Reduction of one-loop amplitudes at the integrand level-nlo QCD calculations

Transcription

1 Reduction of one-loop amplitudes at the integrand level-nlo QCD calculations Costas G. Papadopoulos NCSR Demokritos, Athens Epiphany 2008, Krakow, 3-6 January 2008 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

2 Outline 1 Introduction: Wishlists and Troubles 2 OPP Reduction 3 Numerical Tests 4-photon amplitudes 6-photon amplitudes ZZZ production Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

3 Introduction: LHC needs NLO The experimental programs of LHC require high precision predictions for multi-particle processes (also ILC of course) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

4 Introduction: LHC needs NLO The experimental programs of LHC require high precision predictions for multi-particle processes (also ILC of course) In the last years we have seen a remarkable progress in the theoretical description of multi-particle processes at tree-order, thanks to very efficient recursive algorithms Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

5 Introduction: LHC needs NLO The experimental programs of LHC require high precision predictions for multi-particle processes (also ILC of course) In the last years we have seen a remarkable progress in the theoretical description of multi-particle processes at tree-order, thanks to very efficient recursive algorithms The current need of precision goes beyond tree order. At LHC, most analyses require at least next-to-leading order calculations (NLO) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

6 Introduction: LHC needs NLO The experimental programs of LHC require high precision predictions for multi-particle processes (also ILC of course) In the last years we have seen a remarkable progress in the theoretical description of multi-particle processes at tree-order, thanks to very efficient recursive algorithms The current need of precision goes beyond tree order. At LHC, most analyses require at least next-to-leading order calculations (NLO) As a result, a big effort has been devoted by several groups to the problem of an efficient computation of one-loop corrections for multi-particle processes! Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

7 NLO Wishlist Les Houches [from G. Heinrich s Summary talk] Wishlist Les Houches pp V V + jet 2. pp t t b b 3. pp t t + 2 jets 4. pp W W W 5. pp V V b b 6. pp V V + 2 jets 7. pp V + 3 jets 8. pp t t b b 9. pp 4 jets Processes for which a NLO calculation is both desired and feasible Will we finish in time for LHC? Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

8 What has been done? ( ) Some recent results Cross Sections available pp Z Z Z pp t tz [Lazopoulos, Melnikov, Petriello] pp H + 2 jets [Campbell, et al., J. R. Andersen, et al.] pp VV + 2 jets via VBF [Bozzi, Jäger, Oleari, Zeppenfeld] Mostly 2 3, very few 2 4 complete calculations. e + e 4 fermions [Denner, Dittmaier, Roth] e + e H H ν ν [GRACE group (Boudjema et al.)] This is NOT a complete list (A lot of work has been done at NLO calculations & new methods) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

9 NLO troubles Problems arising in NLO calculations Large Number of Feynman diagrams Reduction to Scalar Integrals (or sets of known integrals) Numerical Instabilities (inverse Gram determinants, spurious phase-space singularities) Extraction of soft and collinear singularities (we need virtual and real corrections) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

10 Methods available - Traditional Method: Feynman Diagrams & Passarino-Veltman Reduction: general applicability major achievements but major problem: not amplitude level Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

11 Methods available - Traditional Method: Feynman Diagrams & Passarino-Veltman Reduction: - Semi-Numerical Approach (Algebraic/Partly Numerical Improved traditional) Reduction to set of well-known integrals - Numerical Approach (Numerical/Partly Algebraic) Compute tensor integrals numerically Ellis, Giele, Glover, Zanderighi; Binoth, Guillet, Heinrich, Schubert; Denner, Dittmaier; Del Aguila, Pittau; Ferroglia, Passera, Passarino, Uccirati; Nagy, Soper; van Hameren, Vollinga, Weinzierl; Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

12 Methods available - Traditional Method: Feynman Diagrams & Passarino-Veltman Reduction: - Semi-Numerical Approach (Algebraic/Partly Numerical Improved traditional) Reduction to set of well-known integrals - Numerical Approach (Numerical/Partly Algebraic) Compute tensor integrals numerically - Analytic Approach (Twistor-inspired) extract information from lower-loop, lower-point amplitudes determine scattering amplitudes by their poles and cuts major advantage: designed to amplitude level quadruple and triple cuts major simplifications Bern, Dixon, Dunbar, Kosower, Berger, Forde; Anastasiou, Britto, Cachazo, Feng, Kunszt, Mastrolia; Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

13 Methods available - Traditional Method: Feynman Diagrams & Passarino-Veltman Reduction: - Semi-Numerical Approach (Algebraic/Partly Numerical Improved traditional) Reduction to set of well-known integrals - Numerical Approach (Numerical/Partly Algebraic) Compute tensor integrals numerically - Analytic Approach (Twistor-inspired) extract information from lower-loop, lower-point amplitudes determine scattering amplitudes by their poles and cuts OPP Integrand-level reduction combine: PV@integrand + n-particle cuts Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

14 OPP Reduction - Intro G. Ossola., C. G. Papadopoulos and R. Pittau, Nucl. Phys. B 763, 147 (2007) arxiv:hep-ph/ and JHEP 0707 (2007) 085 arxiv: [hep-ph] Any m-point one-loop amplitude can be written, before integration, as A( q) = N(q) D 0 D1 D m 1 A bar denotes objects living in n = 4 + ǫ dimensions D i = ( q + p i ) 2 m 2 i q 2 = q 2 + q 2 D i = D i + q 2 External momenta p i are 4-dimensional objects Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

15 The old master formula A = + + m 1 i 0 <i 1 <i 2 <i 3 d(i 0 i 1 i 2 i 3 )D 0 (i 0 i 1 i 2 i 3 ) m 1 i 0 <i 1 <i 2 c(i 0 i 1 i 2 )C 0 (i 0 i 1 i 2 ) m 1 i 0 <i 1 b(i 0 i 1 )B 0 (i 0 i 1 ) m 1 + a(i 0 )A 0 (i 0 ) i 0 + rational terms Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

16 OPP master formula - I General expression for the 4-dim N(q) at the integrand level in terms of D i N(q) = m 1 i 0 <i 1 <i 2 <i 3 m 1 [ ] m 1 d(i 0 i 1 i 2 i 3 ) + d(q;i 0 i 1 i 2 i 3 ) D i i i 0,i 1,i 2,i 3 i 0 <i 1 <i 2 [c(i 0 i 1 i 2 ) + c(q;i 0 i 1 i 2 )] m 1 i 0 <i 1 m 1 [ b(i 0 i 1 ) + b(q;i 0 i 1 ) i 0 [a(i 0 ) + ã(q;i 0 )] m 1 ] m 1 i i 0 D i i i 0,i 1 D i m 1 i i 0,i 1,i 2 D i Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

17 OPP master formula - II N(q) = + m 1 i 0 <i 1 <i 2 <i 3 m 1 i 0 <i 1 [ b(i 0 i 1 ) [ d(i 0 i 1 i 2 i 3 ) ] m 1 i i 0,i 1 D i + ] m 1 m 1 m 1 D i + [c(i 0 i 1 i 2 ) ] D i i i 0,i 1,i 2,i 3 i 0 <i 1 <i 2 i i 0,i 1,i 2 m 1 m 1 [a(i 0 ) ] D i i 0 i i 0 The quantities d(i 0 i 1 i 2 i 3 ) are the coefficients of 4-point functions with denominators labeled by i 0, i 1, i 2, and i 3. c(i 0 i 1 i 2 ), b(i 0 i 1 ), a(i 0 ) are the coefficients of all possible 3-point, 2-point and 1-point functions, respectively. Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

18 OPP master formula - II N(q) = + m 1 i 0 <i 1 <i 2 <i 3 m 1 i 0 <i 1 [ ] m 1 m 1 m 1 d(i 0 i 1 i 2 i 3 ) + d(q; i 0 i 1 i 2 i 3 ) D i + [c(i 0 i 1 i 2 ) + c(q; i 0 i 1 i 2 )] D i i i 0,i 1,i 2,i 3 i 0 <i 1 <i 2 i i 0,i 1,i 2 [ ] m 1 m 1 m 1 b(i 0 i 1 ) + b(q; i 0 i 1 ) D i + [a(i 0 ) + ã(q; i 0 )] D i i i 0,i 1 i 0 i i 0 The quantities d, c, b, ã are the spurious terms They still depend on q (integration momentum) They should vanish upon integration What is the explicit expression of the spurious term? Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

19 Spurious Terms - I Following F. del Aguila and R. Pittau, arxiv:hep-ph/ Express any q in N(q) as q µ = p µ i=1 G i l µ i, l i 2 = 0 k 1 = l 1 + α 1 l 2, k 2 = l 2 + α 2 l 1, k i = p i p 0 l 3 µ =< l 1 γ µ l 2 ], l 4 µ =< l 2 γ µ l 1 ] The coefficients G i either reconstruct denominators D i They give rise to d, c, b, a coefficients Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

20 Spurious Terms - I Following F. del Aguila and R. Pittau, arxiv:hep-ph/ Express any q in N(q) as q µ = p µ i=1 G i l µ i, l i 2 = 0 k 1 = l 1 + α 1 l 2, k 2 = l 2 + α 2 l 1, k i = p i p 0 l 3 µ =< l 1 γ µ l 2 ], l 4 µ =< l 2 γ µ l 1 ] The coefficients G i either reconstruct denominators D i or vanish upon integration They give rise to d, c, b, a coefficients They form the spurious d, c, b, ã coefficients Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

21 Spurious Terms - II d(q) term (only 1) d(q) = d T(q), where d is a constant (does not depend on q) T(q) Tr[(/q + /p 0 )/l 1 /l 2 /k 3 γ 5 ] Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

22 Spurious Terms - II d(q) term (only 1) d(q) = d T(q), where d is a constant (does not depend on q) T(q) Tr[(/q + /p 0 )/l 1 /l 2 /k 3 γ 5 ] c(q) terms (they are 6) j max c(q) = { c1j [(q + p 0 ) l 3 ] j + c 2j [(q + p 0 ) l 4 ] j} j=1 In the renormalizable gauge, j max = 3 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

23 Spurious Terms - II d(q) term (only 1) d(q) = d T(q), where d is a constant (does not depend on q) c(q) terms (they are 6) j max T(q) Tr[(/q + /p 0 )/l 1 /l 2 /k 3 γ 5 ] c(q) = { c1j [(q + p 0 ) l 3 ] j + c 2j [(q + p 0 ) l 4 ] j} j=1 In the renormalizable gauge, j max = 3 b(q) and ã(q) give rise to 8 and 4 terms, respectively Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

24 A simple example 1 D 0 D 1 D 2 D 3 D 4 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

25 A simple example 1 D 0 D 1 D 2 D 3 D 4 1 = [d(i 0 i 1 i 2 i 3 ) + d(q;i ] 0 i 1 i 2 i 3 ) D i4 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

26 A simple example D 0 D 1 D 2 D 3 D 4 1 = [d(i 0 i 1 i 2 i 3 ) + d(q;i ] 0 i 1 i 2 i 3 ) 1 D i4 1 [ d(i 0 i 1 i 2 i 3 ) + D 0 D 1 D 2 D 3 D d(q;i ] 0 i 1 i 2 i 3 ) D i4 4 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

27 A simple example D 0 D 1 D 2 D 3 D 4 1 = [d(i 0 i 1 i 2 i 3 ) + d(q;i ] 0 i 1 i 2 i 3 ) 1 D i4 1 [ d(i 0 i 1 i 2 i 3 ) + D 0 D 1 D 2 D 3 D d(q;i ] 0 i 1 i 2 i 3 ) 4 1 = d(i 0 i 1 i 2 i 3 )D 0 (i 0 i 1 i 2 i 3 ) D 0 D 1 D 2 D 3 D 4 D i4 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

28 A simple example D 0 D 1 D 2 D 3 D 4 1 = [d(i 0 i 1 i 2 i 3 ) + d(q;i ] 0 i 1 i 2 i 3 ) 1 D i4 1 [ d(i 0 i 1 i 2 i 3 ) + D 0 D 1 D 2 D 3 D d(q;i ] 0 i 1 i 2 i 3 ) 4 1 = d(i 0 i 1 i 2 i 3 )D 0 (i 0 i 1 i 2 i 3 ) D 0 D 1 D 2 D 3 D 4 d(i 0 i 1 i 2 i 3 ) = 1 ( ) 1 2 D i4 (q + ) + 1 D i4 (q ) D i4 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

29 A simple example D 0 D 1 D 2 D 3 D 4 1 = [d(i 0 i 1 i 2 i 3 ) + d(q;i ] 0 i 1 i 2 i 3 ) 1 D i4 1 [ d(i 0 i 1 i 2 i 3 ) + D 0 D 1 D 2 D 3 D d(q;i ] 0 i 1 i 2 i 3 ) 4 1 = d(i 0 i 1 i 2 i 3 )D 0 (i 0 i 1 i 2 i 3 ) D 0 D 1 D 2 D 3 D 4 d(i 0 i 1 i 2 i 3 ) = 1 ( ) 1 2 D i4 (q + ) + 1 D i4 (q ) D i4 Melrose, Nuovo Cim. 40 (1965) 181 G. Källén, J.Toll, J. Math. Phys. 6, 299 (1965) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

30 General strategy Now we know the form of the spurious terms: N(q) = + m 1 i 0 <i 1 <i 2 <i 3 m 1 i 0 <i 1 [ ] m 1 m 1 m 1 d(i 0 i 1 i 2 i 3 ) + d(q; i 0 i 1 i 2 i 3 ) D i + [c(i 0 i 1 i 2 ) + c(q; i 0 i 1 i 2 )] D i i i 0,i 1,i 2,i 3 i 0 <i 1 <i 2 i i 0,i 1,i 2 [ ] m 1 m 1 m 1 b(i 0 i 1 ) + b(q; i 0 i 1 ) D i + [a(i 0 ) + ã(q; i 0 )] D i i i 0,i 1 i 0 i i 0 Our calculation is now reduced to an algebraic problem Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

31 General strategy Now we know the form of the spurious terms: N(q) = + m 1 i 0 <i 1 <i 2 <i 3 m 1 i 0 <i 1 [ ] m 1 m 1 m 1 d(i 0 i 1 i 2 i 3 ) + d(q; i 0 i 1 i 2 i 3 ) D i + [c(i 0 i 1 i 2 ) + c(q; i 0 i 1 i 2 )] D i i i 0,i 1,i 2,i 3 i 0 <i 1 <i 2 i i 0,i 1,i 2 [ ] m 1 m 1 m 1 b(i 0 i 1 ) + b(q; i 0 i 1 ) D i + [a(i 0 ) + ã(q; i 0 )] D i i i 0,i 1 i 0 i i 0 Our calculation is now reduced to an algebraic problem Extract all the coefficients by evaluating N(q) for a set of values of the integration momentum q Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

32 General strategy Now we know the form of the spurious terms: N(q) = + m 1 i 0 <i 1 <i 2 <i 3 m 1 i 0 <i 1 [ ] m 1 m 1 m 1 d(i 0 i 1 i 2 i 3 ) + d(q; i 0 i 1 i 2 i 3 ) D i + [c(i 0 i 1 i 2 ) + c(q; i 0 i 1 i 2 )] D i i i 0,i 1,i 2,i 3 i 0 <i 1 <i 2 i i 0,i 1,i 2 [ ] m 1 m 1 m 1 b(i 0 i 1 ) + b(q; i 0 i 1 ) D i + [a(i 0 ) + ã(q; i 0 )] D i i i 0,i 1 i 0 i i 0 Our calculation is now reduced to an algebraic problem Extract all the coefficients by evaluating N(q) for a set of values of the integration momentum q There is a very good set of such points: Use values of q for which a set of denominators D i vanish The system becomes triangular : solve first for 4-point functions, then 3-point functions and so on Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

33 Example: 4-particles process N(q) = d + d(q) [c(i) + c(q; i)]d i + i=0 3 [a(i 0 ) + ã(q; i 0 )] D i i0 D j i0 D k i0 i 0=0 3 i 0<i 1 [ b(i 0 i 1 ) + b(q; ] i 0 i 1 ) D i0 D i1 We look for a q of the form q µ = p µ 0 + x il µ i such that D 0 = D 1 = D 2 = D 3 = 0 we get a system of equations in x i that has two solutions q ± 0 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

34 Example: 4-particles process N(q) = d + d(q) Our master formula for q = q ± 0 is: N(q ± 0 ) = [d + d T(q ± 0 )] solve to extract the coefficients d and d Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

35 Example: 4-particles process N(q) d d(q) = + 3 [c(i) + c(q; i)]d i + i=0 3 i 0<i 1 3 [a(i 0 ) + ã(q; i 0 )] D i i0 D j i0 D k i0 i 0=0 [ b(i 0 i 1 ) + b(q; ] i 0 i 1 ) D i0 D i1 Then we can move to the extraction of c coefficients using N (q) = N(q) d dt(q) and setting to zero three denominators (ex: D 1 = 0, D 2 = 0, D 3 = 0) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

36 Example: 4-particles process N(q) d d(q) = [c(0) + c(q; 0)]D 0 We have infinite values of q for which D 1 = D 2 = D 3 = 0 and D 0 0 Here we need 7 of them to determine c(0) and c(q; 0) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

37 Rational Terms - I Let s go back to the integrand A( q) = N(q) D 0 D 1 D m 1 Insert the expression for N(q) we know all the coefficients N(q) = m 1 i 0<i 1<i 2<i 3 [ d + d(q) ] m 1 D i + i i 0,i 1,i 2,i 3 Finally rewrite all denominators using D i D i = Z i, with Z i m 1 i 0<i 1<i 2 [c + c(q)] (1 q2 D i ) m 1 i i 0,i 1,i 2 D i + Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

38 Rational Terms - II A( q) = m 1 d(i 0 i 1 i 2 i 3 ) + d(q; i 0 i 1 i 2 i 3 ) D i i0 Di1 Di2 Di3 0<i 1<i 2<i 3 m 1 c(i 0 i 1 i 2 ) + c(q; i 0 i 1 i 2 ) D i i0 Di1 Di2 0<i 1<i 2 m 1 b(i 0 i 1 ) + b(q; i0 i 1 ) D i i0 D i1 0<i 1 m 1 a(i 0 ) + ã(q; i 0 ) D i i0 0 m 1 m 1 i i 0 Zi i i 0,i 1 Zi m 1 i i 0,i 1,i 2 Z i m 1 i i 0,i 1,i 2,i 3 Z i The rational part is produced, after integrating over d n q, by the q 2 dependence in Z i Zi ( 1 q2 D i ) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

39 Rational Terms - III The Extra Integrals are of the form where I s;µ (n;2l) 1 µ r d n q q 2l q µ1 q µr D(k 0 ) D(k s ), D(k i ) ( q + k i ) 2 m 2 i,k i = p i p 0 These integrals: - have dimensionality D = 2(1 + l s) + r - contribute only when D 0, otherwise are of O(ǫ) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

40 Rational Terms - IV Tensor reduction iteratively leads to rank m m-point tensors with 1 m 5, that are ultraviolet divergent when m 4. For this reason, we introduced, the d-dimensional denominators D i, that differs by an amount q 2 from their 4-dimensional counterparts D i = D i + q 2. (1) The result of this is a mismatch in the cancelation of the d-dimensional denominators with the 4-dimensional ones. The rational part of the amplitude, called R 1, comes from such a lack of cancelation. A different source of Rational Terms, called R 2, can also be generated from the ǫ-dimensional part of N(q) R 2 are generated by the difference N( q) N(q) = q 2 N 1 + ǫn 2 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

41 Rational Terms - IV q µ = q µ + q µ γ µ = γ µ + γ µ q µ q 2 γ µ ǫ Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

42 Rational Terms - IV d n q q2 d n q D i Dj q 2 D i D j D k = iπ2 2 = iπ2 2 + O(ǫ), [m 2i + m 2j (p i p j ) 2 3 ] + O(ǫ), d n q q 4 D i D j D k D l = iπ2 6 + O(ǫ).(2) b(ij; q 2 ) = b(ij) + q 2 b (2) (ij), c(ijk; q 2 ) = c(ijk) + q 2 c (2) (ijk). (3) Furthermore, by defining D (m) (q, q 2 ) m 1 i 0<i 1<i 2<i 3 [ ] m 1 d(i 0 i 1 i 2 i 3 ; q 2 ) + d(q; i 0 i 1 i 2 i 3 ; q 2 ) D i, (4) i i 0,i 1,i 2,i 3 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

43 Rational Terms - IV the following expansion holds D (m) (q, q 2 ) = m q (2j 4) d (2j 4) (q), (5) j=2 where the last coefficient is independent on q d (2m 4) (q) = d (2m 4). (6) In practice, once the 4-dimensional coefficients have been determined, one can redo the fits for different values of q 2, in order to determine b (2) (ij), c (2) (ijk) and d (2m 4). Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

44 Summary Calculate N(q) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

45 Summary Calculate N(q) We do not need to repeat this for all Feynman diagrams. We can group them and solve for (sub)amplitudes directly Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

46 Summary Calculate N(q) We do not need to repeat this for all Feynman diagrams. We can group them and solve for (sub)amplitudes directly Calculate N(q) numerically via recursion relations Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

47 Summary Calculate N(q) We do not need to repeat this for all Feynman diagrams. We can group them and solve for (sub)amplitudes directly Calculate N(q) numerically via recursion relations Just specify external momenta, polarization vectors and masses and proceed with the reduction! Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

48 Summary Calculate N(q) We do not need to repeat this for all Feynman diagrams. We can group them and solve for (sub)amplitudes directly Calculate N(q) numerically via recursion relations Just specify external momenta, polarization vectors and masses and proceed with the reduction! Compute all coefficients Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

49 Summary Calculate N(q) We do not need to repeat this for all Feynman diagrams. We can group them and solve for (sub)amplitudes directly Calculate N(q) numerically via recursion relations Just specify external momenta, polarization vectors and masses and proceed with the reduction! Compute all coefficients by evaluating N(q) at certain values of integration momentum Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

50 Summary Calculate N(q) We do not need to repeat this for all Feynman diagrams. We can group them and solve for (sub)amplitudes directly Calculate N(q) numerically via recursion relations Just specify external momenta, polarization vectors and masses and proceed with the reduction! Compute all coefficients by evaluating N(q) at certain values of integration momentum Evaluate scalar integrals Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

51 Summary Calculate N(q) We do not need to repeat this for all Feynman diagrams. We can group them and solve for (sub)amplitudes directly Calculate N(q) numerically via recursion relations Just specify external momenta, polarization vectors and masses and proceed with the reduction! Compute all coefficients by evaluating N(q) at certain values of integration momentum Evaluate scalar integrals massive integrals FF [G. J. van Oldenborgh] massless integrals OneLOop [A. van Hameren] Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

52 What we gain PV: Unitarity Methods: Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

53 What we gain PV: N(q) or A(q) hasn t to be known analytically Unitarity Methods: Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

54 What we gain PV: N(q) or A(q) hasn t to be known analytically No computer algebra Unitarity Methods: Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

55 What we gain PV: N(q) or A(q) hasn t to be known analytically No computer algebra Mathematica Numerica Unitarity Methods: Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

56 What we gain PV: N(q) or A(q) hasn t to be known analytically No computer algebra Mathematica Numerica Unitarity Methods: A more transparent algebraic method Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

57 What we gain PV: N(q) or A(q) hasn t to be known analytically No computer algebra Mathematica Numerica Unitarity Methods: A more transparent algebraic method A solid way to get all rational terms Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

58 The Master Equation Properties of the master equation Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

59 The Master Equation Properties of the master equation Polynomial equation in q Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

60 The Master Equation Properties of the master equation Polynomial equation in q Highly redundant: the a-terms have a degree of m 2 2 compared to m as a function of q Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

61 The Master Equation Properties of the master equation Polynomial equation in q Highly redundant: the a-terms have a degree of m 2 2 compared to m as a function of q Zeros of (a tower of ) polynomial equations Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

62 The Master Equation Properties of the master equation Polynomial equation in q Highly redundant: the a-terms have a degree of m 2 2 compared to m as a function of q Zeros of (a tower of ) polynomial equations Different ways of solving it, besides unitarity method Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

63 The Master Equation Properties of the master equation Polynomial equation in q Highly redundant: the a-terms have a degree of m 2 2 compared to m as a function of q Zeros of (a tower of ) polynomial equations Different ways of solving it, besides unitarity method The N N test A tool to efficiently treat phase-space points with numerical instabilities Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

64 4-photon and 6-photon amplitudes As an example we present 4-photon and 6-photon amplitudes (via fermionic loop of mass m f ) Input parameters for the reduction: External momenta p i Masses of propagators in the loop Polarization vectors Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

65 4-photon and 6-photon amplitudes As an example we present 4-photon and 6-photon amplitudes (via fermionic loop of mass m f ) Input parameters for the reduction: External momenta p i in this example massless, i.e. p 2 i = 0 Masses of propagators in the loop all equal to m f Polarization vectors various helicity configurations Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

66 Four Photons Comparison with Gounaris et al. F f ++++ α 2 Q 4 f = 8 Rational Part Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

67 Four Photons Comparison with Gounaris et al. F f ++++ α 2 Q 4 f = ( 1 + 2û ŝ (ˆt 2 + û 2 8 ŝ 2 [ 4 ) ( B 0 (û) ˆt ŝ ) B 0 (ˆt) ) [ˆtC 0 (ˆt) + ûc 0 (û)] ˆt 2 + û 2 ŝ 2 ] D 0 (ˆt, û) Massless four-photon amplitudes Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

68 Four Photons Comparison with Gounaris et al. F f ++++ α 2 Q 4 f = ( 1 + 2û ŝ (ˆt 2 + û 2 8 ) ( B 0 (û) ˆt ŝ ŝ 2 4m2 f ŝ 4 [4m 4f (2ŝm2f + ˆtû)ˆt 2 + û 2 ) B 0 (ˆt) ) [ˆtC 0 (ˆt) + ûc 0 (û)] ŝ 2 + 8m 2 f (ŝ 2m 2 f )[D 0 (ŝ,ˆt) + D 0 (ŝ, û)] + 4m2ˆtû ] f D 0 (ˆt, û) ŝ Massive four-photon amplitudes Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

69 Four Photons Comparison with Gounaris et al. F f ++++ α 2 Q 4 f = ( 1 + 2û ŝ (ˆt 2 + û 2 8 ) ( B 0 (û) ˆt ŝ ŝ 2 4m2 f ŝ 4 [4m 4f (2ŝm 2f + ˆtû)ˆt 2 + û 2 ) B 0 (ˆt) ) [ˆtC 0 (ˆt) + ûc 0 (û)] ŝ 2 + 8m 2 f (ŝ 2m2 f )[D 0(ŝ,ˆt) + D 0 (ŝ, û)] + 4m2ˆtû ] f D 0 (ˆt, û) ŝ Massive four-photon amplitudes Results also checked for F f +++ and F f ++ Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

70 Six Photons Comparison with Nagy-Soper and Mahlon Massless case: [+ + ] and [+ + + ] Plot presented by Nagy and Soper hep-ph/ (also Binoth et al., hep-ph/ ) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

71 Six Photons Comparison with Nagy-Soper and Mahlon Massless case: [+ + ] and [+ + + ] Analogous plot produced with OPP reduction Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

72 Six Photons Comparison with Binoth, Heinrich, Gehrmann, Mastrolia Massless case: [+ + ] and [+ + + ] Same plot as before for a wider range of θ Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

73 Six Photons Comparison with Mahlon Massless case: [+ + ] and [+ + + ] s M /α θ Same idea for a different set of external momenta Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

74 Six Photons with Massive Fermions Massless result [Mahlon] Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

75 Six Photons with Massive Fermions Massless result [Mahlon] m = 0.5 GeV Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

76 Six Photons with Massive Fermions Massless result [Mahlon] m = 0.5 GeV m = 4.5 GeV Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

77 Six Photons with Massive Fermions Massless result [Mahlon] m = 0.5 GeV m = 4.5 GeV m = 12.0 GeV Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

78 Six Photons with Massive Fermions Massless result [Mahlon] m = 0.5 GeV m = 4.5 GeV m = 12.0 GeV m = 20.0 GeV Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

79 q q ZZZ virtual corrections A. Lazopoulos, K. Melnikov and F. Petriello, [arxiv:hep-ph/ ] Poles 1/ǫ 2 and 1/ǫ σ NLO,virt div = C F α s π ( Γ(1 + ǫ) 1 (4π) ǫ (s 12) ǫ ǫ ) σ LO 2ǫ Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

80 q q ZZZ virtual corrections A still naive implementation Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

81 q q ZZZ virtual corrections A still naive implementation Calculate the N(q) by brute (numerical) force namely multiplying gamma matrices! Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

82 q q ZZZ virtual corrections A still naive implementation Calculate the N(q) by brute (numerical) force namely multiplying gamma matrices! Calculate 4d and rational R 1 terms by CutTools Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

83 q q ZZZ virtual corrections A still naive implementation Calculate the N(q) by brute (numerical) force namely multiplying gamma matrices! Calculate 4d and rational R 1 terms by CutTools R 2 terms added by hand Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

84 q q ZZZ virtual corrections A still naive implementation Calculate the N(q) by brute (numerical) force namely multiplying gamma matrices! Calculate 4d and rational R 1 terms by CutTools R 2 terms added by hand Comparison with LMP Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

85 q q ZZZ virtual corrections A still naive implementation Calculate the N(q) by brute (numerical) force namely multiplying gamma matrices! Calculate 4d and rational R 1 terms by CutTools R 2 terms added by hand Comparison with LMP Of course full agreement for the 1/ǫ 2 and 1/ǫ terms Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

86 q q ZZZ virtual corrections A still naive implementation Calculate the N(q) by brute (numerical) force namely multiplying gamma matrices! Calculate 4d and rational R 1 terms by CutTools R 2 terms added by hand Comparison with LMP Of course full agreement for the 1/ǫ 2 and 1/ǫ terms An easy agreement for all graphs with up to 4-point loop integrals Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

87 q q ZZZ virtual corrections A still naive implementation Calculate the N(q) by brute (numerical) force namely multiplying gamma matrices! Calculate 4d and rational R 1 terms by CutTools R 2 terms added by hand Comparison with LMP Of course full agreement for the 1/ǫ 2 and 1/ǫ terms An easy agreement for all graphs with up to 4-point loop integrals A bit more work to uncover the differences in scalar function normalization that happen to show to order ǫ 2 thus influence only 5-point loop integrals. Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

88 q q ZZZ virtual corrections Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

89 q q ZZZ virtual corrections Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

90 q q ZZZ virtual corrections Typical precision: Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

91 q q ZZZ virtual corrections Typical precision: LMP: 9.573(66) about 1% error Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

92 q q ZZZ virtual corrections Typical precision: LMP: 9.573(66) about 1% error OPP: { Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

93 q q ZZZ virtual corrections Typical precision: LMP: 9.573(66) about 1% error OPP: { Typical time: 10 4 times faster (for non-singular PS-points) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

94 q q ZZZ real corrections σ NLO q q = σ NLO gq = VVV VVV [dσ B q q + dσv q q + dσc q q + [ +dσgq C g dσ A gq ] + VVVg g dσ A q q ] + VVVg [ ] dσgq R dσgq A, [ ] dσq q R dσa q q Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

95 q q ZZZ real corrections σ NLO q q = σ NLO gq = VVV VVV [dσ B q q + dσv q q + dσc q q + [ +dσgq C g dσ A gq ] + VVVg g dσ A q q ] + VVVg [ ] dσgq R dσgq A, [ ] dσq q R dσa q q D q 1g 6, q 2 = 8πα sc F 2 x p 1 p 6 ( 1 + x 2 1 x x = p 1 p 2 p 2 p 6 p 1 p 6 p 1 p 2 ) M B q q({ p}) 2 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

96 q q ZZZ real corrections σ NLO q q = σ NLO gq = VVV VVV [dσ B q q + dσv q q + dσc q q + [ +dσgq C g dσ A gq ] + VVVg g dσ A q q ] + VVVg [ ] dσgq R dσgq A, [ ] dσq q R dσa q q D q 1g 6, q 2 = 8πα sc F 2 x p 1 p 6 ( 1 + x 2 1 x x = p 1 p 2 p 2 p 6 p 1 p 6 p 1 p 2 ) M B q q({ p}) 2 dσq q R dσa q q = [ C F M R q q 6 N 2s ({p j}) 2 D q 1g 6, q 2 D q 2g 6,q 1 ]dφ VVVg 12 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

97 q q ZZZ real corrections dσq q C + g + α sc F 2π + α sc F 2π dσ A q q = α sc F 2π Γ(1 + ǫ) (4π) ǫ ( s12 µ 2 ) ǫ [ 2 ǫ ] ǫ 2π2 dσ B ({p j }) 3 dx K q q (x)dσ B (xp 1,p 2 ;p 3,p 4,p 5 ) F 0 (xp 1,p 2 ;p 3,p 4,p 5 ) dx K q q (x)dσ B (p 1,xp 2 ;p 3,p 4,p 5 ) F 0 (p 1,xp 2 ;p 3,p 4,p 5 ) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

98 q q ZZZ real corrections dσq q C + g + α sc F 2π + α sc F 2π dσ A q q = α sc F 2π Γ(1 + ǫ) (4π) ǫ ( s12 µ 2 ) ǫ [ 2 ǫ ] ǫ 2π2 dσ B ({p j }) 3 dx K q q (x)dσ B (xp 1,p 2 ;p 3,p 4,p 5 ) F 0 (xp 1,p 2 ;p 3,p 4,p 5 ) dx K q q (x)dσ B (p 1,xp 2 ;p 3,p 4,p 5 ) F 0 (p 1,xp 2 ;p 3,p 4,p 5 ) K q q (x) = ( ) 1 + x 2 log 1 x + ( s12 µ 2 F ) ( 4 log(1 x) + 1 x ) + (1 x) 2(1 + x) log(1 x) + Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

99 q q ZZZ real corrections σ NLO gq = [ ] dσgq A + dσgq C + [ ] dσgq R dσgq A V V V g V V V g Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

100 q q ZZZ real corrections σ NLO gq = [ ] dσgq A + dσgq C + [ ] dσgq R dσgq A V V V g V V V g dσ R gq dσ A gq = 1 N 1 [ ] T R M R 2s gq({p j } ) 2 F 1 ({p j } ) D g1q6,q2 F 0 ({ p j }) dφ V V V q 12 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

101 q q ZZZ real corrections σ NLO gq = [ ] dσgq A + dσgq C + [ ] dσgq R dσgq A V V V g V V V g dσ R gq dσ A gq = 1 N 1 [ ] T R M R 2s gq({p j } ) 2 F 1 ({p j } ) D g1q6,q2 F 0 ({ p j }) dφ V V V q 12 D g 1q 6,q 2 = 8πα s T R x 2 p 1 p 6 [1 2 x(1 x)] M B q q({ p j }) 2 Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

102 q q ZZZ real corrections dσgq C + q dσ A gq = α st R 2π 1 K gq (x) = [x 2 + (1 x) 2 ]log 0 dx K gq (x) dσ B (xp 1,p 2 ;p 3,p 4,p 5 ) F 0 (xp 1,p 2 ;p 3,p 4,p 5 ) ( s12 µ 2 F ) + 2x(1 x) + 2[x 2 + (1 x) 2 ]log(1 x) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

103 q q ZZZ real corrections dσgq C + q dσ A gq = α st R 2π 1 K gq (x) = [x 2 + (1 x) 2 ]log 0 dx K gq (x) dσ B (xp 1,p 2 ;p 3,p 4,p 5 ) F 0 (xp 1,p 2 ;p 3,p 4,p 5 ) ( s12 µ 2 F ) + 2x(1 x) + 2[x 2 + (1 x) 2 ]log(1 x) check also with phase-space slicing method Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

104 q q ZZZ NLO scale σ 0 σ V /σ 0 σ R σ NLO µ = M Z 1.481(5) 0.536(1) 0.238(2) 2.512(2) µ = 2M Z 1.487(5) 0.481(1) 0.232(2) 2.434(2) µ = 3M Z 1.477(5) 0.452(1) 0.232(2) 2.376(2) µ = 4M Z 1.479(5) 0.436(1) 0.232(2) 2.355(2) µ = 5M Z 1.479(5) 0.424(1) 0.237(2) 2.343(2) Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

105 q q ZZZ NLO Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

106 Outlook Reduction at the integrand level Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

107 Outlook Reduction at the integrand level changes the computational approach at one loop Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

108 Outlook Reduction at the integrand level changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

109 Outlook Reduction at the integrand level Current changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

110 Outlook Reduction at the integrand level changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Current Understand potential stability problems Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

111 Outlook Reduction at the integrand level changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Current Understand potential stability problems Combine with the real corrections Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

112 Outlook Reduction at the integrand level changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Current Understand potential stability problems Combine with the real corrections Automatize through Dyson-Schwinger equations Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

113 Outlook Reduction at the integrand level changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Current Understand potential stability problems Combine with the real corrections Automatize through Dyson-Schwinger equations A generic NLO calculator seems feasible Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43

114 Outlook Reduction at the integrand level changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Current Understand potential stability problems Combine with the real corrections Automatize through Dyson-Schwinger equations A generic NLO calculator seems feasible CUTTOOLS version 0. is ready! Costas G. Papadopoulos (Athens) OPP Reduction Epiphany / 43