arX

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8v1

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Nov

200

5

Diagrammatic proof of the BCFW recursion relation forgluon amplitudes in QCD.

Petros D. Draggiotis(a,c)1, Ronald H.P. Kleiss(b)2, Achilleas Lazopoulos(b)3

and Costas G. Papadopoulos(c)4

(a)University of Athens, Physics Department, Nuclear & Particle PhysicsSection,GR− 15771, Athens, Greece

(b)IMAPP, Institute of Mathematics, Astrophysics and Particle Physics,Radboud University, Nijmegen, the Netherlands

(c)Institute of Nuclear Physics, NCSR ”Demokritos”, 15310, Athens, Greece

Abstract

We present a proof of the Britto-Cachazo-Feng-Witten tree-level recur-sion relation for gluon amplitudes in QCD, based on a direct equivalence be-tween BCFW decompositions and Feynman diagrams. We demonstrate thatthis equivalence can be made explicit when working in a convenient gauge.We exhibit that gauge invariance and the particular structure of Yang-Millsvertices guarantees the validity of the BCFW construction.

1 Introduction

During the last year much progress has been made in the understanding ofanalytical calculations of dual amplitudes in perturbative Yang-Mills theo-ries. Led by an observation of Witten [1], Britto, Cachazo and Feng (BCFW)have proposed a new recursion relation for tree amplitudes of gluons [2]that naturally arrives at the simplest known expressions for some of thoseamplitudes in terms of Weyl - Van der Waerden spinor products. Explicitcalculations have been performed using this technique [3],[4], extensionsto amplitudes involving particles from the electroweak sector [5] have beenpursued and a new approach to one loop amplitudes has been proposed [6]employing MHV vertices and unitarity arguments.

1pdrag@phys.uoa.gr2R.Kleiss@science.ru.nl3A.Lazopoulos@science.ru.nl4costas.papadopoulos@cern.ch

1

The BCFW recursion relation features some remarkable characteristics,among which the on-shell analytic continuation of selectedoff-shell prop-agators, the analytic continuation of two selected external momenta in thecomplex plane and a decomposition of a color helicity amplitude into smallerhelicity amplitudes with complex external momenta that doesn’t appear tobe in direct connection with the decomposition in Feynman diagrams. More-over, the BCFW relation leads to the use of only three-point (modified) ver-tices as building blocks of the theory, thus raising questions about the fun-damental nature of the Yang-Mills four point vertex.

A proof of the BCFW relation was given by Britto, Chachazo, Feng andWitten [7] which, though it made the connection between the analyticityproperties of the color amplitude and the BCFW decomposition obvious,shed little light towards the way the latter relates to the usual Feynman dia-gram approach.

The simplicity of the final result for tree level gluonic amplitudes im-plies the existence of major cancellations between Feynmangraphs. Sincethe BCFW formalism avoids those cancellations altogether it is advanta-geous to find a diagrammatic proof of the BCFW decomposition in order tounderstand these cancellations in the level of Feynman diagrams. A numberof questions immediately arise. Is a diagrammatic proof gauge dependent?If so, are there preferable gauges where simplifications take place? Whichgauges are those? Is this decomposition exclusively working for pure YMtheories, and if so, what is the key feature of the YM verticesthat makes itwork? Can we find a BCFW type recursion relation for scalar theories?

We intend to address these questions in the present paper as well as giv-ing a generic diagrammatic proof of the BCFW decomposition.The paperis organized as follows: after clarifying some notation issues in section2,we present, in section3 an analysis based on diagram enumeration that hintstowards a connection between BCFW decompositions and Feynman graphs.In section4 we prove some important kinematical identities that will supportthe main parts of the diagrammatic proof (sections6 and7), whereas section5 describes the gauge in which we are working and the consequences of thatchoice. We conclude with some remarks on the generality of the proof andthe role of gauge invariance in this analysis in section8.

2

2 Notation

All amplitudes mentioned in this paper are the color-ordered amplitudes thatresult after the usual trace-based color decomposition [8]. We will conve-niently omit all coupling constants and factors ofi as well as the color an-tenna of all color-ordered amplitudes in what follows.

In non-vanishing tree-level color-ordered amplitudes, one can, withoutloss of generality, pick out a pair of adjacent legs with opposite helicity andlabel the positive helicity one by1 and the negative helicity one byn. Unlessotherwise specified we will use this labeling throughout this paper.

The Dirac four-spinors used in this paper are the usualu±(p), u±(p)

with the important note that the relation betweenu(p) andu(p) is not oneof complex conjugation.

The relation with Weyl - Van der Waerden spinors, when using the Weylrepresentation ofγ-matrices, is

u+(p) = (0, λb(p)ǫba) u+(p) = (λb(p)ǫ

ba, 0) (1)

u−(p) = (λa(p), 0) u−(p) = (0, λa(p)) (2)

Four-momenta can be written in terms of Weyl spinors as

pµ =1

2u−(p)γµu−(p) =

1

2λa(p)σ

aaµ λa(p) (3)

and products of four-vectors can be performed using the identity

σµ,aaσbbµ = 2ǫabǫab (4)

whereσµ = (I,−~σ), and similar identities for theσ’s.Note that, since

σµ,aaσµ,bb = 2δabδab (5)

we can write(p·σ)aa = pµσµ,aa = λa(p)λa(p) (6)

Moreover, the usual conventions for spinor products5 will be adopted:

〈ij〉 ≡ u+(pi)u−(pj) = λa(pi)λb(pj)ǫab (7)

[ij] ≡ −u−(pi)u+(pj) = λa(pi)λb(pj)ǫab (8)

This results to2p·q = 〈pq〉[pq] (9)

A fairly general definition of the external gluons’ polarization vectors6

5Note the minus sign in the[ij] definition with respect to the definition in [9].6that corresponds to the light-like axial gauge

3

is (see [9],[10] for similar definitions.)

ǫµ+(p) =

u+(p)γµu+(p)√

2u+(p)u−(p)=

λa(p)σµ,aaλa(p)√2〈pp〉

ǫµ−(p) =

u−(p)γµu−(p)√

2u−(p)u+(p)= −

λa(p)σµ,aaλa(p)√2[pp]

(10)

wherep is an auxiliary, null four-vector that can be chosen at will for everygluon as long as it is not parallel top itself.

Note that one could write

ǫ+aa(p) ≡ ǫ+µ(p)σµaa =

√2λa(p)λa(p)

〈pp〉 (11)

ǫ−aa(p) ≡ ǫ−µ(p)σµaa = −

√2λa(p)λa(p)

[pp](12)

and perform vector products using

paaqbbǫabǫab = 〈pq〉[pq] = 2p·q (13)

It should be noted that nothing of what follows depends on theprecise con-ventions we adopt for the Van der Waerden or Dirac spinors. Infact thepresent analysis, that deals with pure gluonic amplitudes,could be per-formed exclusively on the level of four-vectors.

The hat symbol over a function of momentaf(q) will denote the func-tion f(q; z) where the argument off is analytically continued (shifted) by ageneral four-vectorzǫµ, with

ǫµ ≡ 1

2u−(p1)γµu−(pn) =

1

2λa(p1)σ

aaµ λa(pn) (14)

This has the effectq → q = q+ zǫ (15)

We will omit the explicit reference to thez-dependence off(p) since thisis signified by the hat symbol.

3 Counting contributions

The BCFW recursive relation for pure gluon color ordered amplitudes isgiven by

4

A(pλ11 , . . . , pλn

n ) = (16)

∑

λX=±1

n−1∑

k=1

A(pλ11 , . . . , p

λkk ,−p

−λXX )

1

P21...k

A(pλXX , p

λk+1

k+1 , . . . , pλnn )

wherep andλ represent momentum and polarization. In this section wewould like to study the diagrammatic content of the BCFW equation. Themost appropriate way to do so, is of course to neglect any reference to po-larization and also drop the propagator factor, while keeping the ordering ofthe momenta. In that case of course we count planar graphs (see [12]) andit is obvious that the equation is not self-consistent. In fact the form of theequation is given by

A =∑

AI AII (17)

or in a more mathematical form

ABCF1→n =

n−1∑

k=1

A1→k+1A1→n−k+1 (18)

The equationABCF1→n = A1→n cannot be true because of the follow-

ing reasons:

• Firstly, contributions where no propagator line exists between the1stand thenth particle, are not included in the BCFW equation

A0 = + (19)

These contributions are given by

A01→n = A1→n−1+

∑

n1+n2=n−1

A1→n1A1→n2

(20)

• Secondly, BCFW is multiple-counting contributions of the form

. . .

1 2 M

5

or

. . .

1 M2

In fact we suggest that this over counting is exactly equal tothe multi-plicity of propagator lines connecting particles1 andn.

To make our arguments more quantitative we start with the Berends-Giele [11] (or Dyson-Schwinger for ordered graphs [13],[14]) recursive equa-tion for a generic theory with3− and4−vertices.

= + (21)

A1→n =∑

n1+n2=n

A1→n1A1→n2

+∑

n1+n2+n3=n

A1→n1A1→n2

A1→n3

(22)and we write the following equation

A1→n = ABCF1→n+A0

1→n−D (23)

whereA01→n are the classes of diagrams in eq.19 and the substracted

termD accounts for the overcounting of diagrams. Since the overcountingrelates to diagrams with2, 3, . . . propagators on the line between the firstand the last leg, it can be seen that

D =

n−1∑

M=3

(M− 2)

M∑

k=0

(

M

k

)

DnM+k (24)

whereM − 1 is the number of propagators of the particular overcountedclass of diagrams and

DnM =

∑

n1+...+nM=n

A1→n1. . .A1→nM

(25)

counts the number of diagrams within that class. As an example forn = 5

(6-leg gluon amplitude), the number of Feynman graphs isA = 38, the

6

number of BCFW graphs isABCF = 29, the number of graphs without anypropagator between the first and the last leg isA0 = 17 so overcountingshould giveD = 8. The following graphs are overcounted

(2)

(26)

with the first diagram doubly overcounted (it contains 3 propagators). Thatdiagram corresponds to theM = 4 while the other graphs come from theM = 3 term of eq.24.

In the table below we give the results for up ton = 11 particles.

1 → n A ABCF A0 ABCF +A0−A D3 3 1 2 0 04 10 6 5 1 15 38 29 17 8 86 154 136 64 46 467 654 636 259 241 2418 2871 2992 1098 1219 12199 12925 14190 4815 6080 608010 59345 67860 21659 30174 3017411 276835 327080 99385 149630 149630

This analysis hints strongly towards the idea that a connection betweenFeynman diagrams and the BCFW decomposition might be achieved bygrouping together BCFW (hatted) diagrams with the same chain structurealong the main line (the same number of propagators hence thesame multi-plicity) but differently placed cuts: The multiplicity of each group is equalto the number of propagators along the main line which in turns equals thenumber of possible cuts (one for each propagator along the main line).

4 Kinematical identities

For any functionf(p; z) of z that has only simple poles inz and vanishes atz → ∞, we can perform an expansion over its poles. In particular wehave

f(p; z) =∑

j

[

f(p; z)(z − zj)]

z=zj

1

z− zj(27)

7

where the sum is over all the simple poleszj of f(p; z). This very generalidentity allows us to analytically continue the functionf(p) to the complexplane, make use of the pole expansion and take the limitz → 0 to return tothe real axis, thus obtaining a relation betweenf(p) and the pole expansionof f(p; z).

Applying this to a momentum antenna we get

1

p21p

22 . . . p

2k

=∑

j=1..k

[

1

p21p

22 . . . p

2j−1pj+1 . . . p

2k

]

z=zj

1

p2j

(28)

with zj such that

pj(zj)2 = (pj+ zjǫ)

2 = p2j + 2zjpj·ǫ = 0 (29)

wherep1 . . . pk are arbitrary off-shell four-momenta. In the next sectionswe will use the above identity withpi being sums of on-shell momenta ofthe formp1k = p1+ p2+ . . . + pk.

Taking the limitz → 0 at both sides we have

1

p21p

22 . . . p

2k

=∑

j=1..k

[

1

p21p

22 . . . p

2j−1p

2j+1 . . . p

2k

]

z=zj

1

p2j

(30)

Further more,

zρ

p21p

22 . . . p

2k

=∑

j=1..k

[

zρ

p21p

22 . . . p

2j−1p

2j+1 . . . p

2k

]

z=zj

1

p2j

(31)

which gives the very useful set of identities, valid for every ρ < k

∑

j=1..k

[

zρ

p21p

22 . . . p

2j−1pj+1 . . . p

2k

]

z=zj

1

p2j

= 0 (32)

Finally, if z = k, the function in the left hand side of eq.31 is no longervanishing atz → ∞. Subtracting its limit at infinity, however, we have anew function that does, so

zk

p21p

22 . . . p

2k

− limz→∞

zk

p21p

22 . . . p

2k

=∑

j=1..k

[

zk

p21p

22 . . . p

2j−1pj+1 . . . p

2k

]

z=zj

1

p2j

(33)and taking the limitz → 0 we get

8

− limz→∞

zk

p21p

22 . . . p

2k

=∑

j=1..k

[

zk

p21p

22 . . . p

2j−1pj+1 . . . p

2k

]

z=zj

1

p2j

(34)

or−1

∏kj=12ǫ·pj

=∑

j=1..k

[

zk

p21p

22 . . . p

2j−1pj+1 . . . p

2k

]

z=zj

1

p2j

(35)

5 Choosing a gauge

We have seen in section 3 that classes of Feynman diagrams should some-how correspond to particular BCFW decompositions. Using a particulargauge for the external gluons one can eliminate whole classes of Feynmandiagrams.

A consistent definition of the external gluons’ polarization vectors wasgiven in eq.10 wherep is an auxiliary, null four-vector that can be chosen atwill for every gluon as long as it is not parallel top itself.

We choose to usep1 = pn andpn = p1

ǫµ+1 =

u+nγµu+1√

2u+nu−1

=λa(p1)σ

µ,aaλa(pn)√2〈n1〉

(36)

ǫµ−n =

u−1γµu−n√

2u−nu+1

= −λa(p1)σ

µ,aaλa(pn)√2 [n1]

(37)

A number of simplifications follow immediately. The productof ǫ1 withǫn or ǫ (defined in eq.14) vanishes

ǫ1·ǫn = 0 = ǫ·ǫ1 = ǫ·ǫn (38)

Moreoverǫ+1·pn = 0 = ǫ−n·p1. (39)

As a consequence, any diagram in which the first and the last leg meet in athree-vertex vanishes.

1 n= 0 (40)

where the blob denotes any Feynman diagram with the particular off-shell

leg.

9

These simplifications are in the correct direction in view ofthe fact thatthere are no BCFW graphs with the first and the last leg in the same vertex.We still have to deal with the case when1 andn are attached in a four-vertexwhose two other lines are off-shell (another situation thatdoesn’t occur in aBCFW decomposition). We will see in the next section how to accommodatethese diagrams.

Finally, we should note that the polarization vectorsǫµ+1 andǫµ−n actu-

ally differ by a complex phase.They also differ by a complex factor from theshifting vectorǫ (see 14). Note that in any expression where all these threevectors appear in scalar products, one is allowed to freely interchange them,without altering the result. In a very real sense there is actually only onepolarization vector in this gauge.

Finally, the polarization vectors are now invariant under the shifting op-eration:

p1 → p1+ zǫ pn → pn− zǫ (41)

If we write

pµ1 =

1

2u−1γ

µu−1 →1

2u−1γ

µu−1+1

2zu−1γ

µu−n (42)

and

pµn =

1

2u−nγ

µu−n →1

2u−nγ

µu−n+1

2zu−1γ

µu−n (43)

we see that the shifting operator effectively sends

u−1 → u−1+ zu−n

u−n → u−n− zu−1 (44)

or, in terms of Weyl - Van der Waerden spinors

λa(p1) → λa(p1) + zλa(pn)

λa(pn) → λa(pn) − zλa(p1) (45)

As a result, the denominators ofǫ+1 andǫ−n become

〈n1〉 → 〈n1〉 + z〈nn〉 = 〈n1〉 (46)

and[n1] → [n1] − z[11] = [n1] (47)

The only restriction that we impose on the polarization vectors of theother gluons is that they remain invariant under the shifting operator. In case

10

their assisting vectorspi involve p1 or pn this should be arranged in a waythat preserves this invariance.

This gauge choice explicitly eliminates anyz-dependence from the po-larization vectors. The shifting operation affects amplitudes only throughthe shift at the momenta four-vectorsp1 andpn, and the induced shift tothe momenta of intermediate propagators. Polarization vectors and verticesthat do not carryp1 andpn are left unchanged. Therefore, a diagram aftershifting will be a complex function with poles coming from three-verticesor propagators.

This is in contrast with a general gauge, where the polarization vectorsǫ+1 andǫ−n get an extra factor involvingz in their denominator. A poleexpansion, in that case, would need to take into account (gauge dependent)poles coming from the gluon polarization vectors.

Counting powers ofz in the general gauge shows (see the concludingdiscussion in [7]) that any diagram vanishes at the limitz → ∞. This isnotin general true in the gauge we are working. Thez → ∞ limit of a diagramis used as a guide to group diagrams in classes where these limits cancel.

6 Review of the BCFW decomposition

According to the simplest version of the BCFW decomposition, the coloramplitudeA(pi;hi), wherehi is the helicity of thei’th gluon, is equal to

A(ph1

1 , . . . , phnn ) =

n−1∑

j=2

∑

h

A(ph1

1 , . . . , phj

j ,−Ph1...j)

1

P21...j

A(P−h1...j, p

h+1j+1 , . . . , p

hnn )

(48)The sum overj extends over all partitions of then−2 gluons in two groups.The amplitude is recovered from the sum ofn− 2 decompositions in lowerlevel amplitudes evaluated at a particularz = zj such thatP1...j(zj)2 = 0.

Let us call ‘hatted’ diagrams graphs of the form

A B C

1 n (49)

denoting the corresponding Feynman graph where we have multiplied byp21A, we have performed the momenta shift, evaluated it at somez = zi and

11

divided back byp21A. The valuezi is defined by the demand that the ‘cut’

propagator, carrying momentump1A, vanishes:p21A = 0.

In terms of ‘hatted’ graphs, the BCFW decomposition consists in twoblobs, one containing the first and one the last leg, the amplitude is hattedand a cut is taken on the propagator that connects the two blobs. A sum overall partitions ofn in two integers is employed. Evidently, every particularBCFW decomposition over some propagatorPi is equivalent to the sum ofall hatted diagrams with that particular propagator cut.

Thus the sum of all possible decompositions is equivalent tothe sum ofall hatted diagrams with all possible cuts along the line of propagators thatconnects the first and the last leg of the original color amplitude.

In what follows we will prove that the sum of all possible hatted dia-grams is equivalent to the sum of Feynman diagrams involved in the com-putation of the particular amplitude.

7 The correspondence of hatted graphs toFeynman diagrams

The special treatment of the two selected external legs1 andn implies aspecial classification of the various Feynman diagrams involved in the givenamplitude. There are classes of diagrams with zero, one,two,. . . propagatorson the line between the first and the last leg. Every class consists of sub-classes defined by the particular partition of the othern− 2 external legs inwhich they are connected to the main line.

A particular diagram withn = 20 would be, for example,

(50)

and would correspond to the class

(51)

12

and more precicely to its subclass with3, 1, 2, 1, 5, 3, 3 on-shell legs in thecorresponding blobs7. Let m1,m2, . . . ,mk+1 be a particular partition ofn− 2, such that

∑mi = n− 2. The blobs denote the full set of all possible

color-ordered diagrams with one off-shell andmi on-shell legs.Let us call this subclass of Feynman diagrams byF(m1, . . . , mj, mj+1, . . . ,mk+1)

where the tildes on top of two consecutivem’s mark the presence of a four-vertex attaching the corresponding blobs to the line.

F(m1, . . . , mj, mj+1, . . . ,mk+1) = . . . . . .

m1 mj mj+1 mn

1 n (52)The particular subclass of diagrams in eq.51 would then be called F(3, 1, 2, 1, 5, 3, 3)

and eq.50 would be one of the actual Feynman diagrams contained in thatsubclass.

In the gauge we are working, the hatting operator would leavethe contri-bution of the blobs, in such a diagram, invariant since the momentum flowinginto the blob doesn’t containp1 or pn. It would only affect the propagatorsand vertices along the main line. After the shifting of the momenta a cut canbe placed on any of the propagators on the line, thus fixing thevalue ofz atwhich the residue should be evaluated. We shall denote such agraph with acut on thej’th propagator by

H(m1, . . . ,mk+1; j) =[

F(m1, . . . ,mk+1)p21j

]

z=zi

1

p21j

(53)

Here and in what followsp1Aidenotes the sum of external momenta

p1Ai= p1+ pAi

+ pA2+ . . . + pAi

(54)

with

pAi=

mi∑

k=1

pk (55)

A closer look at the ”hatted” diagram will reveal that hatting affects,through the momentap1 and pn, only the three-vertices along the mainline. The four-vertices, not carrying any momentum, remainunchanged.The effect on the three-vertices is included in the ”hatted”YM three-gluon

7A blob with mi on-shell legs and one off-shell denotes the corresponding sum of color-ordered diagrams with one off-shell leg andmi on-shell legs.

13

vertices, which we denote with a crossed white blob in the figures. Each ofthese can be decomposed in two pieces:

= Vµνρ = gµν(p1− p2)ρ+ gνρ(p2− p3)µ+ gρµ(p3− p1)ν

= gµν(p1− p2)ρ+ gνρ(p2− p3)µ+ gρµ(p3− p1)ν

−zǫσ(2gµρgνσ− gµσgνρ− gµνgσρ)

= Vµνρ− zǫσ(2gµρgνσ− gµσgνρ− gµνgσρ)

= +ǫσ

(56)

where

ǫσ= −zVµνρσ (57)

with Vµνρσ the QCD four-vertex.Let us concentrate for the moment on the class of diagrams with the

smallest number of propagators along the line between the first and the lastleg: diagrams with no propagators along the line can have a three or a four-vertex joining these two legs. The whole class of diagrams with a three-vertex joining the first and the last leg vanishes identically in the gauge weare using. The remaning class with one four-vertex will be dealt with soon.

Next, we have the class with one propagator along the main line.If we write J

µQ for the current coming from the blobQ, we have

H(m1,m2; 1) ≡A B

=[

ǫ1µJAνVµνρVρκλJ

κBǫ

λn

]

z0

1

p21A

=[

ǫ1µJAν(Vµνρ− zǫσV

σµνρ)(Vρκλ− zǫτVτρκλ)JκBǫ

λn

]

z0

1

p21A

=[

ǫ1µJAνVµνρVρκλJ

κBǫ

λn− zǫ1µJ1νǫσV

σµνρVρκλJκBǫ

λn

−zǫ1µJAνVµνρǫτVτρκλJ

κBǫ

λn

+z2ǫ1µJAνǫσVσµνρǫτVτρκλJ

κBǫ

λn

]

z0

1

p21A

(58)

14

The first, z-independent term in the bracket equals the Feynman diagramwith one propagator. The last term in the bracket vanishes due to eq.38 andwe get

H(m1,m2; 1) ≡A B

=

=A B

− [2(ǫ·p1A)(ǫ1·JA)(ǫn·JB)]z01

p21A

with

z0 = −p21A

2(p1Aǫ)(59)

where we have used momentum conservation and our freedom to inter-change the polarization vectors. So

H(m1,m2; 1) ≡A B

=A B

+ (60)

orH(m1,m2; 1) = F(m1,m2) + F(m1, m2) (61)

In the case that the blobsA andB of the first diagram in eq.60 containexactly one on-shell leg, the BCFW decomposition leads to two hatted QCD3-vertices. We have verified that the usual QCD 3-vertex evaluated at thehatted kinematics reproduces the formulas

A(1+, 2−, q+) =√2

[q1]3

[12][2q](62)

and

A(1+, 2+, q−) =√2

[12]3

[2q][q1](63)

15

wherepq = −p1 − p2 is the off-shell leg andpq = −p1 − p2 − zǫ is itson-shell continuation8. The corresponding formulas involving the last leg(which has negative helicity) work similarly but include angle bracket spinorproducts.

Proceeding to graphs that contain two propagators between the first andthe last leg or graphs that contain one propagator with one four-vertex at-tached to it, we have

A B C

+

A B C

(64)

+

A CB

+

A CB

=

A B C

+

A CB

+

A CB

(65)

The expressions that correspond to the first two hatted graphs on the lefthand side are quadradic inz and are evaluated in different values ofz, z1andz2.

A B C

=

[

H01

p21B

+H1z

p21B

+H2z2

p21B

]

z=z1

1

p21A

(66)

8The other polarizations (A(1+, 2+, q+), A(1+, 2−, q−)) can be shown to vanish.

16

with p1B = p1+ pA+ pB

A B C

=

[

H01

p21A

+H1z

p21A

+H2z2

p21A

]

z=z2

1

p21B

(67)Thez-independent term of all four graphs gives the Feynman graphs of

the right hand side of eq.65, due to the kinematic identity eq.30.Moreover, the part linear inz of the first two graphs cancels identically

due to eq.32 withρ = 1. The quadratic part of the first two graphs cancelsexactly the part linear inz of the two last graphs on the left hand side ofeq.65.

The generalization of the above mechanism goes as follows: Each hatteddiagram withv3 three-vertices along the main line can be decomposed in asum of sub-diagrams having0,1, . . ., v3 white-blob vertices correspondingto 0, 1, . . ., v3 powers ofz (evaluated at somezi). If the diagram also hasv4four-vertices, the number of propagators along the line isv3+ v4− 1.

The sum over all possible cuts for the particular diagram canthen bewritten as

v3+v4−1∑

j=1

H(m1 . . . mk; j) =

=

v3+v4−1∑

j=1

(H0+ zH1+ z2H2+ . . . Hv3zv3)

∏

q6=j

1

p21q

z=zj

1

p21j

(68)

or

v3+v4−1∑

j=1

H(m1 . . .mk; j) = H0

v3+v4−1∑

j=1

[

1∏

q6=j p21q

]

z=zj

1

p21j

+H1

v3+v4−1∑

j=1

[

z∏

q6=j p21q

]

z=zj

1

p21j

+H2

v3+v4−1∑

j=1

[

z2∏

q6=j p21q

]

z=zj

1

p21j

+ . . .

17

+Hv3

v3+v4−1∑

j=1

[

zv3∏

q6=j p21q

]

z=zj

1

p21j

(69)

Due to the set of identities eq.32, all terms involvingzλ with 0 < λ <

v3+ v4− 1 vanish identically.Moreover it is easy to see that thez-independent term involvingH0 will

give the corresponding Feynman diagram with the help of the identity eq.30.Let us further distinguish among three cases: diagrams withno four-

vertex on the line (v3 = k+1), with one four-vertex (v3 = k) and with morethan one four-vertex (v3 < k). In the third case the right hand side of eq.69reduces to thez-independent term corresponding to the Feynman diagram.All terms involving z vanish. In other words the sum over all possible cutsof the hatted diagrams with two or more four-vertices on the line, is equal tothe corresponding Feynman diagrams.

In the first two cases the sum of all hatted diagrams over all possible cutsgives the corresponding Feynman diagrams plus a number of terms. We willnow show that these terms cancel exactly each other due to thestructure ofthe YM vertices.

Diagrams with no four vertex on the line havev3 = k+ 1 three-vertices.They are of the form

H(m1, . . . ,mj,mj+1, . . . ,mk+1; j) = . . . . . .

m1 mj mj+1 mn

1 n (70)The highest order term involvesHk+1 that vanishes identically: it corre-sponds to diagrams with only crossed white blobs on the line,hence it con-sists of contractions ofǫ with itself or ǫ1, ǫn. The next to leading orderterm,Hk is a sum of terms with one QCD vertex andk crossed white-blobvertices.

We have

k∑

j=1

H(m1 . . .mk+1; j) = F(m1 . . .mk+1) +

+Hk

k∑

j=1

[

zk∏

q6=j p21q

]

z=zj

1

p21j

(71)

18

and using the kinematical identity eq.35 we get

k∑

j=1

H(m1 . . .mk+1; j) = F(m1 . . .mk+1) −Hk1

∏kj=12ǫ·p1j

(72)

Let us denoteHk,r the term where the QCD vertex is coming from ther’th hatted vertex.

Hk,r is the numerator of a diagram containing blobs connected with themain line by white-blob verticesǫcMµνρc everywhere except the QCDr’thvertex. A generic piece on the line will contribute by

ǫ

. . . . . .µ ρ

Ai

≡ M(Ai)µρ = −JνAiǫc(2gcνgµρ− gcµgνρ− gcρgµν)

= −(2(JAi·ǫ)gµρ− JAi,µǫρ− JAi,ρǫµ) (73)

whereJνAiis the current (subamplitude) coming from thei’th blob. The

first such white-blob vertex, when contracted withǫµ1, gives

ǫ

. . .1 ρ

A1

= ǫµ1M(A1)µρ = −(ǫ1·JA1

)ǫρ (74)

It is easy to see that a chain of consecutive white-blob vertices gives

ǫ ǫ ǫ

. . .

A1 A2 As

1 ρ= (−1)ρ(ǫ1·JA1

)(ǫ·JA2) . . . (ǫ·JAs)ǫρ (75)

SinceHk,r hask such vertices we get

Hk,r = (−1)k∏

j6=r

(ǫ·JAj)ǫ

µ1J

νAr

ǫρn×

×(gµν(p1,r−1− pr)ρ+ gνρ(pr+ p1,r)µ+ gρµ(−p1,r− p1,r−1)ν)

= (−1)k∏

q6=r

(ǫ·JAq )(ǫn·JAr )((p1,r−1+ p1,r)·ǫ1) (76)

where repeated use was made of our ability to interchangeǫ1, ǫn andǫ.Note that for the boundary terms,r = 1 andr = k + 1 the above formulamust be understood with the definitionsp1,0 = p1,k+2 = 0.

19

Summing overr we regainHk, and

k∑

j=1

H(m1 . . .mk+1; j) = F(m1 . . . mk+1)+

+(−1)k+1 1∏

j 2ǫ·p1j

∑

r

(ǫ1·JA1) . . . (ǫn·JAn )[(p1,r−1+ p1,r)·ǫ]

= F(m1 . . .mk) +

+(−1)k+1(ǫ1·JA1) . . . (ǫn·JAn )

∑

r

2p1,r·ǫ(2ǫ·p11)(2ǫ·p12) . . . (2ǫ·p1k)

= F(m1 . . .mk) +

(−1)k+1(ǫ1·JA1) . . . (ǫn·JAn)

∑

r

1∏

q6=r(2ǫ·p1q)(77)

Let us consider now a diagram that could occur from the ones above bycontracting a propagator, thereby merging two of the three-vertices, say ther’th and ther+ 1’th in one four-vertex. Such a diagram has one four-vertexon the line and one propagator less, that is it hask vertices of whichk − 1

three-vertices and one four-vertex, as well ask− 1 propagators.

H(m1, . . . , mr, mr+1, . . . ,mk+1; j) = . . . . . .

m1 mr mr+1 mn

1 n (78)The sum over all cuts of such hatted diagrams is

k∑

j=1

j6=r

Hr(m1, . . . , mr, mr, . . . ,mk; j) =

= F(m1, . . . , mr, mr, . . . ,mk) +

+Hk,r

k∑

j=1

j6=r

[

zk−1

∏q6=j,r p

21q

]

z=zj

1

p21j

(79)

wherer denotes the position of the four-vertex. As before, all contributionsinvolving zλ for 0 < λ < k−1 vanish due to eq.32. Using the identity eq.35

20

we can perform thej sum over cuts and get

k∑

j=1

j6=r

Hr(m1, . . . , mr, mr, . . . ,mk; j) =

= F(m1, . . . , mr, mr, . . . ,mk) − Hk−1,r1

∏q6=r2ǫ·p1q

(80)

HereHk−1,r stands for the vertex contribution of a diagram with one fourvertex andk − 1 white-blob three-vertices. It is not difficult to see that thisis equal to

Hk−1,r = (−1)k−1(ǫ1·JA1)(ǫ·JA2

) . . . (ǫn·JAk) (81)

Summing up over all possible positions of the four vertex we get

k∑

r=1

k∑

j=1

j6=r

Hr(m1 . . .mk; j) = F(m1, . . . , mr, mr, . . . ,mk)+

+(−1)k(ǫ1·JA1) . . . (ǫn·JAn)

∑

r

1∏

q6=r 2ǫ·p1q(82)

which exactly cancels the last term of equation eq.77, thus completing theproof that the sum over all cuts of all possible hatted diagrams is equal tothe sum of Feynman diagrams.

8 Remarks on generality

We have seen in the previous sections how the BCFW decomposition isrelated to Feynman diagrams. In fact, when working in the particular gaugethat we have chosen, every Feynman diagram is broken in pieces (‘hatted’graphs) each of which contains a cut in one of the propagatorsalong the lineconnecting the two ‘special’ legs. Then, hatted graphs withthe same cut areregrouped together in one decomposition, and the sum of decompositionsgives back the whole color amplitude.

In the process of breaking up the Feynman graphs in hatted graphs, someextra terms are produced: those correspond to thez → ∞ limit of the hattedgraph. Thanks to the particular structure of the YM vertices, demonstrated ineq.56, these extra terms cancel exactly among graphs with nofour-vertices

21

and graphs with one four-vertex and one propagator less. In the lowest caseof one propagator along the line, the diagram with two three vertices givesan extra term that is exactly equal to the diagram with no propagator and onefour-vertex. The latter is a diagram which, having no propagator along theline, didn’t fit in an obvious way in any BCFW decomposition.

The absence of a Yang-Mills structure in a scalar theory is what makesa decomposition along the terms of BCFW cumbersome. There one wouldhave to accommodate the diagram in eq.40 by adding an ad hoc term in therecurrence relation which would therefore be less elegant.

The gauge in which we are working explicitly eliminates thez-dependencefrom the polarization vectors. In other gauges the polarization vectorsǫ+1andǫ−n would be affected by the shift. This would complicate significantlythe algebra, as further poles related to vanishing denominators of these po-larization vectors will come to play.

If the identity eq.40 doesn’t hold, for example, contributions from theclass of diagrams in the left hand side of eq.40 would have to be cancelledby contributions from other classes. The appealing cancellations within sub-classes of diagrams with a particular partitioning of the external legs wouldbe lost. Keeping track of the terms in fragmenting and regrouping Feyn-man diagrams into BCFW decompositions would be much harder.Still onecould prove the BCFW decomposition by performing the pole expansionon any Feynman diagram minus itsz → ∞ limit but the algebra would beparticularly cumbersome.

It is becoming increasingly clear that the BCFW decomposition is a re-arrangement of terms of Feynman diagrams within a color ordered gluonicamplitude. A similar impression will presumably occur froman approachof amplitudes with fermionic lines. Under the light of the diagrammaticproof presented here it is seen that the mechanism behind anyapparent can-celations in the BCFW recursion formula does not (unfortunately) divulge adeeper principle or even a contingent effect in gluonic amplitudes, but resultsdirectly from the structure of the Yang-Mills vertices and the cancellationsinduced by gauge invariance.

9 Acknowledgments

Author P.D.D. is currently co-funded by the European SocialFund (75%)and National Resources (25%)-EPEAEK B! - PYTHAGORAS. The researchof authors A.L. and C.G.P. has been funded by the EU program no.HPMD-

22

CT-2001-00105. The author C.G.P. also acknowledges HPC-EUROPA Pro-gramme for its support.

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