Jefferson Lab, May 27, 2009

The energy-momentum tensor form factors of the nucleon

Peter SchweitzerUniversity of Connecticut

work with C. Cebulla, K. Goeke, J. Grabis, J. Ossmann, M. V. Polyakov, A. Silva, D. Urbano

Phys.Rev.D75 (2007) 094021, Phys.Rev.C75 (2007) 055207, Nucl.Phys.A794 (2007) 87

Overview:

• Introduction: hard exclusive reactions, and GPDs

• Form factors of the energy momentum tensor

• Results from chiral soliton models

• Stability and sign of D-term

• Conclusions

1. Introduction: hard exclusive reactions, and GPDs

1.1. Hard meson production eN→e′hB, B = N, . . .Collins, Frankfurt, Strikman, 1997

N(P) B(P’)

h

γ∗ (q)

GPD

DA

+ another diagram + NLO + . . .

+ power corrections

GPD = generalized parton distribution function

DA = meson distribution amplitude

1.2. Deeply virtual Compton scattering (DVCS) eN→e′N ′γRadyushkin 1997, Ji, Osborne 1998, Collins, Freund

N(P) N(P’)

γ∗ (q) real γ

GPD

N(P) N(P’)

γ∗ (q)

real γ

F1,2(t)

+

actually interference: DVCS + Bethe-Heitler process

• deeply virtual Compton scattering

• hard exclusive meson production

• DIS, and semi-inclusive DIS

in the same experiment!

1.3. What we actually can access

real or imaginary parts of amplitudes A(ξ, t|Q2) ∼∫

dxGPD(x, ξ, t)

x± ξ+ iǫ

ξ =xBj

2 + xBj

t = ∆2 with ∆ = P − P ′

xBj =Q2

2Pqq = ’deeply virtual photon’ momentum, Q2 = −q2

not trivial to extract GPDs from that, but

data since 2001: HERMES, CLAS, HERA, Hall A, COMPASS

and many more to come (JLab 12 GeV upgrade!)

Perspective: We will know (something) about GPDs from experiment!

Question: What will we learn from that about the nucleon?

1.4. Definition ’unpolarized GPDs’

in the following outgoing baryon B = N

∫dλ

2πeiλx 〈N(P ′)|ψq(−λn2 )nµγ

µ [−λn2 ,λn2 ]ψq(

λn2 ) |N(P)〉

= u(p′)nµγµ u(p)Hq(x, ξ, t)+ u(p′)iσµνnµ∆ν

2MNu(p)Eq(x, ξ, t)

and analog definitions for gluon GPDs

n = light-like vector

What do x, ξ, t mean?

1.5. Meaning: “microsurgery”

Pav = (P ′+ P)/2

∆ = P ′ − Pt = ∆2

ξ = (n ·∆)/(n · Pav)

k = xPav

lim∆→0

Hq(x, ξ, t) = fq1(x)

N(P av- 2 ∆) N’(P av+ 2

∆)

k - 2 ∆ k + 2

∆

GPD

This is the ’technical meaning’ of the GPD.

What is the ’physical interpretation’ of the GPD?

1.6. Interpretation for ξ → 0 (no momentum transfer along light cone)

momentum transfer perpendicular to light-cone t = ∆2 = − ~∆2⊥

Hq(x, b⊥) =∫

d2∆⊥ ei

~∆⊥~b⊥ Hq(x,0,− ~∆2⊥)

b⊥ = impact parameter

Matthias Burkardt, 2000

exact interpretation:

probability

N(P)

k = xP

b⊥

simultanous “measurement” of longitudinal momentum & transverse position

→ transverse nucleon imaging

has important applications: hadronic final state/event characteristics

in high energy pp collisions (→ . . . , Christian Weiss, . . . )

1.7. What else can we learn from GPDs?

instead of Hq(x, ξ, t) and Eq(x, ξ, t) completely equivalent:

Mellin moments (N = 1, 2, 3, . . .)

for even N :

∫

dx xN−1Hq(x, ξ, t) = hq(N)0 (t) + h

q(N)2 (t) ξ2 + . . .+ h

q(N)N (t) ξN

∫

dx xN−1Eq(x, ξ, t) = eq(N)0 (t) + e

q(N)2 (t) ξ2 + . . .+ e

q(N)N (t) ξN

with hq(N)N (t) = − eq(N)

N (t) for spin 12 particle

(for odd N , highest power is ξN−1)

“polynomiality” ←− Lorentz invariance and time-reversal

(with all respect: always ask whether a model fullfils polynomiality . . .)

i.e. GPD: Hq(x, ξ, t), Eq(x, ξ, t)

⇔

infinitely many form factors:Each form factor:

- related to a local matrix element

- contains different informationN = 1: h

f(1)0 (t), e

f(1)0 (t)

N = 2: hf(2)0 (t), h

f(2)2 (t), e

f(2)0 (t)

N = 3: hf(3)0 (t), h

f(3)2 (t), e

f(3)2 (t), e

f(3)0 (t)

N = 4: hf(4)0 (t), h

f(4)2 (t), h

f(4)4 (t), e

f(4)2 (t), e

f(4)0 (t)

N = 5: hf(5)0 (t), h

f(5)2 (t), h

f(5)4 (t), e

f(5)4 (t), e

f(5)2 (t), e

f(5)0 (t)

N = 6: hf(6)0 (t), h

f(6)2 (t), h

f(6)4 (t), h

f(6)6 (t), e

f(6)4 (t), e

f(6)2 (t), e

f(6)0 (t)

N = 7: . . . . . . . . . . . . . . . . . . . . . . . .

N = 1: electromagnetic form factors Hofstadter et al, 1950s . . .

∑

qeq∫

dx Hq(x, ξ, t) = F1(t)∑

qeq∫

dx Eq(x, ξ, t) = F2(t) what did we learn?

e.g. GE(t) = F1(t) + t4M2

N

F2(t) =∫

d3r ρE(r) ei~q ~r (t = −~q 2, . . .)

GpE(t) ≃ 1/(1− t/m2

dip)2, mdip = 0.84GeV, Gn

E(0) = 0

0

1

2

3

0 0.5 1

r in fm

ρpE(r) in fm-3

proton∝ exp(-mdipr)

0

0.5

1

0 0.5 1

r in fm

ρnE(r) in fm-3

neutron

continued interest: JLab Hall A GnE(t), Hall C Q-weak

e.g. precise data at low t constrain physics at TeV scale

Young, Carlini, Thomas, Roche, PRL99,122003(2007)

n np

π-

pion cloud:

spontanous breaking

of chiral symmetry

N = 2: energy-momentum tensor form factors

∑

q

∫

dx x Hq(x, ξ, t) = +MQ2 (t) + 4

5 dQ1 (t) ξ2

∑

q

∫

dx x Eq(x, ξ, t) = 2JQ(t)−MQ2 (t)− 4

5 dQ1 (t) ξ2 gluons analog

What we know:

MQ2 (0)∼0.5 at few GeV2: quarks carry half of nucleon momentum

(other half gluons)

What we would like to know:

how do quarks + gluons share nucleon spin? We need:

∑

q

∫

dx x (Hq + Eq)(x, ξ, t) = 2JQ(t) and limt→0

JQ(t) = JQ(0) Ji, 1997

Remaining questions:

What will t-dependence of M2(t), J(t) teach us? What is d1(t) good for?

N = 3, 4, 5, . . . : further form factors

of what? Of operators ψγµ1Dµ2Dµ2 . . . DµNψ

scale dependent quantities,

each of them contains information

on some different aspect of nucleon

2. Energy momentum tensor Tµν

Tµν fundamental object in field theory. In QCD TQ,Gµν both gauge invariant

〈P ′|T Q,Gµν |P 〉 = u(P ′)

[

MQ,G2 (t)

PµPν

MN

+ JQ,G(t)i(Pµσνρ + Pνσµρ)∆ρ

2MN

+ dQ,G1 (t)

∆µ∆ν − gµν∆2

5MN± c(t)gµν

]

u(p)

︸ ︷︷ ︸

AQ,G(t)γµPν + γνPµ

2

+ BQ,G(t)i(Pµσνρ + Pνσµρ)∆ρ

4MN

+ CQ,G(t)∆µ∆ν − gµν∆2

MN

using Gordon identity

equiv. decomposition:

M2(t) = A(t)

2J(t) = B(t) +A(t)

d1(t) = 5C(t)

c(t) because TQ,Gµν not conserved separately, drops out from quark+gluon sum.

2.1. Properties of form factors

total Tµν = TQµν + TGµν conserved

M2(t) = MQ(t) +MG2 (t), J(t), d1(t) scale independent (like F1(t), F2(t))

withMQ2 (0) +MG

2 (0) = 1 quarks + gluons carry 100% of momentum

JQ(0) + JG(0) = 12 quarks + gluons make up the nucleon spin

dQ1 (0) + dG1 (0) ≡ d1 what is that?

2.2. ’new characteristics’

being conserved quantity d1 = d1(0) is a characteristic property of nucleon,

like MN , electric charge, anomalous magnetic moment, axial coupling, . . .

d1(t) dictates asymptotics of unpolarized GPDs

(Goeke, Polyakov & Vanderhaeghen, 2001)

What it means? Later.

2.3. What we already know

MQ2 (0) =

∫

dx∑

q xHq(x,0,0) ≡ ∫

dx∑

q xfq1(x) ∼ 0.5 at µ2 ∼ few GeV2,

half of momentum of (fast moving) nucleon carried by quarks, rest by gluons.

asymptotically µ→∞ : MQ2 (0) =

3nf

16 + 3nfand MG

2 (0) =16

16 + 3nf

Gross, Wilczek, 1974

2JQ(0) =∫

dx∑

q x(Hq + Eq)(x,0,0) = ?

i.e. percentage of nucleon spin due to quarks?

asymptotically: 2JQ(0) =3nf

16 + 3nfand 2JG(0) =

16

16 + 3nflike M2

Ji, 1997

2.4. Meaning of d1(t) M.V.Polyakov, PLB 555, 57 (2003)

Go to Breit frame where ~P ′ = − ~P , i.e. E′ = E and ∆µ = (0, ~∆).

Define static EMT: TQµν(~r, ~s) = 1

2E

∫ d3 ~∆(2π)3

ei~∆~r 〈P ′|TQµν|P 〉

with ~s = spin vector of respective nucleon at rest.

Then: JQ(t) +2t

3JQ′(t) =

∫

d3r ei ~r~∆ εijk si rj T

Q0k(~r, ~s)

dQ1 (t) +

4t

3dQ1

′(t) +

4t2

15dQ1

′′(t) = −MN

2

∫

d3r ei ~r~∆

(

rirj − r2

3δij)

TQij (~r)

M2(t)−t

4M2N

(

M2(t)− 2J(t) +4

5d1(t)

)

=1

MN

∫

d3r ei ~r~∆ T00(~r, ~s)

gluon analog (last equation: quark+gluon)

with M2(0) =1

MN

∫

d3r T00(~r, ~s) = 1

J(0) =∫

d3r εijk si rj T0k(~r, ~s) =1

2

d1(0) = −MN

2

∫

d3r

(

rirj − r2

3δij)

Tij(~r) ≡ d1

For spin 0 or 12: Tij(~r) = s(r)

(rirj

r2− 1

3δij

)

+ p(r) δij

p(r) distribution of pressure inside hadron

s(r) related to distribution of shear forces

}

−→ “mechanical properties”

∂µTµν = 0 ⇔ ∇iTij(~r) = 0 implies:2

3s′(r) +

2

rs(r) + p′(r) = 0

−→ “stability condition”∞∫

0dr r2 p(r) = 0

finally d1(t) =15MN

2 t

∫

d3r r2j0(r√−t)p(r) and d1 =

5

4MN

∫

d3r r2p(r)

2.5. What do we know about d1?

• pion: 45 d1 = −M2 soft-pion theorems (M.V.Polyakov and C.Weiss, 1999)

• nucleus liquid drop model d1 < 0 ↔ “surface tension” (M.V.Polyakov, 2003)

(explicit calculations V.Guzey, M.Siddikov, 2006)

• nucleon

in lattice QCD dQ1 < 0 (LHPC 2003, QCDSF 2004)

chiral quark soliton model d1 = −4.75 (Pauli-Villars reg, chiral limit),

also Skyrme model d1 = −6.60 (chiral limit)

Observation: d1 seems to be always negative.

Can we understand why?

possibly: Yes!

(below)

3. Chiral quark-soliton model

Chiral action Seff =∫d4x Ψ(i 6∂−M Uγ5−m)Ψ with Uγ5 = exp(iγ5τ

aπa/fπ)

• derived from instanton model of QCD vacuum Diakonov, Petrov 1984, . . .

(“instanton packing fraction” ρav/Rav ∼ 13 small,

ρav = instanton size, Rav = instanton separation)

• M = 350MeV, Λcut ∼ ρ−1av ∼ 600MeV, no adjustable parameters!

• integrate out quarks: Leff = 14 f

2π∂

µU∂µU† + Gasser-Leutwyler-terms + . . .

with all coefficients known.

• apply to the description of nucleon → chiral quark soliton model

Diakonov, Petrov, Pobylitsa 1988

nucleon = soliton of chiral field U , in limit number of colors Nc → large

Witten 1979

3.1. Description of nucleon

Euclidean correlation function∫ DΨ†

∫ DΨ∫ DUJN(T/2)J

†N(−T/2) e−Seff ∼ e−MNT as T →∞

Solve in limit Nc → large ⇒ minimize action/soliton energy δMM = 0

yields ’self consistent’ field Us = exp(iτaearPs(r))

→ MN = Nc∑

occupied

(

En − En,0)

reg

Hφn = En φn, H = −iγ0γk∂k + γ0MUγ5 + γ0mEn,0 eigenvalues of H0 (H with Uγ5 → 1)

Power:

• Field theory!

• theoretically consistent

• phenomenologically successful

• numerous applications, thanks to:

〈N ′|Ψ(0)ΓΨ(z)|N〉 = A∑

nφn(0)Γφn(z) + . . .

non-occ.continuum

Energy

Elev

occupied continuum

M

0

−M

3.2. Applications of the model

• ’static properties’ (baryon mass splittings, magnetic moments, . . .)√

• electromagnetic (and axial) form factors up to |t| ∼ O(1GeV2)√

• parton distribution functions fa1(x), ga1(x), h

a1(x) at µ ∼ ρ−1

av ∼ 0.6GeV

satisfy all sum rules, positivity, inequalities!√

• GPDs (’discovery of D-term’)

satisfy all requirements including polynomiality!!!√

• form factors of the energy momentum tensor

again consistent: same form factors from Tµν and GPDs!√

as far as known: all observables typical accuracy (10-30)%

(room for higher orders in 1/Nc, in instanton vacuum)

catches properties of nucleon due to chiral physics

Example:

0

0.5

0 0.1 0.2 0.3 x

(f1 d-f1

u)(x)- -

E866 PRD58, 092004 (1998)CQSM PRD59, 034024 (1998)

0.0 0.2 0.4 0.6 0.8 x

−0.10

0.00

0.10

0.20

0.30

x(∆u+∆d)(x)

CQSMGRSV

fq1(x) at Q = 7.35GeV g

q1(x) at low scale vs. GRSV

satisfactory!√

Many more examples. Expected accuracy of model predictions: O(10−30%)

3.3. Form factors of Tµν in model

in model TQµν is total Tµν (gluons suppressed in instanton vacuum)

given by Tµν = 14 ψ(x)

(

iγµ−→∂ν+ iγν

−→∂µ − iγµ←−∂ ν − iγν←−∂ µ

)

ψ(x)

〈p′|Tµν(0)|p〉 =∫

d3xd3y eip′y−ipx∫DψDψDU JN ′(−T2 ,y) T eff

µν (0)J†N(T2 ,x) e

iSeff

results refer to |t| = O(N0c ) < M2

N = O(N2c )

M2(t)−t

5M2N

d1(t) =1

MN

∫

d3r ρE(r) j0(r√−t)

d1(t) =15MN

2

∫

d3r p(r)j0(r√−t)

t

J(t) = 3

∫

d3r ρJ(r)j1(r√−t)

r√−t

c(t) = 0 quark Tµν conserved by itself in model√

with “densities”:

ρE(r) = Nc∑

n,occ

En φ†n(~r)φn(~r)|reg ≡ T00(r) energy density

p(r) =Nc

3

∑

n,occ

φ†n(~r) (γ0~γ ~p )φn(~r)|reg ≡ pressure

ρJ(r) = − Nc

24I

∑

n,occj,non

ǫabcraφ†j(~r)(2p

b + (En +Ej)γ0γb)φn(~r)

〈n|τc|j〉Ej − En

∣∣∣reg

“angular momentum density”

3.4 Consistency

I MN =∫

d3r ρE(r) ⇔ M2(0) = 1√

II J(0) =∫

d3r ρJ(r) = . . . = 12

√

III∞∫

0dr r2p(r) ∝ ∑

n,occ〈n| (γ0~γ ~p ) |n〉 != 0

√but only if evaluated at true minimum Us!

Diakonov, Petrov, Pobylitsa, Polyakov, Weiss, 1996

Remark: possible to solve model with approximate ’profiles’ P(r) ≈ Ps(r).But then strictly speaking unstable nucleon:

if∞∫

0dr r2p(r) < 0 nucleon collapses, if

∞∫

0dr r2p(r) > 0 it explodes!

If you want to be ’mean’,

• ask whether GPD model satisfies polynomiality; if so

• ask about d1(t)↔ p(r) and stability condition∞∫

0dr r2p(r) = 0 . . .

IV same form factors from GPDs√

V∫

dx∑

q xHq(x, ξ, t) = M2(t) + 4

5 d1(t) ξ2

√

VI∫

dx∑

q x(Hq+Eq)(x, ξ, t) = 2J(t)

√Ossmann et al, PRD71, 034011 (2005)

Model is consistent.

Let us see what it predicts.

3.5. Results: energy distribution

0

0.5

1

0 0.5 1 1.5

(a)ρE(r) T00(r) = in fm-3 MN MN

r in fm

mπ = 140 MeV

• ρE(0) = 1.7 GeV fm−3 = 3.0× 1015g/cm3

∼ 13×nuclear matter equilibrium density

• distributed ‘similar’ to electric charge

• chiral limit: ρE(r) ∼ 3(

3gA8πfπ

)2 1r6

at large r

• 〈r2E〉 ≡∫

d3r r2ρE(r)∫

d3r ρE(r)= 0.7 fm2 similar to proton electric charge radius

• leading non-analytic term 〈r2E〉 = 〈◦r2E〉 −

81 g2A64πf2

πMNmπ+ higher orders

• for mπ → 0 nucleon ‘grows’ (range of pion cloud increases)

• reasonable & consistent picture

Compare: energy vs. baryon number density

0

0.5

1

1.5

2

0 0.5 1

Skyrme modeldensity in fm-3

r in fm

T00(r)/MNρel(r)

in Skyrme model

• baryon number density

= isoscalar charge density

• distributions similar at intermediate r

• significant differences at large r

ρE(r) ∝ 1r6

vs. ρel(r) ∝ 1r9

different chiral physics!

Picture in model.

And, in nature?

Angular momentum distribution

0

0.1

0.2

0 0.5 1 1.5

(a)ρJ(r) in fm-3 JN

r in fm

mπ = 140 MeV• ρJ(r) ∝ r2 at small r

• 〈r2J〉 = 1.3 fm2

2 times larger than 〈r2E〉 or 〈r2em〉

• in chiral limit: ρJ(r) ∼ 1r4

at large r

such that 〈r2J〉 diverges

Pressure

0

0.1

0.2

0 0.5 1 1.5

(a)p(r) in GeV fm-3

r in fm

mπ = 140 MeV

• p(0) = 0.23GeV/fm3 = 4 · 1034 N/m2

∼ (10–20) × (pressure in neutron star)

• p(0)× (typical hadronic area 1 fm2)

∼ 0.2GeV/fm ∼ 1

5× {string tension}

• chiral limit: p(r) ∼ −(

3gA8πfπ

)2 1r6

at large r

• consequence: derivative d ′1(0) = − 3 g2AMN

32πf2π mπ

+ . . . diverges in chiral limit

• r < 0.57 fm: p(r) > 0 ⇒ repulsion ↔ quark core, Pauli principle

• r > 0.57 fm: p(r) < 0 ⇒ attraction ↔ pion cloud, binding forces

Intuition on pressure and shear forces

In a ’liquid drop’:

p(r) = p0 θ(Rd − r)− 13 p0Rd δ(Rd − r)

s(r) = γ δ(Rd − r) with surface tension γ = 12 p0Rd (Kelvin, 1858)

-4

-2

0

2

4

6

0 1

(c)p(r) & s(r) in γ Rd-1

r in Rd

liquid drop

p(r)s(r)

0

0.1

0.2

0.3

0 0.5 1 1.5

(a)p(r) in GeV fm-3

r in fm

mπ = 0 mπ = 140 MeV

0

0.05

0.1

0 0.5 1 1.5

(b)s(r) in GeV fm-3

r in fm

mπ = 0 mπ = 140 MeV

nucleon does not resemble much a liquid drop (no ’sharp edge’)

concept more useful for nuclei

3.6. Insights into stability of soliton

MN = minU

Esol[U ], Esol[U ] = Nc(Elev + Econt) Recall:

non-occ.continuum

Energy

Elev

occupied continuum

M

0

−M

-0.05

0

0.05

0 0.5 1 1.5

(c)r2p(r) in GeV fm-1

r in fm

quark corepion cloud

total

level part: “quark core” → repulsion

continuum: “pion cloud” → attraction

we learn:

• strong forces in nucleon really strong

• what we always knew (but now can quantify): nucleon stability due to

subtle balance between repulsive quark core and attractive pion cloud!

3.7. Stability & sign of D-term

-0.005

0

0.005

0.01

0 0.5 1 1.5

(b)r2p(r) in GeV fm-1

r in fm

+

-1.5

-1

-0.5

0

0.5

0 0.5 1 1.5 2 2.5 3 3.5 4

(c)r4p(r) × 54 MN 4π in fm-1

r in fm

+

•∞∫

0dr r2p(r) = 0

√

•∞∫

0dr r4p(r) < 0, of course!

• d1 = 54MN

∫

d3r r2p(r) < 0 natural consequence of stability!

d1 negative, could be a theorem! Remains to be proven in general.

3.8 Results for form factors

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6

(a)M2(t)

-t in GeV2 0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6

(b)2 J(t)

-t in GeV2-4

-3

-2

-1

0

0 0.2 0.4 0.6

(c)d1(t)

-t in GeV2

Mdip(M2) = 0.91GeV Mdip(J) = 0.75GeV Mdip(d1) = 0.65GeV

for |t| . 1GeV2 reasonable approximation F (t) ≈F (0)

(1 − t/M2dip)

2with

vs. electromagnetic form factors, for example, GpE(t) with Mdip ≈ 0.91GeV

⇒ M2(t) similar to em form factors, J(t) and d1(t) differentInstructive: need to extrapolate from data at t < 0 to get JQ(0)!

Conclusions

• GPDs promissing tool to gain new insights on nucleon structure

• form factors of energy momentum tensor fascinating new insights!

but only infinitesimal fraction of information content of GPDs

• what we will learn: energy density, ’distribution of angular momentum’

’distribution of strong forces’, meaning of d1

• illustrated on results from chiral quark soliton model

theoretically consistent model: for GPDs, form factors, etc.√

successful: chiral dynamics important for understanding nucleon√

• interesting lessons from chiral models: form factors, densities,

connection: sign of d1 vs. stability of nucleon

Thank you !!!