A Modern View of Perturbative QCD and Crossings with Mathematics

Y. Sumino (Tohoku Univ.)

☆

Plan of Talk

1. Formulation of pert. QCD Factorization, Effective Field Theories, OPE 2. Foundation by asymptotic expansion of diagrams 3. Nature of radiative corrections in individual parts Theory of multiple zeta values Relation to singularities in Feynman diagrams 4. Summary and future applications

Formulation of pert. QCD 1. Develop methods on how to decompose and systematically organize radiative corrections.

Factorization, EFT, OPE 2. Elucidate nature of radiative corrections in individual parts of the decomposition (which are simplified by the decomposition).

Singularities in amplitudes play key roles.

Formulation of pert. QCD 1. Develop methods on how to decompose and systematically organize radiative corrections.

Factorization, EFT, OPE 2. Elucidate nature of radiative corrections in individual parts of the decomposition (which are simplified by the decomposition).

Separation of scales

Formulation in pert. QCD 1. Develop methods on how to decompose and systematically organize radiative corrections.

Factorization, EFT, OPE

Separation of scales

Wilsonian EFT

in terms of light quarks and IR gluons 𝐸 𝜇 integrate out ℒ

𝑄𝐶𝐷

ℒ EFT

𝜇 = 𝑔

𝑖 𝑖 𝜇 UV 𝒪 𝑖

(𝑞

𝑛

,

𝑛

, 𝐺

𝜇

)

Determine Wilson coeffs

𝑔

𝑖

via pert. QCD.

𝜇

such that physics at

𝐸 < 𝜇

is unchanged,

OPE in Wilsonian EFT

multipole expansion Observable which includes a high scale light quarks and IR gluons 𝜇 𝐸 integrate out

𝑘/𝑃 ≪ 1

non-pert. parameters

Asymptotic Expansion of Diagrams Simplified example: (= 𝑀) Contribution of each scale given by contour integral around singularity

Asymptotic expansion of a diagram and Wilson coeffs in EFT 𝑝 𝑘 𝑝 − 𝑘 𝑞 𝑘 − 𝑞 𝑝 − 𝑞 𝑝 = 𝑑 𝐷 𝑘 𝑑 𝐷 𝑞 𝑘 2 𝑝 − 𝑘 2 𝑘 − 𝑞 1 2 + 𝑀 2 𝑞 2 𝑝 − 𝑞 2 in the case 𝑝 2 ≪ 𝑀 2

Asymptotic expansion of a diagram and Wilson coeffs in EFT 𝑝 𝑘 𝑝 − 𝑘 𝑞 𝑘 − 𝑞 𝑝 − 𝑞 𝑝 = 𝑑 𝐷 𝑘 𝑑 𝐷 𝑞 𝑘 2 𝑝 − 𝑘 2 𝑘 − 𝑞 1 2 + 𝑀 2 𝑞 2 𝑝 − 𝑞 2 in the case 𝑝 2 ≪ 𝑀 2 L H H L L L L L L 𝑝, 𝑘, 𝑞 ≪ 𝑀 L H H L L L 𝑝, 𝑞 ≪ 𝑘, 𝑀 L H H H L H 𝑝 ≪ 𝑘, 𝑞, 𝑀 = 1 = 𝑀 2 = Operators and Wilson coeffs in EFT = 𝑘 4 𝑑 𝐷 𝑘 𝑘 2 + 𝑀 2 = = 𝑘 4 𝑑 𝐷 𝑘 𝑑 𝐷 𝑞 (𝑘 − 𝑞) 2 +𝑀 2 𝑞 4

Formulation of pert. QCD 1. Develop methods on how to decompose and systematically organize radiative corrections.

Factorization, EFT, OPE 2. Elucidate nature of radiative corrections in individual parts of the decomposition (which are simplified by the decomposition).

Numerical and analytical methods

Radiative Corrections and Theory of Multiple Zeta Values Example: Anomalous magnetic moment of electron ( 𝑔 𝑒 − 2 ) ∞ 1 𝜁 𝑛 = 𝑚=1 𝑚 𝑛 ∞ ln 2 = − 𝑚=1 −1 𝑚 𝑚 terms omitted Li 4 1 2 ∞ ~ 𝑚>𝑛>0 −1 𝑚+𝑛 𝑚 3 𝑛

∞ 1 𝜁 𝑛 = 𝑚=1 𝑚 𝑛 ∞ ln 2 = − 𝑚=1 −1 𝑚 𝑚 ☆ Generalized Multiple Zeta Value (MZV) Given as a nested sum Li 4 1 2 ∞ ~ 𝑚>𝑛>0 −1 𝑚+𝑛 𝑚 3 𝑛 , 𝑎 1 ≥ 2 Can also be written in a nested integral form e.g.

0 1 𝑑𝑥 𝑥 0 𝑥 𝑑𝑦 𝑦 − 𝛼 0 𝑦 𝑑𝑧 𝑧 − 𝛽 = −𝑍(∞; 2,1 ; 1 𝛼 , 𝛼 𝛽 )

MZVs can be expressed by a small set of basis (vector space over ℚ ) , 𝑎 1 ≥ 2 weight = 𝑎 1 + ⋯ + 𝑎 𝑁 For 𝜆 𝑖 ∈ {1} : ∞ e.g. Dimension=1 at weight 3: 𝑚>𝑛>0 1 ∞ 𝑚=1 1 = 𝜁 3 𝑑 3 = 1 .

weight dim #(MZVs) Relations to reduce MZVs. (Probably shuffle relations are sufficient for 𝜆 𝑖 ∈ {1} .) New relations for 𝜆 𝑖 ∈ 𝑅𝑜𝑜𝑡𝑠 𝑜𝑓 𝑢𝑛𝑖𝑡𝑦 : Anzai,YS MZV as a period of cohomology, motives

Relation between topology of a Feynman diagram and MZVs? What kind of MZVs are contained in a diagram? Which 𝜆 𝑖 s ?

∞ 𝑍 ∞; 3,1; 𝑒 𝑖𝜋/3 , 1 = 𝑚>𝑛>0 𝑒 𝑖𝑚𝜋/3 𝑚 3 𝑛

Singularities in Feynman Diagrams

𝑝 Complex 𝑞 -plane 𝑞 𝑞 cuts 𝑝 + 𝑞 also log singularity at 𝐼(𝑞) ≡ 𝑑 4 𝑝 𝑝 2 + 1 1 2 [ 𝑝 + 𝑞 2 + 1] +2𝑖 0 −2𝑖

What kind of MZVs are contained in a diagram? Which 𝜆 𝑖 s ?

𝑚 = 1 𝑑 4 𝑞 1 𝑞 2 + 1 2 𝐼(𝑞) 𝑞 𝑚 = 1 𝑚 = 1 𝑞 Singularities map In simple cases all square-roots can be eliminated by (successive) Euler transf. ⟶ Integrals convertible to MZVs

Diagram Computation: Method of Differential Eq.

Analytic evaluation of Feynman diagrams: Many methods but no general one • Glue-and-cut • Mellin-Barnes • Differential eq.

• Gegenbauer polynomial • Unitarity method .

..

.

☆ Evaluation of Cat’s eye diagram 𝑚 = 1 𝑚 = 0

☆ Evaluation of Cat’s eye diagram 𝑚 = 1 𝑚 = 0

☆ Evaluation of Cat’s eye diagram 𝑚 = 1 𝑚 = 0 Some of the lines of Cat’s eye diag. are pinched.

☆ Evaluation of Cat’s eye diagram Some of the lines of Cat’s eye diag. are pinched.

☆ Evaluation of Cat’s eye diagram Solution: ; , etc. : sol. to homogeneous eq.

Using this method recursively, a diagram can be expressed in a nested integral form.

(often MZV as it is.)

Summary A unified view in terms of singularities in physical amplitudes.

(1) Scale separation in Factorization, EFT, OPE by asymptotic exp.

Contour integrals around singularities of amplitudes (2) Unsolved questions in analytic results of individual rad. corr.

Resolution of singularities, Theory of MZVs, singularities and topology of diagrams

Applications in scope

(personal view)

• • • Construction of EFTs from field theoretic approach • Collaboration with lattice

precision physics,

𝛼 𝑠

determination

• IR renormalization of Wilson coeffs. in OPE

𝑚

𝑏

, 𝑚

𝑐 determinations from heavy quarkonium physics

𝑚

𝑡 determination at LHC Kawabata, Shimizu, Yokoya, YS ⋮ ⋮

OPE of QCD potential in Potential-NRQCD EFT 𝐸 integrate out IR gluons and quarks 𝜆 > 𝑟 𝜇 Brambilla,Pineda,Soto,Vairo 𝑟 ≪ Λ −1 𝑄𝐶𝐷

OPE of QCD potential in Potential-NRQCD EFT 𝐸 integrate out IR gluons and quarks 𝜆 > 𝑟 𝜇 Brambilla,Pineda,Soto,Vairo 𝑟 ≪ Λ −1 𝑄𝐶𝐷 QCD potential = Self-energy of singlet bound-state in pNRQCD: 𝑉 𝑄𝐶𝐷 𝑟 = 𝑉 𝑈𝑉 (𝑟) + 𝐸 𝐼𝑅 (𝑟) UV contr.

IR contr.

𝐸 𝐼𝑅 𝑟 ~ 𝑔 2 𝑂𝑆 𝑟 ∙ 𝐸 𝑎 𝑟 ∙ 𝐸 𝑎 ~ 𝑂 Λ 3 𝑄𝐶𝐷 𝑟 2 singlet 𝑉 𝑈𝑉 (𝑟) singlet IR gluon singlet octet singlet

OPE of QCD potential in Potential-NRQCD EFT Brambilla,Pineda,Soto,Vairo QCD potential = Self-energy of singlet bound-state in pNRQCD: 𝑉 𝑄𝐶𝐷 𝑟 = 𝑉 𝑈𝑉 (𝑟) + 𝐸 𝐼𝑅 (𝑟) UV contr.

IR contr.

𝐸 𝐼𝑅 𝑟 ~ 𝑔 2 𝑂𝑆 𝑟 ∙ 𝐸 𝑎 𝑟 ∙ 𝐸 𝑎 ~ 𝑂 Λ Non-pert. Matrix element 3 𝑄𝐶𝐷 𝑟 2 UV gluons 𝜆 < 𝑟 pert. QCD singlet 𝑉 𝑈𝑉 (𝑟) singlet IR gluon singlet octet singlet

✩ Empirically 𝑉 𝑄𝐶𝐷 (𝑟) is approximated well by a Coulomb+linear form.

UV contr.

IR contr.

𝑉 𝑄𝐶𝐷 𝑟 = 𝑉 𝑈𝑉 (𝑟) + 𝐸 𝐼𝑅 (𝑟) ~ 𝒄 −𝟏 𝒓 + 𝒄 𝟏 𝒓 + 𝒄 𝟐 𝒓 𝟐 + ⋯ at 𝒓 ≲ 𝚲 −𝟏 𝑸𝑪𝑫 (naive expansion of 𝑉 𝑄𝐶𝐷 (𝑟) at short-distance) A ‘Coulomb+Linear potential’ is obtained by resummation of logs in pert. QCD

Formulas for Define via then

Key: separate and subtract IR contr.

Comparison of 𝑉 𝐶 𝑟 + 𝜎 𝑟 and lattice comp.

(1) To develop a method on how to decompose and systematically organize the radiative corrections.

Factorization, EFT, OPE (2) to elucidate the nature of the radiative corrections contained in the individual parts of the decomposition (which are simplified by the decomposition).

Singularities in amplitudes play key roles in both of these issues.

1. Review of Pert. QCD ( Round 1, Quick overview )

What’s Pert. QCD?

3 types of so-called “pert. QCD predictions” :

(Confusing without properly distinguishing between them.)

(i) Predict observable in series expansion in 𝛼 𝑠 IR safe obs.

, intrinsic uncertainties ~(Λ 𝑄𝐶𝐷 /𝐸) 𝑛 (ii) Predict observable in the framework of Wilsonian EFT OPE as expansion in (Λ 𝑄𝐶𝐷 /𝐸) 𝑛 , uncertainties of (i) replaced by non-pert. matrix elements

Do not add these non-pert. corr. to (i).

(iii) Predict observable assisted by model predictions High-energy experiments hadronization models, PDFs.

To compare with experimental data

• O(Λ) physics in the heavy quark mass and interquark force 2𝜋 Λ = 𝜇 exp − 𝛽 0 𝛼 𝑠 (𝜇) cannot appear in series expansion in 𝛼 𝑠 (𝜇)

?

Pert. QCD renormalization scale

ℒ

𝑄𝐶𝐷

(𝛼

𝑠

, 𝑚

𝑖

; 𝜇)

Theory of quarks and gluons Same input parameters as full QCD.

Systematic: has its own way of estimating errors.

(Dependence on 𝜇 is used to estimate errors.)

Differs from a model

𝜇 Predictable observables

testable hypothesis

(i) Inclusive observables (hadronic inclusive) ⋯ insensitive to hadronization ∞ e.g. 𝑅

-ratio:

𝑅 𝐸 ≡ 𝜎 𝑒 + 𝑒 − 𝜎 𝑒 + 𝑒 − → ℎ𝑎𝑑𝑟𝑜𝑛𝑠; 𝐸 → 𝜇 + 𝜇 − ; 𝐸 = 𝑞 3𝑄 𝑞 2 1 + 𝑛=1 𝑐 𝑛 (𝐸/𝜇) 𝛼 𝑠 𝑛 (𝜇) (ii) Observables of heavy quarkonium states (the only individual hadronic states) • spectrum, decay width, transition rates

IR

sensitivity at higher-order 𝑅

-ratio:

Renormalon uncertainty (Λ 𝑄𝐶𝐷 /𝐸) 𝑛 𝑅 𝐸 ≡ 𝜎 𝑒 + 𝑒 − 𝜎 𝑒 + 𝑒 → ℎ𝑎𝑑𝑟𝑜𝑛𝑠; 𝐸 − → 𝜇 + 𝜇 − ; 𝐸 𝑞 𝑞 𝑘 𝑞 𝑘 𝑞 𝑘 𝛼 𝑠 (𝜇) Quark self-energy diagrams omitted 𝛼 𝑠 𝜇 × 𝑏 0 𝛼 𝑠 𝜇 log( 𝜇 𝑘 ) 𝛼 𝑠 (𝜇) × 𝑏 2 0 𝛼 𝑠 2 𝜇 log 2 ( 𝜇 𝑘 )

𝑞 𝑘 𝑞 𝑘 𝑞 𝑘 𝛼 𝑠 (𝜇) 𝛼 𝑠 𝜇 × 𝑏 0 𝛼 𝑠 𝜇 log( 𝜇 𝑘 ) 𝛼 𝑠 (𝜇) × 𝑏 2 0 𝛼 𝑠 2 𝜇 log 2 ( 𝜇 𝑘 )

Λ 𝑞 𝑘 𝑞 𝑘 𝑞 𝑘 𝑘 𝛼 𝑠 (𝜇) 𝛼 𝑠 𝜇 × 𝑏 0 𝛼 𝑠 𝜇 log( 𝜇 𝑘 ) 𝛼 𝑠 (𝜇) × 𝑏 2 0 𝛼 𝑠 2 𝜇 log 2 ( 𝜇 𝑘 ) Infinite sum 𝛼 𝑠 𝑘 = 1−𝑏 0 𝛼 𝛼 𝑠 (𝜇) 𝑠 𝜇 log( 𝜇 𝑘 ) = 1 𝑏 0 log( 𝑘 Λ )

Λ 𝑞 𝑘 𝑞 𝑘 𝑞 𝑘 Consequence Renormalon uncertainty 𝑐 𝑛 𝐸/𝜇 𝛼 𝑠 𝑛 𝜇 𝑘 ~ Λ/𝐸 𝑃 Asymptotic series (Empirically good estimate of true corr.) Limited accuracy

Remarkable progress of computational technologies in the last 10-20 years

(i) Higher-loop corrections Resolution of singularities in multi-loop integrals Numerical and analytical methods Intersection with frontiers of mathematics (ii) Lower-order (NLO/NLL) corrections to complicated processes Cope with proliferation of diagrams and many kinematical variables Motivated by LHC physics (iii) Factorization of scales in loop corrections Provide powerful and precise foundation for constructing Wilsonian EFT

Dim. reg.: common theoretical basis

Essentially analytic continuation of loop integrals Contrasting/complementary to cut-off reg.

A ‘Coulomb+Linear potential’ is obtained by resummation of logs in pert. QCD: YS at IR contributions (absorbed into non-pert. matrix elem.) UV contributions

A ‘Coulomb+Linear potential’ is obtained by resummation of logs in pert. QCD: YS UV contributions × Expressed by param. of pert. QCD

Formulas for Define via then In the LL case 𝛼 𝑉 𝑞 = 2𝜋 𝛽 0 log( Λ 𝑞 𝑀𝑆 ) Coulombic pot. with log corr. at short-dist.

Coefficient of linear potential (at short-dist.) 𝜎 𝐿𝐿 = 2𝜋𝐶 𝐹 𝛽 0 Λ 𝑀𝑆 2

Messages: (1) One should carefully examine, from which power of 2𝜋 Λ = 𝜇 exp − non-pert. contributions start, 𝛽 0 𝛼 𝑠 (𝜇) and to which extent pert. QCD is predictable. (as you approach from short-distance region) 𝛼 𝑠 𝑟 𝜇 1 + {𝑏 0 𝛼 𝑠 𝜇 log 𝜇𝑟 + #} + 𝑏 2 0 𝛼 𝑠 2 𝜇 log 2 𝜇𝑟 + ⋯ + ⋯ → (2) IR renormalization of Wilson coeffs.

𝒓

−𝟏 𝑞 𝜇 𝑓

OPE of QCD potential in Potential-NRQCD EFT singlet octet 𝜇 𝐸 integrate out IR gluons and quarks 𝜆 > 𝑟 Brambilla,Pineda,Soto,Vairo singlet IR gluon octet 𝑟 ≪ Λ −1 𝑄𝐶𝐷

OPE of QCD potential in Potential-NRQCD EFT singlet octet UV gluons 𝜆 < 𝑟 pert. QCD Brambilla,Pineda,Soto,Vairo singlet IR gluon octet

OPE of QCD potential in Potential-NRQCD EFT singlet octet Brambilla,Pineda,Soto,Vairo singlet IR gluon octet UV gluons 𝜆 < 𝑟 pert. QCD QCD potential = Self-energy of

𝑺

in pNRQCD: 1 = 𝑺 𝑺 † 𝑺 𝑺 𝑉 𝑄𝐶𝐷 𝑟 = 𝑔 𝑆 𝑟 𝑟 UV contr.

+ 𝐸 𝐼𝑅 (𝑟) IR contr.

𝐸 𝐼𝑅 𝑟 ~ 𝑔 2 𝑂𝑆 𝑟 ∙ 𝐸 𝑎 𝑟 ∙ 𝐸 𝑎 ~ 𝑂 Λ 3 𝑄𝐶𝐷 𝑟 2 singlet 𝑔 𝑆 𝑟 𝑟 singlet IR gluon singlet octet singlet

Formulas for Define via then

Key: separate and subtract IR contr.

In the LL case 𝛼 𝑉 𝑞 = 2𝜋 𝛽 0 log( Λ 𝑞 𝑀𝑆 ) Coulombic pot. with log corr. at short-dist.

( 𝑞 ∗ = Λ 𝑀𝑆 ) Coefficient of linear potential (at short-dist.) 𝜎 𝐿𝐿 = 2𝜋𝐶 𝐹 𝛽 0 Λ 𝑀𝑆 2

Comparison of 𝑉 𝐶 𝑟 + 𝜎 𝑟 and lattice comp.

Summary Today pert. QCD is subdivided and specialized into a wide variety of research fields: jets, DIS,

B

-physics, quarkonium, … A unified view in terms of singularities in physical amplitudes.

(1) Scale separation in Factorization, EFT, OPE.

Contour integrals around singularities of amplitudes (2) Unsolved questions in analytic results of individual rad. corr.

Resolution of singularities, Theory of MZVs, singularities and topology of diagrams