and in Basis Light Front Quanzaon

James P. Vary

Department of Physics and Astronomy Iowa State University Ames, USA

LC2018 JLAB, Newport News, Virginia May 14-18, 2018 Dirac’s forms of relativistic dynamics [Dirac, Rev. Mod. Phys. 21, 392 1949] Dirac’s Instant Forms form is the of well-known Relativistic form of dynamics Dynamics starting with x0 [Dirac,= t = 0 Rev.Mod.Phys. ’49] K i = M 0i , J i = 1 ε ijk M jk , ε ijk = (+1,-1,0) for (cyclic, ant+i-cyclic, repeated) indeces Front form defines2 QCD on the light front (LF) x , t + z =0. Front form defines relativistic dynamics on the light front (LF): x+ = x0+x3 = t+z = 0 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 . instant form front form point form

time variable t = x0 x+ x0 + x3 ⌧ pt2 ~x 2 a2 , ,

surface

0 0 3 µ Hamiltonian H = P P P P P , ~ ~ ~ + ~ + − ~ ~ kinematical P,J P ?,P , E?,E ,J Jz J,K 0 0 dynamical K,P~ F~ ?,P P,P~ dispersion 0 2 2 + µ µ 2 p = p~ 2 + m2 p =(p~ + m )/p p = mv (v =1) relation ? Yang Li, Iowa Statep U, May 20, 2016 2/1 Baryons 2016, Tallahassee,Adapted FL from talk by Yang Li Light Front (LF) Hamiltonian Defined by its Elementary Vertices in LF Gauge

QED & QCD

QCD Light-Front Wavefunctions (LFWFs) ~ + h(P, j, ) = [dµn] n/h( ki ,xi,i n) p~ i ,pi ,i n | i { ? } |{ ? } i n Z LFWFs are frame-independentX (boost invariant) and depend only on the + + relative variables: xi p /P ,~ki p~ i xiP~ ⌘ i ? ⌘ ? ? LFWFs provides intrinsic information of the structure of , 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) “ Physics without LFWFs is like Biology without DNA!” —StanleyJ.Brodsky GTMDs d2b d2k ~k ~r , ~ ~b , Z dx Z ? Z ? ? $ ? ? $ ? ~b = ~r R~ TMFFs ? ? ? TMDs GPDs [Lorc´e& Pasquini ’11]

TMSDs FFs PDFs

charges hadron tomography

9/55 Discretized Quantization [H.C. Pauli & S.J. Brodsky, PRD32 (1985)]

Basis Light Front Quantization [J.P. Vary, et al., PRC81 (2010)] ! ! ! k , x ⎡ f k , x a f * k , x a† ⎤ φ( ⊥ )= ∑ ⎢ α ( ⊥ ) α + α ( ⊥ ) α ⎥ α ⎣ ⎦ where {aα } satisfy usual (anti-) commutation rules. ! Furthermore, fα (x) are arbitrary except for conditions: ! ! 2 * d k⊥dx Orthonormal: fα (k⊥, x) fα ' (k⊥, x) 3 = δαα ' ∫ (2π ) 2x(1− x) ! ! ! ! f k , x f * k′ , x′ 16 3 x(1 x) 2 k k′ x x′ Complete: ∑ α ( ⊥ ) α ( ⊥ )= π − δ ( ⊥ − ⊥)δ ( − ) α For mesons we adopt (later extended to baryons): [Y. Li, et al., PLB758 (2016)] ! ! fα={nml} (k⊥, x)= φnm (k⊥ x(1− x))χl (x)

φnm 2D-HO functions a b χl Jacobi polynomials times x (1− x) Set of Transverse 2D HO Modes for n=4 m=0 m=1 m=2

m=3 m=4

J.P. Vary, H. Honkanen, J. Li, P. Maris, S.J. Brodsky, A. Harindranath, G.F. de Teramond, P. Sternberg, E.G. Ng and C. Yang, PRC 81, 035205 (2010) Normalized longitudinal basis functions

Jacobi polynomials Applications of the BLFQ to Mesons and Baryons at LC2018

Guangyao Chen, Yang Li, Pieter Maris, Kirill Tuchin and James P. Vary, “Diffractive charmonium spectrum in high collisions in the basis light-front quantization approach,” Phys. Letts. B 769, 477 (2017); arXiv: 1610.04945. [Talk on Thursday morning]

Sofia Leitao, Yang Li, Pieter Maris, M.T. Pena, Alfred Stadler, James P. Vary and Elmar P. Biernat, “Comparison of two Minkowski-space approaches to heavy quarkonia”, Eur. Phys. J. C 77, 696 (2017); arXiv: 1705:06178 [Talk on Friday]

Meijian Li, Yang Li, Pieter Maris and James P. Vary, “Radiative transitions between 0-+ and 1-- heavy quarkonia on the light front,” arXiv:1803.11519 [Talk on Thursday in session 3B]

Shaoyang Jia, et al., “Basis Light-Front Quantization for the Light Mesons,” in preparation [Talk on Thursday in session 3B]

Chandan Mondal, et al., “Baryons in a BLFQ approach,” in preparation [Talk on Thursday in session 3B] BLFQ Symmetries & Constraints

All J ≥ Jz states number b B ∑ i = i in one calculation Charge q Q ∑ i = i Angular projection (M-scheme) (m s ) J ∑ i + i = z i k Finite basis Longitudinal momentum (Bjorken sum rule) x i 1 ∑ i = ∑ = i i K regulators Transverse mode regulator (2D HO) (2n m 1) N ∑ i+ i + ≤ max i Longitudinal mode regulator (Jacobi) l L ∑ i ≤ i Global Color Singlets (QCD) Light Front Gauge Optional Fock-Space Truncation Preserve transverse H → H +λHCM boost invariance Light-Front Regularization and Schemes

1. Regulators in BLFQ (Nmax, L) 2. Additional Fock space truncations (if any) 3. Counterterms identified/tested* 4. Sector-dependent renormalization** 5. RGPEP (Glazek, Gomez-Rocha, and others, e.g. arXiv:1805.03436) 6. SRG & OLS in NCSM*** - adapted to BLFQ (future)

*D. Chakrabarti, A. Harindranath and J.P. Vary, Phys. Rev. D 69, 034502 (2004) *P. Wiecki, Y. Li, X. Zhao, P. Maris and J.P. Vary, Phys. Rev. D 91, 105009 (2015)

**V. A. Karmanov, J.-F. Mathiot, and A. V. Smirnov, Phys. Rev. D 77, 085028 (2008); Phys. Rev. D 86, 085006 (2012) **Y. Li, V.A. Karmanov, P. Maris and J.P. Vary, Phys. Letts. B. 748, 278 (2015); arXiv: 1504.05233 **X. Zhao, invited talk at noon on Thursday at LC2018

***B.R. Barrett, P. Navratil and J.P. Vary, Prog. Part. Nucl. Phys. 69, 131 (2013) _ Effective Hamiltonian in the qq sector Adopt a running coupling with regulated IR behavior: improves UV properties of BLFQ applications 1. s(0), IR LO pQCD Nf =4 0.4, 1.03GeV 0.8 0.6, 0.55GeV For 0.8, 0.40GeV results here

2 s (M ) (Q2)= Z s 2 2 0.6 2 Q +IR 1 + s (MZ) 0 ln 2 2 MZ+IR ) Q ( s 0.4

0.2

0. 0.01 0.1 1. 10. 100. 1000. Q [GeV] _ Q = average 4-momentum transfer between q and q

Y. Li, P. Maris and J.P. Vary, PRD96, 016022 (2017) Spectroscopy [YL,[Li, Maris & Vary, PRD ’17; Tang, Li, Maris & Vary, in preparaon] Maris & Vary, PRD ’17; Tang, YL, Maris, & Vary in preparation]

Heavy mesons: rms deviaons 31 – 38 MeV

7.6 7.6 7.47.4 fitting parameters: 7.27.2 mc,mb,qQ = c MqQ ) BD threshold [cf. Dosch ’17] (HQET, ) 7.0 7.0 p ( ) ( ) 6.8( 6.8 rms deviation: 31–38 MeV 6.6 6.6 6.4 BLFQ 6.4 () () BLFQ 6.2 PDG 6.2 PDG Lattice 6.0 6.0 0- 1- 0+ 1+ 2+ 2- 3- 3+ 4+ 0- 1- 0+ 1+ 2+ 2- 3- 3+ 4+ 18

D. Radiative transitions

1. Vector Pseudo-scalar (or Scalar Axial-vector)

Radiave transions between 0According to Equation-+ (C93), and 1 we-- define heavy thequarkonia transition form factor of from a vector to a pseudo- scalar, , V(Q2), as V ! P Meijian Li, et al.; arXiv: 1803.11519 LC2018 talk on Thursday 3:00 pm2 µ 2V(Q ) µ ↵ (P0) J (0) (P, m ) = ✏ P0 P e (P, m ) , (69) hP | |V j i m + m ↵ j P V

2 µ 2V(Q ) µ↵ (P0) J (0) (P, m ) = ✏ P0 P e (P, m ) , (70) hP | |V j i m + m ↵ j P V µ µ µ 2 2 where the momentum transfer q = P0 P , Q q . momentum transfer: ⌘ The transition form factor V(Q2) is Lorentz invariant, or in other words, frame independent. So Impulse approximaon : we first carry out the calculation without choosing any specific frame, while later we would switch to a frame that is convenient for the calculation of helicity matrix. For any transverse vector (kx, ky), we will write its complex form as kR = kx + iky and kL = kx iky.

0, m j = 0 R R 2 8 i P0 P + 2V(Q ) > + + 0 , = > P P0 , m j = 1 (P ) J (0) (P m j) > P + P+ (71) hP | |V i mP + mV > p2 0 >  L L > i + + P0 P < P P0 , m = 1 > + + j > p2 P0 P >  > Light-front wavefuncons: Yang Li, Pieter Maris, James P. Vary. Phys. Rev. D 96, 016022 (2017:> ) R L L R im (P P0 P P0 ), m = 0 V j 2 R R R L 2 R R R L 2 8 i m P + P P0 P0 m P0 + P P P0 2V(Q ) > P V , m = 1 (P0) J(0) (P, m j) = > + + j hP | |V i mP + mV > p2 P0 P >  2 L L R L 2 L L L R > i m P + P P0 P0 m P0 + P P P0 <> P + V , = > + + m j 1 > p2 P0 P >  > (72) :>

R R + P0 P imV P0 + + , m j = 0 P0 P R R 2 8 i  P0 P R 2V(Q ) > R + P0 J P, m = > P P0 , m j = 1 ( ) (0) ( j) > + + hP | |V i mP + mV > p2 P0 P >  2 2 L L > i + + mV mP R P P0 < P P0 ( ) + P0 ( ) , m = 1 > + 2 + 2 + + j > p2 (P ) (P0 ) P P0 >  > (73) :> Radiave transions between 0-+ and 1-- heavy quarkonia Meijian Li, et al.; arXiv: 1803.11519 LC2018 talk on Thursday 3:00 pm Decay width:

BLFQ Predicons

[PDG] C.Patrignani, et al., CPC40,2016. [Lace] J. J. Dudek, et al., PRD73,2006; PRD79, 2009. D. Bečirević, et al., JHEP01,2013; JHEP05,2015. C. Hughes, et al., PRD92,2015. R.Lewis, et al.,PRD86,2012. [relavisc Model (rQM)] D.Ebert, et al., PRD67, 2013. [Godfrey-Isgur Model (GI Model)] T.Barnes, et al., PRD72,2005; S.Godfrey, et al., PRD92, 2015. Bc Meson System in BLFQ

Decay Constants! b l− W − Bc J µ

c¯ ν¯l

0.8

) Decay constants for vector and ! 0.7 × pseudo-scalar Bc mesons. The !

results are obtained with ! ( Nmax=Lmax=32, corresponding ! 0.6 to UV regulators:! ⇤  N m + m UV , max b c¯ 0.5 ×× ⇡ The lengthp of BLFQ error bars: 0.4 fbc¯ =2fbc¯(Nmax = 32) fbc¯(Nmax = 24)

| | which is taken to indicate the ! 0.3 1 1 1 3 3 3 sensitivity to basis truncation.! 1 S0 2 S0 3 S0 1 S1 2 S1 3 S1

BLFQ: S. Tang, et al., in preparation LFQM: H.-M. Cho and C.R. Ji, (2009) Lattice: B. Colquhoun, et al., (2015) Heavy-Light systems (preliminary) S. Tang, et al., in preparation

Decay constants calculated with Nmax = 8 for Ds, Nmax = 32 for Bs, corresponding to UV cutoffs:

)

(

spectrum of heavy-light systems. States under open flavor threshold confirmed by experiments.

* * Hadron Tomography [Adhikari[Adhikari, et al., et al.,Phys. Phys.Rev.C Rev. C 93, 05520293,055202(2016)] (2016); & in preparation]

I Generalized parton distributions (GPDs) [Ji ’97 & ’98] + 1 dz ixP z 1 + 1 H(x, ⇣, t)= e P 0 ( z) (+ z) P 2 2 2 + 2⇡ h | | i z =z?=0 Z + + 2 q = P 0 P , ⇣ = q /P , t = q . I DVCS, SIDIS, ..., physics

I Impact parameter dependent GPDs: [Burkardt ’01]

2 d i~ ~b 2 q(x,~b )= ? e ?· ? H(x, ⇣ =0,t= ). ? (2⇡)2 ? Z 1 ! 22 1 ~ 2 I partonic interpretation: ddbb⊥ dxd qx(xq, (bx,⊥)b=1.) =1 . ∫ ?∫ 0 ? 0 I Light-front wavefunction representationR R [Brodsky ’01, Diehl ’03] 1 1 1 S0 positronium 2 S0 Spin non-flip GPDs H(x,0,t)

1 −+ ' 2 1S 0−+ ηb 1 S0 0 ηb 0 ( ) Yang Li, Iowa State U, May 20, 2016 ( 16/1) Baryons 2016, Tallahassee, FL Charmonium Tomography Spin non-flip GPDs

Impact parameter- dependent GPDs

L. Adhikari, et al., in preparation Moving to light mesons – role of chiral symmetry

Spectroscopy: BLFQ with one- dynamics

Confining strength and quark mass obtained by fitting the lowest PDG excluding pion

BLFQ mass uncertainty due to slight violation of rotational symmetry

r.m.s. deviation (9 states): 202 MeV

Model parameters:

Wenyang Qian, et al., In preparaon Wenyang Qian, et al., In preparaon

Spectroscopy: BLFQ with one-gluon & interactions

Coupling strength λ obtained by fitting the pion mass

r.m.s. deviation (9 states): 119 MeV

Model parameters:

Wenyang Qian, et al., In preparaon

Parton distribution amplitudes for the pion

Exclusive processes at large momentum transfer

DSE: Lei Chang et al, PRL110, 132001(2013) Cloët(2013): Cloët et al, PRL111, 092001(2013) AdS/QCD + IMA: Brodsky et al, PhysRep548, 1(2015)

BLFQ for the light mesons with the Nambu-Jona-Lasinio model

Shaoyang Jia, parallel session 3B, Thursday 14:40 to 15:00 BLFQ NJL Model Parameters 369.5 MeV 8 222.2 MeV 2 398.8 MeV 8

Mass Decay Charge (MeV) Constant Radius (MeV) (fm2)

139.57 148.08 0.263

775.26 121.02 0.773

493.68 177.77 0.250

891.76 106.30 0.608 Baryons Anji Yu, et al., in preparation

P =+,S= 1/2 × N(2220) )

× N(1900)

( N(1710)

× × N(1720) N(1440) N(1680)

× 2 Q

P 1 F / () × N(940) () × () Q2GeV2 Flavor form factor & GPD in BLFQ Chandanwork in progressMondal, talk on Thursday

+ ⇤ Dirac form factor (F1) in light-front [ with q =0],for the J +(0) F ( q2)= P + q; ⇤ P ; ⇤ ; F q( q2)= dx Hq(x, q2). 1 h | 2P + | i 1 Z In terms of overlap of light-front WFs:, Hq(x, q2): X 3 2 2 ⇤ ⇤ dxi d k i(1 xj)(x x1) ( k j) i⇤(xi, k0 i) i (xi, k i) ? ? ? ? i=1 Xi Z Y X X

Nmax =6,Kmax =19, T =0.97 GeV, L =0.7 GeV

1

Kmax =6 0.8 T = L =0.97 GeV

Nmax =6

Nmax =8

0.6 ) ) 2

Nmax =10 t 2 Q Nmax =12 ( d x, 1 Cates et. al. (2011) 1 ( F

0.4 d Qattan & Arrington (2012) 1.2 H Diehl & Kroll (2013) 0 1 0.8 0.2 0 0.2 0.6 0.4 0.4 0.6 0 0.2 0 0.5 1 1.5 2 2.5 3 3.5 0.8 2 1 0 t [GeV ] 2 2 Q [GeV ] x Probing small-x in high-energy nuclear collisions through vector-meson producon ! Ion Collider--high luminosity, wide kinemac range. !Enable precision measurement of VM LFWF,

especially the higher excited states. *() 1. 0.8 10 (2S)/J/ (e+Au)

(2S)/ (1S) (e+Pb) Wp=140 GeV 0.8 0.6 0.6 ) 0.4 (

1

0.4

0.2

0.2

0. 0. 10-1 0 20 40 0 20 40 60 10 102 () () + () ( ) Chen, Li, Maris, Tuchin and Vary, PLB 769, 477, 2017 Guangyao Chen’s talk on Thursday Full Basis Light-Front Quanzaon (FBLFQ) UV and IR regulators but no addional Fock space truncaon

Pure glue sector QCD – Glueballs?

Credit: D. Lineweaver

Retain only the triple gluon vertex in inial test applicaon PRELIMINARY RESULTS 1E+8 ~38 million B 1E+7 B B Projections

J 356 1E+6 Singlets B Reduction factor 1E+5 B J

J Color basis space dimensions of each 1E+4 B J B 105 mul-gluon space-spin configuraon 1E+3 J B J 1E+2 B J ~100 thousand Figure extended from Vary, et al., 2010 (b) 1E+1 B J Numbercolorof states J 1E+0 J 0 1 2 3 4 5 6 7 8 9 10 Number of gluons

0.932 GeV 0.963 GeV 0.968 GeV 0.975 GeV 0.9 Distribuon funcons for 4 lowest mass eigenstates 0.8 0.7 4 gluon Nmax = K = 6 0.6 configs dominate b=0.5 GeV 0.5 (x) gS=0.5 g n 0.4 mg = 0.25 GeV 0.3

Hamiltonian matrix dimension ~ 2000; 0.2 Calculaon runs in 3 mins on laptop. 0.1

0 Next step: add 2 other verces 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Light front momentum x 6 (K)

F-BLFQ for pure glue in color singlet basis Nmax = K = 6, gs = 0.5, b = 0.5 GeV, mg = 0.25 GeV Summary and Outlook

Hamiltonian Light-Front QCD is producing descriptions and predictions with modeled confinement

" Bound states and transitions of the hadrons are described " Time-dependent scattering applications are underway " Future – expand the Fock spaces (e.g. F-BLFQ) " Future – renormalization, counterterms and regulators " Future – additional methods for emergent phenomena " Efficient utilization of supercomputing resources

Thank you for your attention