tr(ABCD), and we shalloften use this same abbreviation of replacing T A simply by A in the rest of this paper.The vertices for QCD and U(N) gauge theory in the background Feynman gauge [23]are exhibited in Fig. 2.

A thin solid line stands for a gluon, a dashed line stands for the7

Fadeev-Popov ghost, and a thick solid line stands for a fermion. The background gaugeis a gauge in which an external gluon is distinguished from an internal gluon; in Fig.

2 acircled ‘A’ at the end of the line signifies an external line. In diagrams where a circled ‘A’makes an appearance, the uncircled gluon lines are meant to be internal lines.

In diagramswhere no circled ‘A’ appear, each of the gluon lines in the diagram may be taken eitheras an internal line or an external line, unless such a combination of external and internalgluon lines have appeared explicitly in a diagram in Fig. 2 in which a circled ‘A’ is present.In that case theFeynman rule (see eq.

(21) below) for the diagram with circled ‘A’sshould be used.The discussion in this section can be applied to any gauge. However for the sakeof application in the next two sections we shall use explicitly the background Feynmangauge.

Although there are more vertices (those with circled ‘A’) in the background gaugethan the usual covariant gauges, nevertheless we shall see in the next section that theuse of background gauge along with the spinor helicity basis simplifies enormously thecalculations.All the momenta in Fig. 2 are understood to be pointing outwards.

It is also un-derstood that the line labelled by 1 carries an outgoing momentum p1, a color index a,and a Lorentz index α. Similarly, line 2 carries the quantum numbers (p2, b, β), etc.

TheFeynman rules for these vertices in the background Feynman gauge are given below, withthe equation numbers corresponding to the diagrams in Fig. 2.

For example, eq. (21a) isthe Feynman rule for the vertex in Fig.

2(a).• igf abc{gαβ(p1 −p2)γ + gβγ(p2 −p3)α + gγα(p3 −p1)β},(21a)• igf abc{gαβ(2p1)γ + gβγ(p2 −p3)α + gγα(−2p1)β},(21b)• −g2f abef ecd(gαγgβδ −gαδgβγ) −g2f acef ebd(gαβgγδ −gαδgβγ)−g2f bcef ead(gβαgγδ −gβδgαγ),(21c)• −g2f abef ecd(gαγgβδ −gαδgβγ + gαβgγδ) −g2f acef ebd(gαβgγδ −gαδgβγ)−g2f bcef ead(gβαgγδ −gβδgαγ −gβγgαδ),(21d)• −igf abc(−p3)α,(21e)• −igf abc(p2 −p3)α,(21f)• −g2f abef ecdgαβ,(21g)• −g2(f abef ecd + f acef ebd)gαβ,(21h)• gT aγα. (21i)The U(N) Feynman rules are the same except that the SU(N) color indices a, b, c, d shouldbe replaced by the U(N) indices A, B, C, D.We shall use (18) and (20) to replace the factors f ABC and f ABEf ECD in (21), andthen proceed to group the terms with the same U(N) traces.

Each of these terms definesa color-oriented vertex in which the U(N) indices of the external lines read clockwisecoincides with the indices in the trace read from left to right.Each color-oriented vertex factor is a product of three quantities: the coupling constantfactor, the color factor, and the spin factor. The coupling constant factor will be taken to8

be g for cubic vertices and g2 for quartic vertices. The color factors will be taken to beT A for a qqg vertex, to be Tr(ABC) for a GGG vertex, and to be Tr(ABCD) for a GGGGvertex.

The rest of the vertex factor will be defined to be the spin factor, the details ofwhich are exhibited in Fig. 3.Fig.

3 should be read in the following way.A gluon line continuing through thevertex indicates a factor of a metric tensor in spacetime.A dot represents the vectorwritten below the diagram; other numerical factors are also written below the diagram.For example, the color factor for diagram (a) is Tr(ABC), its coupling-constant factor isg, and its spin factor is gβγ(p2 −p3)α. The color factor for diagram (e) is Tr(ABCD), itscoupling-constant factor is g2, and its spin factor is 2gαγgβδ.The Feynman diagrams assembled from color-oriented vertices will be called color-oriented Feynman diagrams, or just an oriented diagrams for short.

A color-oriented di-agram can be obtained from an ordinary Feynman diagram by flipping any number ofexternal gluon lines each about the G propagator it emerges from, by interchanging twoidentical external G lines emerging from the same vertex, by interchanging two identicalinternal G lines if this does not alter the topology of the diagram, or a combination ofthese. In general, an ordinary Feynman diagram leads to many color-oriented Feynmandiagrams.

The total contribution to a scattering amplitude is the sum of the contributionsfrom all the color-oriented diagrams.If the Feynman diagram in question can be obtained from the infinite tension limit of astring diagram, then flipping of the gluon line corresponds to a twisting of the string. Thecolor factors for the whole diagram to be discussed below are nothing but the Chan-Patonfactors [24].The total color factor of an oriented diagram is the product of the color factors of itsoriented vertices, summed over intermediate color indices.

Eq. (19) can be used to carryout these sums; the result of which gratifyingly can be read offonce again directly fromthe color-oriented Feynman diagram.Let us start with a tree diagram having n external G particles and no fermion anywherein the diagram.

The color factor of this tree turns out to beTr(C1C2 · · · Cn),(22)where C1, C2, · · ·, Cn are the U(N) color indices of the oriented Feynman diagram readclockwise around the whole tree.For example,the color factor for Fig. 4 isTr[(1)(2)(3)(4) · · ·(14)(15)(16)].From now on, we shall use capital letters near the end of the alphabets to denoteproducts of U(N) generators, e.g., X = C1C2 · · · Cp.Eq.

(22) can be proven by induction. By definition, a color-oriented diagram with asingle GGG or a single GGGG vertex is already of the form of (22).

Suppose now we havetwo trees, the color factor of each is of the form (22). This is illustrated in Fig.

5, wherethe color factors of the two trees are respectively Tr(XA) and Tr(AY ). When we sewthese two trees together at index ‘A’ to obtain a bigger tree, the resulting color factor,using (19), is indeed Tr(XY ), which can be read out directly from Fig.

5 using the rulesof (22). This completes the induction proof of (22).9

The result for SU(N) QCD is equally simple and the color factor is again given by(22), but with the upper case U(N) indices Ci replaced by the corresponding lower caseSU(N) indices ci. This is so because of the absence of coupling between the U(1) andthe SU(N) gauge bosons, viz., f 0BC = 0.

Therefore as long as the external lines of aconnected tree carry an SU(N) indices, the U(1) gluon is decoupled and will never makesits appearance in the internal line either.Next, consider tree diagrams in which a single fermion line is present.The colorfactor for a qqg vertex is T A if A is the U(N) color index of the gluon. If a whole tree of Gparticles with the color factor Tr(XA) is planted at this vertex, then the combined colorfactor is obtained from (19) to be X.

If A1, A2, · · ·, An are the successive qqg vertices aswe go along a fermion line, and if G-trees with color factors Tr(XiAi) are planted at thesevertices, then the combined color factor would beX1X2 · · · Xn. (23)Graphically, this is simply the the multiplication of all the U(N) generators T in clockwiseorder around the whole tree, as shown in Fig.

6.Note the difference between a ggg and a qqg vertex. The former is oriented, in thesense that there are two oriented vertices associated with one ordinary Feynman vertex, butthe oriented vertex in the latter case is the same as the ordinary vertex.

In the ggg vertex,there is no further color specification other than the indices of the external G-lines, but inthe qqg vertex, the color state of the initial and the final quarks must still be specified.The result of these differences is that as we traverse clockwise around an oriented diagramto read out its color factor, we must cover both sides of every G-line, and that the traceof the product of generators must be taken. On the other hand, we should traverse onlythrough the top side of a fermion line and no trace of the product of generators is to betaken.

If we should find it convenient to draw a G-tree downward from a fermion line, as isthe case of the X2-tree in Fig. 6, then clockwise order must still be maintained in the wayindicated in the figure.

In other words, since we only follow the top and not the bottomof the fermion line, the color factor in Fig. 6 is X1X2X3, and not X1X3X2 as we mightthink if we were to follow both sides of the fermion line.The same result (23) is again true if we consider only SU(N) QCD.

Once again thisis due to the lack of coupling between the U(1) and the SU(N) gluons.The situation of having a G-tree connecting two separate quark lines can be obtainedsimilarly from (19), but the paths along which the color generators are multiplied togethernow cross over from one fermion line to another, as in Fig. 7.

This is so becauseTr(AY BV )XAW ⊗UBZ = XY Z ⊗UV W.(24)If on the way one encounters another G-tree connecting to a third fermion line, then onemust cross over to the third tree at that point, etc.This formula is still valid in SU(N) QCD as long as either Y or V factor appearing inthe tree Tr(AY BV ) connecting the two fermion lines contains at least one external SU(N)gluon. In that case, as before, decoupling prevents the U(1) gluon to appear even in theinternal lines.

The situation is different if the tree is simply Tr(AB). In that case an U(1)10

gluon connecting the two fermion lines is present, and its effect must be subtracted awaywhen SU(N) is considered. The result is thenXY ⊗UW −(1/N)XW ⊗′ UZ.

(25)The second term follows the original fermion lines all the way without a cross-over, andthis distinction from the first term is indicated in the formula by using ⊗′ rather than ⊗.Loop diagrams are obtained by joining ends of tree diagrams. Consider first the caseswhen ends of fermions are joined into fermion loops.

In the presence of a single fermionline, or whenever such a fermion line is not attached to another fermion line by gluons,then all we have to do is to take the trace over (23). If two fermion lines are present, as inFig.

7, and if the two ends of the top fermion line are joined together to form a fermionloop, then the color factor for an U(N) gauge theory can be obtained from (24) to beXY ZUV W.(26)Note that this can be read offdirectly from Fig. 7 as long as we remember to cross overat the G-tree.

Other cases involving more fermions can be obtained similarly.Consider next a fermion-G loop, as in Fig. 8, obtained by attaching a G-tree of colorfactor Tr(AY BU) to a fermion line with color factor XAV BZ at points A and B. Againfirst imagine point A to have been attached but not point B.

Then the color factor of thecombined tree is XY BUV BZ, summed over B. This yieldsXY ZTr(UV ).

(27)Note that this factor can again be read offdirectly from Fig. 8: XV Z is the multiplicationof the color generators clockwise order following the outside path of the loop, and Tr(V Y )is the trace factor corresponding to the lines inside of the loop.

This feature about tracingthe outside of a loop and the inside separately will occur again when we discuss G-loops.Note that the outside of the loop, like the original trees, is followed clockwise, whereas theinside of the loop is followed counter-clockwise.As a check, note that Fig. 8 can also be obtained from Fig.

7 by joining the right endof the bottom fermion line to the left end of the top fermion line. In this way one againobtains (26).Lastly, we will consider sewing ends of a G-tree together to form a loop, as in Fig.

9.Before we fuse it together at the point A, the color factor for the tree is Tr(RSAXY ATU).Summing over A yieldsTr(RSTU)Tr(XY ). (28)The result can be read again directly from the graph.

The first trace is taken in clockwiseorder along a closed path around the whole diagram passing through the outside of theloop; and the second trace is taken in counter-clockwise order along the closed path passingthrough the inside of the loop.One can apply (19) to more complicated diagrams with any number of loops. Theresult in the case of an U(N) gauge theory can always be read offsimply from the color-oriented diagram.

The general rule for the color factor Ci is the following. Circle around11

the diagram with continuous ‘color paths’ of the following kind. These paths may startfrom one end of a fermion line and end at another end (of possibly another fermion line), orelse they must be closed.

The upper side of every fermion line and both sides of every gluonand ghost line must be covered once and only once by these paths. Associate each externalgluon with U(N) color index A the generator T A.

Go along the path in clockwise order(counter-clockwise order if it is inside a loop) and multiply these generators successivelyfrom left to right. If the path is an open path, this product is the color factor associatedwith the path.

If the path is closed, then a trace should be taken. The total color factorfor the color-oriented connected diagram is the product of these individual color factors.See Figs.

4–9 for illustrations.The rules for SU(N) can be obtained from the U(N) rules by subtracting out theU(1) gluons which remains coupled in the diagram.Having thus a graphical way to read out the total color factor for an oriented diagram,the next task is to find an equally simple and general graphical method to read out thetotal spin factor of the oriented diagram. This can be done very simply, and the notationadopted in Fig.

3 is actually designed with this in mind.To do so, cover the maximal gluon subdiagram of the oriented diagram in questionwith ‘spin-flow paths’. A spin-flow path is different from a color path discussed above inthat it stays right on the gluon lines of the diagram and not above or below them.

Aspin-flow path is meaningful only for a gluon line, internal or external, and it is simply acontinous path tracing through a portion of the gluon subdiagram. Such a path may be aclosed path, or an open path.

If it is an open path, it must end at an external gluon line,or a cubic vertex. Conversely, there must be one and only one path ending at each cubicvertex.

See Figs. 12 and 13 for examples of these paths.The spin factor associated with a closed path is gµµ = d, and the spin factor associatedwith an open path is the dot product of the vectors at the two ends.

The vector at a cubicvertex is given in Fig. 3, and the vector associated with an external gluon line is simplyits polarization vector ǫ.

The total spin factor of the oriented diagram is the product ofthe spin factors of all the paths, times whatever extra numerical factors appearing at thequartic vertices in Fig. 3, times products of the numerators of fermion propagators (γ ·q) ifpresent, summed over all possible spin-flow path coverings of the maximal gluon subgraph.The coupling constant factor for an oriented diagram is simply the product of thecoupling constant factors of all its vertices.The numerator factor S(q, p) in (1) for a Feynman diagram is then the product of thecolor factor, the coupling-constant factor, and the spin factor, summed over all spin-flowpaths and all color-oriented diagrams.

Extra factors such as the minus sign associated witheach closed fermion loop and the symmetry factor will be absorbed into S(q, p) as well.To be sure, there are many terms present for a complex diagram corresponding to manyspin-paths and many color-oriented diagrams. In fact, all that we have done up to thispoint is to give a graphical interpretation of every term that appears in S(q, p).

We havenot reduced the number of terms there in any way. However, this graphical approach helpsto organize the terms mentally without having to write down a single algebraic formula,so it helps to keep us away from the algebraic jungle.

The real simplification comes in onlywhen we start using the spinor helicity technique and the background gauge in the next12

section.Simplifications can also result from supersymmetry. Consider for example an orienteddiagram containing a quark line.

The spin factor is essentially the same when the quark isreplaced by a gluino. Under such a replacement, the color factor changes simply by havingtraces taken over the original color factor.

So the colored subamplitudes of a quark diagramis the same as that for a gluino diagram. On the other hand, the colored subamplitudeof a gluino diagram is related to that for a pure gluon diagram by supersymmetry.

Thischain of reasoning makes it possible to related pure gluon amplitudes with those with aquark line in it. Such supersymmetry relations [16,17] have been used in tree processes tosimplify calculations, and as we shall see in Sec.

5, it can be used to simplify calculationsfor loop amplitudes as well.4. Spinor helicity basisWe have discussed how to organize graphically the numerator factor S(q, p) in the lastsection.

Nevertheless, there are many terms involved, corresponding to the many color-oriented diagrams and the many spin-flow paths for a given diagram. A clever choice ofgauge and polarization vectors at this point can render many of the terms zero, making itunnecessary to consider some path coverings and/or color-oriented diagrams, and therebyreduce the labour of computation enormously.

We shall see that the use of backgroundgauge together with polarization vectors chosen in the helicity spin basis will accomplishthis purpose.We shall first summarize the known results of the spinor helicity basis taken fromRef. [16], and then go on to discuss further simplifications brought about by the use of thebackground gauge.Let |p±⟩be the incoming wave function of a massless fermion with momentum p andchirality ±1, normalized in such a way that⟨p ± |γµ|p±⟩= 2pµ.

(29)From chirality conservation, one gets⟨p ± |q±⟩= 0(30)valid for any other massless momentum q. This is the central relation that leads to muchof the simplifications.

Let⟨pq⟩= ⟨p −|q+⟩= −⟨q −|p+⟩= −⟨qp⟩,[pq] = ⟨p + |q−⟩= −⟨q + |p−⟩= −[qp]. (31)Then⟨pq⟩∗= sign(p · q)[qp],(32)⟨pq⟩[qp] = 2(p · q),(33)13

⟨p ± |γµ1 · · ·γµ2n+1|q±⟩= ⟨q ∓|γµ2n+1 · · · γµ1|p∓⟩,(34)⟨p ± |γµ1 · · · γµ2n|q∓⟩= −⟨q ± |γµ2n · · · γµ1|p∓⟩,(35)⟨AD⟩⟨CD⟩= ⟨AD⟩⟨CB⟩+ ⟨AC⟩⟨BD⟩,(36)⟨A + |γµ|B+⟩⟨C −|γµ|D−⟩= 2[AD]⟨CB⟩,(37)γp = |p+⟩⟨p + | + |p−⟩⟨p −|. (38)To illustrate how massless momenta and the ensuing chirality conservation can simplifycalculations, consider the calculation of Tr[(γp1)(γp2) · · ·(γp2n−1)(γp2n−1)], where everypi is massless.

Using usual formulas, this is given by a sum of (2n)!/2nn! terms, eachcontaining a product of n pairs of momentum dot products.

Using (38), (30) and (31),this can be reduced to just a sum of two terms:⟨p1p2⟩[p2p3] · · ·[p2n−1p2n]⟨p2np1⟩+ [p1p2]⟨p2p3⟩· · · ⟨p2n−1p2n⟩[p2np1]. (39)The polarization vector for an outgoing photon or gluon with momentum p and helicity±1 can chosen in a multispinor basis to beǫ±µ (p, k) = ±⟨p ± |γµ|k±⟩√2⟨k ∓|p±⟩,(40)where the reference momentum k in (40) is massless but otherwise arbitrary.

The choiceof different k corresponds to the choice of a different gauge, and these different choices arerelated byǫ+µ (p, k) →ǫ+µ (p, k′) −√2⟨kk′⟩⟨kp⟩⟨k′p⟩pµ. (41)These polarization vectors satisfy the following identities:ǫ±µ (p, k) = (ǫ∓µ (p, k))∗,(42)ǫ±(p, k) · p = ǫ±(p, k) · k = 0,(43)ǫ±(p, k) · ǫ±(p, k′) = 0,(44)ǫ±(p, k) · ǫ∓(p, k′) = −1,(45)ǫ±(p, k) · ǫ±(p′, k) = 0,(46)ǫ±(p, k) · ǫ∓(k, k′) = 0,(47)ǫ+µ (p, k)ǫ−ν (p, k) + ǫ−µ (p, k)ǫ+ν (p, k) = −gµν + pµkν + pνkµp · k,(48)γ · ǫ±(p, k) = ±√2⟨k ∓|p±⟩(|p∓⟩⟨k ∓| + |k±⟩⟨p ± |).

(49)This completes the summary of the properties of the spin-helicity basis. The vanishingdot products (43), (44), (46), (47) are what make this basis particularly useful.Background gauge is convenient for loop calculations because (43) – (47) can be usedto eliminate many terms in this gauge.

In this gauge, two of the three terms in the ggg14

vertex with an external line (see Fig. 3) involves only the momentum of the external line.This enables many terms to vanish as we shall see in the following illustration.Consider an n-gluon color-oriented diagram where either all the gluons have the samehelicity, or all but one have the same helicity.

Let us consider the latter, and assumegluon 1 to have a negative helicity while all the other gluons have positive helicities. Letus choose the reference vectors ki for the polarization vector ǫ(pi, ki) to be k1 = p2 andki = p1 for all i ̸= 1.

This choice is designed so that (43) to (47) can be used to show thatǫi · ǫj = 0,(∀i, j),(50)pi · ǫi = p1 · ǫi = 0 (∀i),(51)p2 · ǫ1 = 0.(52)Eq. (50) is particularly useful.

The spin factor consists of products of dot productsof the form ǫ · ǫ′, ǫ · q, q · q′, of degree n in the polarization vectors and of degree min the momenta if there are m GGG vertices. For tree diagrams m ≤n −2, so at leastone ǫ · ǫ′ must be present in every term.

Because of (50) the tree amplitude with thishelicity configuration must vanish [16]. For one loop diagrams, m = n only if no quarticvertices are present.

Otherwise m < n, and the corresponding contribution again vanisheson account of ǫ · ǫ′. This greatly simplifies calculations because no gggg nor ggGG verticesneed ever be considered.

Moreover, in the Feynman-parameter representation, contraction(9) again leads to the presence of ǫ · ǫ′ so current contractions never have to be considered.As a result, all Sk(J, p) = 0 except S0(J, p) = S(J, p).Other simplifications can be seen in Fig. 10.

Paths A and B vanish in the backgroundFeynman gauge because of (51) and Figs. 3(c,d).

Path C vanishes because of (50). As acorollary paths between two dots like C′ are also forbidden because a path C must thenbe present to take up the leftover ǫ factors.

Path D vanishes because of (52).5. One-loop four-gluon amplitudeTo illustrate techniques developed in the last three sections, we compute in this sectiona one-loop four-gluon amplitude in which three of the four gluons have the same helicity.We will first compute the case when quarks are absent because that pure gluon amplitudehas been computed with the string technique [19-21], so a comparison of the efficiency ofthe two methods can be made.

We find that the present method is every bit as efficientas the string technique.Next, we shall compute the same four-gluon amplitude with an internal quark loop.Thanks to chirality conservation the calculation is even simpler than the pure gluon case.Using supersymmetry arguments, this amplitude can be related to the pure gluonic ampli-tude, thereby providing a second method to compute the pure QCD four-gluon amplitude.The result agrees with the first calculation, but the number of steps needed to reach theresult is now even smaller.The reference momenta for the polarization vectors will be chosen as in the last section,so that eqs. (50) – (52) can be used.

As discussed there, quartic vertices do not contribute,and current contractions cannot occur so Sk(J, p) = 0 in (2) for k > 0.15

There is a further simplification when four-point amplitudes are considered.Eachpolarization vector is perpendicular to two momenta: its own gluon momentum and itsreference momentum. That means that its dot product with the other two external mo-menta are equal and opposite, thereby resulting in only one independent dot product perpolarization vector.

LetA1 = ǫ1 · p3 = −ǫ1 · p4 = −⟨13⟩[32]√2[21] ,A2 = ǫ2 · p3 = −ǫ2 · p4 = −[24]⟨41⟩√2⟨12⟩,A3 = ǫ3 · p2 = −ǫ3 · p4 = −[34]⟨41⟩√2⟨13⟩,A4 = ǫ4 · p2 = −ǫ4 · p3 = +[42]⟨21⟩√2⟨14⟩. (53)Then the numerator S0(J, p) = S(J, p) in (2) is proportional to A1A2A3A4.With quartic vertices out of the way, the color-oriented diagrams contributing to thisprocess are the box diagrams, the vertex insertion diagrams, and the self-energy insertiondiagrams.

We shall see that the self-energy diagrams are zero, and three out of the vertex-insertion diagrams also do not contribute.To see that, consider the spin-flow path of Fig. 11(a) originating from gluon 4.

Becauseof eq. (50) this path cannot end at another external gluon line.

It cannot end at the vertexjoining lines 2 and 3 either for then one is forced to have the factor ǫ2 · ǫ3, which is zero.The only other possibility then is for the spin-flow to end at a vertex within the loop, inwhich case a factor ǫ4 · J will result, where J is some combination of the currents flowingthrough the loop. From (7), we see that J is a linear combination of p1 and p4.

Sinceǫ4 · p1 = ǫ4 · p4 = 0, these paths are not allowed either. Consequently diagram 11(a)makes no contribution.

The same argument will hold if instead of 4 and 1 it is gluons 1and 2 which are attached to the loop. Consider now diagram 11(b), which we claim alsomakes no contribution.

To see that, consider a spin-flow path that starts from gluon 1.This path cannot flow on to gluon 2, so it either ends at its own vertex, or goes beyond.In the former case the contribution vanishes because ǫ1 · p1 = ǫ1 · p2 = 0. In the lattercase, the path starting from gluon 2 must end at its own vertex, and this vanishes becauseǫ2 · p1 = ǫ2 · p2 = 0.

Consequently, three of the four vertex-insertion diagrams associatedwith the color factor Tr(abcd) gives no contributions.Essentially the same argument also shows that none of the self-energy insertion graphslike 11(c) makes any contribution. This leaves only the box graph Fig.

12 and the vertexinsertion graph Fig. 13.Let us consider the allowed spin-flow paths in the box diagrams, Fig.

12, rememberingthat paths A,B,C,C′,D of Fig. 10 may not be present.

This means the path starting fromgluon 1 must end at the same vertex, or else there must be another path ending at vertex1 giving rise to some ǫi ·p1 = 0. There are actually altogether nine possible spin flows in aninternal gluon loop and two more in a ghost loop, as shown in Fig.

12. Using (2) and (14),16

the amplitude from the box diagram contributing to the color factor term Tr(abcd)Tr(1)isMB =116π2ZD4αSB(J, p)(α2α4s + α1α3t)2 ,(54)with t = (p1 + p2)2, s = (p1 + p4)2, and u = (p1 + p3)2. The numerator is given bySB(J, p) = Tr(abcd)Tr(1)g4B, with B being the spin factor from all the box diagrams.Using the spin factors for the color-oriented vertices in Fig.

2, and the expression for thecurrents in eq. (14), one getsB1 = gµµ[ǫ1 · (J2 + J1)][ǫ2 · (J3 + J2)][ǫ3 · (J4 + J3)][ǫ4 · (J1 + J4)]/A1A2A3A4= 4(2α4)(2α4)(−2α1 −2α2)(2α3) = −64α24α3(α1 + α2),B2 = [ǫ1 · (J2 + J1)][ǫ2 · (−2p4)][ǫ3 · (−2p2)][ǫ4 · (−2p3)]/A1A2A3A4= (2α4)(2)(−2)(2) = −16α4.Similar calculations show thatB3 = B2,B4 = B5 = 16α24,B6 = B7 = 16α4(α1 + α2),B8 = B9 = 16α4α3,B10 = B11 = −B1/4.

(55)The sum isB =11Xi=1Bi = −32(α1 + α2)α3α24. (56)Consider now the vertex insertion graphs, Fig.

13. Using (2) and (15), the amplitudefrom the box diagram contributing to the color factor term Tr(abcd)Tr(1) isMV =116π2ZD3αSV (J, p)α1α2s2 ,(57)with the numerator given by SV (J, p) = Tr(abcd)Tr(1)g4V , and V to be the spin factorcontribution from all the vertex insertion diagrams.

Using the spin factors for the color-oriented vertices in Fig. 2, and the expression for the currents in eq.

(15), one getsV1 = −64α1α2α3,V2 = V3 = 16α3,V4 = −16(12α1 + α2 + α3),V5 = 16α2(12α1 + α2 + α3),V6 = V9 = V12 = V13 = 8α1α2α3,17

V7 = −16(α1 + 12α2 + α3),V8 = 16α1(α1 + 12α2 + α3),V10 = 16α1(12α1 + α2 + 12α3),V11 = 16α2(α1 + 12α2 + 12α3). (58)The sum isV =13Xi=1Vi = −32α1α2α3.

(59)The final result isMB = Tr(abcd)Tr(1) g412π2 s[24]2[12]⟨23⟩⟨34⟩[41],(60)MV = Tr(abcd)Tr(1) g412π2 t[24]2[12]⟨23⟩⟨34⟩[41],(61)M = MB + MV = Tr(abcd)Tr(1) g412π2 (−u)[24]2[12]⟨23⟩⟨34⟩[41]. (62)This result agrees with the result obtained by the string method [19–21].

The vanishingof the diagrams in Fig. 11 is also a feature shared by the string method.

In fact, it hasbeen observed [21] that the string expression for box diagram corresponds to a calculationin the background Feynman gauge, though a mixture of the background gauge and theNeuveu-Gervais gauge seem to be required for the vertex-insertion diagram. In the presentcase we use the background gauge throughout.

The total number of terms in the boxdiagram (11) and the vertex-insertion diagram (13) is quite comparible with that usingthe string method [20] (2 × 14 each) as well. We conclude therefore that in most respectsthis method is just as efficient as the string method.There is actually another similarity which is telling.

One notes from (55) and (56)that B is proportional to B1, so that many of the 11 terms in (55) combine to cancel oneanother. The same happens in the vertex-insertion diagrams, and the same happens in thestring approach.

This strongly suggests that as simple as the computation is, all together24 non-vanishing terms rather than something of the order of 104 which one has in theordinary approach, there must be even simpler way of calculating things where one canavoid writing down terms that eventually cancel one another. The following calculationshows how this can be attained.We turn now to the computation of the box diagram with an internal quark loop,Fig.

14(a,b). The result per flavor is obtained from (2) to beM′B =116π2ZD4αS′B(J, p)(α2α4s + α1α3t)2 ,(63)18

whereS′B(J, p) = −Tr(abcd)g4{tr[(γǫ1)(γJ1)(γǫ4)(γJ4)(γǫ3)(γJ3)(γǫ2)(γJ2)]+tr[(γǫ1)(γJ2)(γǫ2)(γJ3)(γǫ3)(γJ4)(γǫ4)(γJ1)]}. (64)Like (39), these traces are easily computable using (14), (38), and (49).

The first trace istr[(γǫ1)(γJ1)(γǫ4)(γJ4)(γǫ3)(γJ3)(γǫ2)(γJ2)] =−4(α1 + α2)α3α24⟨12⟩[24]⟨14⟩[43]⟨13⟩[32]⟨13⟩[32] + [23]⟨31⟩[42]⟨21⟩[34]⟨41⟩[23]⟨31⟩[21]⟨12⟩⟨13⟩⟨14⟩=8(α1 + α2)α3α24s2t[24]2[12]⟨23⟩⟨34⟩[41]. (65)The second trace is equal to the first trace.

Therefore,M′B = −2MB/Tr(1). (66)Similarly, one can compute Fig.

14(c,d) to getM′V = −2MV /Tr(1). (67)The equalities in (66) and (67) are easy to understand by using supersymmetry argu-ments[12,16,17,25].

Quarks and gluinos have the same spacetime coupling with the gluon,though they carry different colors. If we should replace the quark loop by a gluino loop,the only change after we replace the color factor Tr(abcd) in the quark loop by the colorfactor Tr(abcd)Tr(1) in the gluino loop would be an extra factor of 1/2, reflecting themajorana nature of the gluino.

Since the four-gluon amplitude in a pure supersymmetricQCD theory (gluinos are present but not quarks) is zero, the gluon/ghost loop contributionis equal and opposite to the gluino loop contribution. Putting these two facts together,the equality of (66) and (67) are obtained.These arguments can be reversed and to be used to compute MB and MV from M′Band M′V .

This simplifies the calculation of the pure gluon amplitudes because the quarkloop box diagram contains only four terms, which are equal, instead of the 11 terms inFig. 12.

This also explains why many of these terms in (55) (similarly (58)) add up to givezero.6. ConclusionsFor high energy scatterings lepton and light-quark masses can be ignored.

Chirality isthen conserved and tremendous simplifications in the calculations of these amplitudes canbe obtained by the use of the spinor-helicity techniques [1–21]. With one exception [19–21],this technique was used only to calculate the tree amplitudes [1–18], because loop graphscontain off-shell momenta where chirality is not conserved and this technique cannot beapplied.

The exception [19–21] makes use of the string theory and is applicable to certain19

one-loop processes.We have developed in this paper a technique, making use of theFeynman-parameter representation of a scattering amplitude to avoid the off-shell internalmomental, to enable to spinor-helicity method to be used for any Feynman diagram withany number of loops.Graphical methods are used throughout to organize the terms and to avoid treadinginto the algebraic tangle. The method was applied to a one-loop four-gluon amplitude toshow that the present method is at least as efficient as the string technique.

Applicationof the method to the calculation of other processes is underway.AcknowledgementsI am grateful to Z. Bern, K. Dienes,D. Kosower, and T.-M. Yan for useful discussions.

This work is supported in partby the Natural Sciences and Engineering Research Council of Canada and the Qu´ebecDepartment of Education. Part of this manuscript was prepared during my visit to theInstitut des Hautes ´Etudes Scientifiques, Bures-sur-Yvette, France.I wish to thank Profs.

L. Michel and M. Berger for their kind hospitality.20

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Figure Captions[Fig. 1] Diagrams used to illustrate the Feynman-parameter rules.

(a) a box diagram; (b) avertex diagram.[Fig. 2] Vertices for QCD in the background Feynman gauge.

A thin solid line represents agluon, a thick solid line represents a quark, and a dashed line represents a Fadeev-Popov ghost. Gluon lines with a circled ‘A’ are external lines; those in the samediagram without a circled ‘A’ are internal lines.

Each gluon line in a diagram withoutany circled ‘A’s present can be taken either as an external or an internal line, providedsuch a combination of external and internal lines has not appeared already in diagramswhere explicit circled ‘A’s appear.[Fig. 3] Color-oriented vertices and their spin factors.

A line continuing through the vertexrepresents a gρσ factor; a line terminated at a heavy dot at the vertex represents avector written below the diagram. Other numerical factors for the vertex also appearbelow the diagrams.

The line labelled ‘1’ carries a momentum p1, a spacetime index α,and a color index a. Similarly, a line labelled ‘2’ carries a momentum p2, a spacetimeindex β, and a color index b, etc.For example, the spin factor for diagram (a) is (p2 −p3)αgβγ; the spin factor fordiagram (e) is +2gαγgβδ.[Fig.

4] The color factor C for a gluon tree diagram.[Fig. 5] A gluon tree diagram and its color factor used to illustrate the proof of eq.

(22).[Fig. 6] The color factor C for a tree diagram containing a number of gluon trees attached toa quark line.[Fig.

7] The U(N) color factor C for a tree diagrams with two quark lines.[Fig. 8] The color factor C for a one-loop diagram with a quark line.[Fig.

9] The color factor C for a one-loop diagram without a quark line.[Fig. 10] One-gluon-loop diagrams and the vanishing spin-flow paths A,B,C,D.

BackgroundFeynman gauge is used; the helicity configuarations as well as e e choice of the referencemomenta are discussed in the text.[Fig. 11] These diagrams make no contributions to the process calculated in Sec.

5.[Fig. 12] Non-vanishing spin-flow paths (diagrams 1 to 11) for the one-gluon-loop box diagramcalculated in Sec.

5.[Fig. 13] Non-vanishing spin-flow paths (diagrams 1 to 13) for the one-gluon-loop vertex-insertiondiagram calculated in Sec, 5.[Fig.

14] One-fermion-loop diagrams for the processes calculated in Sec. 5.22

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출처: arXiv:9207.266 • 원문 보기