Reduction of one-loop amplitudes at the integrand level Costas G. - PowerPoint PPT Presentation

Spurious Terms - I Following F. del Aguila and R. Pittau, arXiv:hep-ph/0404120 Express any q in N ( q ) as q µ = − p µ i , ℓ i 2 = 0 0 + � 4 i =1 G i ℓ µ 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 ˜ c , ˜ d , ˜ b , ˜ a coefficients Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 12 / 33

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 RADCOR 2007 13 / 33

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 1 j [( q + p 0 ) · ℓ 3 ] j + ˜ � c 2 j [( q + p 0 ) · ℓ 4 ] j � � ˜ c ( q ) = ˜ j =1 In the renormalizable gauge, j max = 3 Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 13 / 33

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 1 j [( q + p 0 ) · ℓ 3 ] j + ˜ � c 2 j [( q + p 0 ) · ℓ 4 ] j � � ˜ c ( q ) = ˜ j =1 In the renormalizable gauge, j max = 3 ˜ b (q) and ˜ a (q) give rise to 8 and 4 terms, respectively Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 13 / 33

A simple example � 1 D 0 D 1 D 2 D 3 D 4 Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 14 / 33

A simple example � 1 D 0 D 1 D 2 D 3 D 4 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ 1 = d ( q ; i 0 i 1 i 2 i 3 ) D i 4 Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 14 / 33

A simple example � 1 D 0 D 1 D 2 D 3 D 4 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ 1 = d ( q ; i 0 i 1 i 2 i 3 ) D i 4 � 1 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ d ( q ; i 0 i 1 i 2 i 3 ) D i 4 D 0 D 1 D 2 D 3 D 4 Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 14 / 33

A simple example � 1 D 0 D 1 D 2 D 3 D 4 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ 1 = d ( q ; i 0 i 1 i 2 i 3 ) D i 4 � 1 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ d ( q ; i 0 i 1 i 2 i 3 ) D i 4 D 0 D 1 D 2 D 3 D 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 Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 14 / 33

A simple example � 1 D 0 D 1 D 2 D 3 D 4 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ 1 = d ( q ; i 0 i 1 i 2 i 3 ) D i 4 � 1 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ d ( q ; i 0 i 1 i 2 i 3 ) D i 4 D 0 D 1 D 2 D 3 D 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 1 � D i 4 ( q + ) + 2 D i 4 ( q − ) Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 14 / 33

A simple example � 1 D 0 D 1 D 2 D 3 D 4 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ 1 = d ( q ; i 0 i 1 i 2 i 3 ) D i 4 � 1 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ d ( q ; i 0 i 1 i 2 i 3 ) D i 4 D 0 D 1 D 2 D 3 D 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 1 � D i 4 ( q + ) + 2 D i 4 ( q − ) Melrose, Nuovo Cim. 40 (1965) 181 G. K¨ all´ en, J.Toll, J. Math. Phys. 6, 299 (1965) Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 14 / 33

General strategy Now we know the form of the spurious terms: m − 1 m − 1 m − 1 m − 1 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ � � � N ( q ) = 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 0 < i 1 < i 2 < i 3 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 m − 1 � � b ( i 0 i 1 ) + ˜ � � � � + b ( q ; i 0 i 1 ) D i + [ a ( i 0 ) + ˜ a ( q ; i 0 )] D i i � = i 0 , i 1 i � = i 0 i 0 < i 1 i 0 Our calculation is now reduced to an algebraic problem Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 15 / 33

General strategy Now we know the form of the spurious terms: m − 1 m − 1 m − 1 m − 1 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ � � � N ( q ) = 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 0 < i 1 < i 2 < i 3 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 m − 1 � � b ( i 0 i 1 ) + ˜ � � � � + b ( q ; i 0 i 1 ) D i + [ a ( i 0 ) + ˜ a ( q ; i 0 )] D i i � = i 0 , i 1 i � = i 0 i 0 < i 1 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 RADCOR 2007 15 / 33

General strategy Now we know the form of the spurious terms: m − 1 m − 1 m − 1 m − 1 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ � � � N ( q ) = 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 0 < i 1 < i 2 < i 3 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 m − 1 � � b ( i 0 i 1 ) + ˜ � � � � + b ( q ; i 0 i 1 ) D i + [ a ( i 0 ) + ˜ a ( q ; i 0 )] D i i � = i 0 , i 1 i � = i 0 i 0 < i 1 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 RADCOR 2007 15 / 33

Example: 4-particles process 3 3 � � d + ˜ � � b ( i 0 i 1 ) + ˜ N ( q ) = d ( q ) + [ c ( i ) + ˜ c ( q ; i )] D i + b ( q ; i 0 i 1 ) D i 0 D i 1 i =0 i 0 < i 1 3 � + [ a ( i 0 ) + ˜ a ( q ; i 0 )] D i � = i 0 D j � = i 0 D k � = i 0 i 0 =0 We look for a q of the form q µ = − p µ 0 + x i ℓ 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 RADCOR 2007 16 / 33

Example: 4-particles process d + ˜ N ( q ) = d ( q ) Our “master formula” for q = q ± 0 is: 0 ) = [ d + ˜ N ( q ± d T ( q ± 0 )] → solve to extract the coefficients d and ˜ d Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 16 / 33

Example: 4-particles process 3 3 � � N ( q ) − d − ˜ � � b ( i 0 i 1 ) + ˜ d ( q ) = [ c ( i ) + ˜ c ( q ; i )] D i + b ( q ; i 0 i 1 ) D i 0 D i 1 i =0 i 0 < i 1 3 � + [ a ( i 0 ) + ˜ a ( q ; i 0 )] D i � = i 0 D j � = i 0 D k � = i 0 i 0 =0 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 RADCOR 2007 16 / 33

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 RADCOR 2007 16 / 33

Rational Terms - I Let’s go back to the integrand N ( q ) A (¯ q ) = D 0 ¯ ¯ D 1 · · · ¯ D m − 1 Insert the expression for N ( q ) → we know all the coefficients m − 1 m − 1 m − 1 m − 1 � � � d + ˜ � � � N ( q ) = d ( q ) D i + [ c + ˜ c ( q )] D i + · · · i 0 < i 1 < i 2 < i 3 i � = i 0 , i 1 , i 2 , i 3 i 0 < i 1 < i 2 i � = i 0 , i 1 , i 2 Finally rewrite all denominators using q 2 � � D i 1 − ˜ = ¯ ¯ Z i , Z i ≡ with ¯ ¯ D i D i Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 17 / 33

Rational Terms - II m − 1 m − 1 d ( i 0 i 1 i 2 i 3 ) + ˜ d ( q ; i 0 i 1 i 2 i 3 ) � � ¯ A (¯ q ) = Z i D i 0 ¯ ¯ D i 1 ¯ D i 2 ¯ D i 3 i 0 < i 1 < i 2 < i 3 i � = i 0 , i 1 , i 2 , i 3 m − 1 m − 1 c ( i 0 i 1 i 2 ) + ˜ c ( q ; i 0 i 1 i 2 ) � � ¯ + Z i D i 0 ¯ ¯ D i 1 ¯ D i 2 i � = i 0 , i 1 , i 2 i 0 < i 1 < i 2 m − 1 m − 1 b ( i 0 i 1 ) + ˜ b ( q ; i 0 i 1 ) � � ¯ + Z i D i 0 ¯ ¯ D i 1 i 0 < i 1 i � = i 0 , i 1 m − 1 m − 1 a ( i 0 ) + ˜ a ( q ; i 0 ) � � ¯ + Z i ¯ D i 0 i 0 i � = i 0 q 2 The rational part is produced, after integrating over d n q , by the ˜ dependence in ¯ Z i q 2 � 1 − ˜ � ¯ Z i ≡ ¯ D i Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 18 / 33

Rational Terms - III The “Extra Integrals” are of the form � q µ 1 · · · q µ r I ( n ;2 ℓ ) q 2 ℓ d n q ˜ s ; µ 1 ··· µ r ≡ D ( k s ) , D ( k 0 ) · · · ¯ ¯ where q + k i ) 2 − m 2 ¯ D ( k i ) ≡ (¯ 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 RADCOR 2007 19 / 33

Summary Calculate N ( q ) Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 20 / 33

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 RADCOR 2007 20 / 33

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 RADCOR 2007 20 / 33

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 RADCOR 2007 20 / 33

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 RADCOR 2007 20 / 33

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 RADCOR 2007 20 / 33

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 RADCOR 2007 20 / 33

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 RADCOR 2007 20 / 33

What we gain PV: Unitarity Methods: Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 21 / 33

What we gain PV: N ( q ) or A ( q ) hasn’t to be known analytically Unitarity Methods: Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 21 / 33

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 RADCOR 2007 21 / 33

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 RADCOR 2007 21 / 33

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 RADCOR 2007 21 / 33

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 RADCOR 2007 21 / 33

The Master Equation Properties of the master equation Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 22 / 33

The Master Equation Properties of the master equation Polynomial equation in q Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 22 / 33

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 RADCOR 2007 22 / 33

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 RADCOR 2007 22 / 33

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 RADCOR 2007 22 / 33

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 RADCOR 2007 22 / 33

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 RADCOR 2007 23 / 33

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 RADCOR 2007 23 / 33

Four Photons – Comparison with Gounaris et al. F f ++++ = − 8 α 2 Q 4 f Rational Part Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 24 / 33

Four Photons – Comparison with Gounaris et al. F f 1 + 2ˆ � 1 + 2ˆ u � � t � ++++ B 0 (ˆ = − 8 + 8 B 0 (ˆ u ) + 8 t ) α 2 Q 4 ˆ s ˆ s f t 2 + ˆ � ˆ u 2 � [ˆ tC 0 (ˆ − 8 t ) + ˆ uC 0 (ˆ u )] s 2 ˆ t 2 + ˆ ˆ u 2 � � D 0 (ˆ − 4 t , ˆ u ) s 2 ˆ Massless four-photon amplitudes Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 24 / 33

Four Photons – Comparison with Gounaris et al. F f 1 + 2ˆ � 1 + 2ˆ u � � t � ++++ B 0 (ˆ = − 8 + 8 B 0 (ˆ u ) + 8 t ) α 2 Q 4 ˆ s ˆ s f t 2 + ˆ � ˆ u 2 − 4 m 2 � f [ˆ tC 0 (ˆ − 8 t ) + ˆ uC 0 (ˆ u )] s 2 ˆ s ˆ t 2 + ˆ ˆ u 2 + 4 m 2 f ˆ � t ˆ � u 4 m 4 sm 2 f + ˆ D 0 (ˆ − 4 f − (2ˆ t ˆ u ) t , ˆ u ) s 2 ˆ s ˆ + 8 m 2 s − 2 m 2 s , ˆ f (ˆ f )[ D 0 (ˆ t ) + D 0 (ˆ s , ˆ u )] Massive four-photon amplitudes Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 24 / 33

Four Photons – Comparison with Gounaris et al. F f 1 + 2ˆ � � � � 1 + 2ˆ u t ++++ B 0 (ˆ = − 8 + 8 B 0 (ˆ u ) + 8 t ) α 2 Q 4 s ˆ ˆ s f t 2 + ˆ � ˆ u 2 − 4 m 2 � f [ˆ tC 0 (ˆ − 8 t ) + ˆ uC 0 (ˆ u )] s 2 ˆ ˆ s t 2 + ˆ ˆ u 2 + 4 m 2 f ˆ � t ˆ u � 4 m 4 sm 2 f + ˆ D 0 (ˆ − 4 f − (2ˆ t ˆ u ) t , ˆ u ) s 2 ˆ ˆ s + 8 m 2 s − 2 m 2 s , ˆ f (ˆ f )[ D 0 (ˆ t ) + D 0 (ˆ s , ˆ u )] Massive four-photon amplitudes Results also checked for F f +++ − and F f ++ −− Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 24 / 33

Six Photons – Comparison with Nagy-Soper and Mahlon Massless case: [+ + − − −− ] and [+ − − + + − ] Plot presented by Nagy and Soper hep-ph/0610028 (also Binoth et al., hep-ph/0703311) Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 25 / 33

Six Photons – Comparison with Nagy-Soper and Mahlon Massless case: [+ + − − −− ] and [+ − − + + − ] 25000 20000 15000 10000 5000 0 0.5 1 1.5 2 Analogous plot produced with OPP reduction Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 25 / 33

Six Photons – Comparison with Binoth, Heinrich, Gehrmann, Mastrolia Massless case: [+ + − − −− ] and [+ + − − + − ] 25000 20000 15000 10000 5000 0 0.5 1 1.5 2 2.5 3 Same plot as before for a wider range of θ Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 26 / 33

Six Photons – Comparison with Mahlon Massless case: [+ + − − −− ] and [+ + − − + − ] 18000 16000 14000 12000 s |M| /α 3 10000 8000 6000 4000 0.5 1 1.5 2 2.5 3 θ Same idea for a different set of external momenta Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 27 / 33

Six Photons with Massive Fermions 30000 25000 20000 15000 10000 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Massless result [Mahlon] Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 28 / 33

Six Photons with Massive Fermions 30000 25000 20000 15000 10000 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Massless result [Mahlon] m = 0 . 5 GeV Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 28 / 33

Six Photons with Massive Fermions 30000 25000 20000 15000 10000 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Massless result [Mahlon] m = 0 . 5 GeV m = 4 . 5 GeV Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 28 / 33

Six Photons with Massive Fermions 30000 25000 20000 15000 10000 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Massless result [Mahlon] m = 0 . 5 GeV m = 4 . 5 GeV m = 12 . 0 GeV Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 28 / 33

Six Photons with Massive Fermions 30000 25000 20000 15000 10000 0.25 0.5 0.75 1 1.25 1.5 1.75 2 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 RADCOR 2007 28 / 33

q ¯ q → ZZZ virtual corrections A. Lazopoulos, K. Melnikov and F. Petriello, [arXiv:hep-ph/0703273] Poles 1 /ǫ 2 and 1 /ǫ � 1 � α s Γ(1 + ǫ ) ǫ 2 + 3 σ NLO , virt | div = − C F (4 π ) − ǫ ( s 12 ) − ǫ σ LO π 2 ǫ Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 29 / 33

q ¯ q → ZZZ virtual corrections A very naive implementation Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 30 / 33

q ¯ q → ZZZ virtual corrections A very naive implementation Calculate the N ( q ) by brute (numerical) force namely multiplying gamma matrices ! Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 30 / 33

q ¯ q → ZZZ virtual corrections A very naive implementation Calculate the N ( q ) by brute (numerical) force namely multiplying gamma matrices ! Calculate 4d and rational terms graph by graph Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 30 / 33

q ¯ q → ZZZ virtual corrections A very naive implementation Calculate the N ( q ) by brute (numerical) force namely multiplying gamma matrices ! Calculate 4d and rational terms graph by graph Comparison with LMP Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 30 / 33

q ¯ q → ZZZ virtual corrections A very naive implementation Calculate the N ( q ) by brute (numerical) force namely multiplying gamma matrices ! Calculate 4d and rational terms graph by graph Comparison with LMP Of course full agreement for the 1 /ǫ 2 and 1 /ǫ terms Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 30 / 33

q ¯ q → ZZZ virtual corrections A very naive implementation Calculate the N ( q ) by brute (numerical) force namely multiplying gamma matrices ! Calculate 4d and rational terms graph by graph 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 RADCOR 2007 30 / 33

q ¯ q → ZZZ virtual corrections A very naive implementation Calculate the N ( q ) by brute (numerical) force namely multiplying gamma matrices ! Calculate 4d and rational terms graph by graph 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 RADCOR 2007 30 / 33

q ¯ q → ZZZ virtual corrections Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 31 / 33

q ¯ q → ZZZ virtual corrections Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 31 / 33

q ¯ q → ZZZ virtual corrections Typical precision: Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 32 / 33

q ¯ q → ZZZ virtual corrections Typical precision: LMP: 9.573(66) about 1% error Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 32 / 33

q ¯ q → ZZZ virtual corrections Typical precision: LMP: 9.573(66) about 1% error OPP: � − 26 . 45706742815552 − 26 . 457067428165503661018557937723426 Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 32 / 33

q ¯ q → ZZZ virtual corrections Typical precision: LMP: 9.573(66) about 1% error OPP: � − 26 . 45706742815552 − 26 . 457067428165503661018557937723426 Typical time: 200 times faster (for non-singular PS-points) Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 32 / 33

q ¯ q → ZZZ virtual corrections Typical precision: LMP: 9.573(66) about 1% error OPP: � − 26 . 45706742815552 − 26 . 457067428165503661018557937723426 Typical time: 200 times faster (for non-singular PS-points) Future perspectives: up to 4 orders of magnitude Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 32 / 33

Outlook Reduction at the integrand level Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 33 / 33

Outlook Reduction at the integrand level changes the computational approach at one loop Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 33 / 33

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 RADCOR 2007 33 / 33

Outlook Reduction at the integrand level changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Future Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 33 / 33

Outlook Reduction at the integrand level changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Future Understand potential stability problems Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 33 / 33

Outlook Reduction at the integrand level changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Future Understand potential stability problems Combine with the real corrections Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 33 / 33

Outlook Reduction at the integrand level changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Future Understand potential stability problems Combine with the real corrections Automatize through Dyson-Schwinger equations Costas G. Papadopoulos (Athens) OPP Reduction RADCOR 2007 33 / 33

Outlook Reduction at the integrand level changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Future 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 RADCOR 2007 33 / 33

Outlook Reduction at the integrand level changes the computational approach at one loop Numerical but still algebraic: speed and precision not a problem Future 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 RADCOR 2007 33 / 33

The Outlook � The DS equations to all orders HEP-NCSR DEMOKRITOS October 1, 2007 1

The Outlook � The DS equations to all orders HEP-NCSR DEMOKRITOS October 1, 2007 2