Modern View of Perturbative QCD and
Application to Heavy Quarkonium System
（現在の視点から見る摂動ＱＣＤ及び
重いクォーコニウム系への応用）
Y. Sumino
(Tohoku Univ.)
☆Plan of Talk
1. Review of Pert. QCD (Round 1, Quick overview)
• What’s Pert. QCD?
• Today’s computational technologies
2. Review of Pert. QCD (Round 2, Some details)
3. Application to Heavy Quarkonium System
• O(Λ) physics in the heavy quark mass and interquark force
(4. More details of specific interests, upon request)
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 𝛼𝑠
IR safe obs., intrinsic uncertainties ~(Λ𝑄𝐶𝐷 /𝐸)𝑛
(ii) Predict observable in the framework of Wilsonian EFT
OPE as expansion in (Λ𝑄𝐶𝐷 /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
Many obs in high-energy experiments depend on hadronization models, PDFs.
Necessary (in MC) to compare with experimental data
Systematic uncertainties difficult to control, O(10%) accuracy at LHC
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
Cross-over 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.
Comment on Impacts on Physics Insights:
new interpretations, viewpoints, concepts, …
To date, scattered over specialized fields, yet to frame a general overview
Examples:
• Various EFTs triggered new paradigms, such as HQET for b-physics, SCET for jets
• O(Λ) physics in the heavy quark mass and interquark force
2𝜋
0 𝛼𝑠 (𝜇)
Λ = 𝜇 exp − 𝛽
cannot appear in series expansion in 𝛼𝑠 (𝜇) ?
2. Review of Pert. QCD (Round 2, Some details)
3 types of so-called “pert. QCD predictions” :
(i) Predict observable in series expansion in 𝛼𝑠
(ii) Predict observable in the framework of Wilsonian EFT
(iii) Predict observable assisted by model predictions
2. Review of Pert. QCD (Round 2, Some details)
3 types of so-called “pert. QCD predictions” :
(i) Predict observable in series expansion in 𝛼𝑠
(ii) Predict observable in the framework of Wilsonian EFT
(iii) Predict observable assisted by model predictions
2. Review of Pert. QCD (Round 2, Some details)
3 types of so-called “pert. QCD predictions” :
(i) Predict observable in series expansion in 𝛼𝑠
(ii) Predict observable in the framework of Wilsonian EFT
(iii) Predict observable assisted by model predictions
Pert. QCD
renormalization scale
ℒ𝑄𝐶𝐷 (𝛼𝑠 , 𝑚𝑖 ; 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
𝜎 𝑒 + 𝑒 − → ℎ𝑎𝑑𝑟𝑜𝑛𝑠; 𝐸
• 𝑅-ratio: 𝑅 𝐸 ≡
=
𝜎 𝑒 + 𝑒 − → 𝜇 + 𝜇−; 𝐸
• Inclusive decay widths
∞
3𝑄𝑞2 1 +
𝑞
𝑐𝑛 (𝐸/𝜇) 𝛼𝑠𝑛 (𝜇)
𝑛=1
• Distributions of non-colored particles, ℓ, 𝛾, 𝑊, 𝐻, ⋯
(ii) Observables of heavy quarkonium states (the only individual hadronic states)
• spectrum, leptonic decay width, transition rates
IR sensitivity at higher-order
Renormalon uncertainty (Λ𝑄𝐶𝐷 /𝐸)𝑛
𝜎 𝑒 + 𝑒 − → ℎ𝑎𝑑𝑟𝑜𝑛𝑠; 𝐸
𝑅 𝐸 ≡
𝜎 𝑒 + 𝑒 − → 𝜇 + 𝜇− ; 𝐸
𝑅-ratio:
𝑞
𝑞
𝑞
𝑘
𝛼𝑠 (𝜇)
𝑞
𝑞
𝑘
𝜇
𝛼𝑠 𝜇 × 𝑏0 𝛼𝑠 𝜇 log( )
𝑘
𝑘
𝜇
𝛼𝑠 (𝜇) × 𝑏02 𝛼𝑠2 𝜇 log 2 ( )
𝑘
𝑞
𝑞
𝑞
𝑞
𝑘
𝛼𝑠 (𝜇)
𝑞
𝑞
𝑘
𝜇
𝛼𝑠 𝜇 × 𝑏0 𝛼𝑠 𝜇 log( )
𝑘
𝑘
𝜇
𝛼𝑠 (𝜇) × 𝑏02 𝛼𝑠2 𝜇 log 2 ( )
𝑘
𝑞
𝑞
𝑞
𝑞
𝑘
𝛼𝑠 (𝜇)
𝑞
𝑞
𝑘
𝜇
𝛼𝑠 𝜇 × 𝑏0 𝛼𝑠 𝜇 log( )
𝑘
𝑘
𝜇
𝛼𝑠 (𝜇) × 𝑏02 𝛼𝑠2 𝜇 log 2 ( )
𝑘
𝑞
𝑞
𝑞
Infinite sum
𝛼𝑠 𝑘 =
Λ
𝑘
𝛼𝑠 (𝜇)
1−𝑏0 𝛼𝑠 𝜇
𝜇
log( 𝑘 )
=
1
𝑘
Λ
𝑏0 log( )
𝑞
𝑘
Consequence
Renormalon uncertainty
𝑞
𝑞
𝑐𝑛 𝐸/𝜇 𝛼𝑠𝑛 𝜇
𝑘
𝑞
𝑞
𝑘
𝑞
~ Λ/𝐸
𝑃
Asymptotic series
(Empirically good estimate of true corr.)
Limited accuracy
Λ
𝑘
2. Review of Pert. QCD (Round 2, Some details)
3 types of so-called “pert. QCD predictions” :
(i) Predict observable in series expansion in 𝛼𝑠
(ii) Predict observable in the framework of Wilsonian EFT
(iii) Predict observable assisted by model predictions
𝐸
integrate
out
Wilsonian EFT
in terms of light quarks and IR gluons
𝜇
ℒ𝑄𝐶𝐷
ℒ EFT 𝜇
=
𝑖
𝑔𝑖 𝜇
𝒪𝑖 (𝑞𝑛 , 𝑞𝑛 , 𝐺𝜇 )
less d.o.f.
Determine Wilson coeffs 𝑔𝑖 𝜇 such that physics at 𝐸 < 𝜇 is unchanged,
via pert. QCD:
1. Matching
2. Asymptotic expansion of diagrams
𝑔𝑖 𝜇 include only UV contr.
Free from IR renormalon uncertainties
𝐸
OPE in Wilsonian EFT
integrate
out
multipole expansion
𝜇
Observable which includes a high scale
light quarks and IR gluons
replace renormalons
𝑘/𝑃 ≪ 1
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
Cross-over 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.
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
Cross-over 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.
Dim. reg.
Advantages
• Preserves important symmetries (Lorentz sym, gauge sym)
• In a single step, all loop integrals are rendered finite; both UV and IR.
(cf. Pauli-Villars reg.)
• Many useful computational techniques
Disadvantages
• Not defined as a quantum field theory (cf. lattice reg.)
Nevertheless, well-defined and uniquely defined in pert. computations.
• Difficult to interpret physically
Does 1/𝜖 𝑛 represent IR or UV divergence? Unphysical equalities?
Is only UV part of the theory modified?
𝑑𝐷 𝑘
1
𝑘2
=0
(I can give an argument why I believe dim. reg. leads to correct predictions.)
Integration-by-parts (IBP) Identities
Chetyrkin, Tkachov
Most powerful application of Dim. Reg.
;
Standard technology used to reduce a large number of loop integrals to
a small set of integrals (master integrals).
𝑘+𝑝+𝑞
𝑝+𝑞
Example:
0=
=
=
𝑑𝐷 𝑝𝑑𝐷 𝑘
𝑑𝐷 𝑝𝑑𝐷 𝑘
𝐷
𝐷
𝑑 𝑝𝑑 𝑘
𝜕
𝜕𝑘𝜇 𝑝2 𝑘 2 𝑘 + 𝑝
𝑝2 𝑘 2 𝑘 + 𝑝
𝑝2 𝑘 2 𝑘 + 𝑝
𝜇
2
2
1
𝑝+𝑞
2
1
𝑝+𝑞
𝑘
𝑝+𝑞
2
2
𝑘
𝑞
2
𝑘+𝑝+𝑞
𝑘+𝑝+𝑞
𝑘+𝑝+𝑞
2
𝑝
𝑞
𝑘+𝑝
2𝑘 ∙ 𝑘 2𝑘 ∙ 𝑘 + 𝑝
2𝑘 ∙ (𝑘 + 𝑝 + 𝑞)
−
−
𝑘2
𝑘+𝑝 2
𝑘+𝑝+𝑞 2
2
𝐷−
2
𝑝2 − 𝑘 2
𝑝 + 𝑞 2 − 𝑘2
𝐷−4+
+
𝑘+𝑝 2
𝑘+𝑝+𝑞 2
Integration-by-parts (IBP) Identities
Chetyrkin, Tkachov
Most powerful application of Dim. Reg.
;
Standard technology used to reduce a large number of loop integrals to
a small set of integrals (master integrals).
𝑘+𝑝+𝑞
𝑝+𝑞
Example:
0=
=
=
𝑑𝐷 𝑝𝑑𝐷 𝑘
𝑑𝐷 𝑝𝑑𝐷 𝑘
𝐷
𝐷
𝑑 𝑝𝑑 𝑘
𝜕
𝜕𝑘𝜇 𝑝2 𝑘 2 𝑘 + 𝑝
𝑝2 𝑘 2 𝑘 + 𝑝
𝑝2 𝑘 2 𝑘 + 𝑝
𝜇
2
2
1
𝑝+𝑞
2
1
𝑝+𝑞
𝑘
𝑝+𝑞
2
2
𝑘
𝑞
2
𝑘+𝑝+𝑞
𝑘+𝑝+𝑞
𝑘+𝑝+𝑞
2
𝑝
𝑞
𝑘+𝑝
2𝑘 ∙ 𝑘 2𝑘 ∙ 𝑘 + 𝑝
2𝑘 ∙ (𝑘 + 𝑝 + 𝑞)
−
−
𝑘2
𝑘+𝑝 2
𝑘+𝑝+𝑞 2
2
𝐷−
2
𝑝2 − 𝑘 2
𝑝 + 𝑞 2 − 𝑘2
𝐷−4+
+
𝑘+𝑝 2
𝑘+𝑝+𝑞 2
Remarkable progress of computational technologies in the last 10-20 years
(i) Higher-loop corrections
Resolution of singularities in higher-loop integrals
⟺ cross-over with frontiers of mathematics
(ii) Lower-order (NLO/NNLO/NLL) corrections to complicated processes
Cope with proliferation of diagrams and many variables
Strongly motivated by LHC physics
(iii) Factorization of scales in loop corrections
Provide powerful and precise foundation for constructing Wilsonian EFT
Dim. reg. as the common theoretical basis to all of them
Essentially analytic continuation of loop integrals
Contrasting to cut-off reg.
Asymptotic Expansion of Diagrams
Simplified example:
(= 𝑀)
Asymptotic expansion of a diagram and Wilson coeffs in EFT
𝑘
𝑞
𝑝
𝑝
𝑘−𝑞
𝑝−𝑘
𝑝−𝑞
=
𝑑𝐷 𝑘 𝑑𝐷 𝑞
1
𝑘2 𝑝 − 𝑘
2
𝑘−𝑞
2
+ 𝑀2 𝑞2 𝑝 − 𝑞
2
in the case 𝑝2 ≪ 𝑀2
Asymptotic expansion of a diagram and Wilson coeffs in EFT
𝑞
𝑘
𝑝
𝑝
𝑘−𝑞
L
𝑘2 𝑝 − 𝑘
H
2
𝑘−𝑞
+ 𝑀2 𝑞2 𝑝 − 𝑞
2
H
=
Vertices and Wilson coeffs in EFT
=
H
L
𝑝, 𝑞 ≪ 𝑘, 𝑀
𝑑𝐷 𝑘
𝑘 4 𝑘 2 + 𝑀2
H
L
L
H
L
𝑝, 𝑘, 𝑞 ≪ 𝑀
1
= 2
𝑀
2
H
L
L
L
L
=
1
in the case 𝑝2 ≪ 𝑀2
L
L
𝑑𝐷 𝑘 𝑑𝐷 𝑞
𝑝−𝑞
𝑝−𝑘
L
=
H
=
L
H
𝑝 ≪ 𝑘, 𝑞, 𝑀
𝑑𝐷 𝑘 𝑑𝐷 𝑞
= 4
𝑘 (𝑘 − 𝑞)2 +𝑀2 𝑞 4
Remarkable progress of computational technologies in the last 10-20 years
(i) Higher-loop corrections
Resolution of singularities in higher-loop integrals
⟺ cross-over with frontiers of mathematics
Theory of Multiple Zeta Values (MZV)
(ii) Lower-order (NLO/NNLO/NLL) corrections to complicated processes
Cope with proliferation of diagrams and many variables
Strongly motivated by LHC physics
(iii) Factorizing and separating scales in loop corrections
Provide solid and precise foundation for constructing Wilsonian EFT
Dim. reg. as the common theoretical basis to all of them
Essentially analytic continuation of loop integrals
Contrasting to cut-off reg.
Example: Anomalous magnetic moment of electron (𝑔𝑒 − 2)
terms omitted
∞
𝜁 𝑛 =
𝑚=1
1
𝑚𝑛
∞
ln 2 = −
𝑚=1
−1 𝑚
𝑚
Li4
1
=
2
∞
𝑚>𝑛>0
−1 𝑚+𝑛
𝑚3 𝑛
∞
∞
1
𝑚𝑛
𝜁 𝑛 =
𝑚=1
ln 2 = −
𝑚=1
−1 𝑚
𝑚
Li4
1
=
2
∞
𝑚>𝑛>0
−1 𝑚+𝑛
𝑚3 𝑛
☆ Generalized Multiple Zeta Value (MZV)
Given as a nested sum
, 𝑎1 ≥ 2
Can also be written in a nested integral form
e.g.
1
0
𝑑𝑥
𝑥
𝑥
0
𝑑𝑦
𝑦−𝛼
𝑦
0
𝑑𝑧
1 𝛼
= −𝑍(∞; 2,1 ; 𝛼 , 𝛽 )
𝑧−𝛽
MZVs can be expressed by a small set of basis (vector space over ℚ)
, 𝑎1 ≥ 2
weight = 𝑎1 + ⋯ + 𝑎𝑁
For 𝜆𝑖 ∈ {1}:
∞
e.g.
𝑚>𝑛>0
1
=
𝑚2 𝑛
∞
𝑚=1
1
=𝜁 3
𝑚3
Dimension=1 at weight 3: 𝑑3 = 1.
weight
dim
#(MZVs)
Shuffle relations are powerful in reducing MZVs. (Probably sufficient for 𝜆𝑖 ∈ {1}.)
New relations for 𝜆𝑖 ∈ 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 𝜆𝑖 s ?
𝑚=1
∞
𝑍 ∞; 3,1; 𝑒
𝑖𝜋/3
,1 =
𝑚>𝑛>0
𝑒 𝑖𝑚𝜋/3
𝑚3 𝑛
Singularities in Feynman Diagrams
☆ Classes of singularities in a Feynman diagram
•
IR singularity 𝑞 → 0
•
UV singularity 𝑞 → ∞
•
Mass singularity 𝑚𝑖 → 0
•
Threshold singularity 𝑞 → 𝑖
𝑚𝑖
𝑖∈𝐼
Complex 𝑞 -plane
𝑝
𝑞
𝑞
cuts
0
−2𝑖
𝑝+𝑞
also log singularity at
𝐼(𝑞) ≡
𝑑4 𝑝
1
𝑝2 + 1 2 [ 𝑝 + 𝑞
+2𝑖
2
+ 1]
What kind of MZVs are contained in a diagram? Which 𝜆𝑖 s ?
𝑚=1
𝑑4 𝑞
𝑞
𝑚=1
1
𝑞2 + 1
2
𝐼(𝑞)
𝑞
𝑚=1
Singularities map
In simple cases all square-roots can be eliminated by (successive)
Euler transf. ⟶ Integrals convertible to MZVs
Summary of Overview
Pert. QCD
Higher-order computations
IR renormalons ~(Λ𝑄𝐶𝐷 /𝐸)𝑛 ⟺ increase of 𝛼𝑠 (𝑘) at IR
Summary of Overview
Pert. QCD
Higher-order computations
IR renormalons ~(Λ𝑄𝐶𝐷 /𝐸)𝑛
OPE in Wilsonian EFT
Separation of UV & IR contr.
Wilson coeffs vs. non-pert. matrix elements
Summary of Overview
Pert. QCD
Higher-order computations
IR renormalons ~(Λ𝑄𝐶𝐷 /𝐸)𝑛
OPE in Wilsonian EFT
Separation of UV & IR contr.
Wilson coeffs vs. non-pert. matrix elements
replaced
Summary of Overview
OPE in Wilsonian EFT
Pert. QCD
Higher-order computations
IR renormalons ~(Λ𝑄𝐶𝐷 /𝐸)𝑛
Separation of UV & IR contr.
Wilson coeffs vs. non-pert. matrix elements
replaced
scale separation using analyticity
Dim. reg.
Asymptotic expansion ⋯ integration by region
Summary of Overview
OPE in Wilsonian EFT
Pert. QCD
Higher-order computations
IR renormalons ~(Λ𝑄𝐶𝐷 /𝐸)𝑛
Separation of UV & IR contr.
Wilson coeffs vs. non-pert. matrix elements
replaced
scale separation using analyticity
Dim. reg.
Asymptotic expansion ⋯ integration by regions
Reduction by IBP identities
Resolution of singularities
Summary of Overview
OPE in Wilsonian EFT
Pert. QCD
Higher-order computations
IR renormalons ~(Λ𝑄𝐶𝐷 /𝐸)𝑛
Separation of UV & IR contr.
Wilson coeffs vs. non-pert. matrix elements
replaced
scale separation using analyticity
Dim. reg.
Asymptotic expansion ⋯ integration by region
Reduction by IBP identities
Resolution of singularities
Singularities
of a diagram
Topology
short-cut ?
tough intermediate comp.
MZVs
final results very simple
Pert. QCD: Today’s benchmarks
More than 10 digits!
Universality
Precisions
𝛼𝑠 𝑀𝑍 = 0.1184(7)
0.6% accuracy
𝑚𝑏 (𝑚𝑏 ) = 4.18(3)
0.8% accuracy
𝑚𝑐 (𝑚𝑐 ) = 1.275(25)
2% accuracy
𝑚𝑡 (𝑚𝑡 ) = 160+5
−4
3% accuracy (→ 0.06% at ILC)
3. Application to Heavy Quarkonium System
• O(Λ) physics in the heavy quark mass and interquark force
• IR renormalization of Wilson coeffs in EFT
Static QCD Potential
3-loop pert. QCD vs. lattice comp.
Anzai, Kiyo, YS
𝑛𝑓 = 0
Consider (naively) a “short-distance expansion”
𝒄−𝟏
𝑽𝑸𝑪𝑫 𝒓 ~
+ 𝒄𝟎 + 𝒄𝟏 𝒓 + 𝒄 𝟐 𝒓 𝟐 + ⋯
at 𝒓 ≪ 𝚲−𝟏
𝒓
According to renormalon analysis in pert. QCD, constant and
𝒓𝟐 term contain uncertainties
𝒄𝟎
~𝑶 𝚲
𝒄𝟐 𝒓𝟐 ~ 𝑶(𝚲3 𝒓𝟐 )
IR renormalon in 𝒄𝟎 is canceled in the total energy
if we express the quark pole mass (𝑚𝑝𝑜𝑙𝑒 ) by the MS mass (𝑚).
2𝑚𝑝𝑜𝑙𝑒 = 2𝑚 (1 + 𝑐1 𝛼𝑠 + 𝑐2 𝛼𝑠2 + 𝑐3 𝛼𝑠3 + ⋯ )
Drastic improvement of convergence of pert. series
𝑽𝑸𝑪𝑫
𝒄−𝟏
𝒓 ~
+ 𝒄𝟎 + 𝒄𝟏 𝒓 + 𝒄 𝟐 𝒓 𝟐 + ⋯
𝒓
𝒄𝟎
at 𝒓 ≪ 𝚲−𝟏
~𝑶 𝚲
𝒄𝟐 𝒓𝟐 ~ 𝑶(𝚲3 𝒓𝟐 )
IR renormalon in 𝒄𝟎 is canceled in the total energy
if we express the quark pole mass (𝑚𝑝𝑜𝑙𝑒 ) by the MS mass (𝑚).
2𝑚𝑝𝑜𝑙𝑒 = 2𝑚 (1 + 𝑐1 𝛼𝑠 + 𝑐2 𝛼𝑠2 + 𝑐3 𝛼𝑠3 + ⋯ )
Drastic improvement of convergence of pert. series
𝑽𝑸𝑪𝑫
𝒄
𝒓 ~ −𝟏 + 𝒄𝟎 + 𝒄𝟏 𝒓 + 𝒄𝟐 𝒓𝟐 + ⋯
𝒓
𝒄𝟎
−𝟏
at 𝒓 ≪ 𝚲
~𝑶 𝚲
𝒄𝟐 𝒓𝟐 ~ 𝑶(𝚲3 𝒓𝟐 )
IR renormalon in 𝒄𝟎 is canceled in the total energy
if we express the quark pole mass (𝑚𝑝𝑜𝑙𝑒 ) by the MS mass (𝑚).
2𝑚𝑝𝑜𝑙𝑒 = 2𝑚 (1 + 𝑐1 𝛼𝑠 + 𝑐2 𝛼𝑠2 + 𝑐3 𝛼𝑠3 + ⋯ )
Drastic improvement of convergence of pert. series
𝑟 [GeV-1]
N=3
N=0
N=0
N=3
𝑟 [GeV-1]
Exact pert. potential up to 3 loops
General feature of gauge theory
𝐴𝜇 𝑞 𝑗𝜇 (−𝑞)
𝑞
𝑗𝜇 𝑥 = 𝛿 𝜇0 𝛿 3 (𝑥 − 𝑟/2)
Couples to total charge as 𝑞 → 0.
General feature of gauge theory
𝐴𝜇 𝑞 𝑗𝜇 (−𝑞)
𝑞
𝑗𝜇 𝑥 = 𝛿 𝜇0 𝛿 3 (𝑥 − 𝑟/2)
Couples to total charge as 𝑞 → 0.
What are UV contributions?
𝑽𝑸𝑪𝑫 𝒓 ~
𝒄−𝟏
+ 𝒄𝟎 + 𝒄𝟏 𝒓 + 𝒄𝟐 𝒓 𝟐 + ⋯
𝒓
at 𝒓 ≪ 𝚲−𝟏
IR contributions
𝒄𝟎 ~ 𝑶 𝚲
→
cancel against
𝒄𝟐 𝒓𝟐 ~ 𝑶(𝚲3 𝒓𝟐 )
OPE of QCD potential in Potential-NRQCD EFT
US gluon
Uncetainty in 𝒄𝟐 𝒓𝟐 replaced by
a non-local gluon condensate within pNRQCD
singlet
octet
singlet
A ‘Coulomb+Linear potential’ is obtained by
resummation of logs in pert. QCD:
YS
IR contributions
at
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
2𝜋
In the LL case 𝛼𝑉 𝑞 =
𝑞
𝛽0 log(
)
Λ 𝑀𝑆
Coulombic pot. with log corr. at short-dist.
Coefficient of linear potential (at short-dist.)
𝜎𝐿𝐿 =
2𝜋𝐶𝐹
Λ 𝑀𝑆
𝛽0
2
𝑞
To see nature of
, define Wilson coeff.
𝒓−𝟏
in Potential-NRQCD for 𝑟 −1 ≫ 𝜇𝑓 ≫ 𝛬 as
𝛼𝑉 (𝑞)
accurately
predictable
𝜇𝑓
It can be proven that
This shows that, in pert. QCD, the “Coulomb” and linear parts of 𝑉𝑄𝐶𝐷 (𝑟) are determined
by UV contributions and are independent of the factorization scale 𝜇𝑓 .
Proof of
,
Hence,
Since 𝜇𝑓 𝑟 ≪ 1, along 𝐶3 we can expand
These terms are canceled and
𝑐𝑜𝑛𝑠𝑡. +𝑂(𝜇𝑓3 𝑟 2 ) remain.
Subtraction of IR contributions in 𝑉𝑄𝐶𝐷 𝑟 as contour integral around 𝑞 = 𝑞∗ .
Implications
𝐸𝑡𝑜𝑡 𝑟 ≈ 2𝑚 + 𝑐𝑜𝑛𝑠𝑡 + 𝑉𝐶 𝑟 + 𝜎 𝑟 + 𝑂(Λ3 𝑟 2 )
Heavy quarkonium spectrum ≈ Energy eigenvalues of 𝐻 ≈
𝑝2
2𝑚𝑝𝑜𝑙𝑒
+ 𝐸𝑡𝑜𝑡 (𝑟)
c.f. Rigorous computation in potential-NRQCD
up to NNNLO
3𝑆
Λ×
2𝑆
1𝑆
𝑟
Λ 1/3
𝑚
from linear pot. (predictable part)
0.4 Λ for 𝑚 = 𝑚𝑏 ≳ Coulomb splitting 𝛼𝑠2 𝑚𝑏
0.1 Λ for 𝑚 = 𝑚𝑡 < Coulomb splitting 𝛼𝑠2 𝑚𝑡
Rapid growth of masses of excited states originates from
rapid growth of self-energies of Q & Q due to IR gluons.
Brambilla, Y.S., Vairo
𝑎𝑋
good convergence
𝐸𝑋 ≈ 2𝑚𝑏 +
𝑚𝑏
0
𝑑𝑞 𝑓𝑋 𝑞 𝛼𝑠 (𝑞)
Rapid growth of masses of excited states originates from
rapid growth of self-energies of Q & Q due to IR gluons.
Brambilla, Y.S., Vairo
𝑎𝑋
Mass
__of a bottomonium state mainly consists of
(i) MS masses of 𝑏 and 𝑏
(ii) Contr. to the self-energies of 𝑏 and 𝑏 from
gluons with wave-length 1/𝑚𝑏 ≲ 𝜆 ≲ 𝑎𝑋
Resemble difference of (state-dependent)
__
constituent quark masses and MS masses.
Messages:
(1) One should carefully examine, from which power of
2𝜋
Λ = 𝜇 exp − 𝛽 𝛼 (𝜇) non-pert. contributions start,
0 𝑠
and to which extent pert. QCD is predictable.
(as you approach from short-distance region)
𝛼𝑠 𝜇
1 + {𝑏0 𝛼𝑠 𝜇 log 𝜇𝑟 + #} + 𝑏02 𝛼𝑠2 𝜇 log 2 𝜇𝑟 + ⋯ + ⋯
𝑟
→
𝑞
(2) IR renormalization of Wilson coeffs.
𝒓−𝟏
𝜇𝑓
Spectroscopy
Bottomonium spectrum at NNNLO
𝑙𝑎𝑟𝑔𝑒−𝛽0
𝑑3 = 0.95 × 𝑑3
(𝑚𝑐 = 0)
𝜇 fixed at minimal-sensitivity scale for each level
Kiyo, YS
•
•
•
Highly sensitive to 𝑑3 .
Stability practically determined by
Dependence on
is minor.
Minimal-sensitivity scales 𝜇 are generally larger than at NNLO.
Integration-by-parts (IBP) Identities
Standard form of loop integrals
Express each diagram in terms of standard integrals
NB:
is negative, when
representing a numerator.
Each
1 loop
2 loop
3 loop
can be represented by a lattice site in N-dim. space
Integration-by-parts (IBP) Identities
Chetyrkin, Tkachov
In dim. reg.
𝑐
e.g.
at 1-loop:
𝑎
Reduction to Master Integrals (a small set of simple integrals)
𝑏
In simple cases all square-roots can be eliminated by (successive) Euler
transf. ⟶ Integrals convertible to MZVs
Cause, however, proliferation of 𝜆𝑖 s
Another example
𝑚𝑖 = 2
𝑚𝑖 = 4
𝑖
𝑖
𝑚=1
𝑞
𝑞
𝑚=0
IR
Singularities at
MZVs with singularities at
UV
or
Proof of
,
Hence,
Since 𝜇𝑓 𝑟 ≪ 1, along 𝐶3 we can expand
These terms are canceled and
𝑐𝑜𝑛𝑠𝑡. +𝑂(𝜇𝑓3 𝑟 2 ) remain.
Summary of Overview
3 types of so-called “pert. QCD predictions” :
(i) Predict observable in series expansion in 𝛼𝑠
inclusive obs./heavy quarkonium obs.
uncertainties ~(Λ𝑄𝐶𝐷 /𝐸)𝑛 by higher-order corrections
⟺ increase of 𝛼𝑠 (𝑘) at IR
(ii) Predict observable in the framework of Wilsonian EFT
separation of UV & IR contr.
OPE: uncertainties of (i) replaced by non-pert. matrix elements
UV → Wilson coeffs. (pert. QCD with IR renormalization)
(iii) Predict observable assisted by model predictions
Many obs in high-energy experiments depend on hadronization models, PDFs.
Necessary (in MC) to compare with experimental data
Systematic uncertainties difficult to control, O(10%) accuracy at LHC
Remarkable progress of computational technologies
Dim. reg. as the common theoretical basis
Essentially analytic continuation of loop integrals
Contrasting/complementary to cut-off reg.
e.g. IBP id.
(i) Higher-loop corrections
Resolution of singularities in higher-loop integrals
⟺ Theory of MZVs in mathematics
(ii) Lower-order (NLO/NLL) corrections to complicated processes
Active development motivated by LHC physics
pragmatic but no general (systematic) formulations as yet
(iii) Factorization of scales in loop corrections
Provide powerful and precise foundation for constructing Wilsonian EFT
may lead to new interpretation as substitute for cut-off reg.
Microscopic View
2𝑚𝑝𝑜𝑙𝑒 = 2𝑚 (1 + 𝑐1 𝛼𝑠 + 𝑐2 𝛼𝑠2 + 𝑐3 𝛼𝑠3 + ⋯ )
good convergence
𝐸𝑋 ≈ 2𝑚𝑏 +
𝑚𝑏
0
𝑑𝑞 𝑓𝑋 𝑞 𝛼𝑠 (𝑞)
𝐸
OPE in Wilsonian EFT
integrate
out
multipole expansion
𝜇
Observable which includes a high scale
light quarks and IR gluons
replace renormalons
ー
+
+
ー
1/𝑃 (≪
gluon
gluon wave-length)
Dim. reg. as the common theoretical basis to all of them
Essentially analytic continuation of loop integrals
Contrasting to cut-off reg.
Relation between topology of a Feynman diagram and MZVs?
What kind of MZVs are contained in a diagram? Which 𝜆𝑖 s ?
𝑚=0
𝑚=1
∞
𝑍 ∞; 3,1; 𝑒
𝑖𝜋/3
,1 =
𝑚>𝑛>0
𝑒 𝑖𝑚𝜋/3
𝑚3 𝑛
𝜁 5 = 𝑍(∞; 5; 1)
Example: Anomalous magnetic moment of electron (𝑔𝑒 − 2)
terms omitted
∞
𝜁 𝑛 =
𝑚=1
1
𝑚𝑛
∞
ln 2 = −
𝑚=1
−1 𝑚
𝑚
Li4
1
=
2
∞
𝑚>𝑛>0
−1 𝑚+𝑛
𝑚3 𝑛
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