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Introduction to Loop Calculations Contents Introduction to Loop Calculations May 2010 Contents 1 One-loop integrals 2 1.1 Dimensionalregularisation . ....... 2 1.2 Feynmanparameters ............................... .. 3 1.3 Momentumintegration . .. .. .. ... 3 1.4 Moreaboutsingularities . ..... 7 1.5 Regularisationschemes . ..... 9 1.6 Reductionofone-loopintegrals . ....... 10 1.7 Unitaritycuts................................... .. 14 2 Beyond one loop 18 2.1 Generalformofmulti-loopintegrals . ......... 18 2.2 Construction of the functions and fromtopologicalrules. 18 2.3 ReductiontomasterintegralsF . .U . ...... 19 2.4 Calculationofmasterintegrals. ........ 20 2.4.1 Mellin-Barnes representation . ...... 21 2.4.2 Sectordecomposition . .. 22 A Appendix 27 A.1 Usefulformulae .................................. .. 27 A.2 Multi-looptensorintegrals . ....... 27 A.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 loop a momentum, the propagators are qa = k + ra, where ra = i=1 pi. If we define all momenta as incoming, momentum conservation implies N p = 0 and hence r = 0. i=1 i P N P p2 p1 pN pN−1 If the vertices in the diagram above are non-scalar, this diagram will contain a Lorentz tensor structure in the numerator, leading to tensor integrals of the form ∞ D µ1 µr D, µ1...µr d k k ...k IN (S) = D 2 2 , (1) −∞ iπ 2 (qi mi + iδ) Z i∈S − but we will first consider the scalar integral only, i.e.Q the case where the numerator is equal to one. S is the set of propagator labels, which can be used to characterise the integral, in our D example S = 1,...,N . We use the integration measure dDk/iπ 2 dk¯ to avoid ubiquitous D { } ≡ factors of iπ 2 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 for 2 2 k (ultraviolet divergences) or for qi mi 0 (infrared divergences) and therefore need a 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 Bollini and 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 2ǫ space-time dimensions. Divergences for D 4 will thus appear as poles in 1/ǫ. − → 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. Ultraviolet divergences occur if the loop momentum k , so in general the UV behaviour becomes bet- ter for ǫ> 0, while the IR behaviour becomes→∞ better for ǫ< 0. Certainly we cannot have D < 4 and D > 4 at the same time. What is formally done is to first assume the IR divergences are regulated in some other way, e.g. by assuming all external legs are off-shell or by introducing a small mass for all massless particles. Assuming ǫ > 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 show up as 1/ǫ poles. The only change to the Feynman rules to be made is to replace the couplings in the Lagrangian g gµǫ, where µ is an arbitrary mass scale. This ensures that each term in the Lagrangian has→ the correct mass dimension. 1.2 Feynman parameters 2 2 νi To combine products of denominators of the type Di = [(k + ri) mi + iδ] into one single denominator, we can use the identity − N ∞ N N 1 Γ( ν ) δ(1 zj) i=1 i νi−1 − j=1 ν1 ν2 νN = N dzi zi N (2) D D ...D Pi=1 νi 1 2 N i=1 Γ(νi) 0 i=1 [z1D1 + z2D2 + P. + zN DN ] P Z Y The integration parametersQzi are called Feynman parameters. For a generic one-loop diagram as shown above we have ν =1 i. i ∀ Simple example: one-loop two-point function k p k + p The corresponding integral is given by ∞ dDk 1 I = 2 (2π)D [k2 m2 + iδ][(k + p)2 m2 + iδ] Z−∞ − − ∞ ∞ dDk δ(1 z z ) = Γ(2) dz dz − 1 − 2 1 2 (2π)D [z (k2 m2)+ z ((k + p)2 m2)+ iδ]2 Z0 Z−∞ 1 − 2 − 1 ∞ dDk 1 = Γ(2) dz2 D 2 2 (3) 0 −∞ (2π) [k +2 k Q + A + iδ] µ µ Z Z · Q = z2 p A = z p2 m2 2 − where the δ-constraint has been used to eliminate z1. 1.3 Momentum integration Our general integral, after Feynman parametrisation, is of the following form −N ∞ N N ∞ N ID = Γ(N) dz δ(1 z ) dk¯ k2 +2k Q + z (r2 m2)+ iδ N i − l · i i − i Z0 i=1 l=1 Z−∞ " i=1 # N Y X X µ µ Q = zi ri . (4) i=1 X 3 Now we perform the shift l = k + Q to eliminate the term linear in k in the square bracket to arrive at ∞ N N ∞ ID = Γ(N) dz δ(1 z ) d¯l l2 R2 + iδ −N (5) N i − l − Z0 i=1 l=1 Z−∞ Y X The general form of R2 is N R2 = Q2 z (r2 m2) − i i − i i=1 X N 1 N N 1 N N = z z r r z (r2 m2) z z (r2 m2) z i j i · j − 2 i i − i j − 2 j j − j i i,j=1 i=1 j=1 j=1 i=1 X X X X X 1 N = z z r2 + r2 2 r r m2 m2 −2 i j i j − i · j − i − j i,j=1 X 1 N = z z −2 i j Sij i,j=1 X = (r r )2 m2 m2 (6) Sij i − j − i − j The matrix ij, sometimes also called Cayley matrix is an important quantity encoding all the kinematic dependenceS of the integral. It plays the main role in algebraic reduction as well as in the analysis of so-called Landau singularities, which are singularities where det or a sub-determinant of is vanishing (see below for more details). S S Remember that we are in Minkowski space, where l2 = l2 ~l2, so temporal and spatial com- 0 − ponents are not on equal footing. Note that the poles of the denominator are located at 2 2 ~2 ± 2 ~2 l0 = R + l iδ l0 R + l i δ. Thus the iδ term shifts the poles away from the real axis. − ⇒ ≃ ± ∓ p 2 For the integration over the loop momentum, we better work in Euclidean space where lE = 4 2 2 2 2 ~2 i=1 li . Hence we make the transformation l0 i l4, such that l lE = l4 + l , which → → − ◦ implies that the integration contour in the complex l0-plane is rotated by 90 such that the P contour in the complex l4-plane looks as shown below. The is called Wick rotation. We see that the iδ prescription is exactly such that the contour does not enclose any poles. Therefore the integral over the closed contour is zero, and we an use the identity ∞ −i∞ ∞ dl f(l )= dl f(l )= i dl f(l ) (7) 0 0 − 0 0 4 4 −∞Z iZ∞ −∞Z Re l4 Im l4 4 Our integral now reads ∞ N N ∞ D D N d lE 2 2 −N IN = ( 1) Γ(N) dzi δ(1 zl) D lE + R iδ (8) − 0 − −∞ π 2 − Z i=1 l=1 Z Y X Now we can introduce polar coordinates in D dimensions to evaluate the integral: Using 1 ∞ ∞ 4 2 D D−1 2 2 d l = drr dΩD−1 , r = lE = li (9) −∞ 0 ! Z Z Z q i=1 D X 2π 2 dΩD−1 = V (D)= (10) Γ( D ) Z 2 where V (D) is the volume of a unit sphere in D dimensions: 2π π π D−2 V (D) = dθ1 dθ2 sin θ2 . dθD−1(sin θD−1) Z0 Z0 Z0 Thus we have Γ(N) ∞ N N ∞ 1 ID = 2( 1)N dz δ(1 z ) drrD−1 N − D i − l [r2 + R2 iδ]N Γ( 2 ) 0 i=1 0 Z Y Xl=1 Z − Substituting r2 = x : ⇒ ∞ 1 1 ∞ 1 drrD−1 = dx xD/2−1 (11) [r2 + R2 iδ]N 2 [x + R2 iδ]N Z0 − Z0 − Now the x-integral can be identified as the Euler Beta-function B(a, b), defined as ∞ za−1 1 Γ(a)Γ(b) B(a, b)= dz = dyya−1(1 y)b−1 = (12) (1 + z)a+b − Γ(a + b) Z0 Z0 and after normalising with respect to R2 we finally arrive at ∞ N N D D ID = ( 1)N Γ(N ) dz δ(1 z ) R2 iδ 2 −N . (13) N − − 2 i − l − Z0 i=1 l=1 Y X The integration over the Feynman parameters remains to be done, but we will show below that for one-loop applications, the integrals we need to know explicitly have maximally N =4 external 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 “master integrals” are well-known. The most complicated analytic functions at one loop (appearing in the 4-point integrals) are dilogarithms.
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