Transverse momentum in hadrons, Duke University
2010-03-13
TMD PDFs on the Lattice Bernhard Musch (Jefferson Lab) in collaboration with Philipp H¨agler (TU M¨ unchen), John Negele (MIT), Andreas Sch¨afer (Univ. Regensburg), and the LHP Collaboration ¨ gler et al. EPL88 61001 (2009)] [Ha [Musch arXiv:0907.2381]
overview: parton densities
2 TMD PDFs transverse momentum dependent parton distribution f unctions e.g., f1 (x, k2⊥ ) ⇒ quark density ρ(k⊥ ).
x (longitudinal momentum fraction)
⇒
PDFs
x, b⊥ (impact parameter)
⇒
GPDs
x, k⊥ (intrinsic transverse momentum)
⇒
TMD PDFs
“basic” factorization (example SIDIS)
3
e.g., [Collins PLB 93], [Bacchetta et al. JHEP 07]
...
s gluon ...
quark
ton
pho
ha dr on
lepto n
k
P
P
nucleon
d3 P
dσ ∝ Lµν 3 |{z} h d Pl 0
W µν | {z }
lepton
hadron
tensor
tensor
µν Wunpol.,LO ∝ H(Q2 , . . .) | {z } hard part
Z
d2 k⊥ f1 (x, k⊥ , . . .) Dh (z, k⊥ + q ⊥ , . . .) | {z } | {z } TMD PDF
fragmentation f.
definition of TMD PDFs (“basic” version) Dirac matrix Γ selects quark spin
quark
Γ
P
k P
nucleon
Φ[Γ] (k, P, S) ≡ “ hP, S| q¯(k) Γ q(k) |P, Si ” lightcone coor. w± = √12 (w0 ±w3 ), so w = w+ n ˆ + + w− n ˆ − + w⊥ + proton flies along z-axis: P large, P⊥ = 0 parametrization in terms of TMD PDFs, example Z + dk − Φ[γ ] (k, P, S) + = f1 (x, k2⊥ ) + hspin dep. termsi k =xP + [Ralston, Soper NPB 1979], [Mulders, Tangerman NPB 1996], [Goeke, Metz, Schlegel PLB 2005]
4
definition of TMD PDFs (“basic” version) Dirac matrix Γ selects quark spin
quark
Γ k
P
P
nucleon
Φ[Γ] (k, P, S) ≡
1 2
Z
d4 ` −ik·` e hP, S| q¯(`) Γ U q(0) |P, Si (2π)4
ˆ + + w− n ˆ − + w⊥ lightcone coor. w± = √12 (w0 ±w3 ), so w = w+ n proton flies along z-axis: P + large, P⊥ = 0 parametrization in terms of TMD PDFs, example Z + dk − Φ[γ ] (k, P, S) + = f1 (x, k2⊥ ) + hspin dep. termsi k =xP + [Ralston, Soper NPB 1979], [Mulders, Tangerman NPB 1996], [Goeke, Metz, Schlegel PLB 2005]
4
gauge link operator U !
`
Z
dξ µ Aµ (ξ)
U ≡ P exp −ig
5
along path from 0 to `
0
=⇒
hP | q(`) Γ U q(0) |P i is gauge invariant.
SIDIS / Drell Yan
? ` 0 v=n ˆ − (lightlike),
v 1 or slightly off, v − v +
our lattice method
¨ gler, Musch arXiv:0908.1283, arXiv:0907.2381]6 [Ha
nucleon source (fixed position)
nucleon sink (fixed momentum)
u u d
u u d d
xsrc
U
gauge link on lattice
P
lat d Euclidean
tsrc
tsnk
time
For now, approximate direct gauge link, no soft factor. =⇒ no T -odd structuers (Sivers, Boer-Mulders fcn.)
extract Lorentz-invariant amplitudes A˜i (`2 , `·P ) hP, S| q(`) γµ U q(0) |P, Si = 4 A˜2 Pµ + 4i mN 2 A˜3 `µ ⇒ f1 (x, k2⊥ ) Amplitudes are complex and fulfill [A˜i (`2 , `·P )]∗ = A˜i (`2 , −`·P ) . Operator must not have temporal extent: `0 = `4 = 0 .
link renormalization
7
continuum renormalization of gauge links smooth path q U q¯
[Craigie, Dorn NPB185,204 (1981)]
l [¯ q U q] a l : the total length of the gauge link, δm ˆ : removes the power divergence ∼ 1/a [¯ q U q]ren = Z −1 exp
−δ m ˆ
static quark potential 1.5
string
[L¨ uscher,Symanzik,Weisz (1980)]
at large r: Vren (r) ≈ Vstring (r) = σr −π/12r + C
VHrL HGeVL
Vren (r) = V (r) + 2 δ m/a ˆ
1.0 a = 0.06 fm 0.5
a = 0.09 fm a = 0.12 fm a = 0.18 fm
0.0
method [Cheng PRD77,014511 (2008)] determine δ m ˆ from ! Vren (0.7 fm) = Vstring (0.7 fm)
0.0
0.2
0.4
0.6 r HfmL
0.8
1.0
link renormalization
7
continuum renormalization of gauge links smooth path q U q¯
[Craigie, Dorn NPB185,204 (1981)]
l [¯ q U q] a l : the total length of the gauge link, δm ˆ : removes the power divergence ∼ 1/a [¯ q U q]ren = Z −1 exp
−δ m ˆ
static quark potential 1.0
Vren (r) = V (r) + 2 δ m/a ˆ [L¨ uscher,Symanzik,Weisz (1980)]
at large r: Vren (r) ≈ Vstring (r) = σr −π/12r + C
VHrL HGeVL
0.5
string
a = 0.06 fm 0.0
a = 0.09 fm a = 0.12 fm
-0.5
a = 0.18 fm -1.0
method [Cheng PRD77,014511 (2008)] determine δ m ˆ from ! Vren (0.7 fm) = Vstring (0.7 fm)
0.0
string, linear 0.2
0.4
0.6 r HfmL
0.8
1.0
link renormalization
7
continuum renormalization of gauge links smooth path q U q¯
[Craigie, Dorn NPB185,204 (1981)]
l [¯ q U q] a l : the total length of the gauge link, δm ˆ : removes the power divergence ∼ 1/a [¯ q U q]ren = Z −1 exp
static quark potential
−δ m ˆ
renormalization condition C ren = 0 1.0
Vren (r) = V (r) + 2 δ m/a ˆ [L¨ uscher,Symanzik,Weisz (1980)]
at large r: Vren (r) ≈ Vstring (r) = σr − π/12r + 0
VHrL HGeVL
0.5
string
a = 0.06 fm 0.0
a = 0.09 fm a = 0.12 fm
-0.5
a = 0.18 fm -1.0
method [Cheng PRD77,014511 (2008)] determine δ m ˆ from ! Vren (0.7 fm) = Vstring (0.7 fm)
0.0
string, linear 0.2
0.4
0.6 r HfmL
0.8
1.0
link renormalization
7
continuum renormalization of gauge links smooth path q U q¯
l [¯ q U q] a l : the total length of the gauge link, δm ˆ : removes the power divergence ∼ 1/a [¯ q U q]ren = Z −1 exp
static quark potential
[L¨ uscher,Symanzik,Weisz (1980)]
at large r: Vren (r) ≈ Vstring (r) = σr − π/12r + 0 method [Cheng PRD77,014511 (2008)] determine δ m ˆ from ! Vren (0.7 fm) = Vstring (0.7 fm)
−δ m ˆ
0.4 Yline + ∆m HGeVL
Vren (r) = V (r) + 2 δ m/a ˆ string
[Craigie, Dorn NPB185,204 (1981)]
0.2
lattice cutoff effects
0.0
a = 0.06 fm
-0.2
a = 0.09 fm
-0.4
a = 0.12 fm
-0.6 0.0
a = 0.18 fm
0.2
0.4
0.6
0.8
1.0
1.2
l HfmL
Yline (l) ≡
d lnhtr Ui(Landau gauge) dl
technical details of the lattice data used We employ the Chroma library [Edwards, Joo (2005)] to process ' $ '
8
$
MILC gauge configurations staggered Asqtad action, 2+1 flavors, a ≈ 0.124 fm, mπ ≈ 500, 610, and 760 MeV [Orginos, Toussaint PRD (1999)]
+ finer MILC lattices to test renormalization [Aubin et al. PRD (2004)] [Bazavov et al. 0903.3598]
&
LHPC propagators domain wall valence fermions, mπ adjusted to staggered sea, nucleon momenta: P = 0 and |P | = 500 MeV
& %
e.g., [H¨ agler et al. PRD (2008)]
%
from amplitudes to TMD PDFs Φ[Γ] (k, P, S) ≡
1 2
Z
d4 ` −ik·` e hP, S| q¯(`) Γ U q(0) |P, Si (2π)4
Z ∞ Φ[Γ] (x, k⊥ ; P, S) ≡ dk − Φ[Γ] (k; P, S) + k =xP + −∞ Z 1 − 2 ik·` = d` d ` e hP, S| q ¯ (`) Γ U q(0) |P, Si + ⊥ 2(2π)3 ` =0 Z 2 Z d `⊥ −ik⊥ ·`⊥ d(`·P ) ix(`·P ) e e hP, S| q ¯ (`) Γ U q(0) |P, Si = + + 2 4πP (2π) ` =0 Note: `2 = −`2⊥ . x ←→ `·P k2⊥ ←→ `2 `+ =0
example: unpolarized case +
f1 (x, k2⊥ ) ≡ Φ[γ ] (x, k⊥ ; P, S) Z Z 2 d(` · P ) ix(`·P ) d `⊥ −ik⊥ ·`⊥ ˜ 2 = e e 2A2 (` , ` · P ) + 2 2π (2π) ` =0
9
Lorentz invariant amplitudes from the lattice
10
+
f1 (x, k2⊥ ) ≡ Φ[γ ] (x, k⊥ ; P, S) Z Z 2 d(` · P ) ix(`·P ) d `⊥ −ik⊥ ·`⊥ ˜ 2 = e e 2 A (` , ` · P ) + 2 2π (2π)2 ` =0 unrenormalized lattice data: `2
FT
←−→ k2⊥ FT
`·P ←−→
x
Lorentz invariant amplitudes from the lattice
10
+
f1 (x, k2⊥ ) ≡ Φ[γ ] (x, k⊥ ; P, S) Z Z 2 d(` · P ) ix(`·P ) d `⊥ −ik⊥ ·`⊥ ˜ 2 = e e 2 A (` , ` · P ) + 2 2π (2π)2 ` =0 unrenormalized lattice data: `2
FT
←−→ k2⊥ FT
`·P ←−→
x
Euclidean lattice `4 = 0 ⇓ 2
` ≤ 0,√ |`·P | ≤ |P | −`2
Lorentz invariant amplitudes from the lattice
10
+
f1 (x, k2⊥ ) ≡ Φ[γ ] (x, k⊥ ; P, S) Z Z 2 d(` · P ) ix(`·P ) d `⊥ −ik⊥ ·`⊥ ˜ 2 = e e 2 A (` , ` · P ) + 2 2π (2π)2 ` =0 unrenormalized lattice data: `2
FT
←−→ k2⊥ FT
`·P ←−→
x
Euclidean lattice `4 = 0 ⇓ 2
` ≤ 0,√ |`·P | ≤ |P | −`2
Lowest x-moment of TMD PDFs
lowest x-moment of f1 (x, k2⊥ ) (0x )
f1
(k2⊥ ) ≡
Z
1
dx f1 (x, k2⊥ ) ≡
12
Z
Z dx
+
dk − Φ[γ ] (k, P, S)
−1
Z =
2.0
ìì ìì ç ì
2 Re A2
ììì ç ç
1.0
ì ìì ì ìì
ç ç
1.5
0.5
d2 `⊥ ik⊥ ·`⊥ ˜ e 2 A2 (−`⊥2 , 0) (2π)2
ç
renormalized unrenorm.
ì ì ì ììì ì çç ì ç ìì ì ç ìì ì çç ì ìì çç ìì ç ìì ç ç ìì ççç çç ììì çç ç çç çç ììì çç ì ìììì çç çç ì ìì ç ç ççç ççç ç çç ççç ç ç
up quarks mπ = 500 MeV
0.0 0.0
0.5
1.0 -{2 HfmL
1.5
fit function C1 exp(−|`|2 /σ12 ) Z-factor !
Z −1 C1up-down = 1 ì ç
multiplicative 2.0 renormalization based on quark counting ì ç
lowest x-moment of f1 (x, k2⊥ ) (0x )
f1
(k2⊥ ) ≡
Z
1
dx f1 (x, k2⊥ ) ≡
12
Z
Z dx
+
dk − Φ[γ ] (k, P, S)
−1
Z
4
(0 H1L ) sW
-2
f1 fx1 (k2⊥ ) HGeV (GeV−2L)
=
3 2 1
renormalized unrenorm. up quarks mπ = 500 MeV
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Èk¦ È HGeVL
d2 `⊥ ik⊥ ·`⊥ ˜ e 2 A2 (−`⊥2 , 0) (2π)2
width of the distribution (RMS momentum): hk2⊥ i1/2 = (391 ± 8stat ± 27sys ) MeV compare phenomenology [Anselmino et al., PRD71, 074006 (2005)]:
hk2⊥ i1/2 ≈ 500 MeV (estimate, Gaussian Ansatz)
lowest x-moment of f1 (x, k2⊥ ) (0x )
f1
(k2⊥ ) ≡
Z
1
dx f1 (x, k2⊥ ) ≡
12
Z
Z dx
+
dk − Φ[γ ] (k, P, S)
−1
Z =
d2 `⊥ ik⊥ ·`⊥ ˜ e 2 A2 (−`⊥2 , 0) (2π)2
2
width of the distribution correlator with straight (RMS Wilsonmomentum): line renormalized renormalized to string potential with C = 0 unrenorm. hk2⊥ i1/2 = Gaussian fit ansatz (391 ± 8stat ± 27sys ) MeV (“wrong” up quarksat large-k⊥ [Diehl, arXiv:0811.0774])
1
500 MeV MeV mm π π≈=500
4
(0 H1L ) sW
-2
f1 fx1 (k2⊥ ) HGeV (GeV−2L)
keep in mind
3
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Èk¦ È HGeVL
compare phenomenology [Anselmino et al., PRD71, 074006 (2005)]:
hk2⊥ i1/2 ≈ 500 MeV (estimate, Gaussian Ansatz)
from A˜2
2.0
ìì ìì ç ì
ììì
2 Re A2
ì ìì ì ìì
ç ç
1.5
ç ç
1.0 0.5
13 renormalized unrenorm.
ç ì ì ì ììì ì çç ì ç ìì ì ç ìì ì çç ì ìì çç ìì ç ìì ç ç ìì ççç çç ììì çç ç çç çç ììì çç ì ìììì çç çç ì ìì ç ç ççç ççç ç çç ççç ç ç
up quarks mπ = 500 MeV
0.0 0.0
0.5
1.0
1.5
ì ç
f1H1L sW HGeV-2 L
(0x )
f1
4 3 2
2.0
2 Re A2
0.6 0.4
ììì ç
ì ì ì ì ìì
ç ç ç
renormalized unrenorm.
ç ì ì ì ììì ì çç ì ç ìì ç ì ìì ì çç ì çç ìì ç ç ìì ìì ç ìì ççç çç ìì çç ì ìì ç çç çç ì ìì çç çç ì ììì ì çç ç ç ççç ì ççç ç çç ççç ç ç
0.2 down quarks mπ = 500 MeV 0.0 0.0 0.5 1.0
-{2 HfmL
up quarks mπ = 500 MeV
1
Èk¦ È HGeVL
1.5
f1H1L sW HGeV-2 L
ìì ìì ç ì
0.8
unrenorm.
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2
ì ç
-{2 HfmL
1.0
renormalized
2.0 renormalized
1.5
Fourier-
======⇒1.0 transform
ì ç
ì ç
2.0
0.5
unrenorm. down quarks mπ = 500 MeV
0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Èk¦ È HGeVL
from A˜2
2.0
ìì ìì ç ì
ììì ç ç ç
1.0
renormalized unrenorm.
ç ì ì ì ììì ì çç ì ç ìì ì ç ìì ì çç ì ìì çç ìì ç ìì ç ç ìì ççç çç ììì çç ç çç çç ììì çç ì ìììì çç çç ì ìì ç ç ççç ççç ç çç ççç ç ç
4 3 2
multiplicative renormalizaup quarks tion constant Z adjusted to 1 mπ = 500 MeV number of valence quarks 0 0.0 0.0 0.0 0.5 R 21.0 1.5 2.0 (0) d k⊥ f1 (k2⊥ ) = 2A˜2 (0, 0), 2 -{fixed HfmL in u − d channel 0.5
1.0
ìì ìì ç ì
0.8 0.6 0.4
ììì ç
ì ì ì ì ìì
ç ç ç
-{2 HfmL
1.5
unrenorm. up quarks mπ = 500 MeV
transform
2.0
Èk¦ È HGeVL
renormalized
1.5
Fourier-
ì ç
0.2 0.4 0.6 0.8 1.0 1.2
2.0
======⇒1.0
ì ç
renormalized
ì ç
renormalized unrenorm.
ç ì ì ì ììì ì çç ì ç ìì ç ì ìì ì çç ì çç ìì ç ç ìì ìì ç ìì ççç çç ìì çç ì ìì ç çç çç ì ìì çç çç ì ììì ì çç ç ç ççç ì ççç ç çç ççç ç ç
0.2 down quarks mπ = 500 MeV 0.0 0.0 0.5 1.0
ì ç
f1H1L sW HGeV-2 L
2 Re A2
ì ìì ì ìì
ç
1.5
2 Re A2
13 f1H1L sW HGeV-2 L
(0x )
f1
0.5
unrenorm. down quarks mπ = 500 MeV
0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Èk¦ È HGeVL
(0 ) g1Tx from A˜7
-0.05 -0.10 -0.15 0.0
0.5
renormalized unrenorm.
1.0 2
-{
2 mN Re A7 HGeVL
ç ç çç ç ç ì ççççç ç ç ì ì ì ì ìì ì ì ì ì ì ì
1.5
6 renormalized 5 unrenorm. 4 3 up quarks mπ = 500 MeV 2 1 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2
sW gH1L mN HGeV-3 L 1T
ç ç çç ç ç çç ç çç çç ç çç ç ì çç ç ç ì ç çç ìì ç ç ìì ì ì ç ì ì ççç ìì ì ì ìì ìì ç ìì ì çç ç ìì ì ì ì ìì ì ììì ì ì ì ç
up quarks
2.0
Èk¦ È HGeVL
HfmL
0.035 ì renormalized 0.030 ì ì ìì ìì ç unrenorm. ç ì ì 0.025 ì ì ìì ìç ìì ì ììì ì ì 0.020 ç ì çç ì ìì ç ç ç ìììì ì 0.015 çç ì ç çç ç ì çççç 0.010 ì çç ì ç ç 0.005 down quarksççççççç ç ç ì ìì ì ç ç ç ì ì ç çç ç ç ì ç ç 0.000 ç ç 0.0 0.5 1.0 1.5ç 2.0 ç ç
ì
2
-{ç HfmL ì
sW gH1L mN HGeV-3 L 1T
2 mN Re A7 HGeVL
0.00
14
0.0
Fourier-
======⇒-0.5 transform
down quarks mπ = 500 MeV
-1.0 -1.5
renormalized unrenorm.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 Èk¦ È HGeVL
a polarized k⊥ -dependent quark density
15
Density of quarks with positive helicity, λ = 1, in a transversely polarized nucleon, S ⊥ = (1, 0): Z Z 5 +1 1 ρT L (k⊥ ; S ⊥ , λ) ≡ dx dk − Φ[γ 2 (1+γ )] (k, P, S⊥ ) 2 λ k⊥ · S ⊥ (0x ) 2 1 (0 ) = f1 x (k2⊥ ) + g1T (k⊥ ) 2 2 mN kx (GeV)
kx (GeV) down
up
ky
ky
λ
u
d u
S
λ
u
d u
S
mπ ≈ 500 MeV, straight gauge link operator, renormalization condition C ren = 0, Gaussian fit
a polarized k⊥ -dependent quark density
15
Density of quarks with positive helicity, λ = 1, in a transversely polarized nucleon, S ⊥ = (1, 0): Z Z 5 +1 1 ρT L (k⊥ ; S ⊥ , λ) ≡ dx dk − Φ[γ 2 (1+γ )] (k, P, S⊥ ) 2 λ k⊥ · S ⊥ (0x ) 2 1 (0 ) = f1 x (k2⊥ ) + g1T (k⊥ ) 2 2 mN kx (GeV)
kx (GeV) down
up hkx i = (67 ± 5stat ± 3sys ) MeV
hkx i = (−30 ± 5stat ± 1.3sys ) MeV
ky
ky
λ
u
d u
model [Pasquini et. al 0912.1761] hkx iu = 55.8 MeV S λ u d hkSx id = −27.9 MeV u
mπ ≈ 500 MeV, straight gauge link operator, renormalization condition C ren = 0, Gaussian fit
k⊥ -moments, weighted asymmetries f (mx ,n⊥ ) ≡
Z
1
dx xm
−1
Z
d2 k⊥
k2⊥ 2m2N
n
16
f (x, k2⊥ )
Let us assume the amplitudes A˜i are sufficiently regular at `2 = 0. (0 ,1⊥ )
hk⊥ iρT L = λS ⊥ mN λS ⊥ mN
g1Tx
=
(0x ,0⊥ )
f1
A˜7 (0, 0) ? A˜7 (`2 , 0) = lim λS m ⊥ N `2 →0 A˜2 (0, 0) A˜2 (`2 , 0)
All self-energies from the gauge link cancel on the RHS (⇒ no dependence on the renormalization condition). Similar to weighted asymmetries from experiment (→ EIC): QT mN
ALT
cos(φh −φS )
=2
QT hm cos(φh − φS )iU T N
h1iU U
P
(1 )
q ∝ P
⊥ e2q x g1T,q (x) D1,q (z)
q
e2q x f1,q (x) D1,q (z)
[Boer, Mulders PRD 1998], [Bacchetta et al. arXiv:1003.1328]
testing Gaussian parametrization
17 (0x )
(0 ) f1 x (k2⊥ ) (0 ) g1 x (k2⊥ )
=
C0 exp(−k2⊥ /µ20 )
=
C2 exp(−k2⊥ /µ22 )
1 ρ± LL (k⊥ ) ≡ 2 f1
vs.
(0x )
(k2⊥ ) ± 21 g1
(k2⊥ )
2 2 ρ+ LL (k⊥ ) = C+ exp(−k⊥ /µ+ ) 2 2 ρ− LL (k⊥ ) = C− exp(−k⊥ /µ− )
1.0
0.5
statistical error
0.0
g1
(0xg) f (0x ) 1 /f 1 1
f1 and g1 Gaussian fit f1 +g1 and f1 -g1 Gaussian fit
-0.5
-1.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
|k⊥Èk¦|È HGeVL (GeV) ⇒ Asymptotic behavior at large k⊥ imposed by Gaussian ansatz; not a “lattice result”. Similar issues in analysis of experimental data.
x-dependence
`·P -dependence in A˜2
19
real part `2
FT
←−→ k2⊥ FT
`·P ←−→
x
(x, k⊥ )-factorization hypothesis factorization hypothesis (0x )
f1 (x, k2⊥ ) ≈ f1 (x) f1
(k2⊥ ) / N
as in phenomenological applications, e.g., Monte Carlo event generators
Then A˜2 factorizes, too: (`·P ) A˜2 (`2 , 0). A˜2 (`2 , `·P ) = A˜norm 2 To test this, we define A˜2 (`2 , `·P ) 2 A˜norm (` , `·P ) ≡ 2 Re A˜2 (`2 , 0) (needs no renormalization!) If factorization holds, A˜norm should be 2 `2 -independent.
20
(x, k⊥ )-factorization hypothesis
20 15
factorization hypothesis
{×P =
(0x )
f1 (x, k2⊥ ) ≈ f1 (x) f1
(k2⊥ ) / N
10
as in phenomenological applications, e.g., Monte Carlo event generators
(`·P ) A˜2 (`2 , 0). A˜2 (`2 , `·P ) = A˜norm 2 To test this, we define A˜2 (`2 , `·P ) 2 A˜norm (` , `·P ) ≡ 2 Re A˜2 (`2 , 0)
norm Im 2A2 H+offsetL
5
Then A˜2 factorizes, too:
0
-5
(needs no renormalization!) -10
If factorization holds, A˜norm should be 2 `2 -independent.
X within statistics
3.77 3.46 3.14 2.83 2.51 2.20 1.88 1.57 1.26 0.94 0.63 0.31 0.00 -0.31 -0.63 -0.94 -1.26 -1.57 -1.88 -2.20 -2.51 -2.83 -3.14 -3.46 -3.77
-0.5
0
0.5 -{2 HfmL
1
1.5
global `·P -behavior
21
(`2 , `·P ) at mπ ≈ 610 MeV All our data for A˜norm 2 qualitative comparison to a Fourier transform of f1 (x) from CTEQ5 [Lai et al., EPJ C12, 375 (2000)] √ a scalar diquark model at −`2 = 0 and 1 fm [JMR, NPA626, 937 (1997)]
1.2 1.0
norm Re 2A2
0.8 0.6 0.4 lattice data 0.2
CTEQ
0.0
diquark model
-0.2 -3
-2
-1
0 {×P + small offsets
1
2
3
Towards Extended Gauge Links
SIDIS beyond the “basic” ansatz
23
hadr on
e.g., [Ji, Ma, Yuan PRD (2005)] :
µν Wunpol.,LO
∝
γ*
H × f1 ⊗ Dh ⊗ |{z} S
soft factor
~ H
P nucleon
~ Dh,q ~ S
~ fq
~ H
P
modified definition of TMD PDF correlator: Z d4 ` −ik·` hP, S| q¯(`) Γ U q(0) |P, Si 1 e Φ[Γ] (k, P, S) ≡ e ⊥ , . . .) 2 (2π)4 S(` ?
v
v `
+ 0
gauge links slightly off lightcone: v 6= n ˆ−
?
+ –
`? –
0 ~ v
⇒ evolution eqn. in ζ ≡ (v·P )2 /v 2 ˜ vacuum soft factor S: expectation value of gauge link structure
subtraction factor
24
How to get rid of the gauge link self engergy exp(δm L)? Soft factor in TMD PDF correlator? Suggestion
[Collins arXiv:0808.2665]
`
` ´v
:
2´v
Is this a meaningful definition of TMD PDFs? prerequisite for quantitative lattice predictions “To allow non-perturbative methods in QCD to be used to estimate parton densities, operator definitions of parton densities are needed that can be taken literally.” [Collins arXiv:0808.2665 (2008)] k⊥ -moments from ratios of amplitudes ... ... may bridge the gap until we know more. Example Sivers effect: hk⊥ iρT U from A˜12 /A˜2 . Self-energies cancel, no explicit subtraction factor needed.
Conclusion Summary: We have explored ways to calculate intrinsic transverse momentum distributions in the nucleon with lattice QCD. We directly implement non-local operators on the lattice. First results are based on a simplified operator geometry (direct gauge link) and a Gaussian fit model, at mπ ≈ 500 MeV: We calculate the first Mellin moment of leading twist TMD PDFs (0) (0) ⊥(1) f1 (k2⊥ ), g1T (k2⊥ ), h1L (k2⊥ ) etc. k⊥ -densities of longitudinally polarized quarks in a transversely (0) polarized proton are deformed, due to non-vanishing g1T . So far, no statistically significant incompatibility with (0 ) factorization f1 (x, k2⊥ ) ≈ f1 (x) f1 x (k2⊥ )/N detectable within the limited range of available lattice data. Outlook: Beyond Gaussian fits: Matching to perturbative behavior at small `, i.e., large k⊥ . Study of non-straight gauge links similar as in SIDIS. Focus on selected k⊥ -moments (↔ weighted asymmetries), until appropriate subtraction factors are better understood.
25
Backup Slides
parametrization of the matrix elements 1 Φ (k, P, S) ≡ 2 [Γ]
Z
d4 ` −ik·` e hP, S| q¯(`) Γ U q(0) |P, Si (2π)4
isolation of Lorentz-invariant amplitudes
compare [Mulders, Tangerman NPB (1996)]
hP, S| q(`) γµ U q(0) |P, Si = 4 A˜2 Pµ + 4i mN 2 A˜3 `µ hP, S| q(`) γµ γ 5 U q(0) |P, Si = −4 mN A˜6 Sµ −4i mN A˜7 Pµ (` · S) +4 mN 3 A˜8 `µ (` · S) hP, S| q(`) . . . U q(0) |P, Si = further structures (9 amplitudes in total)
Transformation properties of the matrix element (†, P, T ) limit number of allowed structures. No T -odd structures (Sivers function, . . .) with straight gauge link. i∗ h The amplitudes fulfill A˜i (`2 , ` · P ) = A˜i (`2 , −` · P ) .
28
parametrization of the matrix elements 1 Φ (k, P, S) ≡ 2 [Γ]
Z
28
d4 ` −ik·` e hP, S| q¯(`) Γ U q(0) |P, Si (2π)4
isolation of Lorentz-invariant amplitudes
compare [Mulders, Tangerman NPB (1996)]
hP, S| q(`) γµ U q(0) |P, Si = 4 A˜2 Pµ + 4i mN 2 A˜3 `µ ⇒ f1 (x, k2⊥ ) hP, S| q(`) γµ γ U q(0) |P, Si = −4 mN A˜6 Sµ 5
−4i mN A˜7 Pµ (` · S) +4 mN 3 A˜8 `µ (` · S) ⇒ g1T (x, k2⊥ ) hP, S| q(`) . . . U q(0) |P, Si = further structures (9 amplitudes in total)
Transformation properties of the matrix element (†, P, T ) limit number of allowed structures. No T -odd structures (Sivers function, . . .) with straight gauge link. h i∗ The amplitudes fulfill A˜i (`2 , ` · P ) = A˜i (`2 , −` · P ) .
extracting nucleon structure from the lattice Ingredients
gauge configs.
quark propagators
Output : 3-point correlator C3pt nucleon source (fixed position)
nucleon sink (fixed momentum)
u u d
u u d d
xsrc
U nucleon sequential propagators
29
P
lat d Euclidean
tsrc
tsnk
time
[We neglect “disconnected contributions” (absent for up minus down).]
transfer matrix formalism
30
ratio of correlators far away from nucleon source and sink C3pt (τ ; Γ, `, P ) C2pt (P )
tsrc τ t
sink −−−−−−−− −→
const. (“plateau value”), ⇓ access to hP, S| q(`) Γ U q(0) |P, Si
Γ
1 ren 2 C3pt (τ ; Γ, `, P )/C2pt (P )
1
mN ˜ A1 E(P )
−γ4 γ5
imN A˜7 `z
γ4 1 2 [γ2 , γ4 ]
...
(LHPC projectors)
A˜2 1 ˜ im2N ˜ m2N ˜ A 9 Px + A10 `x + A11 (`z )2 Px E(P ) E(P ) E(P ) ...
transfer matrix formalism
30
ratio of correlators far away from nucleon source and sink C3pt (τ ; Γ, `, P ) C2pt (P )
tsrc τ t
sink −−−−−−−− −→
const. (“plateau value”), ⇓ access to hP, S| q(`) Γ U q(0) |P, Si
example plateau plots at mπ ≈ 600 MeV for Γ = γ4 (⇒ A˜2 ), with HYP smeared gauge link U =
0.20
0.10
0.00
10
0.10 0.05
sink
0.05
0.15
P =0 12
14
16 ` Τ
18
20
0.00
10
P = 2π L (−1,0,0) 12
14
16 ` Τ
18
sink
0.15
source
C3pt C2pt
0.20
source
C3pt C2pt
0.25
:
20
leading-twist TMD PDFs
q N U
U
L
?
f1T
T ?
f1
L T
31
h1 ?
g1
h1L
g1T
h1 h1T
?
t ime-reversal odd
leading-twist TMD PDFs
q N U
U
L
?
f1T
T ?
f1
L T
31
h1 ?
g1
h1L
g1T
h1 h1T
?
t ime-reversal odd
“genuine” signs of intrinsic quark momentum
32
(d)
(u)
ρT L
ρT L
Diplole deformations
ky d u
u
λ
S
(u) ρLT
λ
u
d u
S
ρT L : ∼ λ k⊥ ·S ⊥ g1T ρT L : ∼ Λ k⊥ ·s⊥ h⊥ 1L
s
The corresponding dipole structures ∼ λ b⊥ ·S ⊥ , ∼ Λ b⊥ ·s⊥ for impact parameter densities (from GPDs) are ruled out by symmetries.
(d) ρT L
ky Λ
u
d u
s
kx
Λ
u
d u
kx
¨ gler, Musch, Negele, Scha ¨ fer EPL 88, 61001 (2009)] [Ha
`·P - dependence of A˜2 (`2 , `·P )
2 Re A˜2 (`2 , `·P )
33
2 Im A˜2 (`2 , `·P )
`·P - dependence of A˜2 (`2 , `·P )
33
2 Re A˜2 (`2 , `·P )
2 Im A˜2 (`2 , `·P )
1.0 æ
È{È=0. fm
Re 2A2
È{È=0.25 fm 0.6
È{È=0.37 fm È{È=0.5 fm
0.4 È{È=0.62 fm
È{È=0.74 fm
0.2 0.0
È{È=0.99 fm È{È=1.5 fm -6
-4
È{È=1.2 fm -2
0 {×P
È{È=1.5 fm È{È=1.2 fm È{È=0.99 fm È{È=0.74 fm È{È=0.62 fm È{È=0.5 fm È{È=0.37 fm È{È=0.25 fm È{È=0.12 fm È{È=0. fm
0.8
2
4
Im 2A2 H+offsetL
È{È=0.12 fm
0.8
0.6 0.4 0.2 0.0 -4
-2
0
2 {×P
4
6
effect of normalization with amplitude at `·P = 0
Re A˜2 (`2 , `·P ) Re A˜norm = 2 Re A˜2 (`2 , 0)
2 Re A˜2 (`2 , `·P )
1.0
5
È{È=0.12 fm
0.8
æ
È{È=0. fm 4
0.6
È{È=0.37 fm È{È=0.5 fm
0.4 È{È=0.62 fm
È{È=0.74 fm
0.2 È{È=0.99 fm È{È=1.5 fm
` Re 2A2 H+offsetL
Re 2A2
È{È=0.25 fm
0.0
34
í í í
È{È=8 3
í í íííí íí í íí íííííí í íí í íí í íí áá á áá áá á á ááá ááá áá áá á áá á á á á á áá ç ç ç ç ç ç ç ç ç ç È{È=6 ç ç ç í íí
È{È=5
í
ô ô ôô ô ô ô ô ô ô ô ôô ò òò ò òò
2
È{È=3
-4
-2
0 {×P
2
4
ò òò ò òò ò
È{È=0 -15
-10
È{È=4
ì ì ì ì ì ì ì à à à à à
È{È=1.2 fm 1
-6
È{È=10
È{È=2
æ
0
-5 {×P
5
10
15
effect of normalization with amplitude at `·P = 0
34
Im A˜2 (`2 , `·P ) Im A˜norm = 2 Re A˜2 (`2 , 0)
2 Im A˜2 (`2 , `·P )
5
Im 2A2 H+offsetL
0.8 0.6 0.4 0.2 0.0 -4
-2
0
2 {×P
4
6
4 norm Im 2A2 H+offsetL
È{È=1.5 fm È{È=1.2 fm È{È=0.99 fm È{È=0.74 fm È{È=0.62 fm È{È=0.5 fm È{È=0.37 fm È{È=0.25 fm È{È=0.12 fm È{È=0. fm
È{È=1.2 fm È{È=0.95 fm
3
È{È=0.71 fm È{È=0.6 fm
2
È{È=0.48 fm È{È=0.36 fm
1
È{È=0.24 fm È{È=0.12 fm
0 -4
æ
-2
È{È=0. fm
0
2 {×P
4
6
staple-shaped gauge links
35
` ´v
32 Lorentz-invariant amplitudes
[Goeke,Metz,Schlegel PLB618,90 (2005)]
v2 v·P v·k v·k −1 , , Ai k 2 , k·P, = Ai k 2 , k·P, , ζ , sgn(v·P ) 2 |v·P | |v·P | |v·P | |v·P | | {z } ≈x Links approaching light cone: v → n ˆ − ⇒ ζ → ∞. For large ζ, the evolution with ζ is known [Collins,Soper NPB194,445 (1981)].
time reversal T
36
(v 0 , v 1 , v 2 , v 3 ) (−v 0 , v 1 , v 2 , v 3 ) T future pointing v past pointing v − → TMD PDFs for SIDIS TMD PDFs for Drell-Yan The transformation property of the matrix elements under time reversal provides relations: Example of a T -even amplitude: v·k v·k A2 k2 , k·P , v·P , ζ −1 , 1 = A2 k2 , k·P , v·P , ζ −1 , −1) ⇓ (SIDIS)
f1
(Drell-Yan)
(x, k⊥ ; ζ, . . .) = f1
(x, k⊥ ; ζ, . . .)
⊥ Example of a T -odd amplitude: (→ Sivers function f1T ) v·k v·k A12 k2 , k·P , v·P , ζ −1 , 1 = −A12 k2 , k·P , v·P , ζ −1 , −1) ⇓ ⊥(SIDIS)
f1T
⊥(Drell-Yan)
(x, k⊥ ; ζ, . . .) = −f1T
(x, k⊥ ; ζ, . . .)
A˜2 from the lattice for extended gauge links
v·` −1 A˜2 `2 , `·P, |v·P , sgn(v·P ) |, ζ
≡ lim a ˜2 (`2 , `·P, ηv·`, −η 2 , ηv·P ) η→∞
0.8
-{2 0.00 fm
Re RΓ4
0.6
0.12 fm 0.24 fm
0.4
0.59 fm 1.18 fm
0.2 0.0 -5
0
5
Ηv×P
But a ˜2 = Re Rγ4 always vanishes for large η! Reason: power divergence suppresses a ˜2 ∼ exp(−δm η).
37
A T -odd ratio from the lattice Rodd =
a ˜12 + (η
m2N v1 P1
38
) ˜b8
a ˜2 ±ηv·P large
A˜12 (`2 , 0, 0, ζ −1 , ±1) +
−−−−−−−−→
mN P1
2
˜8 (`2 , 0, 0, ζ −1 , ±1) B
A˜2 (`2 , 0, 0, ζ −1 , ±1)
1.0
Rodd
0.5 -{2
0.0
0.12 fm 0.24 fm
-0.5
0.59 fm 1.18 fm
-1.0 -5
0
5
10
Ηv×P ⊥ Part of the effect comes from the Sivers function f1T via A˜12 !
