Parton Distributions from Large-Momentum Effective Theory Yong - PowerPoint PPT Presentation

Parton Distributions from Large-Momentum Effective Theory Yong Zhao Massachusetts Institute of Technology 2018 Santa Fe Jets and Heavy Flavor Workshop, Santa Fe, Jan 29-31, 2018 Santa Fe, NM 1 1/30/18

Outline ò 1. Difficulties of calculating parton distributions ò 2. Large momentum effective theory (LaMET) ò 3. Applications of the LaMET approach Santa Fe, NM 2 1/30/18

Outline ò 1. Difficulties of calculating parton distributions ò 2. Large momentum effective theory (LaMET) ò 3. Applications of the LaMET approach Santa Fe, NM 3 1/30/18

PDFs from global data analysis ò Currently our best knowledge of the PDFs comes from the global analysis of high-energy scattering data 1.0 1. Extensive experimental analysis motivates a u � val Q � 85 GeV d � val first principle calculation for comparison; 0.1 g 0.8 0.1 sea 2. First principle calculation might be able to 0.6 shed light on kinematic regions and flavor structures where experiments cannot constrain so 0.4 precisely; 0.2 3. The cost of improving calculations seems to be 0.0 much smaller than building larger experiments. 10 � 4 0.001 0.01 0.1 1 CT10 NNLO PDF, CTEQ-TEA group, 2014 Santa Fe, NM 4 1/30/18

Operator definition of PDF ò Definition of PDFs in QCD factorization theorems: d ξ − e - ixP + ξ − P ψ ( ξ − ) γ + U ( ξ − ,0) ∑ ∫ σ = f a ( x 1 ) ⊗ f b ( x 2 ) ⊗ σ ab q ( x , µ ) = ψ (0) P 4 π a , b ξ ± = ( t ± z )/ 2 ⎡ ⎤ ξ − U ( ξ − ,0) = P exp − ig ∫ d η − A + ( η − ) ⎢ ⎥ ⎣ ⎦ 0 • Gauge-invariant and boost-invariant light-cone correlation; • In the light-cone gauge A + =0, has a clear interpretation as parton number density, δ ( k + − xP + ) P ˆ n ( k + , k ⊥ ) P ∫ q ( x )~ dk + d 2 k ⊥ Santa Fe, NM 5 1/30/18

Lattice QCD is the only practical method to solve QCD nonperturbatively so far Parton model: • Minkowski space, real time • Emerges in the infinite momentum frame (IMF), or, the proton as seen by an observer moving at the speed of light (on the light-cone) ξ + = ( t +z )/ 2 = 0 e iS → e − S ∫ O = D ψ D ψ DA O ( x ) e − S Lattice QCD: • Euclidean space, imaginary time ( t=i τ ) • Nucleon static or at finite momentum • Cannot calculate time-dependent quantities generally due to difficulty in PDF not directly accessible analytical continuation in time from the lattice! Santa Fe, NM 6 1/30/18

PDF from the Euclidean Lattice ò Computation of PDF moments: n µ = (1,0,0, − 1)/ 2 " " ∫ µ 1 i µ 2 ! i µ n ψ (0) P x n − 1 q ( x , µ ) dx = a n ( µ ) = n µ 1 n µ 2 ! n µ n P ψ (0) γ dx D D • Moments are calculable as matrix elements of local gauge- invariant and frame-independent operators; • Fitting the PDF from the moments; • Operator mixing due to broken Lorentz symmetry limits computation for moments higher than 3. n ≤ 3 , W. Detmold et al., EPJ 2001, PRD 2002; D. Dolgov et al. (LHPC, TXL), PRD 2002; Santa Fe, NM 7 1/30/18

Proposals in recent years ò Restoration of rotational symmetry to calculate higher moments n>3 , Z. Davoudi and M. Savage, PRD 2012. ò Fictitious heavy-to-light current-current correlator D. Lin and W. Detmold, PRD 2006. ò OPE of the Compton amplitude A. J. Chambers et al. (QCDSF), PRL 2017 ò Direct computation of the physical hadronic tensor K.F. Liu (et al.), 1994, 1999, 1998, 2000, 2017. Santa Fe, NM 8 1/30/18

Proposals in recent years ò Large momentum effective theory (LaMET) X. Ji, PRL 2013; Sci.China Phys.Mech.Astron. 2014. Quasi-PDF (Large momentum factorization) Gradient flow method C. Monahan and K. Orginos, JHEP 2017. Pseudo-PDF (Small distance factorization) A. Radyushkin, PRD 2017; K. Orginos, A. Radyushkin, J. Karpie and S. Zafeiropoulos, 2017. ò Lattice cross section Y.-Q. Ma and J. Qiu, 2014, 2017. ò Factorization of Euclidean correlations in coordinate space V. M. Braun and D. Mueller, EPJ C 2008; G. S. Bali, V. M. Braun, A. Schaefer, et al., 2017. Santa Fe, NM 9 1/30/18

Outline ò 1. Difficulties of calculating parton distributions ò 2. Large momentum effective theory (LaMET) ò 3. Applications of the LaMET approach Santa Fe, NM 10 1/30/18

Parton model and the IMF ò Consider one starts from a static proton. The notion of parton does not exist as quarks and gluons are not free; ò Under a Lorentz boost along the z direction (dynamical transformation), the interacting quark or gluon can be transformed into an infinite number of particles, thus a longitudinal momentum density depends on the reference frame and is not physically meaningful; ò Nevertheless, when boosted to the IMF, all interaction effects are suppressed by powers of the infinite momentum, and the parton model emerges as the leading order approximation. Santa Fe, NM 11 1/30/18

Large momentum effective theory ò If one knows the nucleon wavefunction in the IMF, then all parton physics can be solved, but this is highly nontrivial and unknown in an interacting theory like QCD; ò The good thing is that QCD has asymptotic freedom. If there is a large scale, one can formulate an effective theory defined by that scale, and use this effective theory to match full QCD to physics below the scale; ò For example, the heavy-quark effective theory where the heavy quark mass sets the scale. ò In large-momentum effective theory, the nucleon momentum P z sets the scale. Santa Fe, NM 12 1/30/18

Large momentum effective theory Large momentum effective theory (LaMET) is a theory that expands in powers of 1/P z , where P z is the proton momentum ( Ji, PRL 2013, Sci. China Phys. Mech. Astro., 2014 ): 1. Construct a Euclidean quasi-observable Õ which can be calculated in lattice QCD; 2. The IMF limit of Õ is constructed to be a parton observable O at the operator level; 0 = 0 , U ( Λ ( P = ∞ )) − 1 ! P ≠ 0 = U ( Λ ( P )) P OU ( Λ ( P = ∞ )) = O P = ∞ ! O P = ∞ = P 0 = 0 O P 0 = 0 Recall that one does not know the proton wavefunction in the IMF! Santa Fe, NM 13 1/30/18

Large momentum effective theory 3. At finite P z , the matrix element of Õ depends on the cut-off Λ of the theory (if not renormalized) and generally P z , i.e., Õ(P z / Λ ) , while that of O depends on the renormalization scale μ (if in the MSbar scheme), i.e., O( μ ) ; O ( P z / Λ ) = P = P z ! O P = P z , ! O ( µ ) = P = any O P = any 4. Taking the P z —> ∞ ( P z >> Λ ) limit of Õ(P z / Λ ) is generally ill- defined due to the singularities in quantum field theory, O ( P z / Λ ) = ? " lim P z ≫ Λ Santa Fe, NM 14 1/30/18

Large momentum effective theory 5. But it can be related to O( μ ) through a factorization formula: O ( P z / Λ ) = Z ( P z / Λ , µ / Λ ) ⊗ O ( µ ) + c 2 2 + c 4 ! 4 + … P P z z ò P z is much larger than Λ QCD as well as the proton mass M to suppress the power corrections; ò One can regard as the O( μ ) effective theory observable, and Õ(P z / Λ ) as given by full QCD; ò O( μ ) and Õ(P z / Λ ) have the same infrared (IR) physics, and thus can be perturbatively matched to each other through the leading term. Santa Fe, NM 15 1/30/18

Large momentum effective theory 6. Õ(P z / Λ ) satisfies a “renormalization group equation” : γ ( α S ) = 1 d Z Z d ln P z ò The parton observable O( μ ) in the IMF is the “fixed point” of this RG equation; ò Physics near the “fixed point”, i.e., Õ(P z / Λ ) with different large P z , are related by the RG equation. Santa Fe, NM 16 1/30/18

How matching works P z >> Λ >> M , Λ QCD Λ >> P z >> M , Λ QCD Matching UV Perturbative QCD Non-perturbative QCD IR Õ (full QCD) O (LaMET) Santa Fe, NM 17 1/30/18

Large momentum effective theory ò Quasi-PDF: X. Ji, PRL 2013; Sci.China Phys.Mech.Astron. 2014. z µ = (0,0,0, z ) dz q ( x , P z , Λ = a − 1 ) = e ixP z z P ψ ( z ) γ z U ( z ,0) ∫ ψ (0) P ! 4 π ⎡ ⎤ z ∫ U ( z ,0) = P exp − ig dz ' A z ( z ') ⎢ ⎥ ⎣ ⎦ 0 Time-independent correlation along • ξ 0 = t ξ − the z direction, calculable in lattice ξ + QCD when P z << Λ ; √ 2 γ l Under an infinite Lorentz boost along • -l l ξ 3 = z the z direction ( P z >> Λ ), the spatial gauge link approaches the light-cone direction, and the quasi-PDF reduces −√ 2 γ l to the (light-cone) PDF. Santa Fe, NM 18 1/30/18

Large momentum effective theory ò The (renormalized) quasi PDF is related to the PDF through a factorization formula: Z +1 Λ 2 ✓ M 2 dy ✓ x y , ˜ P z , µ µ ◆ ◆ QCD q i ( x, P z , ˜ ˜ µ ) = | y | C ij q j ( y, µ ) + O , , P z P 2 P 2 � 1 z z ò They have the same IR divergences; ò C factor matches their UV difference, and can be calculated in perturbative QCD; ò Higher-twist corrections suppressed by powers of P z . Santa Fe, NM 19 1/30/18

Procedure of Systematic Calculation 1. Simulation of the quasi 3. Subtraction of higher PDF in lattice QCD twist corrections Z +1 Λ 2 ✓ M 2 ✓ x ◆ ◆ dy y , ˜ P z , µ µ QCD q i ( x, P z , ˜ ˜ µ ) = | y | C ij q j ( y, µ ) + O , , P z P 2 P 2 � 1 z z 2. Renormalization of the 4. Matching to the MSbar PDF. lattice quasi PDF, and then taking the continuum limit Santa Fe, NM 20 1/30/18

Outline ò 1. Difficulties of calculating parton distributions ò 2. Large momentum effective theory (LaMET) ò 3. Applications of the LaMET approach Santa Fe, NM 21 1/30/18