Asymptotic expansions and causal representations through the loop-tree duality

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Asymptotic expansions and causal representations through the loop-tree duality

Judith Plenter

in collaboration with Germ´an Rodrigo

Instituto de F´ısica Corpuscular - Universitat de Val`encia/CSIC

21st March 2022

XIII CPAN Days, Huelva

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Motivation

precision calculations:

need hints where exactly the SM is insufficient

one of the points of interest: the Higgs sector→necessarily includes calculation withvarious scales(at the very leastmt,mH and Mandelstam variables)

but: difficulty of solving master integrals within DREG increases significantly with additional energy scales, extra loops or external legs

example: highly-boosted Higgs production. interesting in order to rule out an effective point-likeggH-coupling. first calculations at NLO (two loops) only recently (numerical or using expansions in the IBPs)

[Jonas, Kerner, Luisoni ’18, Lindert et al. ’18]

⇒ to obtain independent NLO result for this and similar processes: develop formalism for asymptotic expansionswithin new regularization technique

Judith Plenter (IFIC-UV/CSIC) Expansions through LTD 21st March 2022 2 / 15

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Motivation

Loop-tree duality:

Loop-tree duality (LTD) is still a new method, opportunity to explore possible applications

formalism allows to conveniently identify and interpret divergences in the integrand

integrands are functions of Euclidean and not Minkowski momenta→allows to determine hierarchies between the scales and scalar products and thus opens the possibility for well-defined asympotic expansions

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Loop-tree duality theorem (LTD)

use Cauchy residue theorem: R

`₀ →P

residues

LTD theorem

[Catani et al. ’08]

Z

`

N(`)Y

i

G_F(q_i) =−X

i

Z

`

N(`)

∼

δ(q_i)Y

j6=i

G_D(q_i;q_j),

∼

δ(qi) = 2πi θ(qi,0)δ(q_i²−m²_i) dual propagator:

GD(qi;qj) = 1

q_j²−m²_j −i0η·(qj−qi

| {z }

k_ji

)

η may be any future-like vector, typically chooseη= (1,0) with the energy-component integrated out, the remaining loop three-momentum is Euclidean:

∼

δ(qi)q_j²=

∼

δ(qi)

k_ji²+ 2kji,0q⁽⁺⁾_i,0 −2kji·qi+m²_i

q_i,0⁽⁺⁾= q

m_i²+ qi2

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divergences in the dual integrand

most important property of amplitudes: their analytic structure LTD amplitude allows to easily identify divergences and their relevance

dual integrand: −X

i

Z

`

N(`)

∼

δ(qi)Y

j6=i

GD(qi;qj)

consider the on-shell dual propagator:

∼

δ(qi)GD(qi;qj)∼ θ(qi,0)δ(q_i²−m²_i)

q_j,0−q⁽⁺⁾_j,0 q_j,0+q_j,0⁽⁺⁾

−i0η·k_ji

, kji=qj−qi

=

δ

qi,0−q⁽⁺⁾_i,0 2q⁽⁺⁾_i,0λ⁺⁻_ij λ⁺⁺_ij −i0η·kji

, λ^±±_ij =±q_i,0⁽⁺⁾±q_j,0⁽⁺⁾+kji,0

causal unitarity threshold represented byλ⁺⁺_ij →0

unphysical thresholds appear atλ⁺⁻_ij →0in two cuts at once andcancel conditions for these limits to occur are easily derived using this notation

[Aguilera-Verdugo, Driencourt-Mangin, JP, Ram´ırez-Uribe, Rodrigo, Sborlini, Torres Bobadilla, Tracz ’19]

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the causal representation

towards the multiloop case: multiple applications of the residue theorem are unnecessarily complicated when a multitude of unphysical singularities is present

found apurely causal LTD representationof the multiloop sunrise diagram:

themaximal loop topology A^(L)_MLT=

Z

1,...,L

G_F(1, . . . ,L+ 1) =

=− Z

~1,...,~L

1 x_1,L

1 λ⁻_1,L + 1

λ⁺_1,L

!

, x1,L=

L+1

Y

i=1

2q⁽⁺⁾_i,0

causal singularities are encoded inλ^±_1,L=PL

i=1q_i,0⁽⁺⁾±p

for the description of amplitudes withL≥3 further topologies needed

⇒ express through convolutions of the MLT and tree-level amplitudes, the causal representation effects a variety of multiloop amplitudes

[Aguilera-Verdugo, Driencourt-Mangin, Hern´andez-Pinto, JP, Ram´ırez-Uribe, Renter´ıa Olivo, Rodrigo, Sborlini, Torres Bobadilla, Tracz ’20]

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Asymptotic expansions through LTD: overview

apply loop-tree duality theorem to amplitude

expand dual integrand: integrand-level convergence

integrate: integral-level convergence

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Expansion of the dual propagator

for expansion dual propagator may be written as G_D(q_i;q_j) = 1

2qi·kji+m²_i −m²_j +k_ji²−i0η·kji

, Γ_ij+ ∆_ij =m_i²−m²_j +k_ji² define parameters s.t.: |2qi·kji+ Γij| |∆ij|

=

∞

X

n=0

(−∆ji)ⁿ

(2qi·kji+ Γji−i0η·kji)ⁿ⁺¹

additional condition on parameters: assure that there areQ_i²andrij s.t.

2q_i·k_ji+ Γ_ij =Q_i²(r_ij+x_i) r_ij+x_i⁻¹

, |q_i|=m_i

2 x_i−x_i⁻¹

⇒ leads to integrals of the formR∞ 1

dxi

xi(xi+rij)(x_i⁻¹+rij) = ^log(r_r2 ^ij⁾

ij−1, with|rij|<1

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the scalar two-point function

whole range of problems in a physically sensible amplitude: many scales,≥2 loops, multiple legs (angular dependence of the integrand), above threshold

⇒ studybenchmark toy amplitude

A⁽¹⁾= Z

`

G_F(`;m)G_F(`−p;M)

`

`−p

various limits to be studied:

I large mass: M²m²,p²

I large external momentum:

p²M²,m²

I threshold:

β= 1−_(m+M)^p² 2 →0

renormalization through local UV counter-term:

A^(1,R)=A⁽¹⁾−A^(1,cnt)_UV =

∞

Z

0

d|`|a(`)

A^(1,cnt)_UV = Z

`

(GF(`;µUV))²

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general asymptotic expansion of the scalar two-point function

using the expansion of the dual propagator one finds:

A^(1,R)= 1 16π²

"

2 + p²+m²−M²

p² logµ_UV m

+p²+M²−m²

p² logµ_UV M

−m² Q₁²

∞

X

n=0

c_r⁽ⁿ⁾

12 log(r₁₂) +c₁⁽ⁿ⁾

−M² Q₂²

∞

X

n=0

c_r⁽ⁿ⁾

21 log(r₂₁) +c₁⁽ⁿ⁾ #

the integrals are of the type

∞

Z

1

dx

(r+x)(rx+ 1) +i0 = log(r+i0)

r²−1 , ∀ |r|<1 so the coefficients are proportional to (r²−1)⁻¹

[JP, Rodrigo ’21]

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general asymptotic expansion of the scalar two-point function

using the expansion of the dual propagator one finds:

A^(1,R)= 1 16π²

"

2 + p²+m²−M²

p² logµUV

m

+p²+M²−m²

p² logµUV

M

−m² Q₁²

∞

X

n=0

c_r⁽ⁿ⁾₁₂ log(r12) +c₁⁽ⁿ⁾

−M² Q₂²

∞

X

n=0

c_r⁽ⁿ⁾₂₁ log(r21) +c₁⁽ⁿ⁾ #

local renormalization required only for the first two terms of the expansion further orders only modify thesums of coefficients(i.e. higher-order terms can be calculated numerically in 4 dimensions)

solution is general without need to specify a limit divergences in a propagator correspond torij <0

[JP, Rodrigo ’21]

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large mass expansion

consider explicitly the large mass limit for the scalar two-point function:

M²m²,p²

`

`−p the two appearing dual propagators are:

G_D(`;q₁) = 1

−2`·p+p²+m²−M²

=

∞

X

n=0

−m²−p²−M²r₁²ⁿ (−2`·p−M²(1 +r₁²))ⁿ⁺¹

G_D(q₁;`) = 1

2q1·p+p²−m²+M²

=

∞

X

n=0

m²ⁿ

(2q1·p+M²(1 +r₂²))ⁿ⁺¹

r₁= m²p² M⁴

1 2 |l|

10-3

10^-4

10-5 rel. error

n=0 n=1

n=2 r₂= p²

M² no need to integrate, just use the formula provided before

relative error of result forA^(1,R)₁ at ordern= 1,2,3 :2.7%,0.04%,0.0008%

forM/m= 10, p²/m²= 3, µ_UV=M

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large external momentum expansion

large external momentum: p² {m²,M²} p

p²/m= 10,µ_UV=M, expanded integrand imitates the singularity: p

p²/M= 2, ı0 =ı10⁻³

full n=0 n=1 n=2

0 5 10 15 20 25 30

-0.002 0.000 0.002 0.004 0.006 0.008 0.010

l/m Re a(l)/m

full n=0 n=1 n=2

0 5 10 15 20 25 30

0 2.×10^-8 4.×10^-8 6.×10^-8 8.×10-8

l/m Im a(l)/m

good convergence of the integrated expansion:

Re Im Re, rel. error Im, rel. error n= 1 0.008 12 0.014 62 2.5 % 0.24 % n= 2 0.007 94 0.014 60 0.20 % 0.061 % n= 3 0.007 92 0.014 59 0.023 % 0.016 %

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alternative: asymptotic expansion by dual regions

loop energy in the LTD integrand is fixed, remainingloop three-momentum is Euclidean⇒no cancellations within scalar products

any Taylor expansion makes implicit assumptions about the size of`

⇒ combine variousdual regions: expansion converges at integrand-level A^(1,R)=

Z ∞

0

d|`|a(`)

= Z λ

0

d|`| Ta(M,∞) + Z ∞

λ

d|`| Ta({`,M},∞)

full n=1 n=2 n=3

m p0 M/2

0.00606 0.00608 0.00610 0.00612 0.00614 0.00616

λ A^(1,R)

full n=1 n=2

m p0 λ M

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005

|l

| a(l

)/m

determine the matching scale λ through extrema in the result

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tested applications of the expanded dual propagator

scalar two-point function in various limits (large mass, large external momentum, threshold limit both below and above)

one more propagator in the loop:

thescalar three-point function

multiloop example: the Maximal-Loop-Topology in the limitp²m²_s

A^(L)_MLT p²

= =−2

∞

X

n=0

(p²)ⁿ Z

~`₁,...,~`_L

λ⁰_L+1−1−2n

xL+1

λ⁰_L+1=PL+1 s=1q⁽⁺⁾_s,0

highly-boosted Higgs production two limits tested:

large mass, large center-of-mass energy

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Conclusion

divergences in LTD:

the dual integrand allows the straight-forward identification of different types of divergences

unphysical divergences cancel between dual contributions

the novel causal representation of LTD allows to write the dual integrand explicitly free of unphysical divergences

well-defined asymptotic expansions of Feynman amplitudes at the integrand-level:

loop-tree duality allows to rewrite loop integrals in terms of a sum of phase-space integrals over the cut amplitude

⇒ the resulting expression can be expanded more straight-forwardly

general result obtained for scalar two-point function, applicable for different types of limits

succesful expansions for scalar three-point function, the maximal-loop-topology andq¯q→Hg

The project that gave rise to these results received the support of a fellowship from ”la Caixa” Foundation (ID 100010434). The fellowship code is LCF/BQ/IN17/11620037.

This project has received funding from the European Union’s 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No. 713673.

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backup slides

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expansion parameters for scale hierarchies

1

2q_i·k_ji+Γ_ji+ ∆_ji−ı0 η·k_ji =

∞

X

n=0

(−∆ji)ⁿ

(2qi·kji+Γji−ı0 η·kji)ⁿ⁺¹ if the chosen process and limit contains ahierarchy between scalesa set of simple rules can be followed to determine the parameters

1 withQ_i²=±Q²andQ the large scale Γji=Qi²

1 +rij2

, ∆ji=mi²−m²j +kji²−Γji, rij=mikji,0

Q_i²

2 fix the sign ofQ_i²: it directly determines the sign ofrij (always reproduce the analytic structure of the original amplitude)

no integrations necessary, just insert the determined parametersrij,Qi in the solution

imaginary part can be absorbed asrij = ^mⁱ_Q^k^ji,02 i

+ı0

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coefficients of the asymptotic expansion for the scalar two-point function in different kinematical limits

M² {m²,p²} p² {m²,M²} β →0^± GD(q1;`)

Γ12 M²+p² p²+M² 2Mpcosh

q

−^m_M^β −ı0

∆12 −m² −m² p²+M²−m²−Γ12

r12

√

p² M

√M

p² exp

q

−^m_M^β −ı0

Q₁² M² p² M pexp

−q

−^m_M^β−ı0

GD(`;q1)

Γ21 −M²−^m_M²^p2² p²+m² 2m p cosh q

−^M_m^β+ı0

∆₂₁ p²+m²+^m_M²^p₂² −M² p²+m²−M²−Γ₂₁ r₂₁ ^m

√

p²

M² −√^m

p² +ı0 −exp

q

−^M_m^β+ı0

Q₂² −M² p² m p exp

−q

−^M_m^β +ı0

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threshold expansion

there are other types of limits that do not consist of having a large difference between the sizes of scales

→ one such limit is thethreshold expansion β= 1− p²

(m+M)² →O

actually two distinct limits: approaching the threshold from below or from above

small parameter of expansion is essentially

|−∆ji|

|2qi·kji+ Γji−i0η·kji|

→ minimize this? →leads to|r1|= 1. coefficients of expansion diverge

⇒ more important to consider behaviour of the physical threshold!

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threshold expansion

x_div^(±) = M+m 2m√

1−β

"

2m

m+M −β± s

−β

4mM (m+M)²−β

#

forβ→0 the pole in the integrand moves towardsx= 1

(integration goes from 1 to∞) also in the case below threshold (β→0⁺) the pole is the most important property of the amplitude⇒complexr1

expandingxdiv gives r₁=−1 +

rM m

p−β+O(β)

β→0−

β→0+

Rex

Imx x_div⁽⁺⁾

x_div⁽⁻⁾

relative error forA^(1,R)₁ : M/m= 3

β n= 1 n= 2 n= 3

−0.1 0.06% 4·10⁻⁶ 5·10⁻⁸ +0.1 0.2% 2·10⁻⁵ 2·10⁻⁷

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scalar three-point function

three-point functionA⁽¹⁾₃ =R

`GF(q1,q12,q3)(p1=−p2, equal massesM)

A⁽¹⁾₃ =− Z

`

( δ˜(q1;M) (−2q1·p₁)(2q₁·p₂)+

δ˜(`;M)

(−2`·p₂)(−2`·p₁₂+s₁₂+ı0)+

δ˜(`;M) (2`·p₁)(2`·p₁₂+s₁₂)

)

angular dependencenot in thepropagators to be expanded simplification through partial fractioning and momentum shift:

=s₁₂ Z

`

˜δ(`;M) (2`·p₁₂)(2`·p₁)

( 1

(−2`·p₁₂+s₁₂+ı0)+ 1 (2`·p₁₂+s₁₂)

)

large mass limitM√ s₁₂

A⁽¹⁾₃ = Z

`

δ˜(`;M) (2`·p₁₂)(`·p₁)

X

n=1

s₁₂ 2`·p₁₂

2n

=− 1 16π²

1 2M² 1 + r

12+r² 90

!

+O(r³),r= s₁₂ M²

relative error,M/√

s12= 3: n= 1 n= 2 n= 3 9·10⁻³ 1·10⁻⁴ 2·10⁻⁶

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scalar three-point function

three-point functionA⁽¹⁾₃ =R

`GF(q1,q12,q3)(p1=−p2, equal massesM)

A⁽¹⁾₃ =− Z

`

( δ˜(q1;M) (−2q1·p₁)(2q₁·p₂)+

δ˜(`;M)

(−2`·p₂)(−2`·p₁₂+s₁₂+ı0)+

δ˜(`;M) (2`·p₁)(2`·p₁₂+s₁₂)

)

angular dependencenot in thepropagators to be expanded simplification through partial fractioning and momentum shift:

=s₁₂ Z

`

˜δ(`;M) (2`·p₁₂)(2`·p₁)

( 1

(−2`·p₁₂+s₁₂+ı0)+ 1 (2`·p₁₂+s₁₂)

)

small mass limitM √

s12, acc. to rules: r23=−^√^M_s

12 +ı0,r32= ^√^M_s

12

A⁽¹⁾₃ = Z

`

s₁₂²δ˜(`;M) (2`·p12)(`·p1)

∞

X

n=0

s₁₂²r₃₂²(2 +r₂₃²)n

−(2`·p₁₂)²+s₁₂²(1 +r₂₃²)²)n+1 = 1 16π²

log²(−r₂₃²) 2(1 +r₂₃²)²s12

+O(r₂₃²)

relative error,√

s12/M = 3:

n= 0 n= 1 n= 2

Re 33% 7.5% 1.5%

Im 7.5% 0.04% 0.26%

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highly-boosted Higgs boson production

effectivepoint-likeggH-couplingnot ruled out [Grojean et al. ’14]

mt

v ¯ttH → −κg

αs

12πvG_µν^a G^µν,aH+κt

mt

v ¯ttH

need toconsider Higgs + jet production atpT sufficiently high for resolving the top loopin order to disentangle possible BSM effects

LO result known for decades

[Ellis et al. ’87]

[Baur, Glover ’89]

NLO ^I fix Higgs-top mass ratio, numerical integration [Jones, Kerner, Luisoni ’18]

I integration-by-parts identities + expansion in _4m^m^H²2

t, ^m_s²^t [Lindert et al. ’18]

goal: obtain independent NLO result using new regularization technique:

loop-tree duality (LTD) [Catani et al. ’08]

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q¯ q → Hg amplitude

LO amplitude: MLO∼ ε^∗_µ(p3) ¯v(p2)γνu(p1)F12

g^µν− p₁₂^µp₃^ν p₃·p₁₂

straight-forward with dim. reg.: solutions for master integrals available F₁₂=

Z

`

N `·p₃, `·p₁₂, `², s, m²_H

[`²−m²_t +i0][(`+p3)²−m²_t +i0][(`+p123)²−m_t²+i0]

while at LO only one kinematic variable s, NLO amplitude depends on: s,t

MNLO∼ F₁₂⁽¹⁾

g^µν− p^µ₁₂p3^ν

p3·p12

+F¯₁₂⁽¹⁾

g^µν− p¯₁₂^µp^ν3

p3·¯p12

⇒ four scales + two loops: simplification through asymptotic expansions useful

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application of LTD to q¯ q → Hg

F12

δ˜(q0)

= Z

`

−˜δ(q₀)

−2p12·q0+s 2p₃·q₀ m²_H−s

×

"

c₃+ m²_t m²_H−s (2p₁₂·q₀) (2p₃·q₀)c₁+c₂

!p12·q0

p₃·q₀

# ,

F12

δ˜(q3)

= Z

`

˜δ(q₃)

m²_H−s −c2− 8

d−2m²_H−4s

+ 1

2h·q₃+m_H²

m²_t(m²_H−s)

(2p₃·q₃)² c₁−c₂+c₃

! (2p₃·q₃)

! ,

F12

δ˜(q12)

=− Z

`

δ˜(q12)s 2p₁₂·q₁₂+s+ı0

"

c2−2c3

(m²_H−s)s(2p12·q12+s) +c3

m²_H−2h·q₁₂

(m_H²−s)s + sm²_t m²_H−2h·q12

m²_H−s s² c1+c2

! 1 (m²_H−s)

+ 1

(m²_H−s)² 1 m²_H−2h·q₁₂

8p12·q12

d−2

2(m²_H−s)²

s +

(d−4)s+ 3dm²_H

!

+ 8m²_H(s+ m²_H

d−2) + (2p12·q12)²(−c2

m²_H s + 4)

# .