Introduction to SecDec Stephen Jones With SecDec collaboration: S. - PowerPoint PPT Presentation

9 February 2016 HiggsTools Journal Club XI Introduction to SecDec Stephen Jones With SecDec collaboration: S. Borowka, G. Heinrich, S. Jahn, M. Kerner, J. Schlenk, T. Zirke

An Apology (First Half) Apology to Theorists: Talk will be slow, basic and will skip a lot of very important details and steps (Second Half) Apology to Experimentalists: Talk will get technical Don’t worry at the end I’ll introduce a tool that handles all the book-keeping. 2

Content Part 1 • From Cross-sections to Amplitudes • Feynman Rules • Loops ↔ Integrals • Dimensional Regularisation Part 2 • Feynman Parameters • Graph Polynomials • Sector Decomposition Part 3 • SecDec Demo (Implements all of the above) 3

Schematics Total CS Order σ F = σ (0) + σ (1) + . . . F F Final State Z 1 Z 1 Z σ (0) X σ (0) = d x i d x j f i ( x i ) f j ( x j ) dˆ m F 0 0 m i,j Z 1 Z 1 Z � Z σ (1) σ (0) X σ (1) = d x i d x j f i ( x i ) f j ( x j ) dˆ m + dˆ m +1 F 0 0 m +1 m i,j 4

Schematics Total CS Order σ F = σ (0) + σ (1) + . . . Phase Space Integral F F (Differential) PDFs Final State Partonic CS Z 1 Z 1 Z σ (0) X σ (0) = d x i d x j f i ( x i ) f j ( x j ) dˆ # legs m F 0 0 m i,j Z 1 Z 1 Z � Z σ (1) σ (0) X σ (1) = d x i d x j f i ( x i ) f j ( x j ) dˆ m + dˆ m +1 F 0 0 m +1 m i,j 4

Schematics Total CS Order σ F = σ (0) + σ (1) + . . . Phase Space Integral F F (Differential) PDFs Final State Partonic CS Z 1 Z 1 Z σ (0) X σ (0) = d x i d x j f i ( x i ) f j ( x j ) dˆ # legs m F 0 0 m i,j Higher Order More Legs Z 1 Z 1 Z � Z σ (1) σ (0) X σ (1) = d x i d x j f i ( x i ) f j ( x j ) dˆ m + dˆ m +1 F 0 0 m +1 m i,j ``Virtuals’’ ``Reals’’ 4

Schematics (II) (Differential) Partonic CS Phase Space Measure Average/Sum σ (0) m = d Φ m h M (0) m M (0) † (Initial/Final) m i dˆ Spin & Colour σ (0) m +1 = d Φ m +1 h M (0) m +1 M (0) † m +1 i dˆ σ (1) m = d Φ m h M (1) m M (0) † + M (1) † m M (0) m i dˆ m Amplitude 5

Feynman Rules Feynman rules allow us to compute an amplitude , , as an M expansion in the coupling, : g Vertex (3-point) Vertex (4-point) Propagator k 4 k k 1 k 3 k 3 k 1 Z ∞ d 4 k 1 ( k 2 − m 2 + i δ ) (2 π ) 4 k 2 −∞ k 2 g δ (4) ( k 1 + k 2 + k 3 ) g 2 δ (4) ( k 1 + k 2 + k 3 + k 4 ) Corresponds to summing over intermediate states Propagators increment # integrations, Vertices decrement Feynman diagram: `Glue’ these pictures together and `factor out’ a delta function for overall momentum conservation 6

Loops & Integrals I = # internal lines, V = # vertices Count the number of unconstrained momenta and call this number L I = 1 I = 2 V = 2 V = 2 L = 0 L = 1 I = 5 I = 6 V = 4 V = 4 L = 2 L = Overall momentum conservation Generally: , We define, to be # loops L = I − ( V − 1) L # Loops ≡ # Unconstrained Momenta ↔ # of Integrations 7

Loops & Integrals I = # internal lines, V = # vertices Count the number of unconstrained momenta and call this number L I = 1 I = 2 V = 2 V = 2 L = 0 L = 1 I = 5 I = 6 V = 4 V = 4 L = 2 L = 3 Overall momentum conservation Generally: , We define, to be # loops L = I − ( V − 1) L # Loops ≡ # Unconstrained Momenta ↔ # of Integrations 7

Constructing Integrals Finding all the integrals ⇒ compute the diagram Nevertheless, can see the denominator of integrals immediately: k p 1 p 4 D 1 = k 2 − m 2 d 4 k 1 D 2 = ( k + p 1 ) 2 − m 2 Z k + p 1 k + p 1 + p 2 − p 3 ∼ D 3 = ( k + p 1 + p 2 ) 2 − m 2 (2 π ) 4 D 1 D 2 D 3 D 4 D 4 = ( k + p 1 + p 2 − p 3 ) 2 − m 2 p 2 p 3 k + p 1 + p 2 d 4 k 1 d 4 k 2 1 Z Z ∼ (2 π ) 4 (2 π ) 4 P 1 P 2 P 3 P 4 P 5 P 6 k 2 k 1 8

Computing Integrals Z ∞ d 4 k 1 k ∼ ( k 2 − m 2 + i δ ) (2 π ) 4 −∞ Z ∞ 1 − i Z d r r 3 Problem: = d Ω 3 ( r 2 + m 2 − i δ ) (2 π ) 4 0 This integral Z ∞ 2 π 2 1 − i d r r 3 = is divergent! ( r 2 + m 2 − i δ ) (2 π ) 4 Γ (2) 0 2 π 2 1 − i r 2 − m 2 ln( r 2 + m 2 ) nonsense ⇤ ∞ ⇥ ∼ (2 π ) 4 0 Γ (2) 2 There are many ways out of this problem! Note: If measure was then for this integral would be finite, d D k D < 2 this observation led to Dimensional Regularisation Aside: Divergence from , called an ultraviolet (UV) divergence | k µ | → ∞ 9

Dimensional Regularisation Dim. Reg. is the current ``standard’’ in perturbation theory. Key Ideas: • Treat number of space-time dimensions ( D = 4 − 2 ✏ ) ∈ C • Reformulate entire QFT in dimensions (start from ) L D • Use to regulate UV, use to regulate infrared (IR) D < 4 D > 4 • Physical observables for are obtained by (analytic D = 4 D → 4 continuation) not always 't Hooft, Veltman 72 easy ( γ 5 ) For this to be consistent we require (1) uniqueness , (2) existence and we need to know (3) properties (linearity, scaling, translation invar.) Recommended: J. Collins, Renormalization See textbook 10

Computing Integrals (Revisited) d D k 1 Z ∼ I = ( k 2 − m 2 + i δ ) k (2 π ) D Z ∞ 1 − i Z d r r D − 1 = d Ω D − 1 ( r 2 + m 2 − i δ ) (2 π ) D 0 Z ∞ D 2 π 1 − i 2 d r r D − 1 = ( r 2 + m 2 − i δ ) Γ ( D (2 π ) D 2 ) 0 Substitute: r 2 = y ( m 2 − i δ ) D Z ∞ 2 π 1 − i 2 D D 2( m 2 − i δ ) 2 − 1 2 − 1 ( y + 1) − 1 I = d y y (2 π ) D Γ ( D 2 ) 0 Euler Beta Function D 2 π 1 ✓ D ◆ − i 2 , 1 − D 2 D 2( m 2 − i δ ) 2 − 1 B = Re( D/ 2) > 0 Γ ( D (2 π ) D 2 2 ) Re(1 − D/ 2) > 0 D 2 − 1 Γ ( D 2 ) Γ (1 − D 2 ) 2 π 1 − i 2 D 2( m 2 − i δ ) = Γ ( D (2 π ) D Γ (1) 2 ) Our problem is solved! How do we do more complicated integrals? 11

Part 2 There are many ways of computing Feynman integrals! What follows is one specific approach. 12

Conventions Loop integral: L loops L 1 Z Y d D k l ⇥ ⇤ G = Q N j =1 P ν j j ( { k } , { p } , m 2 j ) l =1 N propagators = µ 4 − D d D k l 2 d D k l ⇥ ⇤ D i π Propagator: Mass Important: + P j ( { k } , { p } , m 2 j ) = ( q 2 j − m 2 j + i δ ) External Linear combination of loop/ Loop momenta external momenta momenta 13

Feynman Parameterization Previous integral was easy due to spherical symmetry ! Feynman parameterization is one way to cast all loop integrals into this form. Notice that: Z 1 1 du AB = [ uA + (1 − u ) B ] 2 0 Or more generally: N N Z ∞ 1 Γ ( N ν ) 1 d x j x ν j − 1 Y X = δ (1 − x i ) j Q N Q N i N ν j =1 P ν j j =1 Γ ( ν j ) hP N 0 j =1 x j P j j j =1 i =1 Feynman Parameters Product Sum N ν = ν 1 + . . . + ν N 14

Feynman Parameterization (II) Feynman parameterizing our loop integral: L N N Z ∞ Z ∞ 1 Γ ( N ν ) Y Y d x j x ν j − 1 X d D k l ⇥ ⇤ G = = δ (1 − x i ) j Q N Q N j =1 P ν j j =1 Γ ( ν j ) 0 −∞ j j =1 i =1 l =1 − N ν 2 3 L L L Z ∞ Y X X d D k l k T k T ⇥ ⇤ i M ij k j − 2 j · Q j + J + i δ × 4 5 −∞ i,j =1 j =1 l =1 From quadratic (in k) Linear (in k) terms terms of propagators Key Point: In this form we can shift k to eliminate linear terms (obtain spherical symmetry) then do the momentum integrals! 15

Feynman Parameterization (III) After integration over momenta we obtain: Master Formula N N Z ∞ x i ) U N ν − ( L +1) D/ 2 ( ~ G = ( − 1) N ν Γ ( N ν − LD/ 2) x ) d x j x ν j − 1 Y X � (1 − j Q N F N ν − LD/ 2 ( ~ x, s ij ) j =1 Γ ( ⌫ j ) 0 j =1 i =1 Graph Polynomials: 1st Symanzik Polynomial: U ( ~ x ) = det( M ) 2 3 L 2nd Symanzik Polynomial: X Q i M − 1 F ( ~ x, s ij ) = det( M ) ij Q j − J − i � 4 5 i,j =1 We have exchanged momentum integrals for parameter integrals L N Maybe this looks complicated… but wait! 16

Graph Polynomials Properties: • Homogenous polynomials in the Feynman Parameters is degree U ( ~ x ) L is degree F ( ~ x, s ij ) L + 1 x ) P N Internal masses i =1 x i m 2 F ( ~ x, s ij ) = F 0 ( ~ x, s ij ) + U ( ~ i • and are linear in each Feynman Parameter U ( ~ x ) F 0 ( ~ x, s ij ) and can be constructed graphically F 0 ( ~ x, s ij ) U ( ~ x ) We will follow: Bogner, Weinzierl 10 17

Constructing U Draw graph, label edges with Feynman Parameters q 1 q 2 x 1 x 4 Rules for : U ( ~ x ) x 5 q 5 1. Delete edges all possible ways L x 2 x 3 q 4 q 3 2. Throw away disconnected graphs or graphs with L 6 = 0 3. Sum monomials of Feynman parameters of deleted edges q 1 q 2 q 5 q 4 q 3 U ( ~ x ) = 18

Constructing U Draw graph, label edges with Feynman Parameters q 1 q 2 x 1 x 4 Rules for : U ( ~ x ) x 5 q 5 1. Delete edges all possible ways L x 2 x 3 q 4 q 3 2. Throw away disconnected graphs or graphs with L 6 = 0 3. Sum monomials of Feynman parameters of deleted edges Discon. Loop q 1 q 2 q 5 q 4 q 3 U ( ~ x ) = 18

Constructing U Draw graph, label edges with Feynman Parameters q 1 q 2 x 1 x 4 Rules for : U ( ~ x ) x 5 q 5 1. Delete edges all possible ways L x 2 x 3 q 4 q 3 2. Throw away disconnected graphs or graphs with L 6 = 0 3. Sum monomials of Feynman parameters of deleted edges Discon. Loop q 1 q 2 q 5 q 4 q 3 U ( ~ x ) = 18