Quark Matter 2015 poster
1
v n from instabilities of the GLR equation Guillermo Gambini, Giorgio Torrieri Instituto de F´ ısica Gleb Wataghin. Universidade Estadual de Campinas. Phenomenological Introduction Anisotropic coefficients in heavy ion collisions v n dN d 3 p = dN dp T dy 1+2 X n v n (p T , y) cos (n (φ - ψ n )) ! have been a very well-studied observable. Its usual phenomenological interpretation is that of hydrodynamics, suggesting matter formed in heavy ion collisions is a “perfect fluid” but some deep puzzles have emerged... p T Same!!! Energy independent! Rapidity independent! = photon v2 Hadron v2 Rapidity independent! Driven by <pT> CMS PHENIX STAR BRAHMS PHOBOS CMS p T I v n Similar in photons and hadrons This is strange as photon and hadron evolution very different I v n is present in pA,perhaps pp Initial state “Glasma” models exist for small systems, but cannot describe larger AA systems (“many random antennae”). Scaling suggests v n of same origin in small and large systems I It seems v n (p T ) ∼ n F(p T ) where n is geometrical F(p T ) is universal across √ s,rapidity, system size . Integrated v n variations driven by hp T i∼ 1 S dN dy variations. Hydrodynamics doesn’t factorize this way! I This scaling is reminiscent of Bjorken scaling in parton distributions (PDFs , GPDs ), where G(x, Q 2 ) vary weakly with Q 2 ∼ p T , strongly with x I All these features would be natural if somehow G(x, Q 2 ) → G(x, Q 2 ,φ) while maintaining its QCD RG structure . Can v n be an initial condition which obeys QCD scaling rather than a manifestation of hydro ? But we need to justify “somehow”: G(x, Q 2 ) is a property of the hadron wavefunction, and its azimuthal symmetry is protected... but... Calculations The azimuthally symmetric GLR-MQ evolution equation is given by Q 2 ∂ ∂ Q ∂ xG(x, Q 2 ) ∂ ln(1/x) = α s N c π xG(x, Q 2 ) - α 2 s N c π 2C F S ⊥ 1 Q 2 [xG(x, Q 2 )] 2 Its solution (G 0 (x, Q 2 )) is known partially (for two limiting cases: Q s (x)/Q 1 and Q s (x)/Q 1). Therefore, we propose a model for the complete solution as G 0 (x, Q 2 )= x 2λ 2 1 - tanh Q - Q s (x) ζ + (Q s (x)) 2 Q 2 1 + tanh Q - Q s (x) ζ Calculations show ζ ∝ α s N c Λ QCD . The azimuthally asymmetric QLR-MQ equation reads xQ 2 ∂ ∂ Q + 1 Q ∂ ∂φ ∂ ∂ x [xG(x, Q 2 ,φ)] = α s N c π xG(x, Q 2 ,φ) - α 2 s N c π 2C F S ⊥ 1 Q 2 [xG(x, Q 2 ,φ)] 2 We propose a solution based on G 0 (x, Q 2 ): G(x, Q 2 ,φ)=G 0 (x, Q 2 ) 1+ ∞ X n=1 u n (x, Q 2 )cos(nφ + β n ) As a result of plugging this ansatz intothe asymmetric GLR equation we get a constrain for the β n cos(2β n )=0 and for the limiting case Q s (x)/Q 1 we get a set of equations: x ∂ u n (x, Q 2 ) ∂ x = -(2λ + 1)u n (x, Q 2 ) + α 2 s N c π 2C F S ⊥ 1 Q 2 x 2λ+1 1 n n-1 X k u k (x, Q 2 )u n-k (x, Q 2 )sin(β n - β k - β n-k ) +2 X k u k (x, Q 2 )u n+k (x, Q 2 )sin(β n + β k - β n+k ) The GLR equation and possible instabilities θ 2 θ 1 θ 1 θ 1 θ 2 F( , ) f(x,Q) f(x,Q) I Parton distribution functions G(x, Q 2 ) are the probability of finding a quark of longitudinal momentum xE in a nucleon striking a probe at energy E, where the characteristic momentum of the probe is Q. To leading order G(x, Q 2 ) → G(x) but loop corrections bring a Q 2 dependence. Generalized to GPDs ,local in x ⊥ . Azimuthal dependence forbidden by symmetry at leading orders, could arise α 2 s ( v 2 ) for “extended” probes. I In BFKL evolution it is well known that, as we decrease x, the parton density grows and diverges at a rate where the total proton-proton cross-section would eventually break the Froissart bound required by unitarity, σ tot ∼ ln 2 s. A natural way to resolve this is to assume that, at some critical value x, Q → Q s (x) the density of partons becomes so large that their wave functions start to overlap , i.e. they interact. I Gribov, Levin and Ryskin considered a a quadratic correction to the BFKL equation, which slows down the growth of the gluon distribution until saturation of parton distributions occurs The full GLR equation is a 2+1 diffusion equation (x, Q,φ ) , azimuthal symmetry makes it 1+1 (x, Q ) I Saturation and the Color-Glass Condensate formalism are thought to arise when non-linear parton evolution is implemented within a renormalization group approach, with the separation in | ln(1/x)|→ 0 between target/projectile and “wee” gluons x → 0 providing a scale variable. But RG evolution could contain IR operators breaking the fundamental symmetries of the Lagrangian! I 2+1 diffusion equations symmetric solutions can be unstable against perturbations. 2 dimensional turbulence is characterized by inverse cascades , with high frequency low amplitude perturbations growing in low-frequency high amplitude ones (avoiding the “many antennae” limit of Glasma based models ). Thus, for GPDs small transversely local azimuthal perturbations at the target-projectile could seed global breaking of azimuthal symmetry for low x → G(x, Q x , Q y ) In a RG approach the EFT lagrangian , not just boundary conditions, asymmetric . tiny anisotropies at x →±1 grow with Q . v n , with the right scaling,as initial state Preliminary results As an ansatz we propose u n (x → 0, Q 2 )= δ n,2 ∞ X k=0 A k (Bx C ) k k! Q D-2k And solve for A k from Eq. 5. Afterwards integrate Eq. 5. in x direction. Qualitatively reasonable features emerge: Perturbations dependence on Q 2 practically decoupled from x, linear dependence on , near-decoupling of Fourier modes. A detailed phenomenological study, including factorization , dynamics and fragmentation is forthcoming, but this approach could reproduce scaling seen in the data. v n /ridges at the EIC eA collisions? http://www-i6.informatik.rwth-aachen.de [email protected]
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Transcript of Quark Matter 2015 poster
vn from instabilities of the GLR equationGuillermo Gambini, Giorgio Torrieri
Instituto de F́ısica Gleb Wataghin. Universidade Estadual de Campinas.
Phenomenological Introduction
Anisotropic coefficients in heavy ion collisions vn
dN
d3p=
dN
dpTdy
(1 + 2
∑n
vn(pT, y) cos (n (φ− ψn))
)have been a very well-studied observable. Its usual phenomenologicalinterpretation is that of hydrodynamics, suggesting matter formed in heavyion collisions is a “perfect fluid” but some deep puzzles have emerged...
pT
Same!!!
Energy independent!
Rapidity independent!
= photon v2Hadron v2Rapidity independent!
Driven by <pT>
CMS
PHENIX
STAR
BRAHMS
PHOBOS
CMS
������������
������������
������������������
������������������
pT
I vn Similar in photons and hadrons This is strange as photon and hadronevolution very different
I vn is present in pA,perhaps pp Initial state “Glasma” models exist for smallsystems, but cannot describe larger AA systems (“many randomantennae”). Scaling suggests vn of same origin in small and large systems
I It seems vn(pT) ∼ εnF(pT) where εn is geometrical F(pT) is universalacross
√s,rapidity, system size . Integrated vn variations driven by
〈pT〉 ∼ 1S
dNdy
variations. Hydrodynamics doesn’t factorize this way!
I This scaling is reminiscent of Bjorken scaling in parton distributions (PDFs ,
GPDs ), where G(x,Q2) vary weakly with Q2 ∼ pT, strongly with xI All these features would be natural if somehow G(x,Q2)→ G(x,Q2, φ)
while maintaining its QCD RG structure . Can vn be an initial conditionwhich obeys QCD scaling rather than a manifestation of hydro ? But weneed to justify “somehow”: G(x,Q2) is a property of the hadronwavefunction, and its azimuthal symmetry is protected... but...
Calculations
The azimuthally symmetric GLR-MQ evolution equation is given by
Q
2
∂
∂Q
∂xG(x,Q2)
∂ln(1/x)=αsNc
πxG(x,Q2)−
α2sNcπ
2CFS⊥
1
Q2[xG(x,Q2)]2
Its solution (G0(x,Q2)) is known partially (for two limiting cases:Qs(x)/Q� 1 and Qs(x)/Q� 1). Therefore, we propose a model forthe complete solution as
G0(x,Q2) =x2λ
2
[1− tanh
Q− Qs(x)
ζ+
(Qs(x))2
Q2
(1 + tanh
Q− Qs(x)
ζ
)]Calculations show ζ ∝ αs
NcΛQCD. The azimuthally asymmetric QLR-MQ equation reads
xQ
2
(∂
∂Q+
1
Q
∂
∂φ
)∂
∂x[xG(x,Q2, φ)] =
αsNc
πxG(x,Q2, φ)−
α2sNcπ
2CFS⊥
1
Q2[xG(x,Q2, φ)]2
We propose a solution based on G0(x,Q2):
G(x,Q2, φ) = G0(x,Q2)
(1 +
∞∑n=1
un(x,Q2)cos(nφ + βn)
)As a result of plugging this ansatz intothe asymmetric GLR equation we get a constrain forthe βn cos(2βn) = 0 and for the limiting case Qs(x)/Q� 1 we get a set of equations:
x∂un(x,Q2)
∂x= −(2λ + 1)un(x,Q2)
+α2
sNcπ
2CFS⊥
1
Q2x2λ+11
n
[ n−1∑k
uk(x,Q2)un−k(x,Q2)sin(βn − βk − βn−k)
+ 2∑
k
uk(x,Q2)un+k(x,Q2)sin(βn + βk − βn+k)
]
The GLR equation and possible instabilities
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θ2
θ1θ
1
θ1
θ2
F( , )
f(x,Q)
f(x,Q)
I Parton distribution functions G(x,Q2) are the probability of finding a quarkof longitudinal momentum xE in a nucleon striking a probe at energy E,where the characteristic momentum of the probe is Q. To leading orderG(x,Q2)→ G(x) but loop corrections bring a Q2 dependence.Generalized to GPDs,local in x⊥ . Azimuthal dependence forbidden bysymmetry at leading orders, could arise α2
sε (� v2) for “extended” probes.I In BFKL evolution it is well known that, as we decrease x, the parton
density grows and diverges at a rate where the total proton-protoncross-section would eventually break the Froissart bound required byunitarity, σtot ∼ ln2 s. A natural way to resolve this is to assume that, atsome critical value x,Q→ Qs(x) the density of partons becomes so largethat their wave functions start to overlap , i.e. they interact.
I Gribov, Levin and Ryskin considered a a quadratic correction to theBFKL equation, which slows down the growth of the gluon distribution untilsaturation of parton distributions occurs The full GLR equation is a 2+1diffusion equation (x,Q, φ ) , azimuthal symmetry makes it 1+1 (x,Q )
I Saturation and the Color-Glass Condensate formalism are thought to arisewhen non-linear parton evolution is implemented within a renormalizationgroup approach, with the separation in | ln(1/x)| → 0 betweentarget/projectile and “wee” gluons x→ 0 providing a scale variable. ButRG evolution could contain IR operators breaking the fundamentalsymmetries of the Lagrangian!
I 2+1 diffusion equations symmetric solutions can be unstable againstperturbations. 2 dimensional turbulence is characterized byinverse cascades, with high frequency low amplitude perturbations growingin low-frequency high amplitude ones (avoiding the “many antennae” limitof Glasma based models ). Thus, for GPDs small transversely localazimuthal perturbations at the target-projectile could seed global breakingof azimuthal symmetry for low x→ G(x,Qx,Qy) In a RG approach theEFT lagrangian, not just boundary conditions, asymmetric. tiny anisotropiesat x→ ±1 grow with Q . vn, with the right scaling,as initial state
Preliminary results
As an ansatz we propose
un(x→ 0,Q2) = δn,2
∞∑k=0
Ak(BxC)k
k!QD−2k
And solve for Ak from Eq. 5. Afterwards integrate Eq. 5. in x direction.
Qualitatively reasonable features emerge: Perturbations dependence on Q2
practically decoupled from x, linear dependence on ε, near-decoupling ofFourier modes. A detailed phenomenological study, including factorization,dynamics and fragmentation is forthcoming, but this approach couldreproduce scaling seen in the data. vn/ridges at the EIC eA collisions?
Created with LATEXbeamerposter http://www-i6.informatik.rwth-aachen.de/~dreuw/latexbeamerposter.php
http://www-i6.informatik.rwth-aachen.de [email protected]
