Norma Selomit Ramírez UribeIFIC CSIC-UV.

PARTICLEFACE 2021: Working Group Meeting and Management Committee Meeting

July 15, 2021.

2

Outline

q Motivation

q Loop-Tree Dualityq Brief historyq Notation

q 𝑁"MLT universal topology

q A look at quantum

3

Motivation

Improve theoretical predictions

Achieve higher perturbative orders

Quantum fluctuations at high-energy scattering processes= Multiloop scattering amplitudes =

Loop diagrams

…

4

Minkowski space Euclidean space

LTD-brief history

q What does LTD do?

Opens any loop diagram to a forest of non-disjoint trees.

q How does it do?

Exploits the Cauchy residue theorem to reduce the dimensionof the integration domain by one unit:

# #𝑑𝑞&

�

�

(𝐺*(𝑞,).

,/0𝒒= −2𝜋𝑖 # 7𝑅𝑒𝑠{<=?@A&}

�

𝒊

(𝐺*(𝑞,).

,/0?

5

LTD-brief history

oBuchta, Chachamis, Draggiotis, Malamos and Rodrigo, JHEP 1411 (2014) 014

oHernández, Sborlini and Rodrigo, JHEP 1602 (2016) 044

oS. Buchta, G. Chachamis, P. Draggiotis and G. Rodrigo, EPJC 77 (2017) 274

Duality relation (LTD)

Singular behaviourat one-loop

FDU

Applications

oCatani, Gleisberg, Krauss, Rodrigo and Winter, JHEP 0809 (2008) 065

oBierenbaum, Catani, Draggiotis and Rodrigo, JHEP 1010 (2010) 073

oBierenbaum, Buchta, Draggiotis, Malamos and Rodrigo, JHEP 1303 (2013) 025

oDriencourt, Rodrigo and Sborlini, EPJC 78 (2018) no.3, 231

oDriencourt, Rodrigo, Sborlini and Torres, JHEP 1902 (2019) 143

oSborlini, Driencourt, Hernández and Rodrigo, JHEP 1608 (2016) 160

oSborlini, Driencourt and Rodrigo, JHEP 1610 (2016) 162

oAguilera, Driencourt, Plenter, Ramírez, Rodrigo, Sborlini, Torres and Tracz,JHEP 1912 (2019) 163

Causal and anomalous thresholds

oPlenter and Rodrigo, EPJC 81 (2021) 320 Asymptotic expansions

6

LTD-brief historyoRunkel, Szor, Vesga and Weinzierl, Phys. Rev. Lett. 122 no.11, 111603 Erratum:

[Phys. Rev. Lett. 123 no.5, 059902]

oRunkel, Szor, Vesga and Weinzierl, Phys. Rev. D 101 (2020) 116014

oCapatti, Hirschi, Kermanschah and Ruijl, Phys. Rev. Lett. 123 (2019) no.15, 151602

oCapatti, Hirschi, Kermanschah, Pelloni and Ruijl, JHEP 04 (2020) 096

Alternative dual

representation

Reformulation of LTD to all orders

oAguilera, Driencourt, Hernández, Plenter, Ramírez, Rodrigo, Sborlini, Torres, Tracz, “Open loop amplitudes and causality to all orders and powers from the loop-tree duality”, Phys. Rev. Lett. 124 (2020) no.21, 211602

q Can we find explicit and more compact analytic expressions withthe LTD formalism to all orders?

Causality

7

LTD-brief history𝑀𝐿𝑇 𝑁𝑀𝐿𝑇 𝑁G𝑀𝐿𝑇

oS. Ramırez-Uribe, R. J. Hernández-Pinto, G. Rodrigo, G. F. R. Sborlini, and W. J. Torres Bobadilla, “Universal opening of four-loop scattering amplitudes to trees,” JHEP 04, 129 (2021).

Multiloop topologies that first appear at four loops

8

LTD-Notation

q A generic 𝐿-loop scattering amplitude with𝑁 external legs,

𝒜.I 1,… , 𝑛 = # 𝒜*

I 1,… , 𝑛ℓ𝟏,..,ℓ𝑳

#ℓR

= −𝑖𝜇"TU #𝑑Uℓ,2𝜋 U

�

�

𝒩 ℓ, I, 𝑝X .𝐺*(1,… , 𝑛)

( 𝐺* 𝑞,YR

�

,∈0∪⋯∪]

𝑞,G − 𝑚,G + 𝜄0 T0

9

LTD-Notation

q The LTD representation is written in terms of nested residues,

𝒜bI 1,… , 𝑟; 𝑟 + 1,… , 𝐿 + 1 = −2𝜋𝑖 7 𝑅𝑒𝑠 𝒜b

I 1,… , 𝑟 − 1; 𝑟, … , 𝑛 , 𝐼𝑚𝜂 g 𝑞,h < 0�

𝒊𝒓∈𝒓On-shell Off-shell

starting from

𝒜bI 1; 2, … , 𝐿 + 1 = −2𝜋𝑖 7 𝑅𝑒𝑠 𝒜*

I 1,… , 𝐿 + 1 , 𝐼𝑚𝜂 g 𝑞,k < 0�

𝒊𝟏∈𝟏

𝜂l = 1, 𝟎

10

𝑵𝟒𝑴𝑳𝑻 universal topology

12 3

4

5

12

23

123

234

L+ 1

1 2 3

4

5

12

123

34234

L+ 1

1 2 3

4

5

12

123

234

24

L+ 1

• 𝑞,r = ℓs + 𝑘,r, 𝑠 ∈ 1, … , 𝐿• 𝑞,(uvk) = −∑ ℓsI

s/0 + 𝑘, uvk• 𝑞,kx = −ℓ0 −ℓG +𝑘,kx• 𝑞,kxy = −ℓ0 −ℓG −ℓz + 𝑘,kxy• 𝑞,xy{ = −∑ ℓs"

s/G + 𝑘,xy{• 𝑞,hr = −ℓ| − ℓs + 𝑘,hr, 𝑟, 𝑠 ∈ 2, 3, 4

q Multiloop topologies that appear for the first time at four loops:

𝐽 ≡ 23 ∪ 34 ∪ 24

Can we achieve a unified description?

t-, s- and u- kinematic channels

11

𝑵𝟒𝑴𝑳𝑻 universal topology

= 𝒜.{�I�" 1, 2, 3, 4, 12, 123, 234, 𝐽 ⊗𝒜�I�

IT" 5,… , 𝐿 + 1+𝒜.x�I�

z 1 ∪ 234, 2, 3, 4 ∪ 123, 12, 𝐽 ⊗𝒜�I�ITz 5�, … , 𝐿 + 1

Momentum flow reversed

J

1 2 3

4

5

12

123

234

L+ 1

=

J

1

12

2 3

123

234 4⊗

5

6

7

L+ 1

+

J

1

12

2 3 123

234 4

⊗

5

6

7

L+ 1

𝒜.{�I�I 1,… , 𝐿 + 1, 12, 123, 234, 𝐽

On-shell

12

𝑵𝟒𝑴𝑳𝑻 universal topology

𝒜.{�I�" 1, 2, 3, 4, 12, 123, 234, 𝐽 = 𝒜.x�I�

" 1, 2, 3, 4, 12, 123, 234 ⊗𝒜 & 𝐽

+7𝒜b" 1, 2, 3, 4, 12, 123, 234, 𝒔

�

𝒔∈�

Off-shell

J

1

12

2 3

123

234 4

=

1

2

12

234

123

3

4+

1

12

2 3

23

123

234 4

+

1

12

2 3

123

234 4

34+

1

12

2 3

123

4 234

24

{𝟐𝟑, 𝟑𝟒, 𝟐𝟒}

13

𝑵𝟒𝑴𝑳𝑻 universal topology

J

1

12

2 3123

234 4

=

1

12

2 3 123

234 4

+

1

12

2 3 123

234 4

23

+

1

12

2 3123

234 4

34

+

1

12

2

123 3

4

234

24

𝒜.x�I�z 1 ∪ 234, 2, 3, 4 ∪ 123, 12, 𝐽 = 𝒜.�I�

z 1 ∪ 234, 2, 3, 4 ∪ 123, 12 ⊗𝒜 & 𝐽

+7𝒜bz 1, 2, 3, 4, 12, 123, 234, 𝒔

�

𝒔∈�

14

Causal representation

= # 𝒜.y�I�" (1, 2, 3, 4, 5, 12, 123, 234)

ℓ𝟏,..,ℓ𝑳

= #𝒩.y�I� 𝑞s,&

(�), 𝑘X,&

∏ 2𝑞s,&(�)I�"

s/0 ∏ 𝜆,�0z,/0 𝜆,Tℓk,..,ℓu

= # �𝒜.x�I�" 1, 2, 3, 4, 12, 123, 234

ℓk,..,ℓu

+ 𝒜.�I�z 1 ∪ 234, 2, 3, 4 ∪ 123, 12 ⊗𝒜b

0 5� �

Universal opening

Adding them all together

?

15

Causal representation

1

2 3

4

5λ+1

1

12

3

4

234

λ+2

2

12

123

4

234

λ+3

2 3

123234

5λ+4

2 3

4234

λ+5

2 3

1231

λ+6

12 3

4

5λ+7

1 123

234 4

λ+8

122

234

5λ+9

1

2

12

λ+10

123

3

12

λ+11

123

4

5λ+12

1

234

5λ+13

= #1

𝑥𝑳�𝟒ℓk,..,ℓu

7𝒩𝝈(𝒊𝟏,…,𝒊𝟒) 𝑞s,&

(�), 𝑘X,&𝜆𝝈(𝒊𝟏)𝜆𝝈(𝒊𝟐)𝜆𝝈(𝒊𝟑)𝜆𝝈(𝒊𝟒)

�

�

Reinterpreting in terms of four entangled thresholds

( 2𝑞s,&(�)I�"

s/0

= LTD Causal representation =

16

Causal representation

2 4 6 8 10

2

4

6

8

10

q12,0(+)

q 123,0

(+)

5.×10-6

0.000010

0.000015

0.000020

0.000025LTD representation

LTD causal representation

Clever analytical rearrangement

Impact of noncausal singularities

Integrand-level behaviour of the noncusal LTD representation of a four-loop 𝑁z𝑀𝐿𝑇 diagram.

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Causal representation

2

q12,0(+)

1.0620058x10-5

2

q12,0(+)

1.0620058x10-5

Numerical instabilities of the four-loop 𝑁z𝑀𝐿𝑇 intgrand arising due to noncausal singularities (left), which are absent in the causal representation (right).

Noncausal and causal evaluations of the 𝑁z𝑀𝐿𝑇 configurations.

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Causal representation

2 3

23

λ+14

4

23

234

λ+15

1 12323

λ+16

1

3

12

23

λ+17

1

45

23

λ+18

2

12

12323

λ+19

5

123

234

23

λ+20

2

4

5

12

23

λ+21

3

5

12

234

23

λ+22

1

24

4 3

123

λ(u)+16

1

24

5

12

123

λ+23

1

2

123

24

234

λ+24

𝒜.{�I�I 1, … , 𝐿 + 4, 𝐽 = #

1𝑥 �,s,� ,𝑳�𝟓

ℓk,..,ℓu

7𝒩𝝈(𝒊𝟏,…,𝒊𝟓) 𝑞s,&

(�), 𝑘X,&𝜆𝝈(𝒊𝟏)𝜆𝝈(𝒊𝟐)𝜆𝝈(𝒊𝟑)𝜆𝝈(𝒊𝟒)𝜆𝝈(𝒊𝟓)

�

�

2𝑞{Gz,z",G"},&(�) 𝑥I�"Extra causal configurations of the 𝑡 −channel

Extra causal configurations of the 𝑢 −channel

19

A look at quantum

oS. Ramírez-Uribe, A. E. Rentería-Olivo, G. Rodrigo, G. F. R. Sborlini, and L. Vale Silva, “Quantum algorithm for Feynman loop integrals”, arXiv:2105.08703 [hep-ph]

Bootstrap the causal representation in the LTD of representative multiloop topologies.

0 1

2

0 1

2

34

0 1

2

3

4

5

0

1

2

3

4

56

7

0

1

2

3

4

5 6

7

8

0

1

2

3

4

5 6

7

8

0

1

2

3

4

5 6

7

8

Two possible states: |1⟩𝑜𝑟|0⟩

20

A look at quantum1. Superposition 𝑁 = 2]

2. Oracle

3. Diffusion

Grover’s algorithm

|𝑞⟩ =1𝑁�7|𝑥⟩

|𝑞⟩ = cos 𝜃 |𝑞�⟩ + sin 𝜃 |𝑤⟩o Superposition

Orthogonal state

Winning state

Mixing angle

𝜃|𝑞�⟩

|𝑤⟩|𝒒⟩

|𝑤⟩ =1𝑟�7|𝑥⟩�

¢∈£

|𝑞�⟩ =1𝑁 − 𝑟� 7|𝑥⟩

�

¢∈£

θ = arcsin 𝑟/𝑁�

21

A look at quantum

o Oracle 𝑈£ = 𝑰 − 2|𝑤⟩⟨𝑤| 𝑈£|𝑥⟩ = «−|𝑥⟩ 𝑖𝑓𝑥 ∈ 𝑤

|𝑥⟩ 𝑖𝑓𝑥 ∉ 𝑤

𝜃|𝑞�⟩

|𝑤⟩

2𝜃

−|𝑤⟩𝑼𝒘|𝒙⟩

Flips the state |𝑞⟩ if 𝑞 ∈ 𝑤 and leaves it unchanged otherwise.

22

A look at quantum

o Diffusion 𝑈? = 2|𝑞⟩⟨𝑞| − 𝑰

𝜃|𝑞�⟩

|𝑤⟩

2𝜃

−|𝑤⟩

4𝜃

𝑈?𝑈£|𝑞⟩

Performs a reflection around theinitial state |𝑞⟩

The iterative application of the oracle and diffusion operators t times leads to:

𝑈?𝑈£�|𝑞⟩ = cos 𝜃� |𝑞�⟩ + sin 𝜃� |𝑤⟩

23

A look at quantum

4𝜃

𝜃|𝑞�⟩

|𝑤⟩

2𝜃

−|𝑤⟩

𝑈?𝑈£|𝑞⟩

To consider: 𝜃 ≤ 𝜋/6 𝑟/𝑁 ≤ 1/4 ?

78/256 204/512 230/512204/512

39/256 102/512 102/512 115/512

24

A look at quantum

|𝑞⟩

|𝑐⟩

|𝑎⟩

o Oracle

25

A look at quantum|00000111〉 |00101101〉 |01111011〉 |11110101〉

|000001011〉 |100111001〉 |111000111〉 |001111001〉

|000101101〉 |010100111〉 |110111101〉 |111100011〉

|000101001〉 |100001101〉 |000100101〉 |111000111〉

𝟑𝟗/𝟐𝟓𝟔

𝟏𝟎𝟐/𝟓𝟏𝟐

𝟏𝟎𝟐/𝟓𝟏𝟐

Requires 33 qubits > Qiskit capacity

26

Conclusions

q We have obtained a very compact dual representation for selected looptopologies to all orders up to four loops, which exhibits a nested form interms of simpler topologies. We conjecture that this factorization works atall orders.

q The 𝑁"𝑀𝐿𝑇 universal topology allow us to describe any scatteringamplitude up to four-loop.

q The causal LTD representation is interpreted in terms of entangledcausal thresholds and allows a more efficient numerical evaluation ofmultiloop scattering amplitudes.

q Causal configurations of multiloop Feynman integrals have beenidentified with the application of Grover’s quantum algorithm.

27