QCD and statistical physics Stéphane Munier CPHT, École Polytechnique, CNRS Palaiseau, France Florence, February 1
High energy QCD hadron 1 hadron 2 b = impact parameter (proton, nucleus, photon...) Y = relative rapidity r : transverse size of the projectile r 0 : transverse size of the target k : transverse energy scale of the projectile k 0 : transverse energy scale of the target k k A Y ,r = ∫ d 2 b A b ,Y ,r = elastic amplitude k A b ,Y ,r = fixed impact parameter amplitude ≤ 1 (High) energy dependence of QCD amplitudes?
The Balitsky equation Balitsky (1996) Rapidity evolution of the scattering amplitude: = s N c T = 1 Tr U U , 〈 T 〉= A BFKL kernel; acts on transverse coordinates N c Infinite hierarchy, more ∂ Y A =∗ A −〈 T T 〉 complex operators at each step ∂ Y 〈 T T 〉=∗〈 T T 〉−〈 T T T 〉 2 ∗〈 Tr U U U U U U 〉 source terms A ''mean field'' approximation gives the Balitsky-Kovchegov (simpler) equation: 〈 T T 〉=〈 T 〉〈 T 〉= A ⋅ A ⇒ ∂ Y A =∗ A − A ⋅ A Balitsky (1996); Kovchegov (1999) Understand and solve the full high energy evolution equations! See also JIMWLK and further developments Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner
High energy QCD in the field-theory formulation Balitsky (1996) Inside the Balitsky equation: + ... Effective formulation: "Pomeron" diagrams Pomeron = = + + + + ... A ∂ Y A =∗ A ?
Alternative philosophy Breakthrough by Mueller and Shoshi, 3 years ago: "Small x physics beyond the Kovchegov equation" This talk: Subsequent interpretation of their calculation in the light of some models well-known in statistical mechanics (namely reaction-diffusion processes ). go beyond the Mueller-Shoshi results simple picture, based on the parton model connects the QCD problem to more general physics and mathematics Instead of a direct approach, identify the universality class from the physics of the parton model, then apply general results!
Outline High energy QCD and reaction-diffusion Field theory versus statistical methods for a simple particle model Statistical methods and application to QCD
How a high rapidity hadron looks observer r 0 Y 0 = 0 Y 1 Y 0 rapidity in the frame of the observer
How a high rapidity hadron looks k ' ~ k k 1 Y ~ 1 k ? n ≤ N Parton saturation: 1 2 n k T k ~ s N k Number of partons n unitarity: T r ≤ 1 ⇒ N = 1 2 Y lnQ s 2 s 1 2 lnk 2 2 n − n N n ∂ Y n =−∂ lnk 2 BFKL ~ ∂ lnk n n Noise term due to discreteness 2
How a high rapidity hadron looks k ' ~ k k 1 Y ~ 1 k ? n ≤ N Parton saturation: 1 2 n k T k ~ s 1 k unitarity: T r ≤ 1 ⇒ N = 1 1-Fock state amplitude T 2 Y lnQ s 2 s A =〈 T 〉 Physical amplitude: 2 s 2 lnk 2 s T ∂ Y T =−∂ lnk 2 T − T 2 BFKL ~ ∂ lnk T T Noise term due to discreteness 2
How a high rapidity hadron looks k ' ~ k k 1 t ~ Y ~ 1 k ? n ≤ N Parton saturation: 1 2 n k T k ~ s 1 k unitarity: T r ≤ 1 ⇒ N = 1 1-Fock state amplitude T 2 Y ~ X t lnQ s 2 s A =〈 T 〉 Physical amplitude: 2 s 2 ~ x lnk 2 T ∂ t T =−∂ x T − T N 2 T T branching diffusion ~ ∂ x Noise term due to discreteness
Reaction-diffusion t t t proba t proba t n x ,t N proba p x proba p proba 1 − t − t n t N − 2p n x ,t n x ,t t T = n N N N 1 x x 2 x,t t T T x,t t = T x ,t p T x x ,t T x − x,t − 2T x ,t t T x ,t − t T N x,t t T 2 ∂ t T = −∂ x T − T N 2 2 2 T T − T ∂ t T =∂ x N T 1 − T Prototype equation: sFKPP equation Fisher; Kolmogorov, Petrovsky, Piscunov (1937)
Dictionary Reaction-diffusion High energy QCD 2 / k 0 2 ln k Position x Y Time t Particle density T Partonic amplitude T 1 Maximum/equilibrium 2 s number of particles N 2 / k 0 2 ln Q s Position of the wave front X Saturation scale sFKPP equation QCD evolution in the parton model 2 2 2 s T ∂ Y T =−∂ lnk 2 T − T 2 T T − T ∂ t T =∂ x N T 1 − T
Outline High energy QCD and reaction-diffusion Field theory versus statistical methods for a simple particle model Statistical methods and application to QCD
Simple particle model t t t t t t k particles added: k particles split, n-k do not split proba t proba 1 − t n t n t t = n t k t t proba P n k = k t n k 1 − t n − k = k −〈 k 〉 1 〈 k 〉= n t define 〈〉= 0 t 2 〉= 1 〈 2 =〈 k −〈 k 〉 2 〉= n t k t 1 such that ∑ t t ~± 1 〈 k 〉 dn dt = n n n t t = n t t n t n t t t t 0 n What is, in average , the number of particles at time t? d 〈 n 〉 〈 n t 〉 obtained by solving the trivial equation dt =〈 n 〉 4 t e 1 t 1 2 3
Simple particle model t t t t t t k 2 k 1 particles added, particles removed proba t proba t n t N proba 1 − t − t n t N n t t = n t k 1 t t − k 2 t t n t proba P n k 1, k 2 = k 1 k 2 t k 1 t n t N 1 − t − t n t N k 2 n − k 1 − k 2 n N n 1 n N 2 〈〉= 0 dn dt = n − n 2 〉= 1 〈 dt n t 〈 n t 〉 is not obtained by solving a trivial equation! N d 〈 n 〉 dt =〈 n 〉− 1 2 〉 N 〈 n 2 〉 d 〈 n ...infinite hierarchy! = dt similar to the Balitsky equation in 0D 2 d 〈 n 〉 dt =〈 n 〉−〈 n 〉 4 Mean field approximation: N 1 similar to the Balitsky-Kovchegov equation t 1 2 3
Field-theoretical formulation Doi (1975) Mueller (1995) Statistical formulation: Shoshi, Xiao (2005) Evolution of Poissonian states evolution of fixed particle number states n P z n = z − z n! e n t n t z t exp − ∫ dt [ z N zzz zzzz ] dt − 1 z − zzz 1 d 〈 n t 〉=〈 z t 〉 Path integral average, with weight + + + + ... 〈 n t 〉= 6 − 24 − 2 3t 4t + ... t 2 e 3 e e 2t N e N N 〈 n t 〉= N 1 − Ne − Nexp − t b ∞ db − t ∫ 1 b e After Borel resummation: 0
Statistical method proba t N n 1 n N 2 dn dt = n − n proba t n t proba 1 − t − t n t N N n N = 5000 t
Statistical method proba t N n 1 n N 2 dn dt = n − n proba t n t proba 1 − t − t n t N N n N = 5000 2 dn dt = n − n N ? t
Statistical method proba t N n 1 n N 2 dn dt = n − n proba t n t proba 1 − t − t n t N N n N = 5000 2 dn dt = n − n N ? dn dt = n n
Statistical method proba t N n 1 n N 2 dn dt = n − n proba t n t proba 1 − t − t n t N N n N = 5000 2 dn dt = n − n N 〈 n 〉 2 mean field solution d 〈 n 〉 dt =〈 n 〉−〈 n 〉 N 1 ≪ n ≪ N t dn dt = n n t
Statistical method proba t N n 1 n N 2 dn dt = n − n proba t n t proba 1 − t − t n t N N n N = 5000 2 dn dt = n − n N 1 ≪ n ≪ N t dn dt = n n t 2 dn dt = n − n Solution of the mean-field equation N dn dt = n n Solution of for t n t = n with the initial condition Field-theoretical result: ⇔ N ∞ 〈 n t 〉= ∫ − t − nexp − t dt ne 〈 n t 〉= N 1 − Ne − Nexp − t b ∞ db 1 N 0 − t ∫ 1 b e − t − t n e 0 + Well-established systematics + Simple, intuitive _ _ Complex, abstract No systematics
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