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

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

Outline

Introduction

Integrand reduction @ one loop

Integrand reduction @ many loops

multivariate polynomial division

maximum cut theorem

Applications

Conclusions


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


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


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



One loop amplitude

A =

∫ddk I I =

N

D0 · · ·Dn−1aaaaaaaaaaaaaaaaaaaaa



One loop amplitude

A =

∫ddk I I =

N


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



One loop amplitude

A =

∫ddk I I =

N



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



One loop amplitude

A =

∫ddk I I =

N



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


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)


Our idea


The Idea: in principle is simple, e.g. at one loop

N (k)D0D1D2D3

=∆0123(k)

D0D1D2D3+ · · ·


Our idea



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


Our idea



N (k)D0D1D2D3

=∆0123(k)

D0D1D2D3+ · · ·



1. Write N as∑3



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 . . .


Our idea



N (k)D0D1D2D3

=∆0123(k)

D0D1D2D3+ · · ·



1. Write N as∑3




s2

3. . . . otherwise ∆0123 = R(k)i.e. ∆0123 ∈



answered moving to polynomials . . .


Our idea



N (k)D0D1D2D3

=∆0123(k)

D0D1D2D3+ · · ·



1. Write N as∑3




s2

3. . . . otherwise ∆0123 = R(k)i.e. ∆0123 ∈



answered moving to polynomials . . .

using concepts of algebraic geometry [Zhang, ’12; Mastrolia, EM, Ossola, Peraro ’12]


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)





J ={∑

i hi(z) ωi(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





J ={∑

i hi(z) ωi(z)}




f(z) =∑

i hi(z)ωi(z) +R(z)


Gröbner basis {g1(z), . . . , gr(z)}exists (Buchberger’s algorithm) & generates J

unique R(z)





J ={∑

i hi(z) ωi(z)}




f(z) =∑

i hi(z)ωi(z) +R(z)


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 )


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



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) . . .)



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


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 )



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) =


j xj,(i) eµ

j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)


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)



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)


j xj,(i) eµ

j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)




J ∋ Γi1···in =∑

j hj(z)gj(z) =∑





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)


j xj,(i) eµ

j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)




J ∋ Γi1···in =∑

j hj(z)gj(z) =∑


Reducibility: ( N reducible )⇔ ( N/G ∈ J )→ ( no cut solutions )⇒ ( N reducible ) [Weak Nullstellensatz theorem ]



Ii1···in =

n∑j=1

Ii1···ij−1ij+1···in +∆i1···in

Di1 · · ·Din


j=1

Ni1···ij−1ij+1···in(z) Di

Di1(z) · · ·Din(z)


j xj,(i) eµ

j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)




J ∋ Γi1···in =∑

j hj(z)gj(z) =∑



Recursive relation→ The residues are computed iteratively



Ii1···in =

n∑j=1

Ii1···ij−1ij+1···in +∆i1···in

Di1 · · ·Din


j=1


Di1(z) · · ·Din(z)


j xj,(i) eµ

j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)




J ∋ Γi1···in =∑

j hj(z)gj(z) =∑



Recursive relation→ The residues are computed iteratively

↑n-denominator integrand

տ

(n-1)-denominator integrand

← residue



Ii1···in =

n∑j=1

Ii1···ij−1ij+1···in +∆i1···in

Di1 · · ·Din


j=1


Di1(z) · · ·Din(z)


j xj,(i) eµ

j,(i)& (k1, . . .)↔ z = (x1,(1), . . . , x1,(2) . . .)




J ∋ Γi1···in =∑

j hj(z)gj(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!)


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



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 ∆


Maximum cut




Maximum Cut Theorem univariate polynomial representation. [Mastrolia, EM, Ossola, Peraro ’12]

Maximum cut theorem in action:


“Fit-on-the-cut” approach→ use generic N ’s to get the parametric form of ∆’s

→ determine the coefficients sampling on the cuts


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)



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

)



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



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 ).



I0···(n−1)(k) =∑ijkℓ

∆ijkℓ

DiDjDkDℓ

+∑ijk

Iijk(k)I0···(n−1)(q) =∑

i1<<i4

Ii1i2i3i4(q)I0···(n−1)



DiDjDkDℓ


µk+ x3p

µℓ+ x4v

µ⊥ & z = (x1, . . . , x4)

Nijkℓ =∑

~a

b~a za1

1 za2

2 za3

3 za4

4

∑

i

ai ≤ 4


Nijkℓ/Gijkℓ ∆ijkℓ = c0 + c1x4 = c0 + c1 k · v⊥



I0···(n−1)(k) =∑ijkℓ

∆ijkℓ

DiDjDkDℓ

+∑ijk

Iijk(k)I0···(n−1)(q) =∑

i1<<i4

Ii1i2i3i4(q)I0···(n−1)



DiDjDkDℓ 2 coefficients


µk+ x3p

µℓ+ x4v

µ⊥ & z = (x1, . . . , x4)

Nijkℓ =∑

~a

b~a za1

1 za2

2 za3

3 za4

4

∑

i

ai ≤ 4


Nijkℓ/Gijkℓ ∆ijkℓ = c0 + c1x4 = c0 + c1 k · v⊥Maximum Cut Theorem



I0···(n−1)(k) =∑ijkℓ

∆ijkℓ

DiDjDkDℓ

+∑ijk

∆ijk

DiDjDk

+∑ij

Iij(k)I0···(n−1)(q) =∑ijk

Iijk(q) +




Step 3. Reducing the integrands Iijk =Nijk

DiDjDk 7 coefficients



I0···(n−1)(k) =∑ijkℓ

∆ijkℓ

DiDjDkDℓ

+∑ijk

∆ijk

DiDjDk

+∑ij

∆ij

DiDj

+∑i

Ii(k)I0···(n−1)(q) =






Step 4. Reducing the integrands Iij =Nij

DiDj 9 coefficients



I0···(n−1)(k) =∑ijkℓ

∆ijkℓ

DiDjDkDℓ

+∑ijk

∆ijk

DiDjDk

+∑ij

∆ij

DiDj

+∑i

∆i

Di

I0···(n−1)(q) =∑ij







DiDj 9 coefficients

Step 5. Reducing the integrands Ii =Ni

Di{

Ni linearDi quadratic

in z⇒ ∆i = Ni



I0···(n−1)(k) =∑ijkℓ

∆ijkℓ

DiDjDkDℓ

+∑ijk

∆ijk

DiDjDk

+∑ij

∆ij

DiDj

+∑i

∆i

Di

I0···(n−1)(q) =∑ij







DiDj 9 coefficients


Di 5 coefficients

Extension to d-dimensions is easyz = (x1, x2, x3, x4) → z = (x1, x2, x3, x4, µ2)



I0···(n−1)(k) =∑ijkℓ

∆ijkℓ

DiDjDkDℓ

+∑ijk

∆ijk

DiDjDk

+∑ij

∆ij

DiDj

+∑i

∆i

Di

I0···(n−1)(q) =∑ij







DiDj 9 coefficients


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 ]


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]




→(


)

⇒(


)


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

+


Step 1. Reducing the integrand I = ND1···D8




→(


)

⇒(


)


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

+



decompose kµ1 =∑

i xieµi kµ2 =

∑

j yjτµj & z = (x1, . . . , x4, y1, · · · y4)

define J12345678 = 〈D1, . . . , D8〉 & get G12345678 = (g1, . . .).




→(


)

⇒(


)


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



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




→(


)

⇒(


)


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



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




→(


)

⇒(


)


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



Step 2. Reducing the integrand Ii1···i7 =Ni1···i7

Di1···Di7

e.g. (i1 · · · i7) = (1234567)




→(


)

⇒(


)


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




Di1···Di7

, e.g. (i1 · · · i7) = (1234567)

5

4

32

1




→(


)

⇒(


)


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+




Di1···Di7

, e.g. (i1 · · · i7) = (1234567)

5

4

32

1

N1234567/G1234567 ∆1234567 = 32 monomials in x1,2, y1,2




→(


)

⇒(


)


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+




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




→(


)

⇒(


)


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



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




→(


)

⇒(


)


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


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


“divide-and-conquer” approach→ generate the N of the process→ compute the residues iteratively (no cut solutions needed!)


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





p

D3

D1

D4

D2

D1

I =N

D21D2D3D4

N = 16[µ211 − k21 (k1 · p)] + · · ·



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)





p

D3

D1

D4

D2

D1

I =∆11234

D21D2D3D4

+N1234

D1D2D3D4+N1124

D21D2D4

N = 16[µ211 − k21 (k1 · p)] + · · ·




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





p

D3

D1

D4

D2

D1

I =∆11234

D21D2D3D4

+∆1234

D1D2D3D4+N1124

D21D2D4

+N234

D2D3D4

N = 16[µ211 − k21 (k1 · p)] + · · ·




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)





p

D3

D1

D4

D2

D1

I =∆11234

D21D2D3D4

+∆1234

D1D2D3D4+

∆1124

D21D2D4

+N234

D2D3D4+

N124

D1D2D4

N = 16[µ211 − k21 (k1 · p)] + · · ·




1D2D3D4


D1D2D3D4


D21D3D4

define J124 = 〈D1, D2, D4〉 & G124

N1124/G124 N1124 = N124 D1 +∆1124 ∆1124 = 8µ11





p

D3

D1

D4

D2

D1

I =∆11234

D21D2D3D4

+∆1234

D1D2D3D4+

∆1124

D21D2D4

+∆234

D2D3D4+

∆124

D1D2D4

N = 16[µ211 − k21 (k1 · p)] + · · ·




1D2D3D4


D1D2D3D4


D21D3D4

Step 4. / 5. Reducing I124 = N124

D1D2D4I234 = N234

D2D3D4

I124 & I234 irreducible i.e. Nx = ∆x





p

D3

D1

D4

D2

D1

I =∆11234

D21D2D3D4

+∆1234

D1D2D3D4+

∆1124

D21D2D4

+∆234

D2D3D4+

∆124

D1D2D4

N = 16[µ211 − k21 (k1 · p)] + · · ·



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


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]



p

D3

D1

D4

D2

D1

I = I0 + ǫ I1 + ǫ2 I2

Ia =Na

D21D2D3D4


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.



p

D3

D1

D4

D2

D1

I = I0 + ǫ I1 + ǫ2 I2

Ia =Na

D21D2D3D4


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


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


Backup Slides


Maximum cut theorem – proof


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.





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).


Documents

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