Blois workshopK. Itakura (CEA/Saclay)1 Perturbative Odderon in the Color Glass Condensate in...
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Transcript of Blois workshopK. Itakura (CEA/Saclay)1 Perturbative Odderon in the Color Glass Condensate in...
Blois workshop K. Itakura (CEA/Saclay) 1
Perturbative Odderon Perturbative Odderon in the in the ColorColor GlassGlass CondensateCondensate
in collaboration with
E. Iancu (Saclay),
L. McLerran & Y. Hatta (BNL)
Kazunori Itakura
(SPhT, CEA/Saclay
KEK in two weeks)
based on hep-ph/0501171
Blois workshop K. Itakura (CEA/Saclay) 2
OutlineOutlineIntroduction Odderon in Regge theory and in perturbative QCD
Why Odderon in CGC??
C-odd operators in CGC Relevant operators for dipole-CGC & 3quark-CGC scatterings
Odderon evolutions dipole-CGC scattering decomposition of the Balitsky equation,
BFKL equation in weak-field regime
3quark-CGC scattering new equation, reduces to the BKP eq.
in weak-field regime
Summary
Blois workshop K. Itakura (CEA/Saclay) 3
OdderonOdderon = Leading “C-odd” exchange in hadron scatt. at high energies.
“C-odd” counterpart of the Pomeron (see Ewerz’s talk)
Odderon /Odderon / Introduction (I)Introduction (I)
Regge theory “soft” Odderon [Lukaszuk-Nicolescu ’73]
Elastic amplitude odd under “crossing” (a+ba+b vs “crossed” a+b a+b)
A-- : “particle-particle scatt” – “particle-antiparticle scatt”
ODD under charge conjugation pp
}{ ),(),(2
1),( tsAtsAtsA baab
--
-
*
Perturbative QCD “hard” Odderon
three reggeized gluon exchange in C-odd
state (exists only for )
C-odd three gluon operator
3Nccbaabc AAAd
*
* Experimental status not conclusive so far…
Blois workshop K. Itakura (CEA/Saclay) 4
The BKP equation for 3 gluonsThe BKP equation for 3 gluons [Bartels, Kwiecinski-Praszalowicz ‘80]
F: amplitude for exchange of three reggeized gluons in a color singlet C-odd state Pair-wise interaction between two gluons among three
BFKL evolution HBFKL
The physical amplitude is obtained after convoluting
the impact factor of the projectile
Two solutions for BKP eq. with 3 gluons:
Janik-Wosiek (‘99) , Bartels, Lipatov & Vacca (‘00)
Perturbative Odderon /Perturbative Odderon / Introduction (II)Introduction (II)
FHFHFHkkkFY BFKLBFKLBFKL 312312
321 ),,(
1odd 1odd
**
*
*
Y
Blois workshop K. Itakura (CEA/Saclay) 5
Why Odderon in CGC? / Why Odderon in CGC? / Introduction (III)Introduction (III)
Perturbative Pomeron in the Color Glass Condensate
dipole-CGC scattering ( dipole operator + JIMWLK equation)
The relevant operator for the Pomeron (see talks by Venugopalan, Iancu)
Two reggeized gluon exchange in linear regime
two Wilson lines in nonlinear regime BFKL equation
But n-reggeon dynamics (BKP) is also important at high energy Need to investigate n-reggeon dynamics in the CGC which is in princip
le applicable for n-reggeons.
The first step: 3 gluon exchange in linear regime Odderon !
What is the relevant operator for the Odderon exchange???What is the relevant operator for the Odderon exchange???
Can we reproduce the BKP equation in the CGC???Can we reproduce the BKP equation in the CGC???
22
)(4
1)( ay
axyx Nc
gVVtr }{ ),(exp
xxdxigPVx
Blois workshop K. Itakura (CEA/Saclay) 6
Determine the relevant operators for scatt. btw a projectile and the CGC
A projectile traverses a strong “random” gauge field created by the CGC.
- the eikonal approximation
- The operator is evaluated with
averaging over the color field
W[]: weight function randomness
General strategies in CGCGeneral strategies in CGC
ex)Dipole-CGC scattering: the relevant operator leads to the Balitsky eq.
0|)()( yqxq inin 0|)()( yqxq outout
Compute the evolution equations from the JIMWLK equation
JIMWLK eq. = evolution equation for the weight function in the target. easily converted into the equations for operators.can be made simple for gauge invariant operators
IR finiteness manifest
Blois workshop K. Itakura (CEA/Saclay) 7
Transition from C-odd to C-even dipole states
Relevant operator
- anti-symmetric under the exchange of x and y: O(x,y) = - O(y,x)
- imaginary part of the dipole operator.
Weak field expansion leading order is 3 gluons
gauge invariant combination! ( + c)
C-odd operatorC-odd operator in in dipole-CGCdipole-CGC scatt.scatt.
0|)()()()( ][ xqyqyqxq 0|)()()()( ][ xqyqyqxq
Blois workshop K. Itakura (CEA/Saclay) 8
C-odd operator inC-odd operator in 3-quark--CGC3-quark--CGC scatt. scatt.
Consider the scattering of a color singlet “3-quark state” and transition from C-even to C-odd 3 quark states
Relevant operator
“baryonic Wilson lines”
Weak field expansion
3 gluons with d-symbol, gauge invariant
________ ________ all the possible ways of attaching
0|kzjy
ix
ijk qqq
Blois workshop K. Itakura (CEA/Saclay) 9
Evolution of the Evolution of the dipole Odderondipole OdderonEvolution eq. for the dipole Odderon “imaginary part” of the Balitsky eq.
couple to the Pomeron N(x,y) = 1- 1/Nc Re tr(V+xVy)
becomes equivalent to Kovchegov-Szymanowski-Wallon (‘04)
if one assumes factorization <NO> <N><O>.
initial condition computable with a classical gauge field + color averaging
or in an extended McLerran-Venugopalan model (Jeon-Venugopalan ‘05)
linear part = the BFKL eq. (but with C-odd initial condition)
reproduces the BKP solution with the largest intercept
found by Bartels, Lipatov & Vacca (KSW,04)
intercept reduces due to saturation: <O(x,y)> decreasing as <N(x,y)> 1
Evolution of N(x,y) is also modified due to Odderon: 2 Odderons 1 Pomeron
BFKL
**
*
*
*
1odd
)/1ln( x
*
Blois workshop K. Itakura (CEA/Saclay) 10
Evolution of the Evolution of the dipole Odderondipole Odderon (I (II)I)
The presence of imaginary part (odderon) affects the evolution equation for the scattering amplitude N(x,y).
Balitsky equation new contribution!
- Two Odderons can merge into one Pomeron!
N=1, O=0 is the stable fixed point.
Blois workshop K. Itakura (CEA/Saclay) 11
3-quark--Odderon operator3-quark--Odderon operatorBaryonic Wilson line operatorBaryonic Wilson line operator
multiplying the identity
One can rewrite 3quark-Odderon operator as manifestly gauge invariant
reduces to dipole-Odderon operator when two coordinates are the same
Oproton(x,z,z) = O(x,z) diquark ~ antiquark
can compute nonlinear evolution equation for Oproton(x,y,z) complicated
Blois workshop K. Itakura (CEA/Saclay) 12
: weak field limit of
The BKP equation appears as the equation for 3 point Green function
with infra-red singularities removed
Evolution of Evolution of 3quark-Odderon3quark-Odderon operator operatorin the weak-field limitin the weak-field limit
),,( zyxOproton
cz
by
ax
abcdzyxf ),,(
xyzB
Blois workshop K. Itakura (CEA/Saclay) 13
Relation to the traditional approachRelation to the traditional approachTraditional description CGC formalism
Our operator partly contains the
information of the impact factor
Gauge invariant
impact factorgauge invariance
BKP equation
}{}{
...),,(2
...),,(),,(3),,(12
),,(
xxxf
zxxfyxxfzyxf
zyxOproton
LC wavefunction
Blois workshop K. Itakura (CEA/Saclay) 14
SummarySummary• Identified the relevant operator for C-odd Odderon exchange in dipole-CGC scattering imaginary part of the dipole operator (2pt fnc),
O(x,y) = [ tr(Vx+ Vy) – tr(Vy
+ Vx) ] / 2iNc. in 3-quark--CGC scattering a 3 point fnc constructed from baryonic Wilson line operator Both reduce to 3 gluons with d-symbol in the weak-field limit
• Evolution equations for these operators JIMWLK eq. dipole--CGC scattering Imaginary part of the Balitsky eq. Nonlinear terms represent coupling to the Pomeron. 3-quark--CGC scattering Complicated in the nonlinear (strong field) regime Reproduce the BKP equation in the weak-field limit