confronting nucleon strangeness and charm

Tim Hobbs, Southern Methodist University and CTEQ

2018 CTEQ Workshop @ JLab: ‘Parton Distributions as a Bridge from Low to High Energies’

Friday, November 9, 2018

and EIC Center@JLab

nucleon strange/charm remain very open topics

in spite of extensive effort and significant progess in recent years, ambiguities remain in both the magnitude and properties of the strange & charm components of the nucleon.

● the strangeness magnitude comparatively better constrained,

→ detailed x dependence remains uncertain, despite succession of historical improvements:

?

increasing sensitivity to flavor asymmetries in

the light quark sea

● there is much less clarity regarding the nucleon’s charm content

→ in general, only upper limits to

→ very little agreement on the shape of the intrinsic “fitted dists.

main focus of this talk

2

1. Background

charm in perturbative QCD (pQCD)

·c(x,Q2 ≤ m2c) = c(x,Q2 ≤ m2

c) = 0

F. M. Steffens, W. Melnitchouk and A. W. Thomas,Eur. Phys. J. C 11, 673 (1999) [hep-ph/9903441].

→

·intermediate Q2:

F c2, PGF(x,Q

2) = αs(µ2)9π

� z′

xdzz CPGF(z,Q2,m2

c) · xg�

xz , µ

2�

·high Q2:

massless DGLAP (i.e., variable flavor-number schemes)3

1. Background

simplest nonperturbative model calculations

→ original models possessed scalar vertices...·Brodsky et al. (1980):

P (p → uudcc) ∼�

M2 −�5i=1

k2⊥i

+m2i

xi

�−2

→ produces intrinsic PDF, cIC(x) = cIC(x)

·Blumlein (2015):

τlife =1

�

iEi−E

= 2P�

�

5i=1

k2⊥i

+m2i

xi−M2

�

�

�

�

�

jxj=1

vs. τint =1

q0

→ comparison constrains x−Q2 space over which IC is observable

4

2. meson-baryon models nonperturbative charm

meson-baryon models (MBMs)

· we implement a framework which conserves spin/parity

· nonperturbative mechanisms are needed to breakc(x,Q2 ≤ m2

c) = c(x,Q2 ≤ m2c) = 0!

We build an EFT which connects IC to properties of the hadronicspectrum: [TJH, J. T. Londergan and W. Melnitchouk, Phys. Rev. D89, 074008 (2014).]

·|Ni =√Z2 |N i0 +

�

M,B

�

dy fMB(y) |M(y);B(1− y)iy = k+/P+: k meson, P nucleon

c(x) =�

B,M

�

� 1x

dyy fBM (y) cB

�

xy

��

· a similar convolution procedure maybe used for c(x) . . . _

__ _

____

__

J = 0 + 1/2 J = 1 + 1/2 J = 0 + 3/2 J = 1 + 3/24.1

4.2

4.3

4.4

4.5

4.6

mas

s [G

eV]

D0Λ

c

+

D*0Λ

c

+

D−Σ

c

*++

D0Σ

c

*+

D*0

Σc

+

D−Σ

c

++

D0Σ

c

+

D*− Σ

c

++

D*0

Σc

*+

D*−

Σc

*++

5

2. meson-baryon models nonperturbative charm

charm in the nucleon

·tune universal cutoff Λ = Λ to fit ISR pp → ΛcX collider data

multiplicities, momentum sum:

hni(charm)MB = 2.40% +2.47

−1.36; Pc..= hxiIC = 1.34% +1.35

−0.75

0.001 0.01 0.1 1

x

0.0001

0.001

0.01

0.1

1

F2

cc

Q2 = 60 GeV

2

F cc2 (x,Q2) = 4x

9

�

c(x,Q2) + c(x,Q2)�

→ evolve to EMC scale, Q2 = 60 GeV2

low-x H1/ZEUS data check massless DGLAP evolution6

3. QCD global analysis

systematics of global QCD analysis

extract/constrain quark densities:

F γqh(x,Q

2) =�

f

� 1

0

dξ

ξCγfi

�

x

ξ,Q2

µ2,µ2F

µ,αS(µ

2)

�

· φf/h(ξ, µ2F , µ

2)

·Cγfi : pQCD Wilson coefficients

·φf/h(ξ, µ2F , µ

2): universal parton distributions

(...here, µ2F = 4m2

c +Q2)

=⇒ exploit properties of QCD to constrain models:

�

q

� 1

0dx x · {fq+q(x,Q

2) + fg(x,Q2)} ≡ 1 (mom. conserv.)

· DGLAP: couples Q2 evolution of fq(x,Q2), fg(x,Q

2)7

3. QCD global analysis

constraints from global fits...

P. Jimenez-Delgado, TJH, J. T. Londergan and W. Melnitchouk; PRL 114, no. 8, 082002 (2015).

26 sets:Ndat = 4296

Q2 ≥ 1 GeV2

W 2 ≥ 3.5 GeV2

=⇒∗∗ HTs, TMCs,smearing... 0

50

100

150

200

250

300

0 0.2 0.4 0.6 0.8 1

χ2 -

χ2 0

<x>IC(%)

total

E605

E866 pp

E866 pd

SLAC pE866 rat

SLAC dHERA

E665

NMC

BCDMS p

BCDMS dNMC rat

H1 F2

H1 FLdimuon

JLab p

JLab djets ZEUS

jets H1

jets CDF

jets D0

BCDMS F2

BCDMS FLE140x F2E140x FLHERA σc

· constrain: hxiIC =

� 1

0dx x · [c+ c](x) ... ‘total IC momentum’

8

3. QCD global analysis

...and constrained by EMC

-50

0

50

100

150

200

250

300

0 0.2 0.4 0.6 0.8 1

χ2 -

χ2 0

<x>IC(%)

totalSLACrestEMC

EMC alone: hxiIC = 0.3− 0.4%

+ SLAC/‘REST’: hxiIC = 0.13± 0.04%

...but F cc2 poorly fit — χ2 ∼ 4.3 per datum!

9

4. recent developments

new/ongoing global analyses

· NNPDF3: not anchored to specific parametrizations/modelssee: Ball et al. Eur. Phys. J. C76 (2016) no.11, 647

· included EMC:hxiIC = 0.7± 0.3% at Q ∼ 1.5GeV→ drove a very hard c(x) = c(x)distribution· peaked at x ∼ 0.5· AND, required a negative IC

component to describe EMC F cc2 !

· recent CTEQ-TEA IC analysis, extending CT14see: T. J. Hou et al. JHEP02 (2018) 059.

→ found hxiIC � 2%; examined mpolec dependence

10

4. recent developments

future experimental prospects?

· jet hadroproduction: pp → (Zc) +X at LHCbe.g., Boettcher, Ilten, Williams, PRD93, 074008 (2016).

→ a “direct” measure in the forward region, 2 < η < 5. . . sensitive to c(x), x ∼ 1 for one colliding proton

→ can discriminate hxiIC � 0.3% (“valencelike”), 1% (“sealike”)

· prompt atmospheric neutrinos?see: Laha & Brodsky, 1607.08240 (2016).

→ IceCube ν spectra may constrain IC normalization

· possible impact upon hidden charm pentaquark, P+c ?

e.g., Schmidt & Siddikov, PRD93, 094005 (2016).

· AFTER@LHC? . . . fixed-target pp at√s = 115 GeV

Brodsky et al. Adv. High Energy Phys. 2015, 231547 (2015). [Signori]

11

3

σ terms contribute to nucleon’s mass, BSM cross sections

● ALSO: the heavy quark sigma term (esp. for charm) is important to WIMP direct

searches :

Hill and Solon, Phys. Rev. Lett. 112, 211602 (2014).

● in principle, with knowledge of the wave function, there may be a way to correlate σ-terms and DIS-derive quantities, e.g., terms and DIS-terms and DIS-derive quantities, e.g., derive quantities, e.g.,

● the nucleon mass is a matrix element of the QCD energy-terms and DIS-derive quantities, e.g.,

momentum tensor,

Junnarkar, Walker-terms and DIS-derive quantities, e.g., Loud; PRD 2013.

MANY more lattice determinations of … relatively few for

12

… need models for both the charm PDF and σcc

light-front wave functions (LFWFs) are one such approach

they deliver a frame-independent description of hadronic bound state structure

with them, many matrix elements (GPDs, TMDs) are calculable via the same universal objects:

in fact, have already developed this technology for nucleon strangeness!

● the light front represents physics tangent to the light cone:

TJH, M. Alberg, and G. A. Miller; PRC91, 035205 (2015).

1313

DIS and elastic strangeness·predict inelastic and elastic observables?→ requires knowledge of quark-level proton wave function

eN → e′X eN → e′N ′

xS+ =

� 1

0dx x[s(x) + s(x)]

xS− =

� 1

0dx x[s(x)− s(x)]

F1(Q2) ∼ hP ′, ↑ |J+

EM |P, ↑i

F2(Q2) ∼ hP ′, ↓ |J+

EM |P, ↑i

JµEM = γµF1(Q

2) + iσµνqν2M

F2(Q2)

14

hadronic light-front wave functions (LFWFs)·S. J. Brodsky, D. S. Hwang, B. Q. Ma and I. Schmidt; Nucl. Phys. B 593, 311 (2001).

|ΨλP (P

+,P⊥)i =�

n

� n�

i=1

dxid2k⊥i√

xi(16π3)16π3 δ

�

1−n�

i=1

xi

�

× δ(2)

�

n�

i=1

k⊥i

�

ψλn(xi,k⊥i,λi) |n; k+i , xiP⊥ + k⊥i,λii

n = 2P

q

uudq

|ΨλP (P

+,P⊥)i =1

16π3

�

q=s,s

�

dxd2k⊥�

x(1− x)ψλqλq

(x,k⊥)

× |q;xP+, xP⊥ + k⊥i

→ 3D helicity WF ψλqλq

(x,k⊥); light-front fraction: x = k+/P+15

electromagnetic form factors·the quark q contribution from any 5-quark state is then:

F q1 (Q

2) = eq

�

dxd2k⊥

16π3

�

λq

ψ∗λ=+1qλq

(x,k′

⊥) ψλ=+1

qλq(x,k⊥)

F q2 (Q

2) = eq2M

[q1 + iq2]

�

dxd2k⊥

16π3

�

λq

ψ∗λ=−1qλq

(x,k′

⊥) ψλ=+1

qλq(x,k⊥)

·for strangeness, q → s; total strange: s+ s

F ss1,2(Q

2) = F s1,2(Q

2) + F s1,2(Q

2) =⇒

Sachs form : GssE (Q2) = F ss

1 (Q2)− Q2

4M2F ss2 (Q2)

GssM (Q2) = F ss

1 (Q2) + F ss2 (Q2)

µs = GssM (Q2 = 0) ρDs =

dGssE

dτ

�

�

�

τ=0where τ = Q2

�

4M2

16

strangeness wave functions·require a proton → quark + scalar tetraquark LFWF:

pk I. C. Cloet and G. A. Miller; Phys. Rev. C 86, 015208 (2012).

ψλλs(k, p) = uλs

s (k) φ(M20 ) u

λN (p)

φ(M20 ): scalar function → quark-spectator interaction

(M2

0= quark-tetraquark invariant mass2!)

e.g., ψλ=+1sλs=+1(x,k⊥) =

1√1− x

�ms

x+M

�

φs

gaussian : φs =

√Ns

Λ2s

exp�

−M20 (x,k⊥,q⊥)

�

2Λ2s

�

F s

1(Q2) =

esNs

16π2Λ4s

�

dxdk2⊥

x2(1− x)

�

k2⊥+ (ms + xM)2 − 1

4(1− x)2Q2

�

× exp(−ss/Λ2

s) ss = (M2

0+M ′2

0)/2 sim. for F s

2(Q2)!

17

ss distribution functions·s quark distribution ≡ x-unintegrated F s1 (Q

2 = 0) form factor(up to es!):

s(x) =

�

d2k⊥

16π3

�

λs

ψ∗λ=+1sλs

(x,k⊥) ψλ=+1sλs

(x,k⊥)

→ again inserting helicity wave functions ψλ=+1qλq

(x,k⊥)

(Q2 = 0 =⇒ k′⊥ = k⊥):

s(x) =Ns

16π2Λ4s

�

dk2⊥

x2(1− x)

�

k2⊥+ (ms + xM)2

�

exp(−ss/Λ2s)

ss =1

x(1− x)

�

k2⊥+ (1− x)m2

s + xm2

Sp+

1

4(1− x)2Q2

�

→ total of eight model parameters!(Ns, Λs, ms, and mSp

... AND anti-strange) 18

limits from DIS measurements·DIS measurements have placed limits on the PDF-level totalstrange momentum xS+ and asymmetry xS−

CTEQ6.5S:

0.018 ≤ xS+ ≤ 0.040 −0.001 ≤ xS− ≤ 0.005

·SCAN the available parameter space subject to the DIS limits;SEARCH for extremal values of µs, ρ

Ds

0 0.2 0.4 0.6 0.8 1 x

0

0.05

0.1

0.15

0.2

0.25

0.3

s(x)

s(x)

(a)

0 0.2 0.4 0.6 0.8 1 x

-0.2

-0.1

0

0.1

0.2

s(x) - s(x)

x [s - s](x)

(b)

19

constraints on elastic form factors

0 0.2 0.4 0.6 0.8 1

Q2

-0.1

-0.05

0

0.05

0.1

0.15

0.2

GE

s s (

Q2)

LFWFE

G0 PVA4

(a)

0 0.2 0.4 0.6 0.8 1

Q2

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

GM

ss (

Q2)

LFWFM

(b)

·DIS-driven limits to elasticFFs are significantly morestringent than currentexperimental precision

η(Q2) ∼ 0.94 Q2 → 0 0.2 0.4 0.6 0.8 1

Q2

-0.05

0

0.05

0.1

0.15

GE

ss +

ηG

M

ss

G0, 2005

PVA4

HAPPEX-III

HAPPEX-I, -II

20

see talks by C.-P. Yuan; E. Nocera

preliminary!

models may also enlighten PDF parametrization dependence

● results of fits with different

parametric forms for symmetric strangeness

motivated by models, e.g., on the QCD light

front

● fitted parameters in this instance are identified

with quantities like constituent

quark masses

***

*

21

6

we build a model for the charm wave funcn… 1st the PDF

● use a scalar spectator picture; details in helicity wave funcns :

use a power-terms and DIS-derive quantities, e.g., law (γ=3) covariant vertex function,

invariant mass

covariant k2

TJH, Alberg and Miller, Phys. Rev. D96 (2017) no.7, 074023.

22

then, a covariant formalism gives the sigma term:

…we determine probability distribution functions (p.d.f.s) for this quantity

● IF the LFWFs can be constrained with information from the DIS sector, we may evaluate σ

cc

● this formalism is required because the LFWFs contain noncovariant parts:

it remains to determine the (free) parameters of the light-front model,

23

(input data normalizations are inspired by the just-terms and DIS-derive quantities, e.g., described global analysis)

[ upper limit tolerated by the full fit/dataset ]

[ central value preferred by EMC data alone ]

● rather than traditional χ2 minimization, the model space is instead explored using Bayesian methods

● we constrain the model with hypothetical pseudo-data (taken from the `confining’ MBM) of a given

24

model simulations with markov chain monte carlo (MCMC)

● specifically, use a Delayed-terms and DIS-derive quantities, e.g., Rejection Adaptive Metropolis (DRAM) algorithm

construct a Markov chain consisting of nsim

≈ 105 – 106 simulations, sampling the

joint posterior distribution

BROAD gaussian priors

likelihood function

: input data

: parameters

● asymptotically, the MCMC chain fully explores the joint posterior distribution

from this, we extract probability distribution functions (p.d.f.s) for the model

parameters and derived quantities, including ¾cc

✔.

Haario et al., Stat. Comput. (2006) 16: 339–354.

25

γ =

3 in

tera

ctio

nγ

= 1

in

tera

cti

on

p.d.f.s

corre

latio

ns

MC

MC

Jo

int

po

ste

rio

r d

istr

ibu

tio

n

26

the charm TMD is consistent with deeply internal quarks

prob. dist. func.

knowledge of the wave function may help with building a give-and-take between efforts to describe collinear PDFs and TMDs for detailed tomography

(see talk by F. Yuan…)

27

(χQCD)1

● we find better concordance cf. existing lattice determinations, for somewhat larger IC magnitudes; also, close correlation with the DIS sector –

(MILC)2

1Gong et al., Phys. Rev. D88, 014503 (2013).

2Freeman and Toussaint, Phys. Rev. D88, 054503 (2013).

3Abdel-terms and DIS-derive quantities, e.g., Rehim et al., Phys. Rev. Lett. 116, 252001 (2016).

(AR)3

prob. dist. func.

are directly correlated.

28

● e.g., for a ‘generic’ (slightly higher energy) EIC scenario,

● EIC can access the crucial nonpert. region of the charm distribution!

an intrinsic charm component with a small normalization is would likely need very substantial precision to unambiguously disentagle…

LHeC, EIC sensitivity to charm PDF

LHeC

EIC

… but an EIC would be well-adapted to try.

EIC Whitepaper, Eur. Phys. J. A (2016) 52: 268

B.T. Wang et al., arXiv:1803.02777 [hep-ph]: accepted, Phys. Rev. D

PDFSense: see talk tomorrow by Bo-Ting Wang.

NLO DGLAP

29

closing observations

→ to exploit these connection, more experimental information is required, but diverse channels are/will be available (e.g., at EIC)

● understanding the nucleon’s non-terms and DIS-derive quantities, e.g., valence structure remains an important challenge for the field

→ can construct interpolating models that can help unravel the flavor structure of the proton wave function and unify, inter alia,

– for strangeness, form factors and structure function (GPDs)

– for charm, σ-terms and DIS-derive quantities, e.g., terms and IC PDFs:

– in general, the collinear PDFs and transverse momentum dists.

thanks!

– exploring these connections will be inseparable from the task of mapping the nucleon’s internal tomography

30