Rational terms in two-loop calculations

M. F. Zollerin collaboration with

S. Pozzorini and H. Zhang

RADCOR 2019 – Avignon – 10.09.2019

Motivation: Towards two-loop automation

Example of a full NNLO calculation: gg → tt̄g

︸ ︷︷ ︸LO

q1

︸ ︷︷ ︸NLO

q1 q2

︸ ︷︷ ︸NNLO

Tree and one-loop contributions fully automated in OpenLoops 2 [arxiv:1907.13071] for allLHC processes, including real emission → strong numerical stability and CPU efficiencyDownload: https://gitlab.com/openloops/OpenLoops or https://openloops.hepforge.org

Challenges for numerical two-loop calculation:• Numerical amplitude construction → Numerator of Feynman integral in 4 dimensions• Tensor integral reduction and computation of Master integrals→ computed in d = 4− 2ε dimensions

• Rational terms stemming from discrepancy between Feynman integrands with 4 and d dimensionalnumerators ⇒ one loop: Rational terms of type R2 [Ossola, Papadopoulos, Pittau]this talk: two-loop rational terms

1

https://arxiv.org/pdf/1907.13071.pdfhttps://openloops.hepforge.org/downloads

Rational terms at one loopGeneric one-loop diagram (d-dimensional quantities marked with bar)

Ā1 =∫

dq̄ N̄ (q̄)D̄0(q̄) · · · D̄N−1(q̄)

=

wN−1wN

w1 w2

D̄0

D̄1

D̄2

D̄N−1

q

(scalar propagators D̄i(q̄) = (q̄ + pi)2 −m2i with loop momentum q̄, mass mi and external momentum pi)

Splitting 4 and (d− 4) dimensional parts of the numerator yields

Ā1 = A1 + ∆A1 with ∆A1 = Ā1 −A1 =∫

dq̄ N̄ (q̄)−N (q)D̄0(q̄) · · · D̄N−1(q̄)

,

where N (q) is constructed from 4-dimensional q, γµ, gµν and N̄ (q̄) from d-dimensionalq̄ = q + q̃, γ̄µ̄ = γµ + γ̃µ̃, ḡµ̄ν̄ = gµν + g̃µ̃ν̃

Numerator Ñ = N̄ − N is of order ε⇒ only interaction with 1ε UV poles contributes to finite result [Binoth, Guillet, Heinrich]⇒ ∆A1 is a (finite) rational term, that can be written as a local counterterm[Ossola, Papadopoulos, Pittau]

2

Renormalised amplitude from subtraction of divergencies (R-operation [Caswell, Kennedy])R Ā1 = Ā1 + δZĀ1TA1 (counterterm δZĀ1, tree-level structure TA1)

Rational term can be written as a (local) counterterm ∆A1 = δZR2Ā1TA1

⇒ R Ā1 = A1 + (δZĀ1 + δZR2Ā1

)TA1 =: R4A1

One-loop diagram A1 with 4-dim numerator ⇒ suitable for numerical implementation

Generalisation to two loopsRenormalisation procedure: Subtraction of sub-divergencies from one-loop sub-diagrams γ̄,then of local two-loop divergence:

R Ā2 = Ā2 +∑γ̄

(δZγ̄

)· Ā2/γ̄ + δZĀ2TĀ2

We show that a renormalised d-dim two-loop amplitude can be computed as

R Ā2 = A2 + ∑γ(δZγ̄ + δZR2γ̄

)· A2/γ +

δZĀ2 + δZR2Ā2

TĀ2 =: R4A2.with (extended) one-loop rational terms δZR2γ̄ and a two-loop rational term δZ

R2Ā2⇒ Two and one-loop diagrams A2 and A2/γ constructed with 4-dim numerators

3

Outline

I. Computation of rational terms from tadpole integrals with one scale→ Rational terms of one-loop diagrams and sub-diagrams

II. Rational terms of two-loop diagrams

– without a global divergence

– with a global divergence

III. Full set of QED rational terms at two loops

IV. Summary and Outlook

4

Rational terms from tadpole integrals: The one-loop case

Ā1 =∫

dq̄ N̄ (q̄)D̄0(q̄) · · · D̄N−1(q̄)

=

wN−1wN

w1 w2

D̄0

D̄1

D̄2

D̄N−1

q R Ā1 = Ā1 + δZĀ1TA1

Define difference between diagram with d-dim and 4-dim numerator: ∆A1 = Ā1−A1

Use exact decomposition of propagator denominators D̄i = (q̄ + pi)2 −m2i [Chetyrkin, Misiak, Münz]

1D̄i

= 1q̄2 −M2

+ ∆iq̄2 −M2

1D̄i

= +

with ∆i = −p2i − 2q̄ · pi + m2i −M2 recursively up to a fixed order Z + 1:

1D̄i

=Z∑n=0

(∆i)n

(q̄2 −M2)n+1+ (∆i)

Z+1

(q̄2 −M2)Z+11D̄i

e.g. for Z = 2: = + + +

⇒ Isolate divergencies + rational terms in tadpole integrals with one auxiliary scale M2

Terms with a negative degree of divergence X (naive power counting) cancel in ∆A1.

5

One-loop exampleD = dimension of the loop momenta, metric and γ-matrices in the numerator ⇒ D ∈ {4, d}

∆A1 = D = d − D = 4 = D = d − D = 4 + D = d − D = 4

︸ ︷︷ ︸=O(ε)

+ D = d − D = 4

︸ ︷︷ ︸=O(ε)

+ D = d − D = 4

︸ ︷︷ ︸=O(ε)

+ D = d − D = 4

︸ ︷︷ ︸=O(ε)

+ D = d − D = 4

︸ ︷︷ ︸=O(ε)

+ D = d − D = 4

︸ ︷︷ ︸=O(ε)

+ D = d − D = 4

︸ ︷︷ ︸=O(ε)

=∫

dq̄ N̄ (q̄)−N (q)(q̄2 −M2)3

+O(ε)

6

One-loop rational terms from tadpoles

In general: ∆A1 =N+X∑n=N

∫dq̄

(N̄ (q̄)−N (q)

)Pn(M2, p2i , q̄ · pj)

(q̄2 −M2)npolynomial Pn from decom-position of denominators

Decompose numerators:N̄ (q̄)Pn(M2, p2i , q̄ · pj) =

R∑r=0N̄µ̄1···µ̄r q̄

µ̄1 · · · q̄µ̄r

N (q)Pn(M2, p2i , q̄ · pj) =R∑r=0Nµ1···µrq

µ1 · · · qµr

using q̄ · pj = q · pj for external 4-momenta pj. With AµiBµ̄i = AµiBµi for any vectors A,B

⇒ ∆A1 =N+X∑n=N

R∑r=0

(N̄µ̄1···µ̄r −Nµ1···µr

)I µ̄1···µ̄rn with I µ̄1···µ̄rn =

∫dq̄ q̄

µ̄1 · · · q̄µ̄r(q̄2 −M2)n

• Generic method to compute ∆A1 from tadpole integrals with one (auxiliary) scale M2.• Result independent of M2 since the denominator decomposition is exact and only terms which

cancel in ∆A1 are discarded.• Dependence on external momenta and masses resides only in N̄µ̄1···µ̄r and Nµ1···µr⇒ ∆A1 is a rational term

7

One-loop sub-diagrams in two-loop diagramsOne loop: All external momenta in 4 dimensionsTwo loop: External momentum to a sub-diagram γ̄ can be d-dim loop momentum q̄1• Decomposition of d-dim denominators yields

1(q̄1 + q̄2)2 −m2

= 1q̄22 −M2

+ −q̄21 − 2q̄1 · q̄2 + m2 −M2

q̄22 −M21

(q̄1 + q̄2)2 −m2

⇒∫

dq̄2[N (q2, q1)Pn(q̄21, q̄1q̄2)

]/(q̄22 −M2)n has terms ∝ q21 (from N construction in 4-dim),

∝ q̄21 (from denominator decomposition)

• Difference of d-dim and 4-dim tensor structures t̄ᾱ(q̄1)− tα(q1) multiplied by one-loop UVcounterterm δZγ̄ not negligible due to insertion into a loop diagram

Extended one-loop rational term for one-loop sub-diagram γ̄ with external indices α = (α1α2):

(∆γ)α :=[γ̄ᾱ − γα +

(t̄ᾱ − tα

)δZγ̄

]ᾱ→α

This has the structure ∆γ = R2(γ) + ∆R2(γ)ε with the well-known one-loop R2(γ).

Additional rational term ∆R2(γ) ∝ q̃21 with q̃1 = q̄1 − q1 requires superficial degree ofdivergence = 2 in sub-diagram (e.g. gauge boson self-energy).

8

Example: One-loop self-energy of the photonOne-loop amplitude in d and 4 dimensions from denominator decomposition and tadpole method:

γ̄µ̄ν̄ =∫

dq̄2N̄ µ̄ν̄sub(q̄1, q̄2)

D̄(2)0 (q̄2)D̄

(3)0 (q̄1 + q̄2)

= ie2

16π2

−(4−2ε) q̄21

3εgµ̄ν̄ + 4

3εq̄µ̄1 q̄ν̄1 + C

µ̄ν̄finite

,

γµν =∫

dq̄2N µνsub(q1, q2)

D̄(2)0 (q̄2)D̄

(3)0 (q̄1 + q̄2)

= ie2

16π2

−4q2 − 2q̃213ε

gµν + 43εqµ1 qν1 + C

µνfinite

,

where the constant parts Cµ̄ν̄finite − Cµνfinite = O(ε).

For the difference of the UV counterterms(t̄µ̄ν̄ − tµν

)δZγ̄ =

[(q̄µ̄1 q̄ν̄1 − q

µ1 qν1)−

(q̄21g

µ̄ν̄ − q21gµν)] ie2

16π2−4

3ε

∝ q̃21εgµν for µ̄→ µ, ν̄ → ν

This yield the extended rational term

q̄1 q1 q̄1 q1

(∆γ)µν =

µ̄

ν̄

D = d −

µ

ν

D = 4

+

µ̄

ν̄

−

µ

ν

δZγ̄

µ̄→ µν̄ → ν

= ie2

16π2

q21

23︸ ︷︷ ︸

known R2

+ q̃212

3 ε︸ ︷︷ ︸new ∆R2(γ)/ε

gµν

9

Two-loop diagramsGeneric irreducible two-loop diagram consists of three chains and two connecting vertices V0, V1:

Ā2 =

w(1)1

w(1)2

w(1)N1

D̄(1)0

D̄(1)1

D̄(1)2

D̄(1)N1

w(3)1

w(3)N3

D̄(3)0

D̄(3)1

D̄(3)N3

w(2)1

w(2)2

w(2)N2

D̄(2)0

D̄(2)1

D̄(2)2

D̄(2)N2

V0

V1

q1 q2−q2−q1

=∫

dq̄1∫

dq̄2N̄ ᾱ11 (q̄1) N̄

ᾱ22 (q̄2) N̄

ᾱ33 (q̄3) Γ

V0V1ᾱ1ᾱ2ᾱ3 N1∏

i1=0D̄

(1)i1 (q̄1)

N1∏i2=0

D̄(2)i2 (q̄2)

N1∏i3=0

D̄(3)i3 (q̄3)

with q̄3 = −(q̄1 + q̄2)

Renormalised diagram in d dimensions(what we need)

Splitting in pieces with 4-dim numerators(what we can implement in OpenLoops)∆γ: one-loop rational term insertion

R Ā2 = Ā2 +∑γ̄

(δZγ̄

)· Ā2/γ̄

︸ ︷︷ ︸=:R Ā2

+δZĀ2

R Ā2 = A2 +∑γ

(δZγ̄ + ∆γ

)· A2/γ

︸ ︷︷ ︸=:R4A2

+δZĀ2 + ∆A2

Our task: Compute ∆A2 = R Ā2 − R4A2 and show that it is rational.

10

The case with no global but one subdivergenceUse factorisation and split sub-diagram (chains 2+3) and rest (chain 1) into d-dim and 4-dim parts:

α1

α2

N̄γN̄1

ᾱ = (ᾱ1ᾱ2)Ñ = N̄ − Nᾱ = d-dim indexα = 4-dim indextα = tensor, e.g.qα11 q

α21 − q21gα1α2

R Ā2 =∫

dq̄1N1α(q1) + Ñ1 ᾱ(q̄1)

D̄(1)0 · · · D̄

(1)N1︸ ︷︷ ︸

no divergence

∫

dq̄2N̄ ᾱγ (q̄1, q̄2)

D̄(2)0 · · · D̄

(2)N2D̄

(3)0 · · · D̄

(3)N3

+ t̄ᾱδZγ̄

︸ ︷︷ ︸

finite → discard Ñ1 ᾱ

=∫

dq̄1N1α(q1)

D̄(1)0 · · · D̄

(1)N1

∫

dq̄2Nαγ (q1, q2)

D̄(2)0 · · · D̄

(3)N3

+ tαδZγ̄ + (∆γ)α +O(ε)

with (∆γ)α =∫

dq̄2Ñαγ (q1, q2)

D̄(2)0 · · · D̄

(3)N3

+(t̄ᾱ − tα

)δZγ̄ = R2(γ) +

∆R2(γ)ε

Diagram can be computed with 4-dim numerators:R Ā2 = A2 +A2/γ

(δZγ̄ + ∆γ

)=: R4A2

• No two-loop rational term, i.e. ∆A2 = 0• One-loop rational terms R2(γ) needed only to order ε0

• Additional q̃21 term only in one-loop integrals → need to be computed

11

The case with a global divergence

• Define for each chain i = 1, 2, 3 the maximum degree of divergence of the full diagram(X ≤ 0) and any sub-diagram constructed from chains i, j 6= i (degree of sub-divergence Xij):

Z1 = Max (X,X12, X13) , Z2 = Max (X,X12, X23) , Z3 = Max (X,X23, X13)

• Use exact decomposition of propagator denominators on each chain i up to order Zi+ 1:1

D(i)0 (q̄i) · · ·D

(i)Ni

(q̄i)=

S(i)Zi + F(i)Zi

1D

(i)0 (q̄i) · · ·D

(i)Ni

(q̄i),

with S(i)Zi selecting all terms contributing to a (sub-)divergence,

S(i)Z1

D(i)0 (q̄i) · · ·D

(i)Ni

(q̄i)=

Zi∑n=0

P(i)n (q̄i, p(i)j )(

q̄2i −M2)N+1+n ← tadpoles with scale M2

and F(i)Zi selecting all other terms, especially those with at least one original propagator D(i)j

• In particular, terms selected by F(i)Zi have no global divergence ⇒ Apply previous case!

12

The case with a global divergenceExact decomposition ⇒ isolate divergent parts ⇒ Discard terms which cancel from calculation:

∆A2 =S(1)Z1 + F

(1)Z1

S(2)Z2 + F(2)Z2

S(3)Z3 + F(3)Z3

∆A2 = ∆A2,rat + 3∑i=1

∆A(i)2,sub + ∆A2,fin ,

with ∆A2,rat = S(1)Z1S

(2)Z2S

(3)Z3∆A2 ,

∆A(i)2,sub = F(i)ZiS(j)ZjS

(k)Zk

∆A2 ,

∆A2,fin = F(1)Z1F

(2)Z2F

(3)Z3∆A2

+3∑i=1

S(i)ZiF(j)ZjF(k)Zk∆A2 .

← only one-scale tadpoles

← no global divergence, at most onesub-divergence ⇒ O(ε)

← no global or sub-divergence ⇒ O(ε)

⇒ ∆A = ∆A2,rat is rational! Compute as in one-loop case from

∆A2 =Ni+1+Zi∑ni=Ni+1

∫dq̄1

∫dq̄2

(N̄ (q̄1, q̄2)−N (q1, q2)

)Pn1n2n3(M2, p2i , qi · pj)

(q̄21 −M2)n1 (q̄22 −M2)n2 (q̄23 −M2)n3

=Ni+1+Zi∑ni=Ni+1

R∑r=0

S∑s=0

(N̄µ̄1···µ̄rν̄1···ν̄s −Nµ1···µrν1···νs

)I µ̄1···µ̄rν̄1···ν̄sn1n2n3

with I µ̄1···µ̄rν̄1···ν̄sn1n2n3 =∫dq̄1∫dq̄2 q̄

µ̄11 ···q̄

µ̄r1 q̄

ν̄12 ···q̄

ν̄s2

(q̄21−M2)n1 (q̄22−M2)n2 (q̄23−M2)n3

13

Example: QED vertex correctionLet D ∈ {4, d} be the numerator dimension. Decomposing chain 1 (with external fermion line):

∆A2 =

D = d + D = d δZγ

−

D = 4 + D = 4 (δZγ + ∆γ)

=

D = d + D = d δZγ

−

D = 4 + D = 4 (δZγ + ∆γ)

+

D = d + D = d δZγ

−

D = 4 + D = 4 (δZγ + ∆γ)

︸ ︷︷ ︸

=0+O(ε) (case without global divergence)

+more terms with original propagator denominators along outer chain︸ ︷︷ ︸=0+O(ε) (case without global divergence)

black lines: original propagators D̄(i)j ; red lines: factors in denominator decomposition ∝ (q̄2i −M2).

14

Example: QED vertex correctionDecomposing chains 2,3 (photon self-energy sub-diagram):

∆A2 =

D = d + D = d + D = d + D = d + D = d

+ D = d + D = d δZγ

−

D = 4 + D = 4 + D = 4

+ D = 4 + D = 4 D = 4 + D = 4 (δZγ + ∆γ)

+ D = d − D = 4

︸ ︷︷ ︸=0 (case without global divergence)

+ D = d − D = 4

︸ ︷︷ ︸=0 (case without global divergence)

+ . . .︸ ︷︷ ︸=0

Only massive tadpoles contribute to difference ∆A2.

15

Rational terms for QED in MS scheme (ξ = 0, m = 0)Calculation in the GEXCOM [Chetyrkin, M.Z.] framework: QGRAF [Noguira] → Q2E+EXP[Seidesticker, Harlander, Steinhauser] → FORM [Vermaseren] code → MATAD [Steinhauser]

p

∆A1,e = ie2

16π2 [−1] /p, ∆A2,e = ie4

(16π2)2[ 1918ε +

247108

]/p

p

µ ν ∆Aµν1,γ = ie2

16π2[23p

2 + 23εp̃2]gµν, ∆Aµν2,γ = ie

4

(16π2)2[(pµpν − gµνp2)

( 23ε −

7118)

+ gµνp2(−1112

)]

µ

∆Aµ1,eeγ = ie3

16π2 [−2] γµ, ∆Aν2,eeγ = ie

5

(16π2)2[139ε +

19127

]γµ

µ ν

ρσ ∆Aµνρσ1,4γ = ie4

16π2[43]tµνρσ, ∆Aµνρσ2,4γ = ie

6

(16π2)2 [−3] tµνρσ

with tµνρσ = gµνgρσ + gµρgνσ + gµσgνρ

• ∆Al are polynomial in p and independent of M 2 ⇒ Rational terms

• One-loop rational term for photon self-energy extended with term ∝ p̃2/ε

• Full dependence on ξ and electron mass m in upcoming paper16

Summary and Outlook

• Renormalised diagrams in d-dimensions can be split into objects with 4-dimensional numerators:

R Ā2 =A2 + ∑γ

(δZγ̄ + ∆γ

)· A2/γ

+ ∆A2with (extended) one-loop rational terms ∆γ and a two-loop rational term ∆A2.⇒ Numerical implementation in automated tools, e.g. OpenLoops, possible

• We present a generic method to compute ∆Al from tadpoles with one auxiliary scale M2,which also serves as a proof that ∆Al is rational

• Full set of QED rational terms at two-loop level

• Further tasks:

– Compute necessary one-loop integrals involving q̃2 to order ε

– Implementation in OpenLoops

17

Backup

18

Reducible two-loop diagrams

w(1)1

w(1)2

w(1)N1−2 w(1)N1−1

D̄(1)2

D̄(1)N1−2

D̄(1)N1−1

D̄(1)0

D̄(1)1

q1Pσ1σ2

w(2)N2−2

w(2)N2−1

w(2)1

w(2)2

D̄(2)N2−1

D̄(2)0

D̄(2)1

D̄(2)2

D̄(2)N2−2

q2

The one-loop procedure is directly applicable for reducible two-loop diagrams (two one-loop diagramsconnected by a bridge P ) due to the factorisation

RĀ2 = R

∫

dq̄1Tr

N̄ (1)(q̄1)σ1

D̄(1)0 (q̄1) · · · D̄

(1)N1(q̄1)

Pσ1σ2R∫

dq̄2Tr

N̄ (2)(q̄2)σ2

D̄(2)0 (q̄2) · · · D̄

(2)N2(q̄2)

= R4

∫

dq̄1Tr

N (1)(q1)σ1

D̄(1)0 (q̄1) · · · D̄

(1)N1(q̄1)

Pσ1σ2R4∫

dq̄2Tr

N (2)(q2)σ2

D̄(2)0 (q̄2) · · · D̄

(2)N2(q̄2)

and the fact that both terms R [...] are finite.

19

Example with two sub-divergencesγ1: upper photon-fermion loop; γ2: lower photon-fermion loop.Contributions with one original propagator D̄(i)j (black) in sub-diagram γ̄2:

0 =

D = d + D = d + D = d + D = d

+ D = d δZγ̄1

−

D = 4 + D = 4

+ D = 4 + D = 4 + D = 4 (δZγ̄1 + ∆γ1)

,

20

Example with two sub-divergences

Contributions with one original propagator D̄(i)j (black) in sub-diagram γ̄2:

0 =

D = d + D = d + D = d + D = d

+ D = d δZγ̄2

−

D = 4 + D = 4

+ D = 4 + D = 4 + D = 4 (δZγ̄2 + ∆γ2)

,

Contributions with one original propagator D̄(i)j (black) in both sub-diagrams γ̄1, γ̄2:

0 = D = d − D = 4 .

21