Non-Perturbative Quantum Field Theories on the Light Front

Yang Li [email protected] @AichiLee

Department of Physics and Astronomy, Iowa State University, Ames, IA

October 15, 2017 Guest Lectures at PHYS 625 Physics of Strong Interactions Outline

I: Light-Front Hamiltonian approach to quantum ﬁeld theory

II: Quantum as a relativistic bound state on the light front

Lecture slides will be made available online: http://leeyoung.public.iastate.edu Talks →

1/53 Outline

I: Light-Front Hamiltonian approach to quantum ﬁeld theory

II: Quantum as a relativistic bound state on the light front

Lecture slides will be made available online: http://leeyoung.public.iastate.edu Talks →

1/53 Introduction The core mission of Nuclear Physics is to study, explore and manipulate nuclear matter.

QCD is the underlying theory that describes all forms of nuclear matter Quarks and gluons are the most fundamental d.o.f.’s Perturbation theory fails at large coupling or bound-state problems

Solving QCD in the nonperturbative regime is one of the fundamental challenges in Nuclear Physics. Physics drivers: hadron spectroscopy, hadron structures, hadron production and decay, deriving realistic nuclear forces, QCD phase diagram, ... 2/53 Quantum Field Theory Quantum ﬁeld theory = quantum mechanics + special relativity

many-body continuum I q(t) −−−−−−→ qi(t) −−−−−→ q(t, ~x) µa example: quark and gluon field operators ψiα(x),A (x) continuum 0 3 0 I CCR: [qi(t), pj (t)] = iδij −−−−−→ [ϕ(x), π(x )]t=t0 = iδ (x − x ) continuum I Heisenberg equation: ∂tq = i[H, q] −−−−−→ ∂tψ = i[H, ψ] Z h R 4 i I functional integral: Z[J] = Dϕ exp iS + i d x J(x)ϕ(x) Poincarésymmetry: Lorentz symmetry plus translational invariance † µ µ ν I Poincarétransformations: U (Λ, a)V (x)U(Λ, a) = Λ ν V (Λ · x + a) µ µν I 10 generators: {P ,J } = {H, P,~ J,~ K~ } (or their linear combinations) † I unitary representation: U U = 1 ⇒ generators are Hermitian

I No ﬁnite dimensional unitary representation → physically, ∞ particles Gauge symmetry, quantum chromodynamics and Standard Model

QFT = ⇒

3/53 What is a particle in QFT? Wigner classiﬁcation

I particles are eigenstates of Hamiltonian: H ψ = E ψ | ii i| ii I identiﬁed with quant. numbers of mutually commuting operators: H, P,~ S~2,S { z} µ ψ (P, j, λ) = P µ ψ (P, j, λ) P | h i | h i ~2 ψ (P, j, λ) = j(j + 1) ψ (P, j, λ) S | h i | h i ψ (P, j, λ) = λ ψ (P, j, λ) . Sz| h i | h i I dynamical (diﬃcult) vs kinematic operators (simple), quantization 2 2 2 I 2 Casimirs: P M , S~ j(j + 1) Lorentz scalars → h → † (j) 0 I unitary representation: U (Λ) P, j, λ = 0 D 0 (W ) Λ P, j, λ | i λ λ,λ P | · i Perturbation theory P I scattering problem: S-matrix I each particle appears as an S-matrix pole

I however, any ﬁnite order perturbation amplitude does not have poles

I asymptotic series

4/53 What is a particle in QFT? Wigner classiﬁcation

I particles are eigenstates of Hamiltonian: H ψ = E ψ | ii i| ii I identiﬁed with quant. numbers of mutually commuting operators: H, P~, S~2,S { z} µ ψ (P, j, λ) = P µ ψ (P, j, λ) P | h i | h i ~2 ψ (P, j, λ) = j(j + 1) ψ (P, j, λ) S | h i | h i ψ (P, j, λ) = λ ψ (P, j, λ) . Sz| h i | h i I dynamical (diﬃcult) vs kinematic operators (simple), quantization 2 2 2 I 2 Casimirs: P M , S~ j(j + 1) Lorentz scalars → h → † (j) 0 I unitary representation: U (Λ) P, j, λ = 0 D 0 (W ) Λ P, j, λ | i λ λ,λ P | · i Perturbation theory P I scattering problem: S-matrix I each particle appears as an S-matrix pole

I however, any ﬁnite order perturbation amplitude does not have poles

I asymptotic series

4/53 Non-Perturbative approaches Lagrangian formalism Hamiltonian formalism Euclidean space-time Minkowski space-time

two complementary pictures of QCD Ab initio calculation of QCD remains one of the most challenging problems in theoretical and computational physics.

5/53 Hamiltonian Formalism

Schr¨odingerEquation H ψ = E ψ | i | i

nonrelativistic nonrelativistic relativistic few-body many-body many-body

Wave function provides the full information of the system.

proton density neutron minus proton physical picture

9Be gs density: emergence of a neutron ring binding two α clusters [Cockrell ’12]

6/53 Dirac’s forms of relativistic dynamics [Dirac, Rev.Mod.Phys. ’49] In relativity, t = x0 is not the only choice of “time”, which dictates the direction of the dynamical evolution.

instant form front form point form √ 0 + 0 3 time variable t = x x , x + x τ , t2 − ~x2 − a2

quantization surface

0 − 0 3 µ Hamiltonian H = P P , P − P P kinematical ~ ~ ~ ⊥ + ~ ⊥ + ~ ~ P, J P ,P , E ,E ,Jz J, K dynamical K,P~ 0 F~ ⊥,P − P,P~ 0 dispersion p0 = p~p2 + m2 p− = (~p2 + m2)/p+ pµ = mvµ (v2 = 1) relation ⊥

± 0 3 ~ ⊥ 1 2 ± 0 3 ⊥ 1 2 i +i P , P ± P , P , (P ,P ), x , x ± x , ~x , (x , x ), E = M , + +− i −i i 0i i 1 ijk jk E = M , F = M , K = M , J = 2 M . 7/53 Light-Front dynamics [Reviews: e.g. Burkardt ’96, Carbonell ’98, Brodsky ’98]

Light-front quantization deﬁnes a system on the light front t + z/c = 0. LF time x+ = x0 + x3, LF energy p−, LF 3-momentum: (p+, p1, p2), where p∓ = p0 ∓ p3 dispersion relation (cf. non-relativistic dispersion relation) ( p0 = p~p2 + m2, equal-time pµp = m2 ⇒ µ − 2 2 + p = (~p⊥ + m )/p , light-front + − Spectral condition: p 0, p 0 [Leutwyler ’78]

p1 ≥ ≥ k + + + Implication: δ(p1 + p2 + k ) ∝ ∗ p2 light-front vacuum is simple !! x+ Schr¨odingerequation: dynamical evolution ∂ 1 i ψ(x+) = Pˆ− ψ(x+) . ∂x+ | i 2 | i Einstein equation: hadron spectrum and wave functions + − 2 2 (P Pˆ P~ ) ψ = M ψ . x+ =0 − ⊥ | hi h| hi LFQCD connects Quark Model and Parton Model to QCD. [Bakker ’13]

8/53 Light-Front Wavefunctions (LFWFs) + ψ (P, j, λ) = [dµ ] ψ ( ~k ⊥, x , λ ) ~p ⊥, p , λ | h i n n/h { i i i}n |{ i i i}ni n Z LFWFs are frame-independentX (boost invariant) and depend only on the + + ~ ~ relative variables: xi ≡ pi /P , ki⊥ ≡ ~pi⊥ − xiP⊥ LFWFs provides intrinsic information of the structure of hadrons, and are indispensable for exclusive processes in DIS [Lepage ’80]

I Overlap of LFWFs: structure functions (e.g. PDFs), form factors, ... I Integrating out LFWFs: light-cone distributions (e.g. DAs) “Hadron Physics without LFWFs is like Biology without DNA!” — Stanley J. Brodsky GTMDs 2 2 ~ ~ ~ Z d b Z d k k⊥ ↔ ~r⊥, ∆⊥ ↔ b⊥, Z ⊥ dx Z ⊥ Z ~ ~ TMFFs b⊥ = ~r⊥ − R⊥ TMDs GPDs [Lorc´e& Pasquini ’11]

TMSDs FFs PDFs

charges hadron tomography

9/53 + Light-Front QCD with Light-Cone Gauge (A = 0) [Brodsky ’98] 1 L = F F µνc + ψ iD/ m ψ ym −4 µνc − 1 (i∂⊥)2 + m2 Pˆ− = dx−d2x⊥ ψγ+ ψ + Aia(i∂⊥)2Aia LFqcd 2 i∂+ Z − 2 ⊥ µa a + gs dx d x ψγµA T ψ Z g2 γ+ + s dx−d2x⊥ ψγ AµaT a γ AνbT bψ 2 µ i∂+ ν Z g2 1 + s dx−d2x⊥ ψγ+T aψ ψγ+T aψ 2 (i∂+)2 Z 1 g2 dx−d2x⊥ if abc ψγ+T cψ i∂+AµaAb − s (i∂+)2 µ Z − 2 ⊥ abc µ νa b c + gs dx d x if i∂ A AµAν Z g2 1 + s dx−d2x⊥ if abc if ade i∂+AµbAc i∂+AνdAe 2 µ (i∂+)2 ν Z g2 s dx−d2x⊥ if abc if adeAµbAνcAd Ae . − 4 µ ν Z 10/53 Mode expansions:

ψ (x) = b (x) + d† (x) , i αi Uα αiVα x+=0 α X Aµc(x) = a µ(x) + a† µ∗(x) . γcEγ γcEγ x+=0 γ X † † † aγ , aγ0 =δγγ0 , bα, bα0 = δαα0 , dα, dα0 = δαα0

Basis functions: (x), (x), (x), subject to orthogonality and completeness (forU dynamicalV d.o.f.):E µ µ −ik·x I plane-wave basis: e.g., aγ aλ(k), γ (x) ελ(k)e Discretize Light Cone Quantization→ (discretizedE → plane-wave basis) d3k I wave packet: aγ = (2π)3 φγ (k)aλ(k) Basis Light-Front Quantization (soft-wall AdS/QCD basis) R Regularization, gauge ﬁxing, renormalization [e.g., Chabysheva ’15, cf. Ji ’03] In principle, bound-state and scattering problems can be solved from LFQCD Hamiltonian dynamics. [Zhao ’13] ∂ 1 i ψ(x+) = P − ψ(x+) ∂x+ | i 2 | i

11/53 Basis Light-Front Quantization (BLFQ) [Vary et al, Phys.Rev.C ’10] Second quantized Hamiltonian in a basis representation

† † † Hlc = Hfi a a a a{ρ aρ2 aρ } × {γ1 γ2 ··· γf } 1 ··· i Xf,i I basis can be chosen to encode the analytical approximation to the solution: e.g., transverse soft-wall AdS/QCD basis plus longitudinal discretized momentum basis:

− + i k+x− − 1 ρ2 |m| |m| imθ q + f (~x , x ) = N k e 2 e 2 ρ L (ρ)e , (θ = arg ~x , ρ = k x Ω) α ⊥ n ⊥ P + ⊥ I basis regularization (continuum limit: Nmax → ∞,L → ∞), K → ∞ − L ≤ x− ≤ +L, with (anti-)periodic boundary conditions X eﬀective Hamiltonian: 2ni + |mi| + 1 ≤ Nmax OSL (Nmax,K,Ω) i Hlc Hlc ←−→SRG I optional: additional Fock sector truncation Choice of basis reﬂects symmetries of the Hamiltonian P + + i ki = K;( pi = 2πki/L, P = 2πK/L) P i mi + si = mJ

H → H + λHcm; (separate the c.m. motion) B, L, Q conservation and global color singlet (QCD);

12/53 What motivates the BLFQ approach? BLFQ yields large sparse matrix eigenvalue problems, in parallel with the Conﬁguration Interaction method in nonrelativistic many-body theory.

I exact treatment of the kinematical symmetries [YL ’13, Maris ’13]

I success in No-Core Full Conﬁguration approach [Barrett ’13]

I progresses in diagonalizing sparse matrix on parallel computers [Vary ’09]

I Moore’s law: growth in capacity of supercomputers Key insight:“ nonrelativistic and light-front Hamiltonian problems have much in common” — James P. Vary

��

�� number of Fock sectors

� �� =� 13/53 Eﬀective Hamiltonian How to obtain information of the system from smaller model spaces?

† Heff =S HS d ←−−−−−−−−−→dλ Hλ=[ηλ,Hλ]

Bloch eﬀective Hamiltonian (OSL) [Wilson ’76] Similarity renormalization group (SRG) [G lazek ’94, Wegner ’94]

RGPEP calculation of the eﬀective tri-gluon vertex: [G´omez-Rocha ’15] 3 5 gλ = g0 − g0 β0 ln(λ/λ0) + O(g )

14/53 Diagrammatic Representation [Weinberg ’62, Carbonell ’98, Brodsky ’98]

Hamiltonian perturbation theory: x+-ordered diagrams 2 2 I all particles are on their mass-shells pi = mi ; I longitudinal and transverse momenta are conserved at each vertex; P + + P 2 P ~ 2 P ~ δ( i pi − P ) ⇔ δ( i −1); δ ( i ~pi⊥ − P⊥) ⇔ δ ( i ki⊥) I the light-front energy is not conserved (“oﬀ the energy shell”);

I energy denominator for intermediate states; n ~ 2 2 1 2 k ⊥ + m k1 k1′ , s (k + + k ) = a a , q 2 n 1 n sn M ≡ ··· xa k2 k2′ a=1 − X 2 is the mass eigenvalue s3 =(k1 + q + k2′ ) M

Extended to non-perturbative regime by introducing vertex functions Γn (n 1) 2 Γn = (sn M )ψn. − Γ are also− boost invariants. n Γn

15/53 Fock sector dependent renormalization Systematic Fock sector expansion with a Fock sector dependent renormalization Counterterms are classiﬁed by the number of spectators instead of the coupling constant Key insight: The counterterms are the same regardless of the reference frame of the sub-cluster, thanks to the kinematical nature of light-front boosts.

† Example: scalar model Lint = gχ χϕ χχ¯ph = χχ¯ + χχϕ¯ + χχϕϕ¯ + χχϕϕϕ¯ + | i | i | i | i | i ···

g03 2 δm3 Z(2) = + g03 = + Γ3 Γ2 Γ2

Γ2 Γ2 Γ2 δm2 3 + + +

Γ3 s4a Γ3 s4b Γ3 s +g03 + 4c g03 Γ Γ 3 3 + + +

Γ3 Γ3 Γ3 s4e s4d s4f Renormalization parameters are ﬁxed in the one-body sector [YL, Karmanov, Maris, Vary, in progress]

16/53 Fock sector expansion up to four particles: χph = χ + χϕ + χϕϕ + χϕϕϕ + χϕϕϕϕ | i | i | i | i | i | i · · · Diagrams in the one-body sector:

Γ1 Γ1 Γ2 = + 2 δm gb

Γ2 Γ1 Γ2 = + g b δm2

Γ3 g + b

a a a b b b Γ3 Γ2 Γ2 = + gb gb a a b b Γ3 Γ4 g + + b δm2 a b b c c a Γ4 Γ3 = + a b + a c gb ↔ ↔ 17/53 Solutions of the charge-one sector [YL et al, Phys.Lett.B758,118 (2015), Karmanov, PRD94, 096008 (2016)] Obtain solutions of the charge-one sector up to four-body: χ + ϕϕϕ Study the convergence of Fock sector expansion by comparing diﬀerent Fock sector truncations Fock sector convergence of the electromagnetic form factor:

P P + q P P + q P P + q P P + q + + +

q q q q

(a) one-body (b) two-body (c) three-body (d) four-body

1. 1.

0.8 0.8

0.6 0.6   2 2

FQ 0.4 FQ 0.4 two-body two-body 0.2 three-body 0.2 three-body four-body α = 1.0 four-body α = 2.0 0. 0. 0 5 10 15 20 25 0 5 10 15 20 25 2 2 Q Rapid Fock sector convergence! Q 18/53 Outline

I: Light-Front Hamiltonian approach to quantum ﬁeld theory

II: Quantum as a relativistic bound state on the light front

Lecture slides will be made available online: http://leeyoung.public.iastate.edu Talks →

19/53 Light-Front Tamm-Dancoﬀ equation LF eigenvalue equation can be represented as an inﬁnite numbers of coupled integral equations

2 2 dτ Pˆ ψ = M ψ G0 ψ (P, j, λ) = V (ωτ)G0 ψ (P, j, λ) | hi h| hi ⇒ | h i − 2π | h i Z

I In practical calculations, the inﬁnite tower has to bee truncated. Tamm-Dancoﬀ Approximation: leading Fock sector truncation

= + + ···

~ 2 2 ~ 2 2 2 λ k⊥ + mq k⊥ + mq¯ λ M ψ (x,~k⊥) = + ψ (x,~k⊥) h ss/h¯ x 1 x ss/h¯ h ¯ − i ¯ u¯s0 (k)Σr(k)vs¯(k) λ u¯sΣr(k)vs¯0 λ + ψ 0 (x,~k⊥) + ψ 0 (x,~k⊥) x s s/h¯ 1 x ss¯ /h s0 s¯0 − X 1 0 2 0 X dx d k⊥ 0 0 λ 0 0 + V 0 ¯0 ¯(x,~k⊥, x ,~k )ψ 0 0 (x ,~k ). 2x0(1 x0) (2π)3 s s ,ss ⊥ s s¯ /h ⊥ 0 ¯0 0 sX,s Z − Z 20/53 Light-Front Holography [Review: Brodsky et al, Phys.Rep. (2015)] What is the eﬀective relativistic interaction V for QCD?

I pQCD, Hamiltonian renormalization group, holography, Lattice QCD, relativized NR potentials, ... 4 2 Answer (ﬁrst approximation): κ x(1 x)~r [Brodsky ’06] − ⊥ I AdS/CFT, soft-wall AdS/QCD [Maldacena ’98, Erlich ’05, Karch ’06]

I Holographic variable ζ~⊥ = x(1 x)~r⊥: − ζ~2 = (4∂/∂Q2)T ++(0) = (2/3) r2 r.m.s. matter radius h ⊥i − p h mi I In LFH, ζ⊥ is mapped to the ﬁfth dimension z in anti-de Sitter space

I Superconformal algebra unifying mesons and baryons [Brodsky ’16]

I AdS/QCD running coupling: [Deur ’14] (AdS) 2 (AdS) 2 2 αs (Q ) = αs (0) exp Q /(4κ ) − I Successful applications: spectroscopy, elastic form factors, transition form factor, GPDs, β-function, diﬀractive VM productions, ... Can we go further based on the success of LFH?

I Finite quark masses: LFH is massless

I Short-distance physics: LFH is long-distance

I Spin structure and self-energy: LFH is semi-classical 21/53 Quark Mass and Longitudinal Amplitude 2 2 2 2 k2 k +m k +m Invariant mass ansatz: ⊥ ⊥ q + ⊥ q¯ in LFWFs [Brodsky ’08] x(1−x) → x 1−x Longitudinal conﬁnement ansatz: add quark masses AND an interaction σ2∂ x(1 x)∂ − x − x I Provide independent longitudinal modes

I Agree with IMA in both the chiral limit and the heavy quark limit, except endpoints a b I Solutions power-law-like: x (1 x) ∼ − 2 2 I In heavy quarkonium, choose σ = 4mq/κ to match transverse conﬁnement in NR limit 2.0 1.5 � �� κ solid: IMA ��/κ solid: IMA � � �� /�� dashed: LCA ��/�� dashed: LCA �� /�� 1.5 �� /�� ∫ |�(�) �=� �� (�-�) 1.0 �� ∫ |�(�) �=� ) ) �(�-�)

� ��

( 1.0 ( �� 0.5 0.5 ��

0.0 0.0 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 � � In the light sector, mq-independent σ reproduces GMOR [Gutsche, ’13] 22/53 Krautgärtner-Pauli-WölzOne-Gluon Exchange [Krautgärtner’92] KPW OGE interaction gives the leading effective short-distance physics: 2 (kpw) CF 4παs(Q ) µ 0 0 0 ν V = u¯ (k)γ u (k )¯v¯0 (k¯ )γ v¯(k¯)d , − Q2 s s s s µν 2 0 2 ¯ ¯0 2 P λ∗ λ where Q = −(1/2)(k − k ) − (1/2)(k − k ) ; dµν = λ εµ (k)εν (k).

I Can be obtained from perturbative expansion or leading Fock sector truncation [Chakrabarti ’01, Pauli ’97]

I Extensively tested in QED to high precision [Krautg¨artner’92, Trittmann ’93, Wiecki ’15, Lamm ’16]

I KPW counterterm vs running coupling

I Myth Burster: why solving bound states with a perturbative kernel is still nonperturbative? p0 − m = O(α), 2 O(1/α) O(1/α) |~q| ∼ rb = O(α ) ∼ ∼

In a bound state, each ladder diagram is ∼ O(1/α). Summing over all ladder diagrams gives rise to the bound-state pole. [e.g. Hoyer ’11] Beyond KPW: FSDR, LFCC, RGPEP, ... [e.g. G lazek ’17] 23/53 Self-Energy The self-energy is included in the running mass: 2 2 m ψ ¯ m ψ ¯ + u¯ 0 Σ v¯ψ 0 ¯. q ss → q ss s r s s s s0 The self-energy can also be obtained fromX one-body part of the eigenvalue equation:

p = g0 Σ(p) Γ f

1 2 Z dx Z d k p/ − k/ + mq Σ(p) = − ⊥ C 4πα (s)d γµ 1 Γν (x,~k ) 3 F s µν 2 qg/q ⊥ 0 2x(1 − x) (2π) s − p Here Γ is related to the quark-gluon wave function in the single-quark sector and is not 1PI. In practice, we express Γ in terms of 1PI vertex Γ and model the latter. Γµ = γµ + ··· In one-gluon-exchange, non-perturbative physicse in Γ can be absorbed in the running coupling. e Dynamical chiral symmetry breaking on the light fronte [e.g. Burkardt ’97, Wu & W.M. Zhang ’04, Beane ’13] 24/53 Basis Light-Front Quantization [Li, Maris & Vary, PRD 96, 016022 (2017)] All these improvements can be combined in the BLFQ approach

I Adopt LFH conﬁnement and one-gluon exchange (KPW) for long- & short-distance physics 2 2 2 I pQCD running coupling: αs(Q ) = 4π/[β0 ln(Q /Λqcd + τ) I alternative eﬀective interactions? → connection with full BLFQ?

I Incorporate quark mass and longitudinal dynamics

I Free parameters: κ and mq are ﬁtted to experimental spectrum

Adopt basis function approach [Vary, ’10, Li ’16] 0 ψ 0 (x,~k⊥) = ψ (n, m, l, s, s ) φ (~k⊥/ x(1 x))χ (x). ss /h h nm − l n,m,l X p a b I φ HO functions; χ Jacobi polynomial weighed by x (1 x) . nm l − I Basis truncation: 2n + m + 1 Nmax, l Lmax | | √≤ ≤ √ I UV and IR cutoﬀs: Λuv ≈ κ Nmax, λir ≈ κ/ Nmax −1 I light-cone resolution: ∆x ≈ Lmax I continuum limit: Nmax → ∞, Lmax → ∞ First application: heavy quarkonium Ignore self-energy correction for heavy quarkonium.

25/53 Spectroscopy [YL, Maris & Vary, PRD 96, 016022 (2017); arXiv:1704.06968]

κ (GeV) mq (GeV) rms (MeV) δJ M (MeV) Nmax basis dim.

cc¯ 0.966 1.603 31 17 32 1812 b¯b 1.389 4.902 38 8 32 1812 √ κ determined from ﬁts to spectrum follows the HQET trajectory κh ∝ Mh, in agreement with recent LFH result [Dosch et al, PRD95 (2017)]

26/53 Comparison with the ﬁx-coupling results [YL, Maris, Zhao & Vary Phys. Lett. B 758, 118 (2016)]

ﬁxed coupling running coupling αs = 0.36

Improvement from ﬁxed coupling: violation of rotational symmetry is reduced due to omission of the KPW counterterm.

MeV δMcc¯ δJ M cc¯ δMb¯b δJ M b¯b Nmax basis dim.

ﬁx-αs 52 49 58 17 24 1200

running-αs 31 17 39 7 24 1200 27/53 Unequal-mass heavy quarkonium Bc [Tang, YL, Maris, and Vary, in preparation]

7.6 � � � �� 7.4 � � �� 7.2 � �� 7.0 � � �� (��) � �� 6.8 � � �� 6.6 � ��

6.4 � � �� (��) BLFQ 6.2 preliminary PDG 6.0 0- 1- 0+ 1+ 2+ 2- 3- 3+ 4+

I No free parameter: b, c masses from quarkonia ﬁts; 2 2 κbc = (κcc + κbb)/2 from HQET;

κ (GeV)p mc (GeV) mb (GeV) rms (MeV) δJ M (MeV) Nmax

cc¯ 0.966 1.603 — 31 17 32 b¯b 1.389 — 4.902 38 8 32 bc¯ 1.196 1.603 4.902 37 6 32 28/53 Decay constants [YL, Maris & Vary, PRD 96, 016022 (2017); arXiv:1704.06968] NR picture: wave function at the origin — short-distance physics. Decay constants are deﬁned as the vacuum-to-hadron matrix elements: + + 0 ψ(0)γ γ5ψ(0) P (p) = ip f , h | | i P 0 ψ(0)γ+ψ(0) V (p, λ) = e+M f . h | | i λ V V LFWF representation Λ 1 uv 2 fP,V dx d k⊥ (λ=0) ~ = lim Z2(µ, Λuv) 3 ψ↑↓∓↓↑(x, k⊥). 2√2Nc Λuv→∞ 0 2 x(1 x) (2π) √ Z − Z Λuv = κ Nmax ≈ 1.7mq, Z2 → 1. p

29/53 UV asymptotics of the wave function

I Build-up of a physical asymptotic region as UV regulators increase 2 2 γ p ψ ∼ 1/ q⊥(ln q⊥) ,(q⊥ = k⊥/ x(1 − x)) √ I Beyond Λuv = κ Nmax, the asymptotics is controlled by the basis

��

κ �� ) ⟂ ��-�

( � �=��

↑↓ - ↓↑ -� �� � �� ψ �� 8 16 24 ��-� 32 � � � � � ��

�⟂/ �(�-�)(��) The physical asymptotic tail induces a UV divergence that can be cancelled by Z2 2 2 −γS ln(Λuv/Λqcd) Z2 = 2 2 ln(µ /Λqcd) Low cutoff at Mh 2mq effectivelyh terminatei the physical asymptotics ≈ Since no renormalization is performed, the effective RS µ ≈ Mh ⇒ Z2 ≈ 1

30/53 Radii [YL, Maris & Vary, PRD 96, 016022 (2017); arXiv:1704.06968]

The r.m.s. charge and mass radii – long-distance physics: [YL, FBS ’17] 3 3 r2 6F 0 (0) = b2 , r2 6F 0 (0) = ζ2 h c i , − em 2h ⊥i h mi , − gr 2h ⊥i

~2 P ~ 2 ~ I b⊥ = i ei(~r⊥ − R⊥) ⇒ b⊥ = (1 − x)~r⊥: Burkardt’s impact parameter ~2 P ~ 2 ~ p I ζ⊥ = i xi(~ri⊥ − R⊥) ⇒ ζ⊥ = x(1 − x)~r⊥: Brodsky & de T´eramond’sholographic coordinate [Burkardt ’01, Brodsky ’06]

I Diﬀerence between charge and mass radii is a relativistic eﬀect

~ P R⊥ = i xi~r⊥ is the transverse center of

[DSE: Maris ’07; Lattice: Dudeck ’06; BLFQ15: YL, PLB ’16] the system. 31/53 Transition form factors [M.-j. Li et al, in progress] ∗ M1 transition J/ψ ηcγ : → 2 + 2V ( q ) µαβρ λ P (p + q) J (0) V (p, λ) = − ε (p + q)αpβeρ (p). h | | i MP + MV Radiative decay width: 3 2 1 2 MP ΓV →P γ = αeme 1 (MV + MP ) V (0) 6 f − MV

[Lattice: Dudek ’06; CB: Gaiser ’86;] More radiative transitions: [M.-j. Li et al, in progress]

I E1,C1: J/ψ → χc0γ I E1,M2,C1: χc1 → J/ψγ I E1,C1: hc → ηcγ 32/53 Wave functions [YL, Maris & Vary, PRD 96, 016022 (2017); arXiv:1704.06968] Lorentz structure of the wave function, e.g. pseudo scalar + γ γ5 ψ ¯(x,~k⊥) =u ¯ (k1) φ1(x, k⊥)γ5 + φ2(x, k⊥) v¯(k2). ss s P + s Similarly, vector mesons haveh 6 structure functions φ1–6. i In heavy quarkonium, non-relativistic spin conﬁguration emerges as the dominant component:

ηc(1S)

33/53 Wave functions of angularly excited states

J/ψ vs ψ(1D), dominant components

ψ(1D), λ = +1, dominant component

34/53 Wave functions of radially excited states

ηb(1S) ηb(2S) ηb(3S)

Υ(2D)

35/53 Distribution amplitudes [YL, Maris & Vary, PRD 96, 016022 (2017); arXiv:1704.06968] Distribution amplitudes (DAs) control the exclusive processes at large momentum transfer [Lepage & Brodsky ’80] DAs are deﬁned from nonlocal matrix elements, e.g.: 1 + + ip+z−(x− 1 ) 0 ψ(z)γ γ5ψ( z) P (p) µ = ip fP dx e 2 φP (x; µ) + h | − | i 0 z ,~z⊥=0, Z LFWF representation: 2 .µ 2 fP,V 1 d k⊥ λ=0 ~ φP,V (x; µ) = 3 ψ↑↓∓↓↑(x, k⊥). 2√2Nc x(1 x) 2(2π) − Z p

36/53 DAs (top panels) vs LFWFs in coordinate space (bottom panels):

37/53 Comparison with AdS/QCD models

AdS/QCD with longitudinal dynamics for heavy quarkonium: m2 invariant mass ansatz: φ(x) = Npx(1 − x) exp(− q ) I 2κ2x(1−x) 1 α 2 + 2 2 2 I longitudinal conﬁnement: φ(x) = N x(1 − x) , α = 4mq/κ . Baselines:

I pQCD asymptotics (µ → ∞): 6x(1 − x) p I AdS/QCD of Brodsky and de T´eramond(mq → 0): (8/π) x(1 − x)

38/53 Scale evolution (preliminary) We can compute DAs with diﬀerent scales φ(x; µ), namely access the scale dependent of the DAs. Q1 How does it compare with the ERBL evolution equation for DAs? Q2 How does the DAs at large µ compare with the pQCD asymptotics? Computational challenge: in BLFQ, the basis size is tied together with the UV coverage via µ = Λuv , κ√Nmax (ln 32 1.7 ln 8) What if we extrapolate DAs point-wisely as a function≈ of µ using simple polynomials in µ−1? Probably huge uncertainty. �� μ=�� η� μ=�� μ=�� μ=�� μ=∞(��) �� ∫��ϕ(�)=�

�� ϕ ( � )

��

�� 39/53 Parton distribution functions Parton distribution function q(x; µ) is the probability of ﬁnding a collinear quark carrying momentum fraction x up to scale µ. LFWF representation

2 .µ 2 1 d k⊥ 2 q(x; µ) = ψ ¯(x,~k⊥) . x(1 x) 2(2π)3 ss s,s¯ Z − X

PDF vs DA of excited states (amplitude vs probability)

40/53 Generalized parton distributions [Burkardt ’00&’02] ~b (1 − x)~r 2 ⊥ , ⊥ d ∆⊥ i∆~ ⊥·~b⊥ 2 1 2 q(x,~b⊥) = e H(x,ζ = 0, ∆ ) = ψ 0 (~r⊥, x) . (2π)2 − ⊥ (1 x)2 | ss | Z − ss0 2 X I partonic interpretation: q(x,~b) 0, dx d b⊥ q(x,~b) = 1 e ≥ I unintegrating the form factors Z [Adhikari et al. in preparation]

0 ηc ηc χc

0 ηb η χb 41/53 b Comparison with CST Gross’ Covariant Spectator Theory is based on Bethe-Salpeter equation in Minkowski space, and is recently revived for mesons. [Gross ’69, Leitão’17] Use Brodsky-Huang-Lepage ansatz to obtain light-front wave functions Both reproduce mass spectrum and decay constants [Leitão,arXiv:1705.06178v1] Comparison of wave functions helps to validate the BLFQ state identification

�� =� �� =� �� /ψ(��) �� ψ(��) �� ( � ) ( � ) || ||

ϕ ϕ ��

��

�� 42/53 Diﬀractive vector meson production [Chen, Phys.Lett.B 769 (2017)] Diﬀractive VM production in DIS is an important tool for studying the small-x gluon distribution at a future Electron-Ion Collider.

Color dipole model [Mueller ’94, Nikolaev ’91, Kowalski ’03 & ’06] 1 γ∗p→V p 2 2 dz ∗ −i[b−(1−z)r]·∆ dσqq¯→qq¯ = i d rd b Ψ Ψ ∗ e AT,L 4π V γ T,L d2b Z Z0 Comparison with phenomenologically successful wave functions

�� [��] �� [��] �/ψ �� /ψ �� =�� =�� =�� =��

[ �� ] �� [ �� ] �� ψ ) ψ ) * � * �

�� / � ) ∫ ( ψ / � ) ∫ ( ψ ⟂ ⟂ ( � ( �

��

�� [��] � [��] ⟂ 43/53 ⟂ * γ �→�/ψ� γ*�→�/ψ� γ*�→�/ψ� 102

W=90GeV �=�� =�� (a) (b) 3 (c) �=�� =�� 10 � � 102 0.05 �=�� =�� =�� =�� 3.2 ��

10 ) 2 ��Ψ � 7.0 � 10 ) ) �� / �� ( ( �� 10 (

�� 10 ��

σ 22.4 σ ��

1 σ / �� Q2 (GeV2) � �� Ψ� � �� 1 �� Ψ� �� H1,ZEUS �� Ψ� 1 �� Ψ -1 � 10 2 10 10 100 200 300 0. 0.2 0.4 0.6 0.8 1. 1.2 � � � � � +��/ψ (�� ) � (��) � (�� ) [Chen, Phys.Lett.B 769 (2017)] Model independent ratio: Ultra-peripheral heavy ion collision: 1.4 ��+��→��+��+�/Ψ �� =�� σψ(� �) /σ�/ψ� �� 6. 1.2 �� Ψ � �

�� Ψ � �� Ψ �� 1. � ��Ψ � �� Ψ� 4. �� 0.8 ]

�� [ �

0.6 �� σ /

� 2. 0.4

0.2 0. 0. -4. -2. 0. 2. 4. 0 20 40 60 � �� (��) 44/53 p What’s next? = g0 Unequal-mass Bc and heavy-light Σ(p[S.) Tang in progress]Γ f I only one parameter κ to be ﬁtted

I spectroscopy, form factors (measurable!), decay constants are important for experiments Light mesons π, ρ, K, ... [W.-y. Qian in progress]

I ρ-π splitting and the chiral symmetry breaking?

I renormalization and better treatment of the UV asymptotics Baryons [A.-j. Yu in progress]

I generalizing the model to three-body via the Jacobi coordinates:

I emergence of the AdS/QCD models, quark-diquark model? Running coupling plus self-energy correction

I dynamical mass generation on the light front [Burkardt ’97] Towards systematic light-front Hamiltonian approaches:

I Full Basis Light-Front Quantization [Vary ’10] I Fock sector expansion w. sector dependent renormalization [Karmanov ’08] I Hamiltonian Renormalization Group [e.g. G´omez-Rocha ’16] I AdS/QCD [e.g. Gherghetta ’09] 45/53 Suggested Readings

Weinberg, S., 1995. The quantum theory of fields (Vol. 1&2). Cambridge • university press. • Brodsky, S.J., Pauli, H.C. and Pinsky, S.S., 1998. Quantum chromodynamics and other field theories on the light cone. Physics Reports, 301(4), pp.299-486. • Lepage, G.P. and Brodsky, S.J., 1980. Exclusive processes in perturbative quantum chromodynamics. Physical Review D, 22(9), p.2157. • Brodsky, S.J., de Téramond,G.F., Dosch, H.G. and Erlich, J., 2015. Light-front holographic QCD and emerging confinement. Physics Reports, 584, pp.1-105. • Wilson, K.G., Walhout, T.S., Harindranath, A., Zhang, W.M., Perry, R.J. and G lazek, S.D., 1994. Nonperturbative QCD: A weak-coupling treatment on the light front. Physical Review D, 49(12), p.6720. • Burkardt, M., 2002. Light front quantization. In Advances in Nuclear Physics (pp. 1-74). Springer US.

Lecture slides will be made available online: http://leeyoung.public.iastate.edu Talks →

46/53 Other Aspects of the Strong Interaction

I Perturbative QCD

I Lattice QCD

I Eﬀective Field Theory

I QCD at high temperature and/or high density

I Nuclear Structure Theory

I Dyson-Schwinger equations, Bethe-Salpeter equations and Faddeev equations

I Functional Methods: e.g. Functional Renormalization Group

I ...... Thank you!

47/53 Acknowledgements

Collaborators: Iowa State University, Ames, Iowa: James Vary, Kirill Tuchin, Pieter Maris, Lekha Adhikari, Guangyao Chen, Paul Wiecki, Meijian Li, Shuo Tang, Wenyang Qian, Anji Yu Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou: Xingbo Zhao, Chandan Mondal, Siqi Xu, Hui Liu, Hengfei Zhao Lebedev Physical Institute, Moscow: Vladimir Karmanov, Alexander Smirnov CFTP, Instituto Superior Técnico,Universidade de Lisboa, Lisbon: M.T. Peña,Alfred Stadler, Elmar P. Biernat, Sofia Leitão

bold: graduate students Funding Sources:

48/53 No-Core Conﬁguration Interaction calculations

Barrett, Navrátil, Vary, Ab initio no-core shell model,PPNP69,131(2013) Given a Hamiltonian operator

(p⃗ p⃗ )2 Hˆ = i − j + V + V + ... 2 mA ij ijk !i

Hˆ Ψ(r1,...,rA)=λ Ψ(r1,...,rA) Expand wavefunctioneigenstates in in basis basis states states Ψ = a Φ | ⟩ i| i⟩ ˆ Diagonalize Hamiltonian matrix Hij = Φj H"Φi ⟨ | | ⟩ No-Core CI: all A nucleons are treated the same Complete basis exact result −→ In practice truncate basis study behavior of observables as function of truncation

Progress in Ab Initio Techniques in Nuclear Physics, Feb. 2015, TRIUMF, Vancouver – p. 2/50

49/53 Ground state energy of p-shell nuclei with JISP16 Compare theory and experiment for 24 nuclei Maris,Maris, Vary, Vary, IJMPE22, IJMPE22, 1330016 1330016 (2013)(2013) 0 2 H 3 H -20 6 8 He He 4 6 He Li 9 -40 7 Li Li 10 Be 11 -60 8 9 Be 12 Be Be Be

10 11 13 -80 B B B

12 14 -100 C 13 C C 14 N Ground state energy (MeV) -120 15 expt N 16 JISP16 O -140 246 8 10 12 14 16 A 10B–mostlikelyJISP16producescorrect3+ ground state, but extrapolation of 1+ states not reliable due to mixing of two 1+ states 11Be – expt. observed parity inversion within error estimates of extrapolation 12 12 + + Band N–unclearwhethergsis1 or 2 (expt. at Ex =1MeV) with JISP16

Progress in Ab Initio Techniques in Nuclear Physics, Feb. 2015, TRIUMF, Vancouver – p. 7/50

50/53 Ground state magnetic moments with JISP16 Compare theory and experiment for 22 magnetic moments Maris, Maris,Vary, IJMPE22, Vary, IJMPE, 1330016 in press (2013) 1 µ = J Lp +5.586 J Sp 3.826 J Sn µ0 J +1 ⟨ · ⟩ ⟨ · ⟩− ⟨ · ⟩ ! "

7 9 Li 9 Li 3 B 13 3 B H ? 11 B ) 0 2 8

µ Li 10 B 12 2 B 1 H 6 8 13 Li B C 15 14 O 12 N N 0 13 N 15 expt N magnetic moment ( 11 JISP16 9 C 13 -1 7 9 Be O Be C ? 11 3 Be -2 He 246 8 10 12 14 16 A Good agreement with data, given that we do not have any meson-exchange currents

Pieter Maris SciDAC NUCLEI workshop, June 2013, Bloomington, IN – p. 13/37

51/53 Beyond One-Gluon-Exchange

= + + ···

diagrams included in OGE

diagrams not included in OGE. The leading Fock sector truncation gives rise to the OGE. OGE kernel coincides with O(α) amplitude in perturbation theory Systematic approaches beyond OGE

I Ab initio simulation: basis light-front quantization [e.g. Vary ’10]

I Systematic Fock sector expansion with sector dependent renormalization [e.g. Karmanov ’08]

I Coherent basis: light-front coupled cluster [e.g. Chabysheva ’12]

I Hamiltonian renormalization group: RGPEP [e.g. G lazek ’12, cf. hep-ph/1705.07629]

I Light-Front projection of Minkowski Bethe-Salpeter equations [e.g., Karmanov ’06, Frederico ’09, Salm`e’17] 52/53 Running couplings

LFH Richardson �� Cornell Maris-Tandy(IR×1/50) HO+LO pQCD(IR×1/50)

��  �  � � α

��

�� (��)

I AdS/QCD running coupling: Q2 2 − (pQCD) 2 α (Q ) = α (0)e 4κ2 ϑ(Q0 Q) + αs (Q )ϑ(Q Q0). s s − − I Relativized Cornell:

I Cornell I: 2 2 2 (pQCD) 2 αs(Q ) = 2σ/(Q + µ )ϑ(Q0 − Q) + αs (Q )ϑ(Q − Q0) 2 2 2 I Cornell II (Richardson-like): αs(Q ) = (4π)/[β0 ln(Q /Λ + τ)]; I Cornell III: 2 − Q 2 2 2 2 2 4m2 αs(Q ) = 2σ/(Q + µ ) + (4π)/[β0 ln(Q /Λ + τ)](1 − e t )

53/53