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QCD evolution equations at small x
(A simple physical picture)
Wei ZhuEast China Normal University
KITPC 2012.07.
A simple physical picture
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Two strange things that you have never heard
I. Strange behavior of QGP
II. A lost equation
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At small x, beyond impulse approximation
DGLAP amplitude (for gluon)
Impulse approximation
Review of QCD evolution equations at small x in a unified partonic framework
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Impulse app. Beyond Impulse app.
Small x
What will be happen?
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The correlations among the initial partons are neglected in the derivation of the DGLAP equation. This assumption is invalid in the higher density region of partons, where the parton wave functions begin tospatially overlap.
The corrections of the correlations among initial gluons to the elementary amplitude at small x should be considered.
We add a possible initial gluon to Fig.1a step by step.
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DGLAP
BFKL
GLR-MQ-ZRSGLR-MQ-ZRS
Nonperturbative correlation
Perturbative correlation
Balitsky, Fadin, Kuraev and Lipatov
Gribov, Levin and RyskinMueller and QiuZhu, Ruan Shen
Dokshitzer, Gribov, Lipatov, Altarelli and Parisi
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DGLAPDGLAP
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Real part
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Infrared divergences
The evolution kernel has singularities ,which relate to the emission or absorption of quantawith zero momentum.
Since a correct theory is IR safe, the IR divergences are cancelled by combining real-and virtual-soft gluon emissions.
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TOPT Cutting Rules
F.E. Close, J. Qiu and R.G. Roberts, Phys. Rev. D 40 (1989) 2820.
W. Zhu, Nucl. Phys. B551, 245 (1999).W. Zhu and J.H. Ruan, Nucl. Phys. B559,
378(1999).W. Zhu, Z.Q. Shen and J.H. Ruan, Nucl.
Phys. B692, 417 (2004);
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TOPT-Cutting rule
1. List all possible TOPT diagrams with different cuts.
2. The contributions of the cut diagrams
have the identical integral kernel with only the following different factors R:
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(a)The sign in the first factor is determined by the energy deficits;
(b)The second factor takes a value of 1/2 if the probe-vertex inserts in the initial line;
(c) function relates to the probe vertex.
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BFKL
2
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BFKL
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Comparing with the dipole picture
A strong assumption in the dipole approach is that the transverse size of the dipole is "frozen" during the interacting time.
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DGLAP----BFKL
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DGLAP BFKL
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DGLAP----BFKL
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DGLAP
BFKL
GLR-MQ-ZRSGLR-MQ-ZRS
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GLR-MQ-ZRS
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TOPT Cutting Rule
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GLR-MQ-ZRS
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Gribov, Levin and Ryskin , Mueller and Qiu
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GLR-MQ vs ZRS
AGK cutting rule vs TOPT cutting rule
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Abramovsky, Gribov and Kancheli, Cutting rule (1973)
2 -4 1
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Different predictions
Only shadowing effect Shadosing
and
Antishadowing effects
Looking for the antishadowing effect
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Test 1: EMC Effect
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An alternative form of the GLR-MQ-ZRS equation
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Test 2: Cronin Effect
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Nuclear modification factor
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Test 3:
Nuclear suppression factor
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Independent of any energy loss models!
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?
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???
?
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arXiv:1012.4224W. Zhu, J.H. Ruan and F.Y. Hou
A rapid crossover from week energy
loss to strong energy loss at a universal critical energy of gluon jet Ec ∼ 10GeV
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Predictions
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DGLAP BFKL
GLR-MQ-ZRS
??????
II. Looking for a lost equation
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Physical Pictures of
Present QCD Evolution Equations
• DGLAP
• BFKL
• GLR-MQ-ZRS=DGLAP+gluon fusion
• BK=BFKL+gluon fusion???
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BK in target rest frame and impact space
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BK in Bjorken frame and impact space
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DGLAP BFKL
GLR-MQ-ZRS BK
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BK in impact space and scattering amplitude
BK in momentum space and UPDF
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DGLAP BFKL
GLR-MQ-ZRS BK
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Beautiful Nature
Beautiful evolution equations
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We try to derive a new modified BFKL equation, which is
consistent with DGLAP, BFKL and GLR-MQ-ZRS.
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DGLAP BFKL
GLR-MQ-ZRS
???
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DGLAP BFKL
GLR-MQ-ZRS
???
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DGLAP-like----BFKL-like
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New
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DGLAP BFKL
GLR-MQ-ZRS NEW MD-BFKL
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DGLAP
BFKL
GLR-MQ-ZRS
New
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MD-BFKL
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MD-BFKL Equation NEWUsing TOPT-cutting rule
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DGLAP BFKL
GLR-MQ-ZRS NEW MD-BFKL
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Once the DGLAP, BFKL and GLR-MQ-ZRS equations are determined,
the form of the MD-BFKL equation is fixed.
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Solutions of the MD-BFKL equation
• A stronger shadowing suppresses the gluon density and even leads to the gluon disappearance below the saturation region.
• This unexpected effect is caused by a chaotic solution of the new equation
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Input distribution
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A unexpected solution
The unintegrated gluon distribution function F(x,k^2) in the MD-BFKL equation begins its smooth evolution under suppression of gluon recombination like the solution of the BK equation.
When x comes to a critical x_c, F(x,k^2) will oscillate aperiodically and the shadowing effect suddenly increases .
This stronger shadowing breaks the balance between the gluon fusion and splitting.
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93
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Gluon disappearance at x<x_c
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95
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Lyapunov exponents
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97
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Why we haven’t found Chaosin
previous nonlinear QCD evolution equations?
General structure of QCD evolution equations
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GLR-MQ-ZRS
Nonlinear and Non-singular
k_t is ordered and any oscillations in k_t space are suppressed.
BK
Nonlinear and Non-singular in its nonlinear part
Random and oscillations in k_t space are partly suppressed,
MD-BFKL
Nonlinear and Singular
Random and oscillations in k_t space are strong
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1. Chaos solution is a general property in any nonlinear and regularized evolutionequations by virtual processes.
2. QCD evolution equations beyond DGLAP, BFKL, BK……at next order have nonlinear and singural structures.
3. We can meet Chaos in future evolution equations.
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