Introduction to Loop Calculations

May 2010

Contents

1 One-loop integrals 2

1.1 Dimensional regularisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Feynman parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Momentum integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 More about singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Regularisation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Reduction of one-loop integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Unitarity cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Beyond one loop 18

2.1 General form of multi-loop integrals . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Construction of the functions F and U from topological rules . . . . . . . . . . . 182.3 Reduction to master integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Calculation of master integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.1 Mellin-Barnes representation . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.2 Sector decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

A Appendix 27

A.1 Useful formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27A.2 Multi-loop tensor integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27A.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1

1 One-loop integrals

Consider a generic one-loop diagram with N external legs and N propagators. If k is the loopmomentum, the propagators are qa = k + ra, where ra =

ai=1 pi. If we define all momenta as

incoming, momentum conservation impliesN

i=1 pi = 0 and hence rN = 0.

pN1 pN

p1

p2

If the vertices in the diagram above are non-scalar, this diagram will contain a Lorentz tensorstructure in the numerator, leading to tensor integrals of the form

ID,1...rN (S) =

dDk

iD2

k1 . . . kriS(q

2i m2i + i)

, (1)

but we will first consider the scalar integral only, i.e. the case where the numerator is equal toone. S is the set of propagator labels, which can be used to characterise the integral, in ourexample S = {1, . . . , N}. We use the integration measure dDk/iD2 dk to avoid ubiquitousfactors of i

D2 which will arise upon momentum integration. D is the space-time dimension the

loop momentum k lives in. In D = 4 dimensions, the loop integrals may be divergent either fork (ultraviolet divergences) or for q2i m2i 0 (infrared divergences) and therefore needa regulator. A convenient regularisation method is dimensional regularisation.

1.1 Dimensional regularisation

Dimensional regularisation has been introduced in 1972 by t Hooft and Veltman (and by Bolliniand Gambiagi) as a method to regularise ultraviolet (UV) divergences in a gauge invariant way,thus completing the proof of renormalizability.The idea is to work in D = 4 2 space-time dimensions. Divergences for D 4 will thusappear as poles in 1/.An important feature of dimensional regularisation is that it regulates infrared (IR) singu-larities, i.e. soft and/or collinear divergences due to massless particles, as well. Ultravioletdivergences occur if the loop momentum k , so in general the UV behaviour becomes bet-ter for > 0, while the IR behaviour becomes better for < 0. Certainly we cannot have D < 4and D > 4 at the same time. What is formally done is to first assume the IR divergences areregulated in some other way, e.g. by assuming all external legs are off-shell or by introducing asmall mass for all massless particles. Assuming > 0 we obtain a result which is well-defined

2

(UV convergent), which we can analytically continue to the whole complex D-plane, in partic-ular to Re(D) > 4. if we now remove the auxiliary IR regulator, the IR divergences will showup as 1/ poles.

The only change to the Feynman rules to be made is to replace the couplings in the Lagrangiang g, where is an arbitrary mass scale. This ensures that each term in the Lagrangianhas the correct mass dimension.

1.2 Feynman parameters

To combine products of denominators of the type Di = [(k + ri)2 m2i + i]i into one single

denominator, we can use the identity

1

D11 D22 . . .D

NN

=(N

i=1 i)Ni=1 (i)

0

Ni=1

dzi zi1i

(1Nj=1 zj)[z1D1 + z2D2 + . . .+ zNDN ]

PNi=1 i

(2)

The integration parameters zi are called Feynman parameters. For a generic one-loop diagramas shown above we have i = 1 i.

Simple example: one-loop two-point function

k

k + p

p

The corresponding integral is given by

I2 =

dDk

(2)D1

[k2 m2 + i][(k + p)2 m2 + i]= (2)

0

dz1dz2

dDk

(2)D(1 z1 z2)

[z1 (k2 m2) + z2 ((k + p)2 m2) + i]2

= (2)

10

dz2

dDk

(2)D1

[k2 + 2 k Q+ A+ i]2 (3)Q = z2 p

A = z2 p2 m2

where the -constraint has been used to eliminate z1.

1.3 Momentum integration

Our general integral, after Feynman parametrisation, is of the following form

IDN = (N)

0

Ni=1

dzi (1Nl=1

zl)

dk

[k2 + 2k Q+

Ni=1

zi (r2i m2i ) + i

]N

Q =Ni=1

zi ri . (4)

3

Now we perform the shift l = k +Q to eliminate the term linear in k in the square bracket toarrive at

IDN = (N)

0

Ni=1

dzi (1Nl=1

zl)

dl[l2 R2 + i]N (5)

The general form of R2 is

R2 = Q2 Ni=1

zi (r2i m2i )

=

Ni,j=1

zi zj ri rj 12

Ni=1

zi (r2i m2i )

Nj=1

zj 12

Nj=1

zj (r2j m2j )

Ni=1

zi

= 12

Ni,j=1

zi zj(r2i + r

2j 2 ri rj m2i m2j

)

= 12

Ni,j=1

zi zj Sij

Sij = (ri rj)2 m2i m2j (6)The matrix Sij , sometimes also called Cayley matrix is an important quantity encoding all thekinematic dependence of the integral. It plays the main role in algebraic reduction as wellas in the analysis of so-called Landau singularities, which are singularities where detS or asub-determinant of S is vanishing (see below for more details).Remember that we are in Minkowski space, where l2 = l20 ~l2, so temporal and spatial com-ponents are not on equal footing. Note that the poles of the denominator are located at

l20 = R2 + ~l2 i l0

R2 +~l2 i . Thus the i term shifts the poles away from the

real axis.For the integration over the loop momentum, we better work in Euclidean space where l2E =4

i=1 l2i . Hence we make the transformation l0 i l4, such that l2 l2E = l24 + ~l2, which

implies that the integration contour in the complex l0-plane is rotated by 90 such that the

contour in the complex l4-plane looks as shown below. The is called Wick rotation. We seethat the i prescription is exactly such that the contour does not enclose any poles. Thereforethe integral over the closed contour is zero, and we an use the identity

dl0f(l0) = ii

dl0f(l0) = i

dl4f(l4) (7)

Re l4

Im l4

4

Our integral now reads

IDN = (1)N(N) 0

Ni=1

dzi (1Nl=1

zl)

dDlE

D2

[l2E +R

2 i]N (8)Now we can introduce polar coordinates in D dimensions to evaluate the integral: Using

dDl =

0

dr rD1

dD1 , r =l2E =

(4

i=1

l2i

)12

(9)

dD1 = V (D) =

2D2

(D2)

(10)

where V (D) is the volume of a unit sphere in D dimensions:

V (D) =

20

d1

0

d2 sin 2 . . .

0

dD1(sin D1)D2

Thus we have

IDN = 2(1)N(N)

(D2)

0

Ni=1

dzi (1Nl=1

zl)

0

dr rD11

[r2 +R2 i]N

Substituting r2 = x: 0

dr rD11

[r2 +R2 i]N =1

2

0

dx xD/211

[x+R2 i]N (11)

Now the x-integral can be identified as the Euler Beta-function B(a, b), defined as

B(a, b) =

0

dzza1

(1 + z)a+b=

10

dy ya1(1 y)b1 = (a)(b)(a+ b)

(12)

and after normalising with respect to R2 we finally arrive at

IDN = (1)N(N D

2)

0

Ni=1

dzi (1Nl=1

zl)[R2 i ]D2 N . (13)

The integration over the Feynman parameters remains to be done, but we will show belowthat for one-loop applications, the integrals we need to know explicitly have maximally N = 4external legs. Integrals with N > 4 can be expressed in terms of boxes, triangles, bubbles(and tadpoles in the case of massive propagators). The analytic expressions for these masterintegrals are well-known. The most complicated analytic functions at one loop (appearing inthe 4-point integrals) are dilogarithms.

The generic form of the derivation above makes clear that we do not have to go through the pro-cedure of Wick rotation explicitly each time. All we need is to use the following general formulafor D-dimensional momentum integration (in Minkowski space, and after having performed theshift to have a quadratic form in the denominator):

dDl

iD2

(l2)r

[l2 R2 + i]N = (1)N+r(r +

D2)(N r D

2)

(D2)(N)

[R2 i]rN+D2 (14)

5

Example one-loop two-point function

Applying the above procedure to our two-point function, we obtain

I2 = (2)

10

dz

dDl

(2)D1

[l2 J + i]2 (15)J = Q2 A = p2 z (1 z) +m2

I2 = iD2

(2)D(2 D

2)

10

dz [p2 z (1 z) +m2 ]D2 2 , (16)

where we have dropped the i now. For m2 = 0, the result can be expressed in terms of-functions:

I2|m2=0 =i

D2

(2)D[p2]D2 2(2D/2)B(D/2 1, D/2 1)

B(a, b) =(a)(b)

(a + b). (17)

The two-point function has an UV pole which is contained in

(2D/2) = () = 1 E +O() , (18)

where E is Eulers constant,

E = limn

(n

j=1

1

j lnn

)= 0.5772156649...

Tensor integrals

If we have loop momenta in the numerator, as in eq. (1), the procedure is essentially thesame, except for combinatorics and additional Feynman parameters in the numerator: Thesubstitution k = l Q introduces terms of the form (l Q)1 . . . (l Q)r into the numeratorof eq. (5). As the denominator is symmetric under l l, only the terms with even numbersof l in the numerator will give a non-vanishing contribution upon l-integration. Further, weknow that integrals where the Lorentz structure is only carried by loop momenta can onlybe proportional to combinations of metric tensors g . Therefore we have, as the tensor-generalisation of eq. (14),

dDl

iD2

l1 . . . l2m

[l2 R2 + i]N = (1)N[(g..)m

]{1...2m}(12

)m (N D+2m2

)

(N)

(R2 i)N+(D+2m)/2 ,

(19)which can be derived for example by taking derivatives of the unintegrated scalar expressionwith respect to l. (g..)m denotes m occurences of the metric tensor and the sum over all possi-ble distributions of the 2m Lorentz indices i, to the metric tensors is denoted by [ ]{12m}.

6

Thus, for a general tensor integral, combining with numerators containing the vectors Q, onefinds the following formula [7]:

ID,1...rN (S) =

r/2m=0

(12

)m N1j1,...,jr2m=1

[(g..)mrj1 . . . r

jr2m

]{1...r} ID+2mN (j1, . . . , jr2m;S)(20)

IdN (j1, . . . , j;S) = (1)N(N d

2)

Ni=1

dzi (1Nl=1

zl) zj1 . . . zj(R2 i)d/2N (21)

R2 = 12z S z

The distribution of the r Lorentz indices i, to the external vectors rij is denoted by [ ]{1r}.

These are(

r2m

)mk=1(2k1) terms. (g..)m denotesm occurences of the metric tensor and r/2

is the nearest integer less or equal to r/2. Integrals with zj1 . . . zj in eq. (21) are associatedwith external vectors rj1 . . . rj, stemming from factors of Q

in eq. (5).How the higher dimensional integrals ID+2mN in eq. (20), associated with metric tensors (g

..)m,arise, is left as an exercise.

Schwinger parametrisation

An alternative to Feynman parametrisation is the so-called Schwinger parametrisation, basedon

1

A=

1

()

0

dx x1 exp(xA), Re(A) > 0 (22)

In this case the Gaussian integration formula

dDrE exp( r2E) =(

)D2

, > 0 (23)

can be used to integrate over the momenta.

1.4 More about singularities

The overall UV divergence of an integral can be determined by power counting: if wework in D dimensions at L loops, and consider an integral with P propagators and nlfactors of the loop momentum belonging to loop l {1, . . . , L} in the numerator, we have = DL 2P + 2l nl/2, where nl/2 is the nearest integer less or equal to nl/2.We have logarithmic, linear, quadratic,. . . overall divergences for = 0, 1, 2, . . . and noUV divergence for < 0. This means that in 4 dimensions at one loop, we have UVdivergences in rank 0 two-point functions, rank 2 (and rank 3) three-point functions andrank 4 four-point functions.

7

IR divergences: 1/2L at worst. Necessary conditions for IR divergences are given by theLandau equations: { i zi (q2i m2i ) = 0 ,N

i=1 zi qi = 0 .(24)

If eq. (24) has a solution zi > 0 for every i {1, . . . , N}, i.e. all particles in the loop aresimultaneously on-shell, then the integral has a leading Landau singularity. If a solutionexists where some zi = 0 while the other zj are positive, the Landau condition correspondsto a lower-order Landau singularity. Soft and/or collinear singularities appearing as polesin 1/ are always stemming from some zi = 0.

A special type of Landau singularities are scattering singularities, where detG 0 anddetS (detG)2, and Gkl = 2 pk pl is the Gram matrix, see section 1.6. These singulari-ties are not spurious (i.e. an artifact of the choice of the basis integrals), but correspondto physical kinematics. For example, in the 6-photon amplitude, a double parton scat-tering configuration occurs, where two incoming photons split into fermion pairs, thelatter rescattering into photon pairs (p3, p4), (p5, p6) with vanishing relative transversemomentum.

scaleless integrals are zero in dimensional regularisation:

d2m2k (k2) = 0 .

Note that

(2D/2) D2 /(2)D = ()(4)2

=1

(4)2

(1

E + ln(4) +O()

).

The factor in brackets, = 1/ E + ln(4) will always appear in UV divergent loopintegrals. Therefore it is convenient to subtract, upon UV renormalisation, not only the1/ pole, but the whole factor . This is called MS subtraction (modified MinimalSubtraction).

Drawbacks of dimensional regularisation

It is not obvious how to continue the Dirac matrix 5, which is in 4 dimensions defined as

5 = i 0123 (25)

to D dimensions. One could define 5 =i4!1234

1244 but doing so, Ward identitiesrelying on {5, } = 0 break down due to an extra (D 4)-dimensional contribution. Asolution to this problem for practical calculations is to leave 5 in 4 dimensions and to splitthe other Dirac matrices into a 4-dimensional and a (D 4)-dimensional part, = + ,where is 4-dimensional and is (D 4)-dimensional. The Dirac matrices defined in thisway obey the algebra

{, 5} ={

0 {0, 1, 2, 3}25 otherwise.

(25b)

8

The second line above can also be read as [5, ] = 0, which can be interpreted as 5 acting

trivially in the non-physical dimensions. Note that the 4-dimensional and (D 4)-dimensionalspaces are orthogonal.If we use dimension splitting into 2m integer dimensions and the remainig 2-dimensionalspace, k2(D) = k

2(2m) + k

2(2), we will encounter additional integrals with powers of (k

2) in thenumerator. These are related to integrals in higher dimensions by

dDk

iD2

(k2) f(k, k2) = (1)(+D2 2)

(D2 2)

dD+2k

iD2+

f(k, k2) . (26)

Note that 1/(D2 2) is of order . Therefore the integrals with > 0 only contribute if the

k-integral in 4 2+ 2 dimensions is divergent. In this case they contribute a constant part,which forms part of the so-called rational part of the full amplitude. Note that a divergencein 4 2+ 2 dimensions is always of ultraviolet origin.Exercise: derive eq. (26).

1.5 Regularisation schemes

Related to the 5-problem, it is not uniquely defined how we continue the Dirac-algebra to Ddimensions. There are essentially three different schemes:

CDR: Conventional dimensional regularisation: all momenta and all polarisation vec-tors are taken in D dimensions.

HV: t Hooft-Veltman scheme: momenta and helicities of the unobserved particles areD-dim. while momenta and helicities of the observed particles are 4-dimensional.

DR: Dimensional reduction: momenta and helicities of the observed particles are 4-dim., as well as all polarisation vectors. Only the momenta of the unobserved particlesare D-dimensional.

The conventions are summarized in Table 1.

CDR HV DR

= + = + = , 0{5, } 0 eq. (25b) 0internal momenta k = k + k k = k + k k = k + kexternal momenta pi = pi + pi pi = pi, pi = 0 pi = pi, pi = 0int. gluon pol. D 2 D 2 2ext. gluon pol. D 2 2 2

Table 1: Comparison of different regularisation schemes

At one loop, the transition formulae to relate results obtained in one scheme to another schemeare well known [5, 6].

9

1.6 Reduction of one-loop integrals

In the following two subsections we will show that every one-loop amplitude, with an arbitrarynumber N of legs, can be written as a linear combination of simple basis integrals. Thebasis integrals can be chosen such that they do not have more than four external legs.

Reduction of scalar integrals

In this section we will show that scalar one-loop integrals for arbitrary N can be reduced tointegrals with N 4 only. To this aim we make an ansatz where we cancel some denominatorsby writing linear combinations of propagators into the numerator, with yet unknown coefficientsbl.

IDN = Ired + Ifin =

dk

Nl=1 bl (q

2l m2l )N

l=1(q2l m2l + i)

+

dk

[1Nl=1 bl (q2l m2l )]N

l=1(q2l m2l + i)

(27)

We will show in the following that one can find coefficients bl such that Ifin contains no IR poles.After Feynman parametrisation and momentum shift as explained above, we obtain (using theshort-hand notation dNz =

Ni=1 dzi)

Ifin = (N)

0

dNz (1Nl=1

zl)

dl

[1Nl=1 bl (q2l m2l )]

(l2 R2)N (28)

R2 = 12z S z i , qj = l +

Ni=1

(ij zi) ri

Now the term in square brackets in Eq. (28) can be written as[1

Ni=1

bi (q2i m2i )

]= (l2 +R2)

Nj=1

bj +Nj=1

zj

(1 (S b)j

)+ odd in l (29)

If now the equations

(S b)j = 1 , j = 1, . . . , N (30)are fulfilled, the second term on the right-hand-side of (29) vanishes and one finds

Ifin = (N)(

Nl=1

bl

) 0

dNz (1Nl=1

zl)

dl

l2 +R2

(l2 R2)N (31)

Finally the loop momentum integration gives

Ifin =

(Nl=1

bl

)(1)N+1 (N 1 D

2) (N D 1)

0

dNz(1Nl=1 zl)(R2)N(D+2)/2

= (

Nl=1

bl

)(N D 1) ID+2N (32)

10

Therefore, if eq. (30) can be solved for the bl, i.e. if detS 6= 0 we have

IDN (S) =

Nj=1

bj IDN1(S \ {j}) (N D 1)B ID+2N (S) , det(S) 6= 0 (33)

bj =Ni=1

S1ij , B =Nj=1

bj (34)

If detS = 0, one can construct a reduction formula based on a pseudo-inverse or on the singular-value decomposition of S. In these cases the integrals always can be written as combinations oflower-point integrals only, i.e. the ID+2N (S) drop out. For more details see e.g. refs. [14, 15, 13].Further, we have the important relation

Nj=1

bj detS = (1)N+1 detG (35)

where detG is the Gram determinant, the determinant of the Gram matrix Gij = 2 ri rj.Note that, using rN = 0, Gij is an (N 1) (N 1) matrix. If the matrices Gij and Sij areconstructed from 4-dimensional external momenta, they have the following properties:

detG = 0 for N 6 Nj=1

bj = 0 for N 6 (36)

detS = 0 for N 7 (37)

This is due to the fact that the external momenta become linearly dependent for N 6, asone can construct a basis of 4-dimensional Minkowski space from four 4-dimensional momenta.(For N = 5, one of the 5 external momenta can be eliminated by momentum conservation,leaving just 4 linearly independent momenta.) Therefore the coefficient in front of ID+2N (S) ineq. (33) is identically zero for N 6, which means that we can express our N -point integralsrecursively in terms of (N 1)-point integrals until we reach N = 5. The case N = 5 is special:the coefficient in front of ID+2N (S) is (ND1)B, so it is of order for N = 5. As the integralsID+2N (S) are always UV and IR finite, we can drop this term for all one-loop applications, suchthat scalar pentagons can be written as a sum of five boxes, where in each box a differentpropagator is missing (pinched).

Note:

Traditionally, the notation conventions for tadpole, bubble, triangle, box, . . . integralswere ID1 = A0, I

D2 = B0, I

D3 = C0, I

D4 = D0, . . . (cf. ref. [11] and the programs FF [17] and

LoopTools [18] for infrared finite integrals).

A list (and Fortran program) of all IR divergent triangle (there are 6 of them) and box(there are 16) integrals can be found in ref. [19].

The reduction can also be formulated in terms of signed minors, see e.g. [20].

11

Reduction of tensor integrals

Historically, tensor integrals occurring in one-loop amplitudes were reduced to scalarintegrals using so-called Passarino-Veltman reduction [16]. It is based on the fact that atone loop, scalar products of loop momenta with external momenta can always be expressedas combinations of propagators. The problem with Passarino-Veltman reduction is thatit introduces powers of inverse Gram determinants 1/(detG)r for the reduction of a rankr tensor integral. This can lead to numerical instabilities upon phase space integrationin kinematic regions where detG 0.

It has been proven [12, 13, 22] that the reduction from rank r pentagons (N = 5) to boxes(N = 4) can be done without introducing inverse Gram determinants.

Inverse Gram determinants are unavoidable in the reduction of tensor boxes, triangleswhen a scalar integral basis is chosen. However, from physical arguments we expectsingularities which behave like 1/

detG at worst on amplitude level. Higher powers are

spurious, i.e. an artifact of the choice of a scalar integral basis, and should cancel whencombining the integrals to a gauge invariant quantity. However, this is difficult to achievefor one-loop amplitudes with a large number of external legs.

solutions to the problem mentioned above are use on-shell methods (see section 1.7)

semi-numerical: reduction is stopped before dangerous denominators are produced.The non-scalar integrals (parameter integrals with Feynman parameters in the nu-merator) are calculated numerically [21]

fully numerical: dont do any reduction, calculate the full tensor integral numerically

A form factor representation of a tensor integral is a representation where the Lorentz struc-ture has been extracted, each Lorentz tensor multiplying a scalar quantity, the form factor.Distinguishing A,B,C depending on the presence of zero, one or two metric tensors, we canwrite

ID,1...rN (S) =j1jrS

r1j1 . . . rrjrAN,rj1...,jr(S)

+

j1jr2S

[grj1 rjr2

]{1r}BN,rj1...,jr2(S)

+

j1jr4S

[ggrj1 rjr4

]{1r}CN,rj1...,jr4(S) (38)

Note that we never need more than two metric tensors in a renormalisable gauge where the rankr N , because for N > 5, we can express the metric tensor in terms of 4 linearly independentexternal vectors. Three metric tensors would be needed for rank six, i.e. for six-point integralsor higher, but we can immediately reduce those integrals to lower-point ones:

ID,1...rN (S) = jS

C1j ID,2...rN1 (S \ {j}) (N 6) , (39)

12

where Cl =

kS (S1)kl rk if S is invertible, and if not, it can be constructed from thepseudo-inverse [13].

Example for the distribution of indices:

ID,123N (S) =

l1,l2,l3S

r1l1 r2l2r3l3 A

N,3l1l2l3

(S)

+lS

(g12 r3l + g13 r2l + g

23 r1l ) BN,3l (S) .

Example for Passarino-Veltman reduction:Consider a rank one three-point integral

ID,3 (S) =

dkk

[k2 + i][(k + p1)2 + i][(k + p1 + p2)2 + i]= A1 r

1 + A2 r

2

r1 = p1 , r2 = p1 + p2 .

Contracting with r1 and r2 and using the identities

k ri = 12

[(k + ri)

2 k2 r2i], i {1, 2}

we obtain, after cancellation of numerators(2 r1 r1 2 r1 r22 r2 r1 2 r2 r2

)(A1A2

)=

(R1R2

)(40)

R1 = ID2 (r2) ID2 (r2 r1) r21I3(r1, r2)

R2 = ID2 (r1) ID2 (r2 r1) r22I3(r1, r2) .

We see that the solution involves the inverse of the Gram matrix Gij = 2 ri rj .

Recap

The procedure to calculate (one-)loop integrals is the following:

1. Feynman (or Schwinger) parametrisation

2. Shift the loop momentum to eliminate the term in the denominator which is linear in theloop momentum

3. Use formula (14) ( or (23) ) to perform the integration over the loop momentum

4. Integrate over the Feynman parameters.

By using algebraic reduction one can show that every one-loop N -point amplitude with 4-dimensional external legs can, for any N , be expressed in terms of basis integrals with fouror less external legs. The most complicated analytic functions that appear (at order 0) aredilogarithms (contained in box integrals).

13

1.7 Unitarity cuts

The idea is to use the analytic structure of scattering amplitudes to determine their explicitform. Using the unitarity of the S-matrix, where S = 1 + iT , we have

SS = 1 2 Im(T ) = T T . (41)

Inserting a complete set of intermediate states, we obtain

=

fdf

2Imf

The right-hand side can also be considered as all possible cuts of a loop amplitude, wherecutting a loop amplitude basically means putting the cut propagators on-shell, exploiting therelation

i

p2 + i 2 (+)(p2) . (42)

Applied to one-loop amplitudes, we therefore have

Im A1loop cuts

dcuts

The application of unitarity as an on-shell method of calculating loop amplitudes turns thecutting step around: tree amplitudes are fused together to form one-loop amplitudes.

Using the standard Feynman diagrammatic approach we have shown that any one-loop am-plitude with massless internal particles can be decomposed in terms of known scalar integralswith less than five external legs, IN=1,2,3,4, with D-dependent coefficients, or, alternatively, aslinear combinations of coefficients in D = 4 and a rational part R, which comes from terms ofthe form (D 4) IUVdiv in the Feynman diagrammatic approach.As we know the analytic form of the basis integrals, the imaginary parts of the different scalarintegrals can be uniquely attributed to a given integral. Therefore we have

A1loop =

N=1,2,3,4

CN(D) IDN =

N=1,2,3,4

CN(4) IDN +R

ImA1loop =

N=1,2,3,4

CN(4) Im(IDN ) (43)

The coefficients of the integrals, CN , and R, are rational polynomials in terms of Mandelstamvariables sij = (pi + pj)

2 and masses (or spinor products).

14

Figure 1: Multiple cuts can be used to fix integral coefficients of amplitudes.

Note that there are many different types of scalar two-, three- and four-point functions presentin a given process (which can be classified according to the number and location of off-shell/on-shell external legs and massive/massless propagators). Sums over all these different types areimplcit in eq. (43).

Generalized unitarity corresponds to requiring more than two internal particles to be on shell.Cutting four lines in an N -point topology amounts to putting the corresponding four propaga-tors on-shell. For N = 4, this procedure fixes the associated loop momentum completely andthe coefficient of the related box diagram is given as a product of tree diagrams.

Example:Consider Fig. 1(a) with all external momenta Ki are off-shell (four-mass-box). The corre-sponding box integral is finite and reads

I4m =

d4l

1

(l2 + i)((l K1)2 + i)((l K1 K2)2 + i)((l +K4)2 + i) . (44)

Cutting all four propagators, denoted by quad, we obtain:

quadI4m =

d4l (+)(l2) (+)((l K1)2) (+)((l K1 K2)2) (+)((l +K4)2) . (45)

As no other box integral in our amplitude shares the same singularity, we can deduce a re-

lation for the coefficient C(I4m)4 of this particular box integral in our amplitude, having the

representation (43) at hand:d4l (+)(l2) (+)((l K1)2) (+)((l K1 K2)2) (+)((l +K4)2)Atree1 Atree2 Atree3 Atree4

= C(I4m)4 quadI

4m , (46)

where Atreen is the tree-level amplitude at the corner with total external momentum Kn.Note that, since there are four delta functions, and l is a vector in four dimensions, the integralover l is completely frozen and can be solved for l. Therefore we find that

C(I4m)4 =

1

ns

s,J

nJ(Atree1 A

tree2 A

tree3 A

tree4 ) , (47)

where the sum is over the possible spins J of internal particles and the solution set s of theequations constraining l. ns is the number of these solutions, and nJ is the number of particlesof spin J .

15

The MHV (maximal helicity violating) tree amplitudes are given by [9]

Atree(1+, . . . , j, . . . , k, . . . , n+) = ij k4

1 22 3 . . . n 1 . (48)

However, something seems to be wrong when trying to apply this same procedure to boxintegrals with light-like external legs: some of the tree amplitudes will be all-massless three-point amplitudes. Naively, these amplitudes would vanish. The quadruple cuts would thusvanish as well, making the extraction of the coefficients in this way impossible. The solutionto this problem is to use complex momenta, such that one can treat opposite-helicity spinorsas independent variables. For this purpose one should rather use two-component Weyl spinorsa(p), a(p) instead of Dirac spinors, which can be defined as follows

u+(p) = v(p) =12

[a(p)a(p)

], u(p) = v+(p) =

12

[a(p)

a(p)]. (49)

The three-mass triangles can be isolated through triple cuts. The integrands emerging fromtriple cuts in general will also contain contributions to those box integrals sharing the samecuts. These contributions, and box-like terms which vanish upon loop integration, must beremoved in order to extract the coefficient of the three-mass triangle. Analogous argumentshold for the two-point integrals.

As the rational part does not contribute to the imaginary part of the amplitude, unitarity cutsin 4 dimensions cannot extract this part. Apart from being obtained from Feynman diagrams,it can be obtained by on-shell recursion relations [27] or by D-dimensional unitarity [28].

A similar approach, which is particularly well suited for a numerical solution of the cut condi-tions, has been formulated by Ossola, Papadopoulos and Pittau [29], and has seen a number ofprominent applications meanwhile.One can write the amplitude on integrand level as

Aint =i

Ci4di1di2di3di4

+i

Ci3di1di2di3

+i

Ci2di1di2

+i

Ci1di1

, (50)

where CiN = CiN + C

iN , and C

iN will vanish upon integration. Again, the box coefficient is

particularly simple: multiplying by di1di2di3di4 and putting the propagators on-shell we are leftwith Ci4. For N < 4, linear systems of equations have to be solved to separate C

iN from C

iN

using the on-shell constraints. The rational part can be obtained by several procedures takinginto account the D-dimensionality, which will not be described here.

However, we have a general theorem at hand when to expect rational parts in a one-loopamplitude, the BDDK-theorem [30], following basically from UV power counting (rememberthe structure of the k2-integrals (26), being related to tensor integrals with rank > 2): If aone-loop amplitude in a gauge theory has a representation in which all N -point integrals withN > 2 have at most N 2 powers of the loop momentum in the numerator, then there is no

16

rational part, i.e. the amplitude is uniquely determined by its cuts (cut constructible). Thisis fulfilled e.g. for any amplitude in N = 4 supersymmetric Yang-Mills theory.An important feature of on-shell methods is that the basic building blocks are tree level ampli-tudes and thus already incorporate gauge invariance manifestly. In Feynman diagram compu-tations many graphs have to be combined to result in a gauge invariant expression, leading tolarge intermediate expressions for multi-leg amplitudes.

17

2 Beyond one loop

2.1 General form of multi-loop integrals

A scalar Ddimensional Lloop Feynman diagram with N propagators to the power i can bewritten as

G =

Ll=1

dDkl

iD2

Nj=1

1

Pjj ({k}, {p}, m2j)

(51)

After Feynman parametrisation:

G = (N)

Nj=1

dxj xj1j (1

Ni=1

xi)

dk1 . . . dkL

[L

j,l=1

kj klMjl 2Lj=1

kj Qj + J]N

= (1)N (N LD/2)Nj=1 (j)

0

Nj=1

dxj xj1j (1

Ni=1

xi)UN(L+1)D/2FNLD/2 (52)

U = det(M) , N =Nj=1

j ,

F = det(M)[

Li,j=1

QiMijQj J i].

Pjj ({k}, {p}, m2j) are the propagators to the power j, depending on the loop momenta kl{1,...,L},

the external momenta {p1, . . . pE} and (not necessarily nonzero) masses mj . The functions Uand F can be straightforwardly derived from the momentum representation.A necessary condition for the presence of infrared divergences is F = 0. The function U cannotlead to infrared divergences of the graph, since giving a mass to all external legs would notchange U . Apart from the fact that the graph may have an overall UV divergence containedin the overall -function (see Eq. (52)), UV subdivergences may also be present. A necessarycondition for these is that U is vanishing.The functions U and F also can be constructed from the topology of the corresponding Feynmangraph, as explained in the following subsection.

2.2 Construction of the functions F and U from topological rulesCutting L lines of a given connected L-loop graph such that it becomes a connected tree graphT defines a chord C(T ) as being the set of lines not belonging to this tree. The Feynmanparameters associated with each chord define a monomial of degree L. The set of all such trees(or 1-trees) is denoted by T1. The 1-trees T T1 define U as being the sum over all monomialscorresponding to a chord C(T T1). Cutting one more line of a 1-tree leads to two disconnectedtrees, or a 2-tree T . T2 is the set of all such 2-trees. The corresponding chords define monomialsof degree L+1. Each 2-tree of a graph corresponds to a cut defined by cutting the lines which

18

connected the 2 now disconnected trees in the original graph. The momentum flow throughthe lines of such a cut defines a Lorentz invariant sT = (

jCut(T) pj)

2. The function F0 is thesum over all such monomials times minus the corresponding invariant:

U(~x) =TT1

[ jC(T )

xj

],

F0(~x) =TT2

[ jC(T )

xj

](sT ) ,

F(~x) = F0(~x) + U(~x)Nj=1

xjm2j . (53)

p2

1

3

4

6p3

7

p452

p1

Example: planar double box with p21 = p22 = p

23 = 0, p

24 6= 0:

Using k1 = k, k2 = l and propagator number one as 1/(k2+i), the denominator, after Feynman

parametrisation, can be written as

D = x1 k2 + x2 (k p1)2 + x3 (k + p2)2 + x4 (k l)2 + x5 (l p1)2 + x6 (l + p2)2 + x7 (l + p2 + p3)2

= ( k, l )

(x1234 x4x4 x4567

)(kl

) 2 (Q1, Q2)

(kl

)+ x7 (p2 + p3)

2 + i

Q = (Q1, Q2) = (x2p1 x3p2, x5p1 x6p2 x7(p2 + p3)) ,where we have used the short-hand notation xijk... = xi + xj + xk + . . ..Therefore

U = x123x567 + x4x123567F = (s12) (x2x3x4567 + x5x6x1234 + x2x4x6 + x3x4x5)

+(s23) x1x4x7 + (p24) x7(x2x4 + x5x1234) .A general representation for tensor integrals also exists, it can be found e.g. in [10].

2.3 Reduction to master integrals

Integration by parts

Integration-by-part identities [31] are based on the fact that the integral of a total derivativeis zero:

dDk

kvf(k, pi) = 0 , (54)

19

where v can either be a loop momentum or an external momentum. Working out the derivativefor a certain number of numerators yields systems of relations among scalar integrals which canbe solved systematically. The endpoints of the reduction are called master integrals.

Simplest example: massive vacuum bubble with general propagator power:

F () =

dk

1

(k2 m2 + i) (55)

Here we know that the master integral is

F (1) = (1D/2) (m2)D2 1 (56)

Using the integration-by-part identitydk

k

{k

(k2 m2 + i)}= 0

leads to

0 =

dk

{1

(k2 m2 + i)

k(k) k 2k

(k2 m2 + i)+1}

= DF () 2 (F () +m2 F ( + 1)) F ( + 1) = D 2

2 m2F () . (57)

In less trivial cases, to be able to solve the system for a small number of master integrals,an order relation among the integrals has to be introduced. For example, a topology T1 isconsidered to be smaller than a topology T2 if T1 can be obtained from T2 by pinching someof the propagators. Within the same topology, the integrals can be ordered according to thepowers of their propagators.A completely systematic approach has first been formulated by Laporta [34]. Recent imple-mentations and refinements of the algorithm are provided by the programs FIRE [35] andReduze [38]. Other automated reduction programs are MINCER [36] (for two-point integralsonly) and AIR [37].

Note: It is not uniquely defined which integrals are master integrals. For complicated multi-loopexamples, it is in general not clear before the reduction which integrals will be master integrals.Further, it can sometimes be more convenient to define an integral with a loop momentum inthe numerator rather than its scalar parent as a master integral.

2.4 Calculation of master integrals

Once we have reduced our expression for a multi-loop amplitude to a linear combination ofmaster integrals, the task is to evaluate these master integrals. A number of techniques havebeen developed for this task, analytical as well as numerical ones. Simple integrals of course canbe evaluated straightforwardly by integration over the Feynman parameters. More complicatedones require additional tricks. We fill focus only on two of them.

20

2.4.1 Mellin-Barnes representation

The basic formula underlying the Mellin-Barnes representation of a (multi-)loop integral reads

(A1 + A2 + ...+ An) =

1

()

1

(2i)n1

c+ici

dz1...

c+ici

dzn1 (58)

(z1)...(zn1)(z1 + ...+ zn1 + ) Az11 ...Azn1n1 Az1...zn1nEach contour is chosen such that the poles of (zi) are to the right and the poles of (. . .+ z)are to the left. Imz

Rezpoles of (z)poles of (+ z)

possible contour

The representation in eq. (58) can be used to convert the sum of monomials contained in thefunctions U and F into products, such that all Feynman parameter integrals are of the formof simple integrations over -functions. However, we are still left with the complex contourintegrals. The latter are then performed by closing the contour at infinity and summing upall residues which lie inside the contour. In general we will obtain multiple sum over residuesand need techniques to manipulate these sums. In simple cases the contour integrals can beperformed in closed form with the help of two lemmas by Barnes. Barnes first lemma statesthat

1

2i

ii

dz(a+ z)(b + z)(c z)(d z) = (a+ c)(a+ d)(b+ c)(b+ d)(a+ b+ c+ d)

,

(59)

if none of the poles of (a+ z)(b+ z) coincides with the ones from (c z)(d z). Barnessecond lemma reads

1

2i

ii

dz(a + z)(b+ z)(c + z)(d z)(e z)

(a+ b+ c+ d+ e+ z)

=(a + d)(b+ d)(c+ d)(a+ e)(b+ e)(c+ e)

(a + b+ d+ e)(a+ c + d+ e)(b+ c+ d+ e). (60)

21

Example: two-point function with one massive propagatorIn this example the Mellin-Barnes representation allows us to isolate the mass dependence fromthe denominator and to perform the Feynman parameter integration as in the massless case:

F (1, 2) =

dk

1

[k2 m2 + i]1 [(p k)2 + i]2 (61)

=1

2 i

(1)1+2(1)

c+ici

dz(m2)z

[k2 i]1+z[(p k)2 i]2 (1 + z)(z) .

Now we use Feynman parametrisation for the remaining denominator:

1

[k2 i]1+z[(p k)2 i]2 =(1 + 2 + z)

(1 + z)(2)

10

dxx21(1 x)1+z1

[k2 + 2px xp2]1+2+z .

After the substitution l = k x p and integration over l we obtain

F (1, 2) =1

2 i(p2)D2 12 (1)

1+2

(1)(2)

10

dx xD21z1(1 x)D2 21 (62)

c+ici

dz

(m2

p2)z

(z) (1 + 2 + z D/2)

=1

2 i(p2)D2 12 (1)

1+2(D/2 2)(1)(2)

c+ici

dz

(m2

p2)z

(z)(D/2 1 z)(1 + 2 + z D/2)(D 1 2 z) . (63)

We see that the integration over x in (62) was trivial as we factorised out the mass dependencebeforehand. The price to pay is that we are still left with the contour integration in the complexz-plane. Using Cauchys residue theorem one will end up with a series of residues, which inmost cases can be summed in a closed form. Calculations for i {1, 2} are given in example4.1 of Ref. [3].

2.4.2 Sector decomposition

Sector decomposition is a method operating in Feynman parameter space which is useful toextract singularities regulated by dimensional regularisation, converting the integral into aLaurent series in . It is particularly usefule if the singularites are overlapping in the sensespecified below. The coefficients of the poles in 1/ will be finite integrals over Feynmanparameters, which, for most examples beyond one loop, will be too complicated to be integratedanalytically, so have to be integrated numerically.

To introduce the basic concept, let us look at the simple example of a two-dimensional parameterintegral of the following form:

I =

10

dx

10

dy x1a yb(x+ (1 x) y

)1. (64)

22

The integral contains a singular region where x and y vanish simultaneously, i.e. the singularitiesin x and y are overlapping. Our aim is to factorise the singularities for x 0 and y 0.Therefore we divide the integration range into two sectors where x and y are ordered (seeFig. 2)

I =

10

dx

10

dy x1a yb(x+ (1 x) y

)1[(x y)

(1)

+(y x) (2)

] .

Now we substitute y = x t in sector (1) and x = y t in sector (2) to remap the integration rangeto the unit square and obtain

y

x

+ (2)(1)

+

y

x

t

t

Figure 2: Sector decomposition schematically.

I =

10

dx x1(a+b) 10

dt tb(1 + (1 x) t

)1+

10

dy y1(a+b) 10

dt t1a(1 + (1 y) t

)1. (65)

We observe that the singularities are now factorised such that they can be read off from thepowers of simple monomials in the integration variables, while the polynomial denominatorgoes to a constant if the integration variables approach zero.The singularities can then be extracted using

x1+ =1

(x) +

n=0

()n

n!

[lnn(x)

x

]+

,

where 10

dx f(x) [g(x)/x]+ =

10

dxf(x) f(0)

xg(x) , (66)

and f(x) should be a smooth function. This is known under the name plus prescription.After the singularities have been extracted, we can expand in .

The same concept can be applied to N -dimensional parameter integrals over polynomials raisedto some power, as for example the functions F and U appearing in loop integrals, where theprocedure in general has to be iterated to achieve complete factorisation. It also can be appliedto phase space integrals, where (multiple) soft/collinear limits are regulated by dimensionalregularisation.In the case of multi-loop integrals, it is convenient to integrate out the -function constrainingthe sum of all Feynman parameters xi in a special way, such as to preserve the feature that

23

singularities only occur for xi 0 and still having integration limits from 0 to 1: We decomposethe parameter integration range for the N -propagator integral into N sectors, where in eachsector l, xl is larger than all other Feynman parameters (note that the remaining xj 6=l are notfurther ordered), using the identity

0

(Nj=1

d xi

)=

Nl=1

0

(Nj=1

d xi

)Nj=1

j 6=l

(xl xj) . (67)

This is called primary sector decomposition. The integral is now split into N domains cor-responding to N integrals Gl from which we extract a common factor: G = (1)N(N LD/2)

Nl=1Gl. In the integrals Gl we substitute

xj =

xltj for j < lxl for j = lxltj1 for j > l

(68)

and then integrate out xl using the -function. As U ,F are homogeneous of degree L,L + 1,respectively, and xl factorises completely, we have U(~x) Ul(~t ) xLl and F(~x) Fl(~t ) xL+1land thus, using

dxl/xl (1 xl(1 +

N1k=1 tk)) = 1, we obtain

Gl =

10

N1j=1

dtj tj1j

UN(L+1)D/2l (~t )FNLD/2l (~t )

, l = 1, . . . , N . (69)

The primary sector decomposition is in general not sufficient to achieve complete factorisation.Therefore the decomposition into sectors where the Feynman parameters go to zero in anordered way usually has to be iterated.

Iteration

Starting from Eq. (69) we repeat the following steps until a complete separation of overlappingregions is achieved.

II.1: Determine a minimal set of parameters, say S = {t1 , . . . , tr}, such that Ul, respectivelyFl, vanish if the parameters of S are set to zero. S is in general not unique, and thereis no general prescription which defines what set to choose in order to achieve a minimalnumber of iterations. Strategies to choose S such that the algorithm is guaranteed tostop are given in [39, 41, 43, 42]. Using these strategies however in general leads to alarger number of iterations than heuristic strategies to avoid infinite loops, described inmore detail below.

II.2: Decompose the corresponding r-cube into r subsectors by decomposing unity accordingto

rj=1

(1 tj 0) =r

k=1

rj=1

j 6=k

(tk tj 0) . (70)

24

II.3: Remap the variables to the unit hypercube in each new subsector by the substitution

tj {

tktj for j 6= ktk for j = k .

(71)

This gives a Jacobian factor of tr1k . By construction tk factorises from at least one ofthe functions Ul, Fl. The resulting subsector integrals have the general form

Glk =

10

(N1j=1

dtj tajbjj

)UN(L+1)D/2lkFNLD/2lk

, k = 1, . . . , r . (72)

For each subsector the above steps have to be repeated as long as a set S can be found suchthat one of the functions Ul... or Fl... vanishes if the elements of S are set to zero. This waynew subsectors are created in each subsector of the previous iteration, resulting in a tree-likestructure after a certain number of iterations. The iteration stops if the functions Ulk1k2... orFlk1k2... contain a constant term, i.e. if they are of the form

Ulk1k2... = 1 + u(~t ) (73)Flk1k2... = s0 +

(s)f(~t ) ,

where u(~t ) and f(~t ) are polynomials in the variables tj (without a constant term), and s arekinematic invariants defined by the cuts of the diagram as explained above, or internal masses.Thus, after a certain number of iterations, each integral Gl is split into a certain number, say, of subsector integrals, which are of the same form as in Eq. (72).Evidently the singular behaviour of the integrand now can be read off directly from the ex-ponents aj , bj for a given subsector integral. As the singular behaviour is manifestly non-overlapping now, it is straightforward to define subtractions.

Extraction of the poles

The subtraction of the poles can be done implicitly by expanding the singular factors intodistributions, or explicitly by direct integration over the singular factors. In any case, thefollowing procedure has to be worked through for each variable tj=1,...,N1 and each subsectorintegrand:

Let us consider Eq. (72) for a particular tj, i.e. let us focus on

Ij =

10

dtj t(ajbj)j I(tj , {ti6=j}, ) , (74)

where I = UN(L+1)D/2lk /FNLD/2lk in a particular subsector. If aj > 1, the integrationdoes not lead to an pole. In this case no subtraction is needed and one can go to the

25

next variable tj+1. If aj 1, one expands I(tj , {ti6=j}, ) into a Taylor series aroundtj = 0:

I(tj , {ti6=j}, ) =|aj |1p=0

I(p)j (0, {ti6=j}, )tpjp!

+R(~t, ) , where

I(p)j (0, {ti6=j}, ) = pI(tj , {ti6=j}, )/tpjtj=0

. (75)

Now the pole part can be extracted easily, and one obtains

Ij =

|aj |1p=0

1

aj + p+ 1 bjI(p)j (0, {ti6=j}, )

p!+

10

dtj tajbjj R(~t, ) . (76)

By construction, the integral containing the remainder term R(~t, ) does not producepoles in upon tj-integration anymore. For aj = 1, which is the generic case forrenormalisable theories (logarithmic divergence), this simply amounts to

Ij = 1bj

Ij(0, {ti6=j}, ) +1

0

dtj t1bjj

(I(tj , {ti6=j}, ) Ij(0, {ti6=j}, )

),

which is equivalent to applying the plus prescription (see eq. (66)), except that theintegrations over the singular factors have been carried out explicitly. Since, as long asj < N 1, the expression (76) still contains an overall factor taj+1 bj+1j+1 , it is of the sameform as (74) for j j + 1 and the same steps as above can be applied.

After N 1 steps all poles are extracted, such that the resulting expression can be expandedin . This defines a Laurent series in with coefficients Clk,m for each of the (l) subsectorintegrals Glk. Since each loop can contribute at most one soft and collinear 1/

2 term, thehighest possible infrared pole of an Lloop graph is 1/2L. Expanding to order r, one has

Glk =2L

m=r

Clk,mm

+O(r+1) , G = (1)N(N LD/2)Nl=1

(l)k=1

Glk . (77)

Following the steps outlined above one has generated a regular integral representation of thecoefficients Clk,m, consisting of (N 1 m)dimensional finite integrals over parameters tj .We recall that F was non-negative in the Euclidean region where all invariants are negative(see eqs. (53,73)), such that the numerical integrations over the finite parameter integrals arestraightforward in this region. In principle, it is also possible to do at least part of theseparameter integrals analytically, but in most applications such an analytical approach reachesits limits very quickly.

A review on sector decomposition can be found in ref. [10]. Public programs for the calculationof loop integrals are also available [39, 40].

26

A Appendix

A.1 Useful formulae

(x) =

0

et tx1dt (A.1)

x(x) = (x+ 1)

(x)(x+1

2) =

(2x) 212x

(1

2) =

(1 + ) = exp

(E+

n=2

(1)nn

nn

), n =

j=1

1

jn.

0

d(sin )D =

(D+12)

(D2+ 1)

(A.2)

dDl

iD2

(l2)r

[l2 R2 + i]N = (1)N+r(r +

D2)(N r D

2)

(D2)(N)

[R2 i]rN+D2 (A.3)

dDl

iD2

l1 . . . l2m

[l2 R2 + i]N

= (1)N[(g..)m

]{1...2m}(12

)m (N D+2m2

)

(N)

(R2 i)N+(D+2m)/2 , (A.4)

d2m2k

im(k2)f(k(2m), k

2) = (1)( )()

d2m+22k

im+f(k2m) , m integer . (A.5)

A.2 Multi-loop tensor integrals

A general L-loop, rank R Feynman diagram with N propagators can be written as

G[1 . . . R] =

dk1 . . . dkL

k1(l1)l1

. . . kR(lR)lR

Nj=1

Pjj ({k}, {p}, m2j)

(A.6)

where l1, . . . , lR {1, . . . , L} and P jj ({k}, {p}, m2j) are the propagators to the power j .Introducing Feynman parameters, the integral can be expressed in terms of a symmetric (LL)matrix M , an L-vector Q (with 4-vectors in each component) and a scalar function J . The

27

contraction of Lorentz indices is indicated by a dot.

G[1 . . . R] =(N)Nj=1 (j)

Nj=1

dxj xj1j (1

Ni=1

xi)

dk1 . . . dkL

k1(l1)l1

. . . kR(lR)lR

[L

j,l=1

kj klMjl 2Lj=1

kj Qj + J]N

(A.7)

N =Nj=1

j

The factors of ki(l)l in the numerators can be generated from G[1] by partial differentiation

with with respect to Qi(l)l , where l {1, . . . , L} denotes the lth component of the l-vector

Q, corresponding to the lth loop momentum. Therefore it is convenient to define the doubleindices i = (l, i(l)) , l {1, . . . , L}, i {1, . . . , R} denoting the ith Lorentz index, belongingto the lth loop momentum. After having shifted the loop momenta to obtain a quadratic formand differentiated with respect to Q

i(l)l , one obtains after momentum integration the following

parameter representation:

G[1 . . . R] = (1)N 1Nj=1 (j)

0

Nj=1

dxj xj1j (1

Nl=1

xl)

R/2m=0

(12)m(N m LD/2)

[(M1 g)(m)(ls)(R2m)

]1,...,R U

N(L+1)D/2R

FNLD/2m (A.8)

where

F(~x) = det(M)[

Lj,l=1

Qj QlM1jl J]

(A.9)

U(~x) = det(M)M1 = UM1

ls = M1 Q

and R/2 denotes the nearest integer less or equal to R/2. The expression[(M1 g)(m)(ls)(R2m)]1,...,R stands for the sum over all different combinations of R (double-)indices distributed to m metric tensors and (R 2m) vectors ls. The above expression is wellknown, for more details the reader is referred to the literature [3, 4].

28

A.3 Exercises

Problem 1: Higher Dimensional Integrals

To see how the higher dimensional integrals ID+2mN , associated with metric tensors (g..)m, arise

in eq. (20), calculate the simplest non-trivial subpart of eq. (19), a rank two tensor, involvingtwo loop momenta in the numerator:

L12N = (N)

0

Ni=1

dzi (1Nl=1

zl)

dDl

iD2

l1 l2[l2 R2 + i]N .

Solution:As there is no dimensionful object in the integral which could carry the Lorentz structure, itmust be proportional to the metric tensor:

L12N = (N)

0

Ni=1

dzi (1Nl=1

zl)

dl l1 l2[l2 R2]N (A.10)

= K g12

Contracting both sides of eq. (A.10) with g12 , we obtain

g12L12N = (N)

0

Ni=1

dzi (1Nl=1

zl)

dl l2[l2 R2]N = KD (A.11)

= (N)

0

Ni=1

dzi (1Nl=1

zl)

dl{[l2 R2]N+1 +R2 [l2 R2]N}

Now remember the formula for the scalar case:

IDN (S) = (N)

0

Ni=1

dzi (1Nl=1

zl)

dl[l2 R2]N

= (1)N(N D2)

0

Ni=1

dzi (1Nl=1

zl)[R2]D/2N

(A.12)

We see that it can be applied as well to the first term in eq. (A.11) with N N 1. Weobtain:

g12L12N = (1)N1

(N)

(N 1)(N 1D/2) 0

Ni=1

dzi (1Nl=1

zl)[R2]D/2N+1

+ (1)N(N D2)

0

Ni=1

dzi (1Nl=1

zl)[R2]D/2N+1

= (1)N(N D + 22

)

Ni=1

dzi (1Nl=1

zl)[R2](D+2)/2N {(N 1) +N 1D/2}

= D2ID+2N (S) (A.13)

Hence we find K = 12ID+2N (S).

29

Problem 2: k-Integrals

Show that the effect of (k2) in the numerator is to formally shift the integration from D toD + 2 dimensions, i.e. derive eq. (26).

Solution:We use k2(D) = k

2(4) + k

2(2). k and k live in orthogonal spaces. The external vectors live in

four dimensions (i.e. we use the t Hooft-Veltman scheme). Hence all vectors k which arecontracted with external (i.e. 4-dim) vectors will be projected to zero. Therefore, the integralswe encounter after dimension splitting are of the form

ID,r,s,N =

dDk

iD2

k1 . . . kr (k2) (k2)sNi=1(q

2i m2i + i)

. (A.14)

The factors of k in the numerator are entirely in four dimensions and therefore irrelevant tothe treatment of the k-part. Hence we only need to consider the integral

ID,N =

dDk

iD2

(k2)Ni=1(q

2i m2i + i)

. (A.15)

After Feynman parametrisation as usual and after the substitution k = lQ k2 = l2 + l2 2Q l +Q2 , k2 = l2 (as Q lives in 4 dimensions), we obtain

ID,N = (N)

0

Ni=1

dzi (1Nl=1

zl)

d4l

i2d(D4) l

D22

(l2)

[l2 + l2 R2 + i

]N(A.16)

After Wick rotation we can define polar coordinates with radial components = |lE|2, t = |lE|2and obtain (note that d(D4)lE =

12tD6

2 dt and that we use the convention l2 = l2E)

ID,N =1

4(1)N+(N)V (4)V (D 4

2)

0

Ni=1

dzi (1Nl=1

zl)

0

d

2

0

dt

D22

tD6

2+

[+ t+R2 i]N= (1)N+ (N)

(D42)

0

Ni=1

dzi (1Nl=1

zl)

0

d

0

dt tD6

2+

[+ t+R2 i]N (A.17)The integrals over and t can be mapped to the Euler Beta-function (see lecture). Doing first

30

the -integral (subst. v = /(t+R2)) and then the t-integral leads to

ID,N = (1)N+(N 2)(D4

2)

0

Ni=1

dzi (1Nl=1

zl)

0

dt tD6

2+

[t+R2 i]N+2

= (1)N+(D2 2 + )(N D

2 )

(D42)

0

Ni=1

dzi (1Nl=1

zl)[R2 i]D2 +N

= (1)(D2 2 + )

(D42)

ID+2N , (A.18)

where ID+2N is of the form of an ordinary scalar integral in D+2 dimensions, see eq.(A.12).

Problem 3: Generalized Unitarity

43

12 5

Figure 3: Box integral with propagator 1 pinched, p212 6= 0.

Using quadruple cuts, compute the coefficient of a box integral occurring in the pure Yang-Mills theory amplitude A1loop5 (1

, 2, 3+, 4+, 5+), shown in figure 1. The integral is given by(pij = pi + pj , i terms are implicit)

ID4 (S \ {1}) =

dl1

l2(l + p12)2(l + p123)2(l p5)2 . (A.19)

Solution:See Z. Bern, L. J. Dixon and D. A. Kosower, On-Shell Methods in Perturbative QCD, AnnalsPhys. 322 (2007) 1587 [arXiv:0704.2798 [hep-ph]], section 4.4.

Problem 4: Kirchhoff rules for multi-loop graphs

p

1

4

3

2

5

31

Determine the functions F and U for the graph shown in figure 2 using the topological cuttingrules.

Solution:

U = (x1 + x2)(x3 + x4) + x5 (x1 + x2 + x3 + x4)F = (p2) {x1 x2 (x3 + x4 + x5) + x3 x4 (x1 + x2 + x5) + x5 (x1 x4 + x2 x3)}

Problem 5: Sector decomposition

m

m

1

2

3

Using sector decomposition, factorize the singularities of the two-loop vacuum bubble graphwith two massive propagators (see figure)

G =

dk dq

1

[k2 m2 + i] [(q k)2 + i] [q2 m2 + i] .

Solution:

G =

dk dq

1

[k2 m2 + i] [(q k)2 + i] [q2 m2 + i] (A.20)

= (3D) (m2)12 0

3i=1

dxi (13

l=1

xl) (x1 + x3)12 (x1x2 + x1x3 + x2x3)

2+

Now we split the integration domain into three parts and eliminate the distribution in sucha way that the remaining integrations are from 0 to 1 (primary sector decomposition). In ourexample, this means

0

dx1 dx2 dx3 =

0

dx1 dx2 dx3 [ (x1 x2)(x1 x3)+ (x2 x1)(x2 x3)+ (x3 x1)(x3 x2) ] .

Our integral is now split into 3 domains corresponding to 3 integrals Gl from which we extracta common factor: G = (3D)3l=1Gl. In the integrals Gl we substitute

xj =

{xltj for j 6= lxl for j = l

(A.21)

32

and then integrate out xl using the distribution. As U ,F are homogeneous of degree L,L+1,respectively, and xl always factorises completely, and we have U(~x) Ul(~t ) xLl and F(~x) Fl(~t ) xL+1l . Thus, using

dxl/xl (1 xl(1 +

N1k=1 tk)) = 1, we obtain

G1 = =

10

dt2dt3 (1 + t3)12(t2 + t3 + t2 t3)

2+

G2 = =

10

dt1dt3 (t1 + t3)12(t1 + t3 + t1 t3)

2+

G3 = G1 with t1 t3Now we iterate the procedure until the polynomials in the Feynman parameters (which are thefunctions F and U in terms of the new variables) are of the form constant plus polynomial inthe ti. For example, in G1, we decompose in the variables t2, t3:

G1 = =

10

dt2dt3 (1 + t3)12(t2 + t3 + t2 t3)

2+

(t2 t3)

(a)

+ (t3 t2) (b)

Subst. t3 = t2 t3 in (a) ,

t2 = t3 t2 in (b)

G(a)1 = =

10

dt2dt3 t1+2 (1 + t2t3)

12(1 + t3 + t2 t3)2+ (A.22)

G(b)1 = =

10

dt2dt3 t1+3 (1 + t3)

12(1 + t2 + t2 t3)2+ . (A.23)

We see that the singularities have been factored out, residing now simply in factors like t1+2 ,while the remaining polynomials are finite in the limit ti 0.We apply the same procedure to G2 and G3. The final result for each pole coefficient will be asum of finite parameter integrals stemming from the endpoints of the decomposition tree.

Problem 6: Integration by Parts

p

1

4

3

2

5

The integral for the graph shown above with massless propagators and general propagatorpowers is given by (i dropped)

F (1, . . . , 5) =

dk

dl

1

[k2]1 [(k p)2]2 [l2]3 [(l p)2]4 [(l k)2]5 . (A.24)

33

Use the integration-by-parts identitydk

dl

1

[l2]3[(l p)2]4

k

(k l

[k2]1[(k p)2]2 [(l k)2]5)= 0

to express F (1, . . . , 1) in terms of integrals where one of the i is zero.

Disclaimer: The reference list is far from complete.

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