Introduction to Lattice QCD - Challenges and New Opportunities

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Introduction to lattice QCD - Challenges and New Opportunities

Constantia Alexandrou

University of Cyprus and The Cyprus Institute

QCD - old challenges and new opportunities

Bad Honnef, 27 September. 2017

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 1 / 58 Outline

1 Motivation Standard Model of Elementary Particles QCD versus QED

2 Introduction Current status of simulations Low-lying hadron masses Evaluation of matrix elements in lattice QCD

3 Nucleon charges Nucleon axial charge The quark content of the nucleon (σ-terms)

4 The nucleon spin decomposition

5 Nucleon Electromagnetic and axial form factors Electromagnetic form factors Axial form factors

6 Challenges Direct computation of Parton Distribution functions Ab Initio Nuclear Physics

7 Conclusions

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 2 / 58 Standard model The Standard Model (SM) is a synthesis of three of the four forces of nature described by gauge theories with coupling constants:

Strong Interactions: αs 1 ∼ Electromagnetic interactions: αem 1/137 ≈ Weak interactions: G 10−5 GeV −2. F ≈ Basic constituents of matter: Six quarks, u, d, s, c, b, t, each in 3 colors, and six leptons e, νe, µ, νµ, τ, ντ The quarks and leptons are classiﬁed into 3 generations of families. The interactions between the particles are mediated by vector bosons: the 8 gluons mediate strong interactions, the W ± and Z mediate weak interactions, and the electromagnetic interactions are carried by the photon γ. The weak bosons acquire a mass through the Higgs mechanism. The SM is a local gauge ﬁeld theory with the gauge group SU(3) SU(2) U(1) specifying the interactions among these constituents. × ×

Masses in the Standard Model Parameters Number Comments Masses of quarks 6 u, d, s light c, b heavy t = 175 6 GeV ± Masses of leptons 6 e, µ, τ Mνe, νµ, ντ non-zero

Mass of W ± 1 80.3 GeV Mass of Z 1 91.2 GeV Mass of gluons, γ 0 (Gauge symmetry)

Mass of Higgs 1 125.03+0.26 (stat)+0.13 (sys) GeV discovered at LHC, 2012 −0.27i −0.15i C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 3 / 58 C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 4 / 58 C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 5 / 58 QuantumChromoDynamics (QCD) QCD-Gauge theory of the strong interaction Lagrangian: formulated in terms of quarks and gluons

1 a a µν X µ = F F + ψ¯ iγ Dµ m ψ LQCD − 4 µν f − f f f =u,d,s,c,b,t a λ a Dµ = ∂µ ig A − 2 µ

Phys.Lett. 47B (1973) 365-368

This “simple” Lagrangian produces the amazingly rich structure of strongly interacting matter in the universe.

Numerical simulation of QCD provides essential input for a wide class of complex strong interaction phenomena

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 6 / 58 Properties of QCD

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 7 / 58 C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 8 / 58 C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 9 / 58 QCD versus QED

Quantum Electrodynamics (QED): The interaction is due to the exchange of photons. Every time there is an exchange of a photon there is a correction in the interaction of the order of 0.01.  we can apply perturbation theory reaching whatever accuracy we like

QCD: Interaction due to exchange of gluons. In the energy range of ~ 1GeV the coupling constant is ~1  We can no longer use perturbation theory

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 10 / 58 QCD on the lattice

Discrete space-time lattice acts as a non-perturbative regularization scheme with the lattice spacing a providing an ultraviolet cutoff at π/a no inﬁnities. Furthermore, renormalized physical quantities have a ﬁnite well behaved limit as a 0. → → Can be simulated on the computer using methods analogous to those used for Statistical Mechanics systems. These simulations allow us to calculate correlation functions of hadronic operators and matrix elements of any operator between hadronic states in terms of the fundamental quark and gluon degrees of freedom.

Like continuum QCD lattice QCD has as unknown input parameters the coupling constant αs and the masses of the up, down, strange, charm and bottom quarks (the top quark is too short lived). = Lattice QCD provides a well-deﬁned approach to calculate observables non-perturbative starting directly from⇒ the QCD Langragian.

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 11 / 58 LatticeQuantumChromoDynamics (QCD) QCD-Gauge theory of the strong interaction Lagrangian: formulated in terms of quarks and gluons

1 a a µν X µ = F F + ψ¯ iγ Dµ m ψ LQCD − 4 µν f − f f f =u,d,s,c,b,t a λ a Dµ = ∂µ ig A − 2 µ

Choice of fermion discretisation scheme e.g. Clover, Twisted Mass, Staggered, Domain Wall, Overlap Each has its advantages and disadvantages

Eventually, all discretization schemes must agree in the continuum limit a 0 → observables extrapolated to the inﬁnite volume limit L → ∞

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 12 / 58 QCD on the lattice

Lattice QCD: K. Wilson, 1974 provided the formulation; M. Creutz, 1980 per- formed the ﬁrst numerical simulation

Discretization of space-time with lattice spacing a and implement gauge invariance I quark fields ψ(x) and ψ¯(x) on lattice sites I Introduce parallel transporter connecting point x and x + aµˆ: iaAµ(x) Uµ(x) = e i.e. gauge field Uµ(x) is defined on links 1 lattice derivative µψ(x) [Uµ(x)ψ(x + aµˆ) ψ(x)] −→ D → a − Finite a provides an ultraviolet cutoff at π/a non-perturbative regularization; Finite L discrete momenta→ in units of2 π/L if periodic b.c. → Construct an appropriate action S and rotate into imaginary time: P ¯ S = SG + SF where SF = x ψ(x)Dψ(x) i.e. quadratic in the fermions can be integrated out −→ Path integral over gauge fields: R Q −SG[U] Z Uµ(x) f det(Df [U]) e ∼MonteD Carlo simulation to produce a representative ensemble of → Uµ(x) using the largest supercomputers { } P −1 Observables: = O(D , Uµ) → hOi {Uµ}

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 13 / 58 Computational resources

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 14 / 58 With simulations at the physical point lattice QCD can provide essential input for the experimental programs.

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 15 / 58 Questions we would like to address

With simulations at the physical value of the pion mass there is a number of interesting questions we want to address: Can we reproduce known quantities including the excited spectrum of the nucleon and its associated resonances? Can we resolve the long-standing issue of the spin content of the nucleon? Can we determine accurately enough the charge radius of the proton? Can we provide input for experimental searches for new physics?

In this talk I will address two topics: The nucleon scalar content or σ-terms as a probe of new physics The nucleon spin decomposition of the nucleon Nucleon form factors

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 16 / 58 Status of simulations

0.5 BMW, Nf=2+1 0.5 CLS, Nf=2+1

PACS, Nf=2+1 0.4 ETMC, Nf=2 0.4 ETMC, Nf=2+1+1

] MILC, Nf=2+1+1 ] V 0.3 NME, N =2+1 V

e f 0.3 e G QCDSF, N =2 G

[ f [

QCDSF/UKQCD, Nf=2+1 m m 0.2 0.2 RBC/UKQCD, Nf=2+1

0.1 0.1

0.0 0.0 0 2 4 6 0.00 0.05 0.10 0.15 Lm a [fm]

Size of the symbols according to the value of mπ L: smallest value mπ L 3 and largest mπ L 6.7. ∼ ∼

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 17 / 58 Hadron mass First goal: reproduce the low-lying masses Use Euclidean correlation functions:

X −i~xs·~q † G(~q, ts) = e J(~xs, ts)J (0) h i ~xs t →∞ X −En(~q)ts s −E0(~q)ts = Ane A e −→ 0 n=0,··· ,∞ + ¯ Interpolating ﬁeld with the quantum numbers of π : J(x) = d(x)γ5u(x) 0.30

0.25 ] V

Large Euclidean time evolution gives ground e G

[ 0.20

state for given quantum numbers = enables f f ⇒ e

determination of low-lying hadron properties m aEeff(~q, ts) = ln G(~q, ts)/G(~q, ts + a) 0.15 = aE0(~q) + excited states ~q=0 i aE (~q) am → 0 → 0.10 0 1 2 3 4 5 t [fm]

Nf = 2 + 1 + 1 TM fermions at mπ = 210 MeV Nf = 2 TM plus clover fermions at physical pion mass

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 18 / 58 Hadron mass

First goal: reproduce the low-lying masses Use Euclidean correlation functions:

X −i~xs·~q † G(~q, ts) = e J(~xs, ts)J (0) h i ~xs t →∞ X −En(~q)ts s −E0(~q)ts = Ane A e −→ 0 n=0,··· ,∞

abc a> b c Interpolating ﬁeld with the quantum numbers of p: J(x) = u (x)Cγ5d (x) u (x)

2.0

Ω 1.6 * Large Euclidean time evolution gives ground Ξ Ξ

state for given quantum numbers = enables (GeV) 1.2 determination of low-lying hadron properties⇒ eff Λ m N aEeff(~q, ts) = ln G(~q, ts)/G(~q, ts + a) 0.8 = aE0(~q) + excited states ~q=0 i aE (~q) am 0.4 → 0 → 0 4 8 12 16 20 ts/a

Nf = 2 TM plus clover fermions at physical pion mass 3 (mh− mπ )ts Noise to signal increases with ts: e 2 ∼

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 18 / 58 1.75

phys aµs = 0.0259(3) Hadron masses 1.7

phys (GeV) m ⌦ ⌦ m 1.65

1.6 0.023 0.024 0.025 0.026 0.027 0.028 aµs

− + For our computation, the masses of the strange and charm quarks are ﬁxed using the Ω and Λc . 1.75

aµphys = 0.0259(3) aµphys = 0.3319(15) s 2.35 c

1.7

phys m + phys ⇤c (GeV) m (GeV) ⌦ + c ⌦ ⇤ 2.25 m m 1.65

1.6 2.15 0.023 0.024 0.025 0.026 0.027 0.028 0.3 0.31 0.32 0.33 0.34 0.35 0.36 aµs aµc R R ms = 108.6(2.2) MeV and mc = 1392.6(23.5) MeV, in the MS-scheme at 2 GeV.

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 19 / 58

aµphys = 0.3319(15) 2.35 c

phys m + ⇤c (GeV) + c ⇤ 2.25 m

2.15 0.3 0.31 0.32 0.33 0.34 0.35 0.36 aµc Low-lying spectrum

1.8 = 2.0 ETMC N f 2 1.7 ETMC N f = 2 + 1 + 1 QCDSF-UKQCD N = 2 + 1 Ω 1.6 f 1.6 PACS-CS N f = 2 + 1 Ξ* 1.5 BMW N f = 2 + 1 Ξ (GeV) 1.2 eff Λ 1.4 m

N (GeV) 1.3 0.8 M 1.2 0.4 1.1 0 4 8 12 16 20 ts/a 1 . Using Nf = 2 simulations at a physical value of the 0 9 pion mass N ⇤⌃⌅⌃⇤ ⌅⇤ ⌦ C. Alexandrou and C. Kallidonis, Phys. Rev. D96 (2017) 034511, arXiv:1704.02647

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 20 / 58 Low-lying spectrum

4 5 ETMC N f = 2 3.8 ETMC N f = 2 ETMC N f = 2 + 1 + 1 ETMC N f = 2 + 1 + 1 S. Brown et al. N f = 2 + 1 4.5 3.6 S. Brown et al. N f = 2 + 1 G. Bali et al. N f = 2 + 1 G. Bali et al. N = 2 + 1 3.4 f Na et al. N f = 2 + 1 N = + 4 Na et al. f 2 1 Briceno et al. N f = 2 + 1 + 1 3.2 Briceno et al. N f = 2 + 1 + 1 Liu et al. N = 2 + 1 f PACS-CS N f = 2 + 1 (GeV) 3 PACS-CS N = 2 + 1 (GeV) f 3.5 M M 2.8 2.6 3 2.4 2.2 2.5

⇤c ⌃c ⌅c ⌅c0 ⌦c ⌅cc ⌦cc ⌃c⇤ ⌅c⇤ ⌦c⇤ ⌅cc⇤ ⌦cc⇤ ⌦ccc spin-1/2 spin-3/2 C. Alexandrou and C. Kallidonis, Phys. Rev. D96 (2017) 034511, arXiv:1704.02647

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 20 / 58 Evaluation of matrix elements

Three-point functions: ~ ~ µν P ixins·q µν G (Γ,~q, ts, tins) = ~ ~ e Γβα Jα(~xs, ts) (~xins, tins)Jβ (~x0, t0) xs,xins h OΓ i OΓ ~q = ~p′ − ~p OΓ q = p′ − p

(xins, tins) (xs, ts) (x0, t0) (~xins, tins) (~xs, ts) (~x0, t0)

connected contribution disconnected contribution

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 21 / 58 Evaluation of matrix elements

Three-point functions: ~ ~ µν P ixins·q µν G (Γ,~q, ts, tins) = ~ ~ e Γβα Jα(~xs, ts) (~xins, tins)Jβ (~x0, t0) xs,xins h OΓ i

OΓ q = p′ − p

(xins, tins) (xs, ts) (x0, t0)

connected contribution

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 21 / 58 Evaluation of matrix elements Three-point functions: ~ ~ µν P ixins·q µν G (Γ,~q, ts, tins) = ~ ~ e Γβα Jα(~xs, ts) (~xins, tins)Jβ (~x0, t0) xs,xins h OΓ i

OΓ q = p′ − p

(xins, tins) (xs, ts) (x0, t0)

connected contribution

Plateau method:

(t −t )∆1 0 ins 0 −∆(p)(tins−t0) −∆(p )(ts−tins) R(ts, tins, t0) [1 + ... e + ... e ] −(ts−−−tins−)∆−−1−−M→

Summation method: Summing over tins:

ts 0 X −∆(p)(ts−t0) −∆(p )(ts−t0) R(ts, t , t ) = Const. + [(ts t ) + (e ) + (e )]. ins 0 M − 0 O O tins=t0

Excited state contributions are suppressed by exponentials decaying with ts t , rather than ts t − 0 − ins and/or tins t0 However, one− needs to fit the slope rather than to a constant or take differences and then fit to a constant L. Maiani, G. Martinelli, M. L. Paciello, and B. Taglienti, Nucl. Phys. B293, 420 (1987); S. Capitani et al., arXiv:1205.0180 Fit keeping the first excited state, T. Bhattacharya et al., arXiv:1306.5435 All should yield the same answer in the end of the day!

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 21 / 58 Evaluation of matrix elements

Three-point functions: ~ ~ µν P ixins·q µν G (Γ,~q, ts, tins) = ~ ~ e Γβα Jα(~xs, ts) (~xins, tins)Jβ (~x0, t0) xs,xins h OΓ i OΓ q = p′ − p

(xins, tins) (xs, ts) (x0, t0)

connected contribution 102

101

the desired matrix element 100 M gS ts, tins, t0 the sink, insertion and source gA gT time-slices xq 10 1 gE ∆(p) the energy gap with the ﬁrst excited state GM(1) xDq

10 2 10 12 14 16 18 ts/a

To ensure ground state dominance need multiple sink-source time separations ranging from 0.9 fm to 1.5 fm

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 21 / 58 Nucleon isovector charges:g A,g s,g T

a µ τa axial-vector operator: = ψ¯(x)γ γ5 ψ(x) OA 2 a scalar operator: a = ψ¯(x) τ ψ(x) OS 2 a τa pseudoscalar: = ψ¯(x)γ5 ψ(x) Op 2 a tensor operator: a = ψ¯(x)σµν τ ψ(x) OT 2 = extract from matrix element: N(p~0) N(~p) ⇒ h OX i|q2=0 Axial charge g Scalar charge g Pseudoscalar charge gp, Tensor charge g • A • S • • T

(i) isovector combination has no disconnect contributions; (ii) gA well known experimentally, Goldberger-Treiman relation yields gp, gT to be measured at JLab, Predict gS

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 22 / 58 Nucleon axial chargeg A

1.4

1.3

1.2 d

u A 1.1 g

1.0 Nf =2 TMF : a =0.089, 0.07, 0.056 fm Nf =2 +1 clover : a =0.116, fm

Nf =2 +1 +1 TMF : a =0.082, 0.064 fm Nf =2 +1 +1 HISQ/clover : a =0.12, fm 0.9 Nf =2 TMF/Clover : a =0.093 fm Nf =2 +1 +1,HISQ/DWF (PNDME) extrap. Nf =2 +1 MILC/DWF : a =0.124 fm Nf =2 +1 +1,HISQ/DWF, extrap.

Nf =2 clover : a =0.08, 0.07, 0.06 fm experiment 0.00 0.05 0.10 0.15 0.20 0.25 m2 [GeV2 ]

C.g AlexandrouA well known (Univ. experimentally. of Cyprus & Cyprus It Inst.) is an iso-vector quantityLattice QCDextracted at zero momentumBad Honnef, transfer 27 September.straight 2017 23 / 58 forward to compute in lattice QCD. → Nucleon scalar charge

3 N = 2 twisted mass plus clover, 48 96, a = 0.093(1) fm, mπ = 131 MeV f × 9260 statistics for ts/a = 10, 12, 14, 48000 for ts/a = 16 and 70000 for ts/a = 18 ∼ ∼ ∼ 5 sink-source time separations ranging from 0.9 fm to 1.7 fm

1.40 3.5 Plateau 1.35 3.0 Plateau Summation Summation 1.30 Two-state 2.5 Two-state 1.25 2.0 1.5 S A 1.20 g g 1.0 1.15 0.5 1.10 0.0 1.05 0.5 1.00 1.0 0.9 1.0 1.1 1.2 1.3 0.8 0.9 1.0 1.1 1.2 1.0 1.2 1.4 1.6 0.8 0.9 1.0 1.1 1.2 low low ts [fm] ts [fm] ts [fm] ts [fm]

At the physical point we ﬁnd from the plateau method:

isov isov isov gA = 1.21(3)(3) , gS = 0.93(25)(33) , gT = 1.00(2)(1)

gA further study for larger ts. Important to keep constant error A. Abdel-Rehim et al. (ETMC):1507.04936, 1507.05068, 1411.6842, 1311.4522 u−d M. R. A. Courtoy, A. Bacchettad, M. Guagnellia, arXiv: New analysis of COMPASS and Belle data: gT = 0.81(44), 1503.03495 For g increasing the sink-source time separation to 1.5 fm is crucial S ∼ C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 24 / 58 The quark content of the nucleon

σf mf N q¯f qf N : measures the explicit breaking of chiral symmetry Largest≡ uncertaintyh | | i in interpreting experiments for direct dark matter searches - Higgs-nucleon coupling depends on σ, e.g. spin-independent cross-section can vary an order of magnitude if σπN changes from 35 MeV to 60 MeV, J. Ellis, K. Olive, C. Savage, arXiv:0801.3656 In lattice QCD: ∂mN I Feynman-Hellmann theorem: σl = ml ∂ml ∂m Similarly σ = m N , S. Dürr et al. (BMW ) Phys.Rev.Lett. 116 (2016) 172001 s s ∂ms c

I Direct computation of the scalar matrix element G. Bali, et al. (RQCD) Phys.Rev. D93 (2016) 094504, arXiv:1603.00827; Yi-Bo Yang et al. (χQCD) Phys.Rev. D94 (2016) no.5, 054503; A. Abdel-Rehim et al. arXiv:1601.3656, PRL116 (2016) 252001;

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 25 / 58 The quark content of the nucleon via Feynman-Hellmann

BMW Collaboration: 47 lattice ensembles with Nf = 2 + 1 clover fermions, 5 lattice spacings down to 0.054 fm, lattice sizes up to 6 fm and pion masses down to 120 MeV.

σπN = 38(3)(3) MeV σs = 105(41)(37) MeV

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 26 / 58 The quark content of the nucleon via direct determination

Need disconnected contributions

RQCD: Nf = 2 clover fermions with a range of pion masses down to mπ = 150 MeV and a = 0.06 0.08 fm G. Bali, et al., Phys.Rev. D93 (2016) 094504, arXiv:1603.00827 − χQCD: Valence overlap fermions on Nf = 2 + 1 ﬂavor domain-wall fermion (DWF) conﬁgurations, 3 ensembles of mπ = 330 MeV, mπ = 300 MeV and mπ = 139 MeV Yi-Bo Yang et al., Phys.Rev. D94 (2016) no.5, 054503; M/ Gong et al., Phys. Rev. D 88 (2013) 014503 arXiv:1304.1194 3 ETM Collaboration: Nf = 2 twisted mass plus clover, 48 96, a = 0.093(1) fm, mπ = 131 MeV, A. Abdel-Rehim et al., arXiv:1601.3656, PRL116 (2016) 252001 ×

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 27 / 58 The quark content of the nucleon from ETMC

3 N = 2 twisted mass plus clover, 48 96, a = 0.093(1) fm, mπ = 131 MeV f × Connected: t/a = 10, 12, 14 9264 statistics, t/a = 16 47,600 statistics and t/a = 18 70,000 statistics ∼ ∼ Disconnected: 213,700 statistics ∼ A. Abdel-Rehim et al. arXiv:1601.3656, PRL116 (2016) 252001

80 50 Plateau Plateau 70 Summation Summation Two-state 40 Two-state 60 ] ] V 50 V e 30 e M [

M 40

[

n s n

N 20 o 30 c 20 10 10 0 0 1.0 1.2 1.4 1.6 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 1.0 1.2 1.4 1.6 1.8 low ts [fm] ts [fm] low ts [fm] ts [fm] Connected Disconnected strange

Our results are: σπN = 36(2) MeV

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 28 / 58 The quark content of the nucleon from ETMC

80 50 Plateau Plateau 70 Summation Summation Two-state 40 Two-state 60 ] ] V 50 V e 30 e M [

M 40

[

n s n

N 20 o 30 c 20 10 10 0 0 1.0 1.2 1.4 1.6 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 1.0 1.2 1.4 1.6 1.8 low ts [fm] ts [fm] low ts [fm] ts [fm] Connected Disconnected strange

Our results are: σπN = 36(2) MeV σs = 37(8) MeV σc = 83(17) MeV

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 28 / 58 The quark content of the nucleon

Comparison of results

0.3

0.25

0.2

0.15 [GeV] N π a 0.08 fm σ 0.1 a ≈ 0.07 fm a ≈ 0.06 fm ETMC ≈ 0.05 Nf = 2 + 1 + 1 χQCD Nf = 2 + 1 ETMC N = 2 0 f 0 0.05 0.1 0.15 0.2 0.25 2 [GeV2] mπ G. Bali, et al., Phys.Rev. D93 (2016) 094504, arXiv:1603.00827

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 29 / 58 The quark content of the nucleon

Comparison of results

c QCD '13 Pavan '02 ETMC '16 Alarcón '11 s Hoferichter '15 QCDSF-UKQCD '12 QCDSF-UKQCD '12 BMWc '16 ETMC '14 QCDSF '12 BMWc '16 QCD '13 QCDSF '12 QCD '15 QCD '15 RQCD '16 RQCD '16 ETMC '16 ETMC '16

0 25 50 75 100 0 25 50 75 100 125 N [MeV] s, c [MeV] Recent results from lattice QCD at the physical point and from phenomenology. Filled symbols for lattice QCD results include simulations with pion mass close to its physical value, A. Abdel-Rehim et al. arXiv:1601.3656, PRL116 (2016) 252001

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 29 / 58 Nucleon spin

1 Spin sum: 1 = P ∆Σq + Lq +Jg 2 q 2 | {z } Jq q 1 q q q q J = 2 A20(0) + B20(0) and ∆Σ = gA

Need isoscalar gA, which has disconnected contributions

3 Nf = 2 twisted mass fermions with a clover term at a physical value of the pion mass, 48 96 and a• = 0.093(1) fm × Intrinsic quark spin: ∆Σq = gq • A 1.40 0.75 ts=0.9 fm ts=1.1 fm ts=1.3 fm ts=0.9 fm ts=1.1 fm ts=1.3 fm 1.35 0.70 d +

A 1.30 A u g Summation 0.65 Summation 1.25 g → → ) Two-state 0.60 s ) t

1.20 s Two-state , t s , 0.55 n s i 1.15 n t i ( t ( R 1.10 0.50 R 1.05 0.45 1.00 0.40 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.6 0.4 0.2 0.0 0.2 0.4 0.6 tins ts/2 [fm] tins ts/2 [fm] Isovector Connected isoscalar

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 30 / 58 Nucleon spin 1 Spin sum: 1 = P ∆Σq + Lq +Jg 2 q 2 | {z } Jq q 1 q q q q J = 2 A20(0) + B20(0) and ∆Σ = gA

Need isoscalar gA, which has disconnected contributions 0.1 0.00 0.0

) 0.1 c s

i 0.05 s A d (

0.2 g d + u A

g 0.3 0.10 Plateau Plateau 0.4 Summation Summation Two-state Two-state 0.5 0.15 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.0 1.2 1.4 1.2 1.4 1.6 1.8 1.0 1.2 1.4 1.6 low low ts [fm] ts [fm] ts [fm] ts [fm] Isoscalar disconnected Strange

We ﬁnd from the plateau method:

gu+d = 0.15(2) (disconnected only) with 854,400 statistics A −

Combining with the isovector we ﬁnd: gu = 0.828(21), gd = 0.387(21) A A − gs = 0.042(10) with 861,200 statistics A −

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 30 / 58 Quark total spin Jq Generalized parton distributions functions (GPDs) are matrix elements of light cone operators that cannot be computed directly Factorization leads to matrix elements of local operators: → vector operator a µ ···µ ↔ ↔ τ 1 n = ψ¯(x)γ{µ1 i D µ2 ... i D µn} ψ(x) OV a 2

axial-vector operator a µ ···µ ↔ ↔ τ 1 n {µ1 µ2 µn} = ψ¯(x)γ i D ... i D γ5 ψ(x) OAa 2

tensor operator

a µ ···µ ↔ ↔ τ 1 n = ψ¯(x)σ{µ1,µ2 i D µ3 ... i D µn} ψ(x) OT a 2

Special cases: no-derivative nucleon form factors → For Q2 = 0 parton distribution functions one-derivative→ ﬁrst moments e.g. average momentum fraction x Generalized form→ factor decomposition: h i

" {µα ν} {µ ν} 0 0 µν 0 0 2 {µ ν} 2 iσ qαP 2 q q 1 N(p , s ) N(p, s) = u¯N (p , s ) A20(q )γ P +B20(q ) +C20(q ) uN (p, s) h |OV 3 | i 2m m 2

q 1 q q q Total quark spin J = A (0)+ B (0) and x q = A (0) 2 20 20 h i 20

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 31 / 58 Momentum fraction hxiu−d

0.40 Plateau 0.7 0.35 Summation 0.30 Two-state 0.6

d 0.25 d +

u 0.20 u 0.5 ® ® x x

0.15 0.4 0.10 Plateau 0.05 Summation 0.3 Two-state 0.00 1.0 1.2 1.4 1.6 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.0 1.2 1.4 1.6 0.8 0.9 1.0 1.1 1.2 1.3 1.4 low low ts [fm] ts [fm] ts [fm] ts [fm] Results for the isovector in the MS at 2 GeV Results for the connected isoscalar in the MS at 2 GeV

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 32 / 58 Momentum fraction hxiu−d

1.2 0.4 ) ts=0.9 fm ts=0.9 fm .

c ts=1.1 fm s 1.0 ts=1.1 fm 0.3 i s ts=1.3 fm d ts=1.3 fm ® ( x

d 0.8

0.2 + u → ® 0.6 ) x

s 0.1 t

0.4 , s → n i ) 0.0 t s ( t 0.2 , R s n

i 0.1

t 0.0 ( R 0.2 0.2 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.6 0.4 0.2 0.0 0.2 0.4 0.6 tins ts/2 [fm] tins ts/2 [fm] Results for the disconnected isoscalar Results for the strange At the physical point we ﬁnd in the MS at 2 GeV from the plateau method ( (860, 000) statistics): O x − = 0.194(9)(10) h iu d

x + + = 0.80(12) (10) h iu d s stat syst x + + is perturbatively renormalized to one-loop due to its mixing with the gluon operator. h iu d s A. Abdel-Rehim et al. (ETMC):1507.04936, 1507.05068, 1411.6842, 1311.4522

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 32 / 58 Gluon content of the nucleon

Gluons carry a signiﬁcant amount of momentum and spin in the nucleon g I Compute gluon momentum fraction : x g = A20 g 1 g hg i I Compute gluon spin: J = 2 (A20 + B20)

Nucleon matrix of the gluon operator: Oµν = GµρGνρ gluon momentum fraction extracted from − → 1 N(0) O O N(0) = m < x >g h | 44 − 3 jj | i N Disconnected correlation function, known to be very noisy we employ several steps of stout smearing in order to remove ﬂuctuations in the gauge ﬁeld ⇒ Results are computed on the Nf = 2 ensemble at the physical point, mπ = 131 MeV, a = 0.093 fm, 3 V = 48 96, A. Abdel-Rehim et al. (ETMC):1507.04936 × The methodology was tested for Nf = 2 + 1 + 1 twisted mass at mπ = 373 MeV, C. Alexandrou, V. Drach, K. Hadjiyiannakou, K. Jansen, B. Kostrzewa, C. Wiese, PoS LATTICE2013 (2014) 289

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 33 / 58 Nucleon gluon moment-Renormalization Mixing with x + + = Perturbation theory - M. Constantinou and H. Panagopoulos h iu d s ⇒

C.Millions Alexandrou of (Univ. terms of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 34 / 58 Nucleon gluon moment-Renormalization

Mixing with x + + = Perturbation theory - M. Constantinou and H. Panagopoulos h iu d s ⇒

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 34 / 58 Results for the gluon content

2094 gauge conﬁgurations with 100 different source positions each more than 200 000 measurements → Due to mixing with the quark singlet operator, the renormalization and mixing coefﬁcients had to be extracted from a one-loop perturbative lattice calculation, M. Constantinou and H. Panagopoulos Renormalization x = 0.318(24) h ig,bare −−−−−−−−→ R < x >g = Zgg < x >g +Zgq < x >u+d+s = 0.267(12)stat(10)syst . The renormalization is perturbatively done using two-levels of stout smearing. The systematic error is the difference between using one- and two-levels of stout smearing.

P CI DI Momentum sum is satisﬁed: x q + x g = x + x + x g = 1.07(12) (10) q h i h i h iu+d h iu+d+s h i stat syst

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 35 / 58 Nucleon spin

Disconnected contribution using (860000) statistics O 0.5 Up

0.4

Hybrid, N =2+1+1: a=0.060 fm a=0.090 fm . 1 u f 0 4 ⌃ 0.3 2 TMF, Nf=2+1+1: a=0.083 fm a=0.06 fm

0.05 Down TMF, Nf=2: a=0.094 fm a=0.088 fm a=0.071 fm a=0.056 fm 0.2 0.10 q 0.15 1 s 1 2 2⌃ 0 0.20 1 d 2⌃ Strange . 0.01 Contributions to nucleon spin 0 2 0.02 00.05 0.10.15 0.20.25 Hybrid, Nf=2+1, a=0.124 fm 2 2 0.03 m⇡ (GeV ) Clover: Nf=2+1, a=0.074 fm Nf=2, a=0.073 fm 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 m [GeV]

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 36 / 58 0.4

0.3 1⌃u+d 0.2 2

0.1 Lu+d 0

0.1

Contributions to nucleon spin 0.2 00.05 0.10.15 0.20.25 2 2 m⇡ (GeV ) Nucleon spin

1 Spin sum: 1 = P ∆Σq + Lq +Jg 2 q 2 | {z } Jq

1 1 ∆Σu = 0.415(13)(2), 1 ∆Σd = 0.193(8)(3), ∆Σs = 0.021(5)(1) (1) 2 2 − 2 − Ju = 0.308(30)(24), Jd = 0.054(29)(24), Js = 0.046(21) Lu = 0.107(32)(24), Ld = 0.247(30)(24), Ls = 0.067(21)(1) −

We ﬁnd that Bq (0) 0 taking B (0)g 0 we can directly check the nucleon spin sum: 20 ∼ −→ 20 ∼

JN = (0.308)u + (0.054)d + (0.046)s + (0.133)g = 0.54(6)(5)

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 37 / 58 The proton spin puzzle 1987: the European Muon Collaboration showed that only a fraction of the proton spin is carried by the quarks = ETMC has now provided the solution ⇒

Recent results from lattice QCD at the physical point

C.A. et al., Phys. Rev. Lett. (in press) arXiv:1706.02973

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 38 / 58 Electromagnetic form factors

h µν i N(p0, s0) jµ(0) N(p, s) = u¯ (p0, s0) γµF (q2)+ iσ qν F (q2) u (p, s) h | | i N 1 2m 2 N 2 2 q2 2 GE (q ) = F1(q ) + F2(q ) 4m2 N 2 2 2 GM (q ) = F1(q ) + F2(q )

E. J. Downie, EPJ Conf. 113 (2016) 05021 Proton radius extracted from muonic hydrogen is 7.9 σ different from the one extracted from electron scattering, R. Pohl et al., Nature 466 (2010) 213 Muonic measurement is ten times more accurate and a reanalysis of electron scattering data may give agreement with muonic measurement

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 39 / 58 Recent results on the electric and magnetic form factors

Isovector form factors 5 1.0 Kelly parametrization Kelly parametrization ETMC (this work) ETMC (this work) LHPC (2014) LHPC (2014) 0.8 4 ) ) 2 0.6 2 3 Q Q ( ( d d

u E 0.4 u M G G 2

0.2 1 0.0

0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 Q2 [GeV2] Q2 [GeV2]

3 ETMC using N = 2 twisted mass fermions (TMF), a = 0.093 fm, 48 96 G with ts = 1.7 fm and f × E 66,000 statistics, GM with ts = 1.3 fm and 9,300 statistics 4 LHPC using Nf = 2 + 1 clover fermions, a = 0.116 fm, 48 , summation method with 3 values of ts from 0.9 fm to 1.4 fm and 7, 800 statistics, 1404.4029 ∼

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 40 / 58 Recent results on the electric and magnetic form factors

1.0 Lattice Results 2.7 Lattice Results 0.8 Experiment 2.2 Experiment ) ) 2 2 1.7 Q Q

0.6 ( ( p E p M 1.2 G 0.4 G 0.7 0.2 0.2

0.3 ) ) 0.06 0.6 2 2 Q Q ( ( 0.03 0.9 n E n M G G 1.2 0.00 1.5 0.03 1.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Q2[GeV2] Q2[GeV2] ETMC using N = 2 twisted mass fermions (TMF), a = 0.093 fm, 483 96 f × Connected contributions: GE with ts = 1.7 fm and 66,000 statistics, GM with ts = 1.3 fm and 9,300 statistics only<1>C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou and A. Vaquero Aviles-Casco. Phys. Rev. D96 (2017) 034503, arXiv:1706.00469

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 41 / 58 Recent results on the electric and magnetic form factors

0.010

0.008 ) .

c 0.006 s i d (

)

2 0.004 Q ( n , p E 0.002 G

0.000

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Q2[GeV2]

ETMC using N = 2 twisted mass fermions (TMF), a = 0.093 fm, 483 96 f × Connected contributions: GE with ts = 1.7 fm and 66,000 statistics, GM with ts = 1.3 fm and 9,300 statistics Disconnected uses about 200,000 statistics only<1>C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou and A. Vaquero Aviles-Casco. Phys. Rev. D96 (2017) 034503, arXiv:1706.00469

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 41 / 58 Strange Electromagnetic form factors

Experimental determination: Parity violating e N scattering HAPPEX experiment ﬁnds Gs (0.62) = 0.070−(67) M − New methods for disconnected fermion loops: hierarchical probing, A. Stathopoulos, J. Laeuchli, K. Orginos, arXiv:1302.4018

Nf = 2 + 1 clover fermions, mπ 320 MeV, J. Green et al., Phys.Rev. D92 (2015) 3, 031501, arXiv: 1505.01803 ∼

Sampling of the fermion propagator using site colouring schemes

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 42 / 58 Strange Electromagnetic form factors

Experimental determination: Parity violating e N scattering s − HAPPEX experiment ﬁnds GM (0.62) = 0.070(67) R. S. Suﬁan et al. (χQCD Collaboration) 1606.07075 −

3 3 Overlap valence on Nf = 2 + 1 domain wall fermions, 24 64, a = 0.11 fm, mπ = 330 MeV; 32 64, 3 × × a = 0.083 fm, mπ = 300 MeV and 48 , a=0.11 fm, mπ = 139 MeV

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 42 / 58 Recent results on the axial form factors

s 2 m Qµ N(p0 s0) A N(p s) = i N u¯ (p0 s0) G (Q2) i G (Q2) u (p s) , µ , 0 N , γµ A p γ5 N , h | | i EN (~p )EN (~p) − 2mN Isovector

25 1.6 mA = 1. 077(39) GeV Two-state mA = 1. 077(39) GeV mA = 1. 350(170) GeV Fit, mA = 1. 302(42)(56) GeV mA = 1. 350(170) GeV 1.4 mA = 1. 010(240) GeV ETMC, m 130 MeV ' 20 mA = 1. 010(240) GeV 1.2 Pion pole dominance prediction ETMC, m 130 MeV ' ) )

2 1.0 2 15 Q Q ( ( d 0.8 d u A u p

G G 10 0.6

0.4 5 0.2

0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Q2[GeV2] Q2[GeV2]

ETMC using N = 2 twisted mass fermions (TMF), a = 0.093 fm, 483 96 f × Connected contributions: GE with ts = 1.7 fm and 66,000 statistics, GM with ts = 1.3 fm and 9,300 statistics C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou and A. Vaquero Aviles-Casco. Phys. Rev. D, arXiv:1705.03399 [hep-lat]

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 43 / 58 Recent results on the axial form factors

s 2 m Qµ N(p0 s0) A N(p s) = i N u¯ (p0 s0) G (Q2) i G (Q2) u (p s) , µ , 0 N , γµ A p γ5 N , h | | i EN (~p )EN (~p) − 2mN

Isoscalar 0.8 16 u + d u + d u + d u + d 0.7 GA , Conn. GA , Total Gp , Conn. Gp , Total u + d s 12 u + d s G , Disc. GA Gp , Disc. Gp 0.6 A 8 0.5 4 ) 0.4 ) 2 2 Q Q ( 0.3 ( 0 p A G G 0.2 4 0.1 8 0.0 0.1 12 0.2 16 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.2 0.4 0.6 0.8 1.0 Q2[GeV2] Q2[GeV2]

ETMC using N = 2 twisted mass fermions (TMF), a = 0.093 fm, 483 96 f × Connected contributions: GE with ts = 1.7 fm and 66,000 statistics, GM with ts = 1.3 fm and 9,300 statistics C. A.et al. (ETMC), Phys. Rev. D, arXiv:1705.03399 [hep-lat]

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 44 / 58 Recent results on the axial form factors

s 2 m Qµ N(p0 s0) A N(p s) = i N u¯ (p0 s0) G (Q2) i G (Q2) u (p s) , µ , 0 N , γµ A p γ5 N , h | | i EN (~p )EN (~p) − 2mN

Isoscalar 0.05 0 2 0.00 4 )

) 0.05 6 2 2 Q Q ( (

p 8 A G G 0.10 10 12 0.15 u+d (Disc.) u+d (Disc.) 14 s s 0.20 16 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Q2[GeV2] Q2[GeV2]

ETMC using N = 2 twisted mass fermions (TMF), a = 0.093 fm, 483 96 f × Connected contributions: GE with ts = 1.7 fm and 66,000 statistics, GM with ts = 1.3 fm and 9,300 statistics Disconnected uses about 200,000 statistics C. A.et al. (ETMC), Phys. Rev. D, arXiv:1705.03399 [hep-lat]

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 44 / 58 Direct evaluation of parton distribution functions - an exploratory study

Z +∞ n−1 a˜n(x, Λ, P3) = dx x q˜(x, Λ, P3) , −∞ i

Z +∞ dz −izxP3 ¯ q˜(x, Λ, P3) = e P ψ(z, 0) γ3 W (z)ψ(0, 0) P −∞ 4π h| | i {z | }i h(P3,z)→ can be computed in LQCD

is the quasi-distribution deﬁned by X. Ji Phys.Rev.Lett. 110 (2013) 262002, arXiv:1305.1539 Exploratory calculations:

Huey-Wen Lin et al. Phys. Rev. D91 (2015) 054510, Clover on Nf = 2 + 1 + 1 HISQ, mπ = 310MeV and Jiunn-Wei Chen et al., arXiv:1603.06664 C.A., K. Cichy, E. G. Ramos, V. Drach, K. Hadjiyiannakou, K. Jansen, F. Steffens, C. Wiese, Phys.Rev. D92 (2015) 014502 3 N = 2 + 1 + 1, V = 32 64, mπ = 373 MeV, a 0.082 fm f × ≈ 1000 gauge conﬁgurations with 15 source positions each and 2 sets of stochastic samples 30 000 measurements → 5 steps of HYP smearing for the gauge links in the operator Stochastic method for the three-point functions Matching and mass corrections are included

Currently under study for future applications:

I Renormalization I A new smearing method indicates an improvement of errors that can enable us to reach larger momentum

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 45 / 58 The momentum, helicity and transversity parton distributions

P3 =4π/L P3 =6π/L

1.4 u(0) d(0) 1.4 u(0) d(0) − − q˜(x, P3) = 1.2 MSTW 1.2 MSTW CJ12 CJ12 ∞ dz −izk 1 1 R 3 ¯ ABM11 ABM11 −∞ 4π e P ψ(z)γ3W3(z, 0)ψ(0) P 0.8 0.8 h | | i 0.6 0.6 crossing relation: q¯(x) = q( x) 0.4 0.4 − − ¯ 0.2 0.2 negative x d u¯ 0 0 ⇒ − 0.2 0.2 − -1 -0.5 0 0.5 1 − -1 -0.5 0 0.5 1 x x

P3 =4π/L P3 =6π/L 2 2 ↑ ↓ ∆u(0) ∆d(0) ∆u(0) ∆d(0) ∆q(x) = q (x) q (x) − − DSSV08 DSSV08 − 1.5 1.5 JAM15 JAM15 ∆q˜(x, P3) = R ∞ dz −izk 1 1 e 3 P ψ¯(z)γ5γ W (z, 0)ψ(0) P −∞ 4π h | 3 3 | i crossing relation: ∆q¯(x) = ∆q( x) 0.5 0.5 − negative x region ∆u¯ ∆d¯ 0 0 ⇒ − -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x x

P3 =4π/L P3 =6π/L 2 2 > ⊥ δu(0) δd(0) δu(0) δd(0) δq(x) = q (x) q (x) − − − 1.5 1.5 δq˜(x, P3) = R ∞ dz e−izk3 P ψ¯(z)γ γ W (z, 0)ψ(0) P 1 1 −∞ 4π h | j 3 3 | i crossing relation: δq¯(x) = δq( x) 0.5 0.5 − − negative x region δd¯ δu¯ 0 0 ⇒ − -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x x C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, F. Steffens and C. Wiese, arXiv:1610.03689 [hep-lat]

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 46 / 58 Challenges: Ab Initio Nuclear Physics?

From the qq¯ potential to the determination of nuclear forces

K. Schilling, G. Bali and C. Schlichter, 1995

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 47 / 58 Challenges: Ab Initio Nuclear Physics?

From the qq¯ potential to the determination of nuclear forces

K. Schilling, G. Bali and C. Schlichter, 1995 A.I. Signal, F.R.P. Bissey and D. Leinweber, arXiv:0806.0644

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 47 / 58 Challenges: Ab Initio Nuclear Physics?

From the qq¯ potential to the determination of nuclear forces

Two approaches: Determine N-N energy as a function of L extract phase shift - NPQCD → Detemine BS wave function 0 N(~r)N(~0) NN and exract asymptotically the phase shift - HALQCD h | | i study nuclear physics, neutron stars, ... → Only at the begining...

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 47 / 58 Challenges: Ab Initio Nuclear Physics? From the qq¯ potential to the determination of nuclear forces

1 Deuteron and nn ( S0 channel) binding energy, K. Orginos et al. 1508.07583

T.Yamazaki, K. Ishikawa, Y. Kuramashi, A. Ukawa, 1502.04182 Only at the begining...

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 47 / 58 Conclusions and Future Perspectives

Computation of g , x − , etc, at the physical point allows direct comparison with experiment A h iu d Provide predictions for gs, gT , tensor moment, σ-terms, etc. Resolution of the spin decomposition of the nucleon A number of collaborations are now using simulations with close to physical values of the pion mass to: Compute gluonic observables Study excited states and resonances and scattering lengths Hadon-hadron interactions and multi-nucleon systems

···

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 48 / 58 European Joint doctorate

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AND BIOLOGICAL SYSTEMS

APPLY NOW 15 Ph.D. positions for European Joint Doctorates Applications are welcome from candidates worldwide with a scientiﬁc background and a keen interest in simulation and data science.

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3 degrees in 1 The successful candidates will obtain a single joint Ph.D. degree from three academic institutions.

Marie Sklodowska-Curie Ph.D. fellowships An attractive remuneration package is oﬀered including salary, mobility and family allowances. Deadline for applications 15th of December 2017 For further information on admissions, requirements and eligibility criteria please visit our website www.stimulate-ejd.eu and ﬁnd STIMULATE on social media

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 49 / 58 European Twisted Mass Collaboration European Twisted Mass Collaboration (ETMC)

Cyprus (Univ. of Cyprus, Cyprus Inst.), France (Orsay, Grenoble), Germany (Berlin/Zeuthen, Bonn, Frankfurt, Ham- burg, Münster), Italy (Rome I, II, III, Trento), Netherlands (Groningen), Poland (Poznan), Spain (Valencia), Switzerland (Bern),UK (Liverpool)

Collaborators: A. Abdel-Rehim, S. Bacchio, K. Cichy, M. Constantinou, J. Finkenrath, K. Hadjiyian- nakou, K.Jansen, Ch. Kallidonis, G. Kout- sou, K. Ottnad, M. Petschlies, F. Steffens, A. Vaquero, C. Wiese

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 50 / 58 Thank you for your attention Backup slides

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 51 / 58 Bibliography

1 J. W. Negele, QCD and Hadrons on a lattice, NATO ASI Series B: Physics vol. 228, 369, eds. D. Vautherin, F. Lenz and J.W. Negele. 2 M. Lüscher, Advanced Lattice QCD, Les Houches Summer School in Theoretical Physics, Session 68: Probing the Standard Model of Particle Interactions, Les Houches 1997, 229, hep-lat/9802029. 3 R. Gupta, Introduction to Lattice QCD, hep-lat/9807028. 4 P. Lepage, Lattice for Novices, hep-lat/0506036. 5 H. Wittig, Lecture week, SFB/TR16, 3-7 Aug. 2009, Bonn. 6 H. J. Rothe, Quantum Gauge Theories: An introduction, World Scientiﬁc, 1997. 7 I. Montvay and G. Münster, Quantum Fields on a Lattice, Cambridge University Press, 1994. 8 C.Gattringer and C. B. Lange, Quantum Chromodynamics on the Lattice - An Introductory Presentation, Springer, 2009.

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 52 / 58 Metropolis Algorithm (α) e−S[x] We need an algorithm to to create our set of random paths x with probability Z , where Z = R [x(t)]e−S[x]. = a simpleD procedure, though not always the best, is the Metropolis Algorithm: ⇒ Start with an arbitrary path x (0)

Modify by visiting each of the sites on the lattice, and randomizing the xj ’s at those sites, one at a time, in a particular fashion as described below generate a new random path from the old one: x (0) x (1). This is called “updating” the path. → → (1) (2) Apply to x to generate path x , and so on until we have Ncf random paths. th The algorithm for randomizing xj at the j site is: Generate a random number < ζ , with uniform probability; − ≤ Let x x + ζ and compute the change ∆S in the action; j → j If ∆S < 0 retain the new value for xj , and proceed to the next site; If ∆S > 0 accept change with probability exp( ∆S) i.e. generate a random number η uniformly − distributed between 0 and 1; retain the new value for xj if exp( ∆S) > η, otherwise restore the old value; proceed to the next site. − Comments:

Choice of : should be tuned so that 40%–60% of the xj ’s are changed on each pass (or “sweep”) through the lattice. Then is of order the typical quantum fluctuations expected in the theory. Whatever the , successive paths are going to be quite similar and so contain rather similar information about the theory. Thus when we accumulate random paths x (α) for our Monte Carlo estimates we should keep only every Ncor -th path; the intervening sweeps erase correlations, giving us configurations that are statistically independent. The optimal value for Ncor depends upon the theory, and can be found by experimentation. It also depends on the lattice spacing a. Initial configuration: Guess the first configuration discard some number of configurations at the → beginning, before starting to collect x (α)’s. This is called “thermalizing the lattice.”

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 53 / 58 Metropolis Algorithm (α) e−S[x] We need an algorithm to to create our set of random paths x with probability Z , where Z = R [x(t)]e−S[x]. = a simpleD procedure, though not always the best, is the Metropolis Algorithm: ⇒ Start with an arbitrary path x (0)

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 53 / 58 Gauge degrees of freedom

In the continuum a fermion moving from site x to y in the presence of a gauge ﬁeld Aµ(x) picks up a phase factor given by the path ordered product

R y ψ(y) = ei x gAµ(x)dxµ ψ(x) . P

= associate gauge ﬁelds with links that connect sites on the lattice. So, with each link associate a discrete version⇒ of the path ordered product:

iagAµ(x) U(x; x +µ ˆ) Uµ(x) = e , ≡ U is a3 3 unitary matrix with unit determinant. It follows that ×

−iagAµ(x) † U(x; x µˆ) U−µ(x) = e = U (x µˆ; x) . − ≡ −

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 54 / 58 Local gauge symmetry The effect of a local gauge transformation V (x) on the variables ψ(x) and U is deﬁned as

ψ(x) V (x)ψ(x) → ψ¯(x) ψ¯(x)V †(x) → † Uµ(x) V (x)Uµ(x)V (x +µ ˆ) →

where V (x) is in the same representation as the Uµ(x), i.e., it is an SU(3) matrix. With these deﬁnitions there are two types of gauge invariant objects that one can construct on the lattice.

A string consisting of a path-ordered product of links capped by a fermion and an antifermion e.g.

Tr ψ¯(x) Uµ(x) Uν (x +µ ˆ) ... Uρ(y ρˆ) ψ(y) − where the trace is over the color indices. If the string stretches across the lattice and is closed by the periodicity are called Polyakov lines. The simplest example of closed Wilson loops is the plaquette, a1 1 loop, × 1×1 † † Wµν = Pµν (x) = Re Tr Uµ(x) Uν (x +µ ˆ) Uµ(x +ν ˆ) Uν (x) .

Preserve gauge invariance at all a protects from having many more parameters to tune (the zero gluon mass, and the equality of the quark-gluon,→ 3-gluon, and 4-gluon couplings) and there would arise many more operators at any given order in a.

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 55 / 58 U(1) gauge theory ∗ µ ∗ Consider a Lagrangian of a complex ﬁeld φ: L = ∂µφ ∂ φ V (φ , φ). If we require that the Lagrangian is 0 −iα(x−) invariant under a local gauge transformation φ (x) = e φ(x) then we need a ﬁeld Aµ(x) to compensate the change in the derivative ∂µφ that transforms as

0 1 A (x) = Aµ(x) + ∂µα(x) ∂µ Dµ ∂µ + igAµ(x) µ g → ≡

The gauge invariant Lagrangian is written as

1 µν ∗ ∗ L = Fµν F + (Dµφ) Dµφ V (φ , φ) − 4 −

A scalar moving from site x to y in the presence of a gauge ﬁeld Aµ(x) picks up a phase factor given by R y µ U(x; y) = eig x dxµA (x) which removes the phase between the value of the ﬁeld at the two points and yields a gauge invariant result.

n + ν Uµ†(n + ν) n + µ + ν The action is defined in terms of link variables assigned to links between sites of the space-time lattice. The link variable from site n in the µ direction to site n + aeˆµ is defined as the R n+µ Uν†(n) P ig iθµ(n) Uν (n + µ) discrete approximation to the integral e n : Uµ(n) = e with θµ(n) R n+µ µ the approximation of g n dxµA (x). n n + µ The integral over the field variables is the invariant group measure for U(1): Uµ(n) 1 R π 2π −π dθ. † † The action is the sum of all plaquettes Pµν = U(n)µUν (n + µ)Uµ(n + ν)Uν (n). iθµ(n) iθν (n+µ) −iθµ(n+µ) −iθν (n) iBµν a→0 For U(1): Pµν (n) = e e e e e , Bµν = ∆µθν ∆ν θµ Fµν . ≡ − → C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 56 / 58 Lattice action of U(1)

Since a plaquette produces Fµν the action can be constructed by choosing a function of the plaquette such that 2 it generates Fµν in the continuum limit.

X X X X S = β (1 Re Pµν (n)) = β (1 cos Bµν ) , − − n µ>ν n µ>ν

1 a→0 where β = and Bµν = ∆µθν ∆ν θµ Fµν . g2 − →

In the limit a 0 we recover continuum QED: → 2 Taking θµ(n) = agAµ(n) and expanding θν (n + eˆµa) = θν (n) + a∂µθν (n) + (a ) O

" 4 2 # 1 X 1 X h 2 i 1 X X a g 2 S [1 cos(a∂µθν a∂ν θµ)] = 1 cos(a gFµν ) = F + ∼ g2 − − g2 − g2 2 µν ··· P P n µ>ν 1 Z d 4x F 2 (x) → 4 µν

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 57 / 58 Path integral for QED

The Hamiltonian does not constrain the charge state of the system project the states appearing in the path → integral onto the space satisfying ~ .E~ = ρ where ρ is the background charge. This is done by including in the path integral the δ-function: ∇ Z R ~ ~ χei dxdtχ(∇.E−ρ) . D Consider A = 0 and replacing x A and the momentum E in the coordinate path-integral we have 0 → →

h ~ i Z R dxdt i~E.A˙ − 1 (E2+B2)+iχ(∇~ .~E−ρ) Z = χ ~A E~ e 2 D D D h ~ i Z − R dxdt 1 (A˙ −∇~ χ)2+B2−iχρ = χ ~A e 2 D D

P Rename χ = A and take ρ(x) = qnδ(x xn) 0 n − Z R 1 2 R − dxdt Fµν Y −iqn dt A0(xn,t) Z = Aµe 4 e D n

= we obtain the Lagrangian path integral with a line of A0 ﬁelds at the positions of the ﬁxed external charges.⇒ ± ±

C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Lattice QCD Bad Honnef, 27 September. 2017 58 / 58