overview of tmd factorization and evolution (corrected) · 2014-05-13 · overview of tmd...
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Overview of TMD Factorization and Evolution(corrected)
John Collins (Penn State)
• TMD factorization/evolution
• How should non-perturbative part of evolution kernel behave?
• Tool for diagnosis and comparison of formalisms/fits:
– Introduce scheme-independent L(bT) function– Examples
QCD Evolution workshop, May 12, 2014
Basic parton model inspiration: Case of Drell-Yan at qT Q
Ap
Bp
PB
PA
• Lorentz contracted high-energy hadrons
• qT(leptons) =∑
kT(quarks)
• Use parton distribution in x and kT
• But parton model needs to be substantially modified in QCD
Symptom of the QCD complications: qT distribution broadens
0.01
0.1
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E605 Data
Data
Normalized LY-G Fit
Normalized DWS-G Fit
Normalized BLNY Fit
qT (GeV)
Ed d
qy
3
30
03
σp
bG
eV
nu
cleo
nat
-2
=
.
Q = 7-8GeV
Q = 13.5-18GeV
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
0 5 1 0 1 5 2 0
CDF Z Run 1
Data
Normalized LY-G Fit
Normalized DWS-G Fit
Normalized BLNY Fit
qT (GeV)
d dq
Tσp
b
GeV
Q = 91 GeV
Q: 7–18 GeV,√s = 38.8 GeV Q = mZ,
√s = 1800 GeV
(Plot of E dσ/d3q) (Plot of dσ/dqT: has qT factor.)
(Adapted from Landry et al., PRD 67,073016 (2003))
Steps to derive factorization,given typical structure of graphs + momentum regions:
PB
PA Fourier trans. of 〈p|ψ WL ψ|p〉
• Approximations at leading power
• + Ward identities =⇒ Wilson lines for “misattached” glue (Feynman gauge)
• + contour deformation + “unitarity cancellation”, etc
• =⇒ initial-state Wilson lines for DY
• Factorization of contributions of different regions, including central/soft
• Reorganize: Construct subtractions, define TMD pdfs, with glue restricted tocorrect hemisphere, etc. Soft factors somewhere.
=⇒ Broadening from pert. and non-pert. glue into increasing rapidity range.
Full TMD factorization (modernized Collins-Soper form)
dσ
d4q dΩ=
2
s
∑j
dσj(Q,µ)
dΩ
∫eiqT·bT fj/A(xA, bT; ζA, µ) f/B(xB, bT; ζB, µ) d2bT
+ poln. terms + high-qT term + power-suppressed
where can set ζA = ζB = Q2, µ = Q.
Evolution:∂ ln ff/H(x, bT; ζ;µ)
∂ ln√ζ
= K(bT;µ)
dK
d lnµ= −γK(αs(µ))
d ln ff/H(x, bT; ζ;µ)
d lnµ= γf(αs(µ); 1)− 1
2γK(αs(µ)) ln
ζ
µ2
Small-bT:
ff/H(x, bT; ζ;µ) =∑j
∫ 1+
x−Cf/j(x/x, bT; ζ, µ, αs(µ)) fj/H(x;µ)
dx
x+ O[(mbT)p]
Key to predictivity: Universality etc derived from QCD
• All process use the same TMD pdfs (and fragmentation functions)
• Except:
– Scale dependence: Use evolution– Reversed Wilson lines in TMD pdfs between DY and SIDIS– Hence sign reversal for Sivers function etc
• Same evolution kernel K (color triplet) in all cases, including all polarized cases(Sivers, Boer-Mulders, etc)
• But breakdown of TMD factorization in HH → HH +X
• Non-perturbative information:
– Ordinary pdfs– Large bT TMD pdfs: “intrinsic transverse momentum”– Large bT of evolution kernel K(bT, µ): recoil against radiation per unit rapidity
Formalisms used: They don’t all appear compatible
Parton model: QCD complications ignored
Original CSS: non-light-like axial gauge; soft factor
Ji–Ma–Yuan: non-light-like Wilson lines; soft factor; parameter ρ
New CSS: clean up, Wilson lines mostly light-like;
absorb (square roots of) soft factor in TMD pdfs
Becher–Neubert: SCET, but without actual finite TMD pdfs
Echevarrıa–Idilbi–Scimemi: SCET
Mantry–Petriello: SCET
Boer, Sun-Yuan: Approximations on CSS
Disagreement on non-perturbative contribution to evolution (K(bT) at large bT), oreven whether it exists.
Geography of evolution of cross section
qT
0.01
0.1
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E605 Data
Data
Normalized LY-G Fit
Normalized DWS-G Fit
Normalized BLNY Fit
qT (GeV)
Ed d
qy
3
30
03
σp
bG
eV
nu
cleo
nat
-2
=
.
Q = 7-8GeV
Q = 13.5-18GeV
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
0 5 1 0 1 5 2 0
CDF Z Run 1
Data
Normalized LY-G Fit
Normalized DWS-G Fit
Normalized BLNY Fit
qT (GeV)
d dq
Tσpb
GeV
Q = 91 GeV
bT
0.25 0.5 0.75 1 1.25 1.5 1.75 2
b @GeV-1D
0
10
20
30
40
Nfit-
1b
WHb
,Q,x
A,x
BL@
fbG
eVD
p + Cu ® Μ+Μ- + X;!!!
s=38.8 GeV; Q=11 GeV; y=0
bmax=1.5 GeV-1, C3=b0, Nfit=1.19
bmax=1.5 GeV-1, C3=2b0, Nfit=1.05
bmax=0.5 GeV-1, C3=b0, Nfit=1.09
Qiu-Zhang , bmax=0.3 GeV-1, Nfit=1
0.2 0.4 0.6 0.8 1 1.2 1.4
b @GeV-1D
0
0.1
0.2
0.3
0.4
0.5
0.6
bWHb
,Q,x
A,x
BL@
nbG
eVD
p + p-
® Z0+ X;
!!!s=1.96 TeV; Q=MZ; y=0
bmax=1.5 GeV-1, C3=b0
bmax=1.5 GeV-1, C3=2b0
bmax=0.5 GeV-1, C3=b0
Q: 7–18 GeV,√s = 38.8 GeV Q = mZ,
√s = 1800 GeV
(Adapted from Landry et al., PRD 67,073016 (2003), Konychev & Nadolsky, PLB 633, 710 (2006))
Evolution of cross section (a la CSS)
0.25 0.5 0.75 1 1.25 1.5 1.75 2
b !GeV!1"
0
10
20
30
40
Nfit!
1b
W"#b
,Q,x
A,x
B$!
fb%G
eV"
p # Cu $ Μ#Μ! # X;&'''
s&38.8 GeV; Q&11 GeV; y&0
bmax&1.5 GeV!1, C3&b0, Nfit&1.19
bmax&1.5 GeV!1, C3&2b0, Nfit&1.05
bmax&0.5 GeV!1, C3&b0, Nfit&1.09
Qiu!Zhang , bmax&0.3 GeV!1, Nfit&1
Q = 11 GeV Q = 91GeV
dσ
d4q= norm.×
∫eiqT·bTW (bT, s, xA, xB) d2bT
∂W
∂ lnQ2
∣∣∣∣∣fixed xA, xB
=∂W
∂ ln s
∣∣∣∣∣fixed xA, xB
=(K(bT, µ) +G(Q,µ)
)W
• Universal K
• Perturbative: G, K at small bT, with RG
• Non-perturbative K at large bT
Different results for evolution at large bT
With CSS prescription: K(bT, µ) = K(b∗, µ)− gK(bT; bmax) fits at Q up to mZ give:
gK(bT) =0.68+0.01
−0.02
2b2T (BLNY, bmax = 0.5 GeV−1 = 0.1 fm)
gK(bT) =0.158± 0.023
2b2T (KN, bmax = 1.5 GeV−1 = 0.3 fm)
But this implies wrong behavior at large bT, smaller Q:
With this parameterization
W = . . . e−b2[coeff(x)+const ln(Q
2/Q
20)]
2 5 10 20 50-0.1
0.0
0.1
0.2
0.3
0.4
0.5
Blue: BLNY, Red: KN
(Sun & Yuan, PRD 88, 114012 (2013))
Tool to compare different methods: The L function(JCC & Rogers, in preparation)
• Shape change of transverse momentum distribution comes only frombT-dependence of K
• So define scheme independent
L(bT) = − ∂
∂ ln b2T
∂
∂ lnQ2 ln W (bT, Q, xA, xB)CSS= − ∂
∂ ln b2TK(bT, µ)
• QCD predicts it is
– independent of Q, xA, xB– independent of light-quark flavor– RG invariant– perturbatively calculable at small bT
– non-perturbative at large bT
Relation of L function to properties of cross section
If L were constant, then
W (bT, Q) = normalization× W (bT, Q0)×(
1
b2T
)L ln(Q2/Q
20)
L is positive: W decreases at large bT and increases at small bT when Q increases.
Of course, L is not actually constant.
Comparing different results using the L function(Preliminary)
0 1 2 3 4 5bT !GeV
!1"
0.2
0.4
0.6
0.8
1.0
L!bT "
!"#$%&'%
()*+%,%-./%01234%
'#%&'%
()*+%,%4./%01234%
5%6789*:%;<))1=9<%%
%%%%%%%%(91*>8?&%5%6@?A?1)1?=%
"B%C0%D)E9@F1GH%
#@%(I%=*)8?&%
0 1 2 3 4 5bT !GeV
!1"
0.2
0.4
0.6
0.8
1.0
L!bT "
!"#$!#
%&'()&*'##
+#,#-#./0#
%&'()&*'##
+#,#-1#./0#
2#3456*7#89::/;69##
########<6/*=5'$# 2#3>'?'/:/';#
@A#B.#C:D6>E/FG#
">#<H#;*:5'$#
Q Typical bT
2 GeV 3 GeV−1
10 GeV 1.2 GeV−1
mZ 0.5 GeV−1
SY = Sun & Yuan (PRD 88, 114012 (2013)):
LSY = CFαs(Q)
πDepends on Q: contrary to QCD
Implications
• Important to determine actual non-perturbative part of TMD evolution (i.e.,K(bT) at large bT).
• Older fits (e.g., KN) OK for bT up to around 3 GeV−1 = 0.6 fm.
• But their extrapolation to larger bT makes K(bT) too large.
• Use L(bT) to diagnose the issues: It’s a universal scheme independent function inQCD.
• What does this mean physically? . . .
Physical meaning of non-perturbative K(bT)
• Overall principle: Emission of glue is uniform in rapidity
• Old idea/intuition:
– In one unit of rapidity emit Gaussian (??) distribution of kT:
e−k2T/k
20 T
1
πk20 T
– Exponential convolution =⇒ W (bT, Q) = W (bT, Q0)e−b2T ln(Q
2/Q
20)k
20 T/4
– Gives K(bT)NP = −b2Tk20 T/4
• New proposal
– Per unit rapidity: a probability of no emission, and a probability of emitting aparticle (or more)
– So
K(bT)NP = FT of c[−δ(2)(kT) + e−k
2T/k
20 T/(πk2
0 T)]
= c[−1 + e−b
2Tk
20 T/4
]– ¿Change to exponential at large bT instead of Gaussian? (Normal for Euclidean
correlation function)