P Large Momentum Effective Field Theory (LaMET) and
P Large Momentum Effective Field Theory (LaMET) and Parton Physics XIANGDONG JI UNIVERSITY OF MARYLAND/SHANGHAI JIAO TONG U JEFFERSON LAB, NOV. 20, 2017
Outline Parton physics as fixed point of the momentum RG evolution Calculating parton physics at the fixed point Large-momentum effective field theory (LaMET) Practical considerations & applications Summary
Parton physics as fixed point of the momentum RG evolution X. JI, SCI.CHINA PHYS.MECH.ASTRON. 57 (2014) 1407-1412 Center-of-Mass and Internal Motions : nonrelativistic case In non-relativistic systems, the COM
motion is decoupled from the internal motion in the sense that the internal dynamics is independent of the COM momentum. H = Hcom + Hint Hint is independent of P (COM momentum) and R (COM position) Wave function of the H-atom is independent of its speed.
Galilei transformation Center-of-Mass and Internal Motions : Relativistic case In relativistic theory, the internal dynamics DOES depend on the total momentum of the system. The internal wave function of a system is framedependent. Wave functions in the different frame is related by Lorentz boost
|p = U((p)) |p=0>, is related to the boost Ki Bound state properties do depend on the COM momentum p. Momentum distribution of constituents Consider the momentum distribution of the constituent
In relativistic bound state, this becomes a COM momentum-dependent quantity, This is in fact true for any generic properties measured by operator O, unless it commutes with the boost operator Ki, [O,Ki]= 0, i.e. a scalar. Computing the momentum dependence computing the momentum dependence of an
observable O(p) is in principle possible through commutation relation, [O, Ki] = ... However, in relativistic theories, the boost operator K is highly non-trivial, it is interaction-dependent, just like the Hamiltonian. Thus, computing the p-dependence of an observable is just as difficult as studying the dynamical evolution.
Asymptotic freedom (AF) and large momentum case QCD is an asymptotic-free theory. As such, once there is a large scale in the problem, such a scale dependence can be studied in pert. theory. One important example is the heavy-quark physics when the heavy-quark mass mQ is large. Thus, it is expected that AF allows to compute the
large p-dependence in pert. theory (as in HQET) where a is some UV cut-off. Renormalization group equation When power suppressions can be ignored, one has And P-dependence become computable in pert. theory, Define anomalous dimension: We have the renormalization group eq.
Fixed point and parton physics The RG equation has a fixed point at P= This is the infinite momentum limit in which the partons were first introduced by Feynman. Thus the parton physics corresponds to frame-dependent physical obervables at the fixed point of the frame transformations. In principle, there is nothing new here. However, as far as I know, this is the first time when this
notation has been expressed in QCD equations. More importantly, this teaches one how to calculate the parton physics in QCD Parton physics: a unique way to describe the hadron structure The hadron structure is better represented by partons: parton amplitudes, parton correlations, parton distributions, etc
The great advantage is that partons separate the structure of hadrons from that of the vacuum in which all partons have + momentum zero. All high-energy scattering data can be used to learn about parton physics Calculating parton physics at the fixed point (p=)
Traditional approache to parton physics Usually the parton physics is uncovered in perturbation theory by taking the infinite moment limit of Feynman diagrams The rules can be simplified into the so-called lightfront perturbation theory. These rules can also be generated from light-front quantization of Dirac (1949). Parton as light-front
correlations Consider the Lorentz transformation: |p = U((p)) |M(p=0)> Thus, the matrix element in a large p state can be expressed as when p is large, the transformation turn all spatial correlations into light-front correlations. Parton distribution
Can be formulated in as the matrix elements of the boost-invariant light-front correlations . where the light-front coordinates, Partons as light-front correlations Quark and gluon fields are distributed along the light-cone direction
- 0 + 3 Parton physics involves time-dependent dynamics. This is very general, parton physics = light cone physics of
bound states. Light-front quantization Choose a new system of coordinates with as the new time (light-cone time) and as the new space (light-cone space); Also choose the lightcone gauge (Dirac, 1949) Light-cone correlation becomes equal-time correlation. Parton physics is manifest through light-cone
quantization (LCQ) Status of Hamiltonian approach In early 1990s, Ken Wilson and Stan Brodsky among others became strong proponents for LCQ as a nonperturbative approach to solve QCD, and thus a way to calculate parton physics. Despite many years of efforts, light-front quantization has not yielded a systematic approach to calculating non-perturbative parton physics.
Not approximate calculations of parton distributions using QCD LF Hamiltonian have appeared. Monte carlo simulations Real-time correlations in a dynamical system are usually NPC problem, and cannot be solved using known numerical approaches. However, there are spatial classes of problems which can be reduced to P problem when
conditions are appropriate. It is not know how to do this directly with lightfront quantization. Parton physics on lattice? One can form local moments to get rid of the timedependence matrix elements of local operators However, one can only calculate lowest few moments in practice. Higher moments quickly become noisy. Many other parton properties cannot be not
related to local operators, e. g. TMDs, jet functions etc. One must find alternative approach. Large-momentum effective theory (LaMET) Main goal Calculating parton physics by working away from the fixed point (finite p) !
This be done through a clever use of QCD perturbation theory Effective Field Theory In QFT, there is also the UV cut-off, which is often taking to INFINITY as well. Two different limits The first limit : fixed-point parton physics, first, followed by
We can recover the first limit by taking the second limit: first, followed by large p. Example fixed-point limit is just
Anatoly with HQET In physical processes involving heavy quarks, it is often useful to take first before UV cutoff is applied, although the actual is finite and UV cutoff must be larger than in physical case. This result in a HQET in which the symmetric properties are manifest. In this sense, the QFT partons are an effective field theory description of high-energy scattering.
A Euclidean quasidistribution Consider space correlation in a large momentum P in the z-direction. 0 0 Quark fields separated along the z-direction
The gauge-link along the z-direction The matrix element depends on the momentum P. Z 3 The second limit In taking the second limit, the Q-PDFs will have log dependence on the momentum of the nucleon.
Meanwhile, the parton density has support outside of 0 first After renormalizing all the UV divergences, one has the standard quark distribution!
One can prove this using the standard OPE One can also see this by writing |P = U((p)) |p=0> and applying the boost operator on the gauge link. 0 +
3 Finite but large P The distribution at a finite but large P shall be calculable in lattice QCD. Since it differs from the standard PDF by simply an infinite P limit, it shall have the same infrared (collinear) physics. It shall be related to the standard PDF by a
matching factor which is perturbatively calculable. One-loop matching Properties It was done in cut-off regulator so that the result will be similar for lattice perturbation theory. It does not vanish outside 0
divergences appear in 0
Where the matching factor is perturbative. All order proof (Qiu and Ma) Consider A3= 0 gauge (no subtle spurious singularity in this case). There are no soft-singularities in the quasidistribution. All collinear singularities are in ladder diagrams, happen when P large and gluons are collinear with the P. This singularity is the same as that of the
light-distribution. General recipe LaMET is a theory designed to make ab initio computation of light-cone (light-front, parton) physics on a Euclidean lattice! The recipe: Deconstructing (unboosting) the light-cone operator Lattice computation with large momentum Effective field theory interpretation
Step 1: Deconstructing the light-cone operators Consider parton physics described by light-cone operators, o. Construct a Euclidean quasi-operator O such that in the IMF limit, O becomes a lightcone (light-front, parton) operator o. There are many operators leading to the same light-cone operator.
Step 2: Lattice calculations Compute the matrix element of Euclidean operator O on a lattice in a nucleon state with large momentum P. The matrix element may depend on the momentum of the hadron P, O(P,a), and also on the details of the lattice actions (UV specifics). Step 3: Extracting the
light-cone physics from the lattice ME Extract light-front physics o() from O(P,a) at large P through a EFT matching condition or factorization theorem, Where Z is perturbatively calculable. Infrared factorization Infrared physics of O(P,a) is entirely captured by the parton physics o(). In particular, it contains all
the collinear divergence when P gets large. Z contains all the lattice artifact (scheme dependence), but only depends on the UV physics, can be calculated in perturbation theory Why factorization exists? When taking first, before a UV regularization imposed, one recovers from O, the light-cone operator. [This is done through construction.]
The lattice matrix element is obtained at large P, with UV regularization (lattice cut-off) imposed first. Universality class Just like the same parton distribution can be extracted from different hard scattering processes, the same light-cone physics can be extracted from different lattice operators. All operators that yield the same light-cone
physics form a universality class. Universality class allows one exploring different operator O so that a result at finite P can be as close to that at large P as possible. Some remarks Parton physics on lattice Gluon helicity distribution and total
gluon spin Generalized parton distributions Transverse-momentum dependent parton distributions Light-cone wave functions Higher twist observables Practical considerations For a fixed x, large Pz means large kz, thus, as Pz gets
larger, the valence quark distribution in the z-direction get Lorentz contracted, z ~1/kz. Thus one needs increasing resolution in the z-direction for a large-momentum nucleon. Roughly speaking: aL/aT ~ =4 x,y
Large P z Power divergence problem I How does the LaMET theory cure the power divergence problem in moments calculation Local
vs non-local formulation is log divergent at large x However, if one expands One has successive power divergences. Power divergence problem II There are power divergences in qpdf formulation.
It can be shown all power divergences in quark sector can be absorbed into self-energy renormalization In the case of gluon, proof is a bit involved because one needs to find a gauge-invariant regularization scheme that preserves power divergence. Could be done on lattice. Gluon spin laMET formulation provides the first practical
calculation of the gluon helicity contribution to the spin of the proton. Relation to other formulations There are other attempts to relate parton physics to lattice calculations. But either has not been proven successful, or more or less equivalent. large momentum is essential Ioffe-time formalism (A. Radyushkin), part that
rigorous is entirely equivalent. Summary Parton physics demands new ideas to solve nonperturbative QCD New progress recently allows one calculate parton physics using Euclidean lattice. Thus interesting properties like GPDs and TMDs can now be calculated. With 12 GeV upgrade at Jlab and future EIC, there is an exciting era of precision comparison of parton physics between experimental data and lattice QCD
calculations. Papers X. Ji, Parton Physics on a Euclidean Lattice, Phys. Rev. Lett. 110, 262002 (2013). X. Ji, Sci.China Phys.Mech.Astron. 57 (2014) 1407-1412 X. Ji, J. Zhang, and Y. Zhao, Physics of Gluon Helicity Contribution to Proton Spin, Phys. Rev. Lett. 111, 112002 (2013). Y. Hatta, X. Ji, and Y. Zhao, Gluon Helicity G from a Universality Class of Operators on a Lattice, arXiv:1310.4263v1
H. W. Lin, Chen, Cohen and Ji, arXiv:1402.1462 X. Ji, J. Zhang, X. Xiong, Y. Zhao, submitted to JHEP Collaborators Yong Zhao, postdoc at MIT Xiaonu Xiong, postdoc in Bonn Jianhui Zhang, postdoc at Regensburg Huey-wen Lin, Assistant prof at MSU Jiunn-Wei Chen, Prof. at National Taiwan U. Feng Yuan, Senior research scientist at LBL
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