max planck societyintroduction integrand reduction [mastrolia, em, ossola, peraro arxiv:1205.7087,...
TRANSCRIPT
![Page 1: Max Planck SocietyIntroduction Integrand reduction [Mastrolia, EM, Ossola, Peraro arXiv:1205.7087, arXiv:1209.4319] extensively used @ one loop see Luisoni’s talk extremely efficient](https://reader036.vdocument.in/reader036/viewer/2022071007/5fc5091b26e1e56fb82c3678/html5/thumbnails/1.jpg)
Integrand reduction via
multivariate polynomial division
EDOARDO MIRABELLA
in collaboration with P. Mastrolia, G. Ossola, T. Peraro & U. Schubert
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.1/17
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Outline
Introduction
Integrand reduction @ one loop
Integrand reduction @ many loops
multivariate polynomial division
maximum cut theorem
Applications
Conclusions
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.2/17
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Introduction
Integrand reduction[Mastrolia, EM, Ossola, Peraro arXiv:1205.7087, arXiv:1209.4319]
extensively used @ one loop see Luisoni’s talkextremely efficient
easy-to-implement (e.g. CutTools & Samurai )
its multi-loop extensioninteresting & lively field [ Mastrolia, Ossola,’11; Badger, Frellersvig, Zhang ’12; Zhang ’12;
with Mastrolia, Ossola, EM, Peraro ’12; Kleiss, Malamos, Papadopoulos, Verheyen ’12;with Badger, Frellersvig, Zhang ’12; Feng, Huang ’12; Mastrolia, Ossola, EM, Peraro ’12;with Huang, Zhang; ’13 ]
takes a different perspective see also Badger’s & Feng’s talk
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.3/17
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Introduction
Integrand reduction via multivariate polynomial division[Mastrolia, EM, Ossola, Peraro arXiv:1205.7087, arXiv:1209.4319]
applicable to any amplitude at any order
leads to new insight on amplitudesinformation on the structure of the residues
re-interpretation of the one loop results
generates recursively the residue of any cutone ingredient: Feynman denominators
one operation: polynomial division
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.3/17
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Introduction
Integrand reduction via multivariate polynomial division[Mastrolia, EM, Ossola, Peraro arXiv:1205.7087, arXiv:1209.4319]
applicable to any amplitude at any order
leads to new insight on amplitudesinformation on the structure of the residues
re-interpretation of the one loop results
generates recursively the residue of any cutone ingredient: Feynman denominators
one operation: polynomial division
Alternative approach: Maximal Unitarity [Kosower, Larsen ’11; Larsen ’12; Larsen, Caron-Huot ’12
Johansson, Kosower, Larsen ’12]
see Kosower’s talk
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.3/17
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Integrand reduction @ one loop
One loop amplitude
A =
∫ddk I I =
N
D0 · · ·Dn−1aaaaaaaaaaaaaaaaaaaaa
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.4/17
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Integrand reduction @ one loop
One loop amplitude
A =
∫ddk I I =
N
D0 · · ·Dn−1aaaaaaaaaaaaaaaaaaaaa
the multipole decomposition in 4 dimensions: [Ossola, Papadopoulos, Pittau ’07]
N (k)D0 · · ·Dn−1
=∑ijkℓ
∆ijkℓ
DiDjDkDℓ
+∑ijk
∆ijk
DiDjDk
+∑ij
∆ij
DiDj
+∑i
∆i
Di
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.4/17
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Integrand reduction @ one loop
One loop amplitude
A =
∫ddk I I =
N
D0 · · ·Dn−1aaaaaaaaaaaaaaaaaaaaa
the multipole decomposition in 4 dimensions: [Ossola, Papadopoulos, Pittau ’07]
N (k)D0 · · ·Dn−1
=∑ijkℓ
∆ijkℓ
DiDjDkDℓ
+∑ijk
∆ijk
DiDjDk
+∑ij
∆ij
DiDj
+∑i
∆i
Di
∆’s : (known) functions of (unknown) coefficients
N : process-dependent numerator function
Di’s: denominators
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.4/17
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Integrand reduction @ one loop
One loop amplitude
A =
∫ddk I I =
N
D0 · · ·Dn−1aaaaaaaaaaaaaaaaaaaaa
the multipole decomposition in 4 dimensions: [Ossola, Papadopoulos, Pittau ’07]
N (k)D0 · · ·Dn−1
=∑ijkℓ
∆ijkℓ
DiDjDkDℓ
+∑ijk
∆ijk
DiDjDk
+∑ij
∆ij
DiDj
+∑i
∆i
Di
∆’s : (known) functions of (unknown) coefficients
N : process-dependent numerator function
Di’s: denominators
∆ij··· computed by sampling N . . .
. . . on the solutions of the multiple cut Di = Dj = · · · = 0
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.4/17
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Our idea
This Talk: Efficient computation of the residues at any loop
N (k1, . . . , kℓ)
D0(k1, . . . kℓ) · · ·Dn−1(k1, . . . , kℓ)=
∑κ
∑i1···iκ
∆i1···iκ(k1, . . . , kℓ)
Di1(k1, . . . , kℓ) · · ·Diκ(k1, . . . , kℓ)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.5/17
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Our idea
This Talk: Efficient computation of the residues at any loop
The Idea: in principle is simple, e.g. at one loop
N (k)D0D1D2D3
=∆0123(k)
D0D1D2D3+ · · ·
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.5/17
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Our idea
This Talk: Efficient computation of the residues at any loop
The Idea: in principle is simple, e.g. at one loop
N (k)D0D1D2D3
=∆0123(k)
D0D1D2D3+ · · ·
0. Decompose kµ =∑4
i=1 xi eµi and z ≡ (x1, x2, x3, x4)
1. Write N as∑3
j=0 Fj(k)Dj(k) +R(k)q: is it possible? a: Gröbner basis
+ multivariate polynomial division
2. If R(k) = 0 discard it . . .i.e. M∈ the ideal of the Di’s
s2
3. . . . otherwise ∆0123 = R(k)q: how to do that?
a: basis in a polynomial space
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.5/17
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Our idea
This Talk: Efficient computation of the residues at any loop
The Idea: in principle is simple, e.g. at one loop
N (k)D0D1D2D3
=∆0123(k)
D0D1D2D3+ · · ·
0. Decompose kµ =∑4
i=1 xi eµi and z ≡ (x1, x2, x3, x4)
1. Write N as∑3
j=0 Fj(k)Dj(k) +R(k)q: is it possible? a: Gröbner basis
+ multivariate polynomial division
2. If R(k) = 0 discard it . . .i.e. N ∈ the ideal of the Di’s
s2
3. . . . otherwise ∆0123 = R(k)i.e. R ∈
(polynomial space)/(the ideal of the Di’s)
Fundamental question arises . . .
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.5/17
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Our idea
This Talk: Efficient computation of the residues at any loop
The Idea: in principle is simple, e.g. at one loop
N (k)D0D1D2D3
=∆0123(k)
D0D1D2D3+ · · ·
0. Decompose kµ =∑4
i=1 xi eµi and z ≡ (x1, x2, x3, x4)
1. Write N as∑3
j=0 Fj(k)Dj(k) +R(k)q: is it possible? a: Gröbner basis
+ multivariate polynomial division
2. If R(k) = 0 discard it . . .i.e. N ∈ the ideal of the Di’s
s2
3. . . . otherwise ∆0123 = R(k)i.e. ∆0123 ∈
(polynomial space)/(the ideal of the Di’s)
Fundamental question arises . . .
answered moving to polynomials . . .
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.5/17
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Our idea
This Talk: Efficient computation of the residues at any loop
The Idea: in principle is simple, e.g. at one loop
N (k)D0D1D2D3
=∆0123(k)
D0D1D2D3+ · · ·
0. Decompose kµ =∑4
i=1 xi eµi and z ≡ (x1, x2, x3, x4)
1. Write N as∑3
j=0 Fj(k)Dj(k) +R(k)q: is it possible? a: Gröbner basis
+ multivariate polynomial division
2. If R(k) = 0 discard it . . .i.e. N ∈ the ideal of the Di’s
s2
3. . . . otherwise ∆0123 = R(k)i.e. ∆0123 ∈
(polynomial space)/(the ideal of the Di’s)
Fundamental question arises . . .
answered moving to polynomials . . .
using concepts of algebraic geometry [Zhang, ’12; Mastrolia, EM, Ossola, Peraro ’12]
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.5/17
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Algebraic_geometry.tar.gz
deals with multivariate polynomials in z = (z1, z2, . . .) .
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.6/17
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Algebraic_geometry.tar.gz
deals with multivariate polynomials in z = (z1, z2, . . .) .
Ideal J ≡ 〈ω1(z) · · ·ωs(z)〉 generated by ωi
J ={∑
i hi(z) ωi(z)}
polynomial coefficients hi(z)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.6/17
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Algebraic_geometry.tar.gz
deals with multivariate polynomials in z = (z1, z2, . . .) .
Ideal J ≡ 〈ω1(z) · · ·ωs(z)〉 generated by ωi
J ={∑
i hi(z) ωi(z)}
polynomial coefficients hi(z)
Multivariate polynomial division of f(z)/{ω1(z), . . . , ωs(z)}
needs an order, i.e. z1z2?> z21
f(z) =∑
i hi(z)ωi(z) +R(z)
hi(z) & R(z) not unique
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.6/17
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Algebraic_geometry.tar.gz
deals with multivariate polynomials in z = (z1, z2, . . .) .
Ideal J ≡ 〈ω1(z) · · ·ωs(z)〉 generated by ωi
J ={∑
i hi(z) ωi(z)}
polynomial coefficients hi(z)
Multivariate polynomial division of f(z)/{ω1(z), . . . , ωs(z)}
needs an order, i.e. z1z2?> z21
f(z) =∑
i hi(z)ωi(z) +R(z)
hi(z) & R(z) not unique
Gröbner basis {g1(z), . . . , gr(z)}exists (Buchberger’s algorithm) & generates J
unique R(z)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.6/17
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Algebraic_geometry.tar.gz
deals with multivariate polynomials in z = (z1, z2, . . .) .
Ideal J ≡ 〈ω1(z) · · ·ωs(z)〉 generated by ωi
J ={∑
i hi(z) ωi(z)}
polynomial coefficients hi(z)
Multivariate polynomial division of f(z)/{ω1(z), . . . , ωs(z)}
needs an order, i.e. z1z2?> z21
f(z) =∑
i hi(z)ωi(z) +R(z)
hi(z) & R(z) not unique
Gröbner basis {g1(z), . . . , gr(z)}exists (Buchberger’s algorithm) & generates J
unique R(z)
Hilbert’s NullstellensatzV (J ) = set of common zeros of J
( f = 0 in V (J ) )⇒ ( fr ∈ J for some r )
Weak Nullstellensatz: ( V (J ) = ∅ )⇔ ( 1 ∈ J )
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.6/17
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Residue via multivariate division
Ii1···in(k1, . . . , kℓ) =Ni1···in(k1, . . . , kℓ)
Di1(k1, . . . , kℓ) · · ·Din(k1, . . . , kℓ)Ii1···in(z) =
Ni1···in(z)Di1(z) · · ·Din(z)
I
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.7/17
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Residue via multivariate division
Ii1···in(z) =Ni1···in(z)
Di1(z) · · ·Din(z)Ii1···in(q1, . . . , qℓ) =
Ni1···in(q1, . . . , qℓ)
Di1(q1, . . . , qℓ) · · ·Din(q1, . . . , qℓ)I
0. in components kµi =∑
j xj,(i) eµ
j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.7/17
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Residue via multivariate division
Ii1···in(z) =Ni1···in(z)
Di1(z) · · ·Din(z)Ii1···in(q1, . . . , qℓ) =
Ni1···in(q1, . . . , qℓ)
Di1(q1, . . . , qℓ) · · ·Din(q1, . . . , qℓ)I
0. in components kµi =∑
j xj,(i) eµ
j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)
1. build J ≡ 〈Di1 (z) · · ·Din (z)〉 and Gi1···in = {g1(z), . . . , gr(z)}
→ ( Di1 = · · · = Din = 0)⇐⇒ (g1 = · · · = gr = 0 )
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.7/17
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Residue via multivariate division
Ii1···in(z) =Γi1···in(z) + ∆i1···in(z)
Di1(z) · · ·Din(z)Ii1···in(z) =
Ni1···in(z)Di1(z) · · ·Din(z)
Ii1···in(q1, . . . , qℓ) =
0. in components kµi =∑
j xj,(i) eµ
j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)
1. build J ≡ 〈Di1 (z) · · ·Din (z)〉 and Gi1···in = {g1(z), . . . , gr(z)}
2. Ni1···in (z) / Gi1···in Ni1···in = Γi1···in +∆i1···in
∆i1···in is the residue
J ∋ Γi1···in =∑
j hj(z)gj(z) =∑
j Ni1···ij−1ij+1···in (z)Dij (z)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.7/17
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Residue via multivariate division
Ii1···in(z) =n∑
j=1
Ni1···ij−1ij+1···in(z) Dij (z)Di1(z) · · ·Din(z)
+∆i1···in
Di1(z) · · ·Din(z)Ii1···in =
Γi1···ij−1ij+1
Di1(z)
0. in components kµi =∑
j xj,(i) eµ
j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)
1. build J ≡ 〈Di1 (z) · · ·Din (z)〉 and Gi1···in = {g1(z), . . . , gr(z)}
2. Ni1···in (z) / Gi1···in Ni1···in = Γi1···in +∆i1···in
∆i1···in is the residue
J ∋ Γi1···in =∑
j hj(z)gj(z) =∑
j Ni1···ij−1ij+1···in (z)Dij (z)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.7/17
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Residue via multivariate division
Ii1···in(z) =n∑
j=1
Ni1···ij−1ij+1···in(z) Dij (z)Di1(z) · · ·Din(z)
+∆i1···in
Di1(z) · · ·Din(z)Ii1···in =
Γi1···ij−1ij+1
Di1(z)
0. in components kµi =∑
j xj,(i) eµ
j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)
1. build J ≡ 〈Di1 (z) · · ·Din (z)〉 and Gi1···in = {g1(z), . . . , gr(z)}
2. Ni1···in (z) / Gi1···in Ni1···in = Γi1···in +∆i1···in
∆i1···in is the residue
J ∋ Γi1···in =∑
j hj(z)gj(z) =∑
j Ni1···ij−1ij+1···in (z)Dij (z)
Reducibility: ( N reducible )⇔ ( N/G ∈ J )→ ( no cut solutions )⇒ ( N reducible ) [Weak Nullstellensatz theorem ]
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.7/17
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Residue via multivariate division
Ii1···in =
n∑j=1
Ii1···ij−1ij+1···in +∆i1···in
Di1 · · ·Din
Ii1···in(z) =n∑
j=1
Ni1···ij−1ij+1···in(z) Di
Di1(z) · · ·Din(z)
0. in components kµi =∑
j xj,(i) eµ
j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)
1. build J ≡ 〈Di1 (z) · · ·Din (z)〉 and Gi1···in = {g1(z), . . . , gr(z)}
2. Ni1···in (z) / Gi1···in Ni1···in = Γi1···in +∆i1···in
∆i1···in is the residue
J ∋ Γi1···in =∑
j hj(z)gj(z) =∑
j Ni1···ij−1ij+1···in (z)Dij (z)
Reducibility: ( N reducible )⇔ ( N/G ∈ J )→ ( no cut solutions )⇒ ( N reducible ) [Weak Nullstellensatz theorem ]
Recursive relation→ The residues are computed iteratively
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.7/17
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Residue via multivariate division
Ii1···in =
n∑j=1
Ii1···ij−1ij+1···in +∆i1···in
Di1 · · ·Din
Ii1···in(z) =n∑
j=1
Ni1···ij−1ij+1···in(z) Di
Di1(z) · · ·Din(z)
0. in components kµi =∑
j xj,(i) eµ
j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)
1. build J ≡ 〈Di1 (z) · · ·Din (z)〉 and Gi1···in = {g1(z), . . . , gr(z)}
2. Ni1···in (z) / Gi1···in Ni1···in = Γi1···in +∆i1···in
∆i1···in is the residue
J ∋ Γi1···in =∑
j hj(z)gj(z) =∑
j Ni1···ij−1ij+1···in (z)Dij (z)
Reducibility: ( N reducible )⇔ ( N/G ∈ J )→ ( no cut solutions )⇒ ( N reducible ) [Weak Nullstellensatz theorem ]
Recursive relation→ The residues are computed iteratively
↑n-denominator integrand
տ
(n-1)-denominator integrand
← residue
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.7/17
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Residue via multivariate division
Ii1···in =
n∑j=1
Ii1···ij−1ij+1···in +∆i1···in
Di1 · · ·Din
Ii1···in(z) =n∑
j=1
Ni1···ij−1ij+1···in(z) Di
Di1(z) · · ·Din(z)
0. in components kµi =∑
j xj,(i) eµ
j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)
1. build J ≡ 〈Di1 (z) · · ·Din (z)〉 and Gi1···in = {g1(z), . . . , gr(z)}
2. Ni1···in (z) / Gi1···in Ni1···in = Γi1···in +∆i1···in
∆i1···in is the residue
J ∋ Γi1···in =∑
j hj(z)gj(z) =∑
j Ni1···ij−1ij+1···in (z)Dij (z)
↑n-denominator integrand
տ
(n-1)-denominator integrand
← residue
Two Approaches
“fit-on-the-cut"→ use generic N ’s to get the parametric form of ∆’s
→ determines the coefficients sampling on the cuts
“divide-and-conquer"→ generate the N of the process→ compute the residues iteratively (no cut solutions needed!)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.7/17
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Maximum cut
Maximum Cut ≡ cut constraining all the ki’s→@ 1 loop: 4-ple cut in 4 dimensions 5-ple cut in d dimensions
→ Assumption: finite number ns of solutions, all non-degenerate
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.8/17
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Maximum cut
Maximum Cut ≡ cut constraining all the ki’s→@ 1 loop: 4-ple cut in 4 dimensions 5-ple cut in d dimensions
→ Assumption: finite number ns of solutions, all non-degenerate
Maximum Cut Theorem The residue is parametrized by ns coefficients. It exists an
Maximum Cut Theorem univariate polynomial representation. [Mastrolia, EM, Ossola, Peraro ’12]
→ relies on the Finiteness Theorem & Shape Lemma
⇒ cut-constructibility of the Maximum Cut
⇒ ∃ an univariate parametrization of ∆
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.8/17
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Maximum cut
Maximum Cut ≡ cut constraining all the ki’s→@ 1 loop: 4-ple cut in 4 dimensions 5-ple cut in d dimensions
→ Assumption: finite number ns of solutions, all non-degenerate
Maximum Cut Theorem The residue is parametrized by ns coefficients. It exists an
Maximum Cut Theorem univariate polynomial representation. [Mastrolia, EM, Ossola, Peraro ’12]
Maximum cut theorem in action:
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.8/17
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“Fit-on-the-cut” approach→ use generic N ’s to get the parametric form of ∆’s
→ determine the coefficients sampling on the cuts
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.9/17
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Application: one loop in one slide
I0···(n−1)(k) =N0···(n−1)(k)
D0(k) · · ·Dn−1(k)n > 4I0···(n−1)(q) =
∑i1<<i4
Ni1i2i3i4(q)
Di1(q) · · ·Di4(q)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.10/17
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Application: one loop in one slide
I0···(n−1)(k) =N0···(n−1)(k)
D0(k) · · ·Dn−1(k)n > 4I0···(n−1)(q) =
∑i1<<i4
Ni1i2i3i4(q)
Di1(q) · · ·Di4(q)
Step 1. Reducing to 4-point functions(
D0 = · · · = Dn−1 = 0
impossible
)
⇒
(
I0···(n−1) reducible, in termsof (n-2)-integrands
)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.10/17
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Application: one loop in one slide
I0···(n−1)(k) =∑ijkℓ
Iijkℓ(k)I0···(n−1)(q) =N0···(n−1)(q)
D0(q) · · ·Dn−1(q)n > 4I0···(n−1)(q) =
i
Step 1. Reducing to 4-point functions(
D0 = · · · = Dn−1 = 0
impossible
)
⇒
(
I0···(n−1) reducible, in termsof (n-2)-integrands
)
reiterate until 4-point integrands are reached
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.10/17
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Application: one loop in one slide
I0···(n−1)(k) =∑ijkℓ
Iijkℓ(k)I0···(n−1)(q) =N0···(n−1)(q)
D0(q) · · ·Dn−1(q)n > 4I0···(n−1)(q) =
i
Step 1. Reducing to 4-point functions
Step 2. Reducing the integrands Iijkℓ =Nijkℓ
DiDjDkDℓ
decompose kµ = x1pµj + x2p
µk+ x3p
µℓ+ x4v
µ⊥ & z = (x1, . . . , x4)
Nijkℓ =∑
~a
b~a za1
1 za2
2 za3
3 za4
4
∑
i
ai ≤ 4
define Jijkℓ = 〈Di, . . . , Dℓ〉 & get Gijkℓ = (g1, . . . , g4 ).
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.10/17
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Application: one loop in one slide
I0···(n−1)(k) =∑ijkℓ
∆ijkℓ
DiDjDkDℓ
+∑ijk
Iijk(k)I0···(n−1)(q) =∑
i1<<i4
Ii1i2i3i4(q)I0···(n−1)
Step 1. Reducing to 4-point functions
Step 2. Reducing the integrands Iijkℓ =Nijkℓ
DiDjDkDℓ
decompose kµ = x1pµj + x2p
µk+ x3p
µℓ+ x4v
µ⊥ & z = (x1, . . . , x4)
Nijkℓ =∑
~a
b~a za1
1 za2
2 za3
3 za4
4
∑
i
ai ≤ 4
define Jijkℓ = 〈Di, . . . , Dℓ〉 & get Gijkℓ = (g1, . . . , g4 ).
Nijkℓ/Gijkℓ ∆ijkℓ = c0 + c1x4 = c0 + c1 k · v⊥
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.10/17
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Application: one loop in one slide
I0···(n−1)(k) =∑ijkℓ
∆ijkℓ
DiDjDkDℓ
+∑ijk
Iijk(k)I0···(n−1)(q) =∑
i1<<i4
Ii1i2i3i4(q)I0···(n−1)
Step 1. Reducing to 4-point functions
Step 2. Reducing the integrands Iijkℓ =Nijkℓ
DiDjDkDℓ 2 coefficients
decompose kµ = x1pµj + x2p
µk+ x3p
µℓ+ x4v
µ⊥ & z = (x1, . . . , x4)
Nijkℓ =∑
~a
b~a za1
1 za2
2 za3
3 za4
4
∑
i
ai ≤ 4
define Jijkℓ = 〈Di, . . . , Dℓ〉 & get Gijkℓ = (g1, . . . , g4 ).
Nijkℓ/Gijkℓ ∆ijkℓ = c0 + c1x4 = c0 + c1 k · v⊥Maximum Cut Theorem
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.10/17
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Application: one loop in one slide
I0···(n−1)(k) =∑ijkℓ
∆ijkℓ
DiDjDkDℓ
+∑ijk
∆ijk
DiDjDk
+∑ij
Iij(k)I0···(n−1)(q) =∑ijk
Iijk(q) +
Step 1. Reducing to 4-point functions
Step 2. Reducing the integrands Iijkℓ =Nijkℓ
DiDjDkDℓ 2 coefficients
Step 3. Reducing the integrands Iijk =Nijk
DiDjDk 7 coefficients
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.10/17
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Application: one loop in one slide
I0···(n−1)(k) =∑ijkℓ
∆ijkℓ
DiDjDkDℓ
+∑ijk
∆ijk
DiDjDk
+∑ij
∆ij
DiDj
+∑i
Ii(k)I0···(n−1)(q) =
Step 1. Reducing to 4-point functions
Step 2. Reducing the integrands Iijkℓ =Nijkℓ
DiDjDkDℓ 2 coefficients
Step 3. Reducing the integrands Iijk =Nijk
DiDjDk 7 coefficients
Step 4. Reducing the integrands Iij =Nij
DiDj 9 coefficients
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.10/17
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Application: one loop in one slide
I0···(n−1)(k) =∑ijkℓ
∆ijkℓ
DiDjDkDℓ
+∑ijk
∆ijk
DiDjDk
+∑ij
∆ij
DiDj
+∑i
∆i
Di
I0···(n−1)(q) =∑ij
Step 1. Reducing to 4-point functions
Step 2. Reducing the integrands Iijkℓ =Nijkℓ
DiDjDkDℓ 2 coefficients
Step 3. Reducing the integrands Iijk =Nijk
DiDjDk 7 coefficients
Step 4. Reducing the integrands Iij =Nij
DiDj 9 coefficients
Step 5. Reducing the integrands Ii =Ni
Di{
Ni linearDi quadratic
in z⇒ ∆i = Ni
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.10/17
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Application: one loop in one slide
I0···(n−1)(k) =∑ijkℓ
∆ijkℓ
DiDjDkDℓ
+∑ijk
∆ijk
DiDjDk
+∑ij
∆ij
DiDj
+∑i
∆i
Di
I0···(n−1)(q) =∑ij
Step 1. Reducing to 4-point functions
Step 2. Reducing the integrands Iijkℓ =Nijkℓ
DiDjDkDℓ 2 coefficients
Step 3. Reducing the integrands Iijk =Nijk
DiDjDk 7 coefficients
Step 4. Reducing the integrands Iij =Nij
DiDj 9 coefficients
Step 5. Reducing the integrands Ii =Ni
Di 5 coefficients
Extension to d-dimensions is easyz = (x1, x2, x3, x4) → z = (x1, x2, x3, x4, µ2)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.10/17
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Application: one loop in one slide
I0···(n−1)(k) =∑ijkℓ
∆ijkℓ
DiDjDkDℓ
+∑ijk
∆ijk
DiDjDk
+∑ij
∆ij
DiDj
+∑i
∆i
Di
I0···(n−1)(q) =∑ij
Step 1. Reducing to 4-point functions
Step 2. Reducing the integrands Iijkℓ =Nijkℓ
DiDjDkDℓ 2 coefficients
Step 3. Reducing the integrands Iijk =Nijk
DiDjDk 7 coefficients
Step 4. Reducing the integrands Iij =Nij
DiDj 9 coefficients
Step 5. Reducing the integrands Ii =Ni
Di 5 coefficients
Extension to d-dimensions is easyz = (x1, x2, x3, x4) → z = (x1, x2, x3, x4, µ2)
All the ∆’s agree with the literature [Ossola, Papadopoulos, Pittau ’07; Giele, Kunszt, Melnikov ’08 ]
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.10/17
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Application: a two loops example
General result @ two loops:
→(
n− fold cut withn ≥ 9 impossible
)
⇒(
n− denominator integrandswith n ≥ 9 reducible
)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.11/17
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Application: a two loops example
General result @ two loops:
→(
n− fold cut withn ≥ 9 impossible
)
⇒(
n− denominator integrandswith n ≥ 9 reducible
)
Example: five-point N = 4 SYM topology
D3 D5
D2
D7
D6
D8
D1 D4
5
4
32
1
I =N
D1 · · ·D8=
N12345678
D1 · · ·D8+
8∑i=1
N1···(i−1)(i+1)···8∏k 6=i Dk
+
N = α+ v1 · k1 + v2 · k2 [Carrasco, Johansson ’11]
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.11/17
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Application: a two loops example
General result @ two loops:
→(
n− fold cut withn ≥ 9 impossible
)
⇒(
n− denominator integrandswith n ≥ 9 reducible
)
Example: five-point N = 4 SYM topology
D3 D5
D2
D7
D6
D8
D1 D4
5
4
32
1
I =N
D1 · · ·D8=
N12345678
D1 · · ·D8+
8∑i=1
N1···(i−1)(i+1)···8∏k 6=i Dk
+
N = α+ v1 · k1 + v2 · k2 [Carrasco, Johansson ’11]
Step 1. Reducing the integrand I = ND1···D8
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.11/17
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Application: a two loops example
General result @ two loops:
→(
n− fold cut withn ≥ 9 impossible
)
⇒(
n− denominator integrandswith n ≥ 9 reducible
)
Example: five-point N = 4 SYM topology
D3 D5
D2
D7
D6
D8
D1 D4
5
4
32
1
I =N
D1 · · ·D8=
N12345678
D1 · · ·D8+
8∑i=1
N1···(i−1)(i+1)···8∏k 6=i Dk
+
N = α+ v1 · k1 + v2 · k2 [Carrasco, Johansson ’11]
Step 1. Reducing the integrand I = ND1···D8
decompose kµ1 =∑
i xieµi kµ2 =
∑
j yjτµj & z = (x1, . . . , x4, y1, · · · y4)
define J12345678 = 〈D1, . . . , D8〉 & get G12345678 = (g1, . . .).
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.11/17
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Application: a two loops example
General result @ two loops:
→(
n− fold cut withn ≥ 9 impossible
)
⇒(
n− denominator integrandswith n ≥ 9 reducible
)
Example: five-point N = 4 SYM topology
D3 D5
D2
D7
D6
D8
D1 D4
5
4
32
1
I =∆12345678
D1 · · ·D8+
8∑i=1
N1···(i−1)(i+1)···8∏k 6=i Dk
I1···(i−1)(i+1)···8
N = α+ v1 · k1 + v2 · k2 [Carrasco, Johansson ’11]
Step 1. Reducing the integrand I = ND1···D8
decompose kµ1 =∑
i xieµi kµ2 =
∑
j yjτµj & z = (x1, . . . , x4, y1, · · · y4)
define J12345678 = 〈D1, . . . , D8〉 & get G12345678 = (g1, . . .).
N/G12345678 ∆12345678 = c0 + c1x4 + c2y3 + c3y32 + c4y4 + c5x4y4 + c6y42 + c7y43
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.11/17
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Application: a two loops example
General result @ two loops:
→(
n− fold cut withn ≥ 9 impossible
)
⇒(
n− denominator integrandswith n ≥ 9 reducible
)
Example: five-point N = 4 SYM topology
D3 D5
D2
D7
D6
D8
D1 D4
5
4
32
1
I =∆12345678
D1 · · ·D8+
8∑i=1
N1···(i−1)(i+1)···8∏k 6=i Dk
I1···(i−1)(i+1)···8
N = α+ v1 · k1 + v2 · k2 [Carrasco, Johansson ’11]
Step 1. Reducing the integrand I = ND1···D8
decompose kµ1 =∑
i xieµi kµ2 =
∑
j yjτµj & z = (x1, . . . , x4, y1, · · · y4)
define J12345678 = 〈D1, . . . , D8〉 & get G12345678 = (g1, . . .).
N/G12345678 ∆12345678 = c0 + c1x4 + c2y3 + c3y32 + c4y4 + c5x4y4 + c6y42 + c7y43
ci’s obtained by sampling N on the
eight solutions of D1 = · · · = D8 = 0 Maximum Cut Theorem
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.11/17
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Application: a two loops example
General result @ two loops:
→(
n− fold cut withn ≥ 9 impossible
)
⇒(
n− denominator integrandswith n ≥ 9 reducible
)
Example: five-point N = 4 SYM topology
D3 D5
D2
D7
D6
D8
D1 D4
5
4
32
1
I =∆12345678
D1 · · ·D8+
8∑i=1
N1···(i−1)(i+1)···8∏k 6=i Dk
I1···(i−1)(i+1)···8
N = α+ v1 · k1 + v2 · k2 [Carrasco, Johansson ’11]
Step 1. Reducing the integrand I = ND1···D8
Step 2. Reducing the integrand Ii1···i7 =Ni1···i7
Di1···Di7
e.g. (i1 · · · i7) = (1234567)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.11/17
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Application: a two loops example
General result @ two loops:
→(
n− fold cut withn ≥ 9 impossible
)
⇒(
n− denominator integrandswith n ≥ 9 reducible
)
Example: five-point N = 4 SYM topology
D3 D5
D2
D7
D6
D8
D1 D4
5
4
32
1
I =∆12345678
D1 · · ·D8+
8∑i=1
N1···(i−1)(i+1)···8∏k 6=i Dk
I1···(i−1)(i+1)···8
N = α+ v1 · k1 + v2 · k2 [Carrasco, Johansson ’11]
Step 1. Reducing the integrand I = ND1···D8
Step 2. Reducing the integrand Ii1···i7 =Ni1···i7
Di1···Di7
, e.g. (i1 · · · i7) = (1234567)
5
4
32
1
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.11/17
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Application: a two loops example
General result @ two loops:
→(
n− fold cut withn ≥ 9 impossible
)
⇒(
n− denominator integrandswith n ≥ 9 reducible
)
Example: five-point N = 4 SYM topology
D3 D5
D2
D7
D6
D8
D1 D4
5
4
32
1
I =∆12345678
D1 · · ·D8+
8∑i=1
∆1···(i−1)(i+1)···8∏k 6=i Dk
N12345678
D1 · · ·D8+
N = α+ v1 · k1 + v2 · k2 [Carrasco, Johansson ’11]
Step 1. Reducing the integrand I = ND1···D8
Step 2. Reducing the integrand Ii1···i7 =Ni1···i7
Di1···Di7
, e.g. (i1 · · · i7) = (1234567)
5
4
32
1
N1234567/G1234567 ∆1234567 = 32 monomials in x1,2, y1,2
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.11/17
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Application: a two loops example
General result @ two loops:
→(
n− fold cut withn ≥ 9 impossible
)
⇒(
n− denominator integrandswith n ≥ 9 reducible
)
Example: five-point N = 4 SYM topology
D3 D5
D2
D7
D6
D8
D1 D4
5
4
32
1
I =∆12345678
D1 · · ·D8+
8∑i=1
∆1···(i−1)(i+1)···8∏k 6=i Dk
N12345678
D1 · · ·D8+
N = α+ v1 · k1 + v2 · k2 [Carrasco, Johansson ’11]
Step 1. Reducing the integrand I = ND1···D8
Step 2. Reducing the integrand Ii1···i7 =Ni1···i7
Di1···Di7
, e.g. (i1 · · · i7) = (1234567)
5
4
32
1
N1234567/G1234567 ∆1234567 = 32 monomials in x1,2, y1,2
coefficients by sampling ∆1234567 = (N −∆12345678)/D8
on the solutions of D1 = · · · = D7 = 0
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.11/17
![Page 55: Max Planck SocietyIntroduction Integrand reduction [Mastrolia, EM, Ossola, Peraro arXiv:1205.7087, arXiv:1209.4319] extensively used @ one loop see Luisoni’s talk extremely efficient](https://reader036.vdocument.in/reader036/viewer/2022071007/5fc5091b26e1e56fb82c3678/html5/thumbnails/55.jpg)
Application: a two loops example
General result @ two loops:
→(
n− fold cut withn ≥ 9 impossible
)
⇒(
n− denominator integrandswith n ≥ 9 reducible
)
Example: five-point N = 4 SYM topology
D3 D5
D2
D7
D6
D8
D1 D4
5
4
32
1
I =c0 + c1(k1 · p5) + c1(k1 · p1)
D1 · · ·D8+
8∑i=1
ci∏k 6=i Dk
N = α+ v1 · k1 + v2 · k2 [Carrasco, Johansson ’11]
Step 1. Reducing the integrand I = ND1···D8
Step 2. Reducing the integrand Ii1···i7 =Ni1·i7
Di1···Di7
e.g. (i1 · · · i7) = (1234567)
reduction completed after two steps (N = 4 ) (Checked via the N = N test )
other topologies & N = 8 SUGRA reduced
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.11/17
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Application: a two loops example
General result @ two loops:
→(
n− fold cut withn ≥ 9 impossible
)
⇒(
n− denominator integrandswith n ≥ 9 reducible
)
Example: five-point N = 4 SYM topology
D3 D5
D2
D7
D6
D8
D1 D4
5
4
32
1
I =c0 + c1(k1 · p5) + c1(k1 · p1)
D1 · · ·D8+
8∑i=1
ci∏k 6=i Dk
N = α+ v1 · k1 + v2 · k2 [Carrasco, Johansson ’11]
Integrand reduction via unitarity works as well: [Uli Schubert’s Diplomarbeit]
numerator known at the multiple cuts only
different bookkeeping in the top-down approach, e.g.
4
5
2
1
3
=4
5
2
1
3
−
3
1
24
5
−
3
5
42
1
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.11/17
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“divide-and-conquer” approach→ generate the N of the process→ compute the residues iteratively (no cut solutions needed!)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.12/17
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Application: the photon self-energy I
The method does not require on-shell solutions higher powers of propagators are not problematic
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.13/17
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Application: the photon self-energy I
The method does not require on-shell solutions higher powers of propagators are not problematic
Example: QED photon self energy @ two loops
p
D3
D1
D4
D2
D1
I =N
D21D2D3D4
N = 16[µ211 − k21 (k1 · p)] + · · ·
d dimensions: kµi = k
µi + ~µi µij ≡ ~µi · ~µj
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.13/17
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Application: the photon self-energy I
The method does not require on-shell solutions higher powers of propagators are not problematic
Example: QED photon self energy @ two loops
p
D3
D1
D4
D2
D1
I =N
D21D2D3D4
N = 16[µ211 − k21 (k1 · p)] + · · ·
d dimensions: kµi = k
µi + ~µi µij ≡ ~µi · ~µj
Step 1. Reducing the integrand I = ND2
1D2D3D4
decompose kµ1 =∑
i xieµi kµ2 =
∑
i yieµi z = (xi, yj , µ11, µ22, µ12)
define J1234 = 〈D1, D2, D3, D4〉 ( G1234 ) & J11234 = 〈D21 , D2, D3, D4〉
N/G1234 N = α1234 D1 + α1124 D3 +R R = 8µ11(2µ11 − p2)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.13/17
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Application: the photon self-energy I
The method does not require on-shell solutions higher powers of propagators are not problematic
Example: QED photon self energy @ two loops
p
D3
D1
D4
D2
D1
I =∆11234
D21D2D3D4
+N1234
D1D2D3D4+N1124
D21D2D4
N = 16[µ211 − k21 (k1 · p)] + · · ·
d dimensions: kµi = k
µi + ~µi µij ≡ ~µi · ~µj
Step 1. Reducing the integrand I = ND2
1D2D3D4
decompose kµ1 =∑
i xieµi kµ2 =
∑
i yieµi z = (xi, yj , µ11, µ22, µ12)
define J1234 = 〈D1, D2, D3, D4〉 ( G1234 ) & J11234 = 〈D21 , D2, D3, D4〉
N/G1234 N = α1234 D1 + α1124 D3 +R R = 8µ11(2µ11 − p2)
→ R /∈ J1234 ⇒ R /∈ J11234 ⇒ ∆11234 ≡ R
→ N1234 ≡ α1234 & N1124 ≡ α1124
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.13/17
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Application: the photon self-energy I
The method does not require on-shell solutions higher powers of propagators are not problematic
Example: QED photon self energy @ two loops
p
D3
D1
D4
D2
D1
I =∆11234
D21D2D3D4
+∆1234
D1D2D3D4+N1124
D21D2D4
+N234
D2D3D4
N = 16[µ211 − k21 (k1 · p)] + · · ·
d dimensions: kµi = k
µi + ~µi µij ≡ ~µi · ~µj
Step 1. Reducing the integrand I = ND2
1D2D3D4
Step 2. Reducing the integrand I1234 = N1234
D1D2D3D4
use J1234 = 〈D1, D2, D3, D4〉 & G1234
N1234/G1234 N1234 = N234 D1 +∆1234 ∆1234 = −8(µ11 + p2)
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.13/17
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Application: the photon self-energy I
The method does not require on-shell solutions higher powers of propagators are not problematic
Example: QED photon self energy @ two loops
p
D3
D1
D4
D2
D1
I =∆11234
D21D2D3D4
+∆1234
D1D2D3D4+
∆1124
D21D2D4
+N234
D2D3D4+
N124
D1D2D4
N = 16[µ211 − k21 (k1 · p)] + · · ·
d dimensions: kµi = k
µi + ~µi µij ≡ ~µi · ~µj
Step 1. Reducing the integrand I = ND2
1D2D3D4
Step 2. Reducing the integrand I1234 = N1234
D1D2D3D4
Step 3. Reducing the integrand I1124 = N1124
D21D3D4
define J124 = 〈D1, D2, D4〉 & G124
N1124/G124 N1124 = N124 D1 +∆1124 ∆1124 = 8µ11
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.13/17
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Application: the photon self-energy I
The method does not require on-shell solutions higher powers of propagators are not problematic
Example: QED photon self energy @ two loops
p
D3
D1
D4
D2
D1
I =∆11234
D21D2D3D4
+∆1234
D1D2D3D4+
∆1124
D21D2D4
+∆234
D2D3D4+
∆124
D1D2D4
N = 16[µ211 − k21 (k1 · p)] + · · ·
d dimensions: kµi = k
µi + ~µi µij ≡ ~µi · ~µj
Step 1. Reducing the integrand I = ND2
1D2D3D4
Step 2. Reducing the integrand I1234 = N1234
D1D2D3D4
Step 3. Reducing the integrand I1124 = N1124
D21D3D4
Step 4. / 5. Reducing I124 = N124
D1D2D4I234 = N234
D2D3D4
I124 & I234 irreducible i.e. Nx = ∆x
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.13/17
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Application: the photon self-energy I
The method does not require on-shell solutions higher powers of propagators are not problematic
Example: QED photon self energy @ two loops
p
D3
D1
D4
D2
D1
I =∆11234
D21D2D3D4
+∆1234
D1D2D3D4+
∆1124
D21D2D4
+∆234
D2D3D4+
∆124
D1D2D4
N = 16[µ211 − k21 (k1 · p)] + · · ·
d dimensions: kµi = k
µi + ~µi µij ≡ ~µi · ~µj
Reduction completed after five steps . . .
I =8µ11(2µ11 − p2)
D21D2D3D4
−8(µ11 + p2)
D1D2D3D4+
8µ11
D21D2D4
−8
D2D3D4+
8
D1D2D4
. . . performed for the full N . . .
. . . and for the other diagrams
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.13/17
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Application: the photon self-energy II
p
D3
D1
D4
D2
D1
I = I0 + ǫ I1 + ǫ2 I2
Ia =Na
D21D2D3D4
[Mastrolia, Peraro, in progress]
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.14/17
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Application: the photon self-energy II
p
D3
D1
D4
D2
D1
I = I0 + ǫ I1 + ǫ2 I2
Ia =Na
D21D2D3D4
[Mastrolia, Peraro, in progress]
The numerator
N0 = 16µ11µ12 + 16µ211 − 16 (p · k1) k21 − 32 (p · k1) (k1 · k2) + 32 (p · k1)µ12
N0 = +16 (p · k1)µ11 + 16 (p · k2) k21 − 16 (p · k2)µ11 + 16 k21 (k1 · k2)N0 = −16 k21 µ12 − 32 k21 µ11 + 16 (k21)
2 − 16 (k1 · k2)µ11
N1 = −32µ11µ12 − 32µ211 + 32 (p · k1) k21 + 64 (p · k1) (k1 · k2)− 64 (p · k1)µ12
N1 = −32 (p · k1)µ11 − 32 (p · k2) k21 + 32 (p · k2)µ11 − 32 k21 (k1 · k2) + 32 k21 µ12
N1 = +64 k21 µ11 − 32 (k21)2 + 32 (k1 · k2)µ11
N2 = 16µ11µ12 + 16µ211 − 16 (p · k1) k21 − 32 (p · k1) (k1 · k2) + 32 (p · k1)µ12
N2 = +16 (p · k1)µ11 + 16 (p · k2) k21 − 16 (p · k2)µ11 + 16 k21 (k1 · k2)N2 = −16 k21 µ12 − 32 k21 µ11 + 16 (k21)
2 − 16 (k1 · k2)µ11.
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.14/17
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Application: the photon self-energy II
p
D3
D1
D4
D2
D1
I = I0 + ǫ I1 + ǫ2 I2
Ia =Na
D21D2D3D4
[Mastrolia, Peraro, in progress]
The numerator’s decomposition
I0 =−8p2
D21D2D3
+8p2
D21D2D4
+8
D21D3
+−8
D21D4
+16 k2 · p
D1D2D3D4+
8
D2D3D4
I1 =16p2
D21D2D3
+−16p2
D21D2D4
+−16
D21D3
+16
D21D4
+−32 k2 · p
D1D2D3D4+
−16
D2D3D4
I2 =−8p2
D21D2D3
+8p2
D21D2D4
+8
D21D3
+−8
D21D4
+16 k2 · p
D1D2D3D4+
8
D2D3D4
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.14/17
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Conclusions
Integrand reduction via multivariate polynomial division
novel framework for multi-loop computationsconstructs residues iteratively . . .
. . . for any amplitude . . .
. . . at any number of loops
uses a "minimal" frameworkone ingredient: Feynman denominatorsone operation: polynomial division
leads to new insight into amplitudesreducibility criteriamaximum cut theorem
Outlook
Automation
Implement identities to reduce the number of MI’se.g. merge integrand reduction and IBP’s
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.15/17
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Backup Slides
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.16/17
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Maximum cut theorem – proof
Maximum Cut Theorem The residue is parametrized by ns coefficients. It exists an
Maximum Cut Theorem univariate polynomial representation.
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.17/17
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Maximum cut theorem – proof
Maximum Cut Theorem The residue is parametrized by ns coefficients. It exists an
Maximum Cut Theorem univariate polynomial representation.
Proof
We parametrize the loop momenta via z = (z1, . . . , zm). The solutions of the cut are,
z(i) =(
z(i)1 , . . . , z
(i)m
)
i = 1, . . . , ns
finite and non-degenerate solution⇒ J is zero-dimensional and radical.
Finiteness Theorem holds⇒ The remainder is parametrized by ns coefficients.
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.17/17
![Page 73: Max Planck SocietyIntroduction Integrand reduction [Mastrolia, EM, Ossola, Peraro arXiv:1205.7087, arXiv:1209.4319] extensively used @ one loop see Luisoni’s talk extremely efficient](https://reader036.vdocument.in/reader036/viewer/2022071007/5fc5091b26e1e56fb82c3678/html5/thumbnails/73.jpg)
Maximum cut theorem – proof
Maximum Cut Theorem The residue is parametrized by ns coefficients. It exists an
Maximum Cut Theorem univariate polynomial representation.
Proof
We parametrize the loop momenta via z = (z1, . . . , zm). The solutions of the cut are,
z(i) =(
z(i)1 , . . . , z
(i)m
)
i = 1, . . . , ns
finite and non-degenerate solution⇒ J is zero-dimensional and radical.
Finiteness Theorem holds⇒ The remainder is parametrized by ns coefficients.
Moreover:
We can assume z(i)1 6= z
(j)1 ∀ i 6= j.
J and z1 fulfill the Shape Lemma
⇒ Gröbner basis for the lexicographic order z1 < z2 < · · · < zn is
g1(z) = f1(z1)
g2(z) = z2 − f2(z1)
.
.
.
gm(z) = zm − fm(z1)
f1 = rank-ns square-free polynomial.
⇒ Remainder = univariate polynomial in z1 of degree (ns − 1).
E. Mirabella, Amplitudes 2013, Ringberg Castle, 05.02.2013 – p.17/17