the energy-momentum tensor form factors of the nucleon ... · pdf filethe energy-momentum...
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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 !!!