Lattice DIS Structure Functions

TL;DR

This paper reports on lattice QCD calculations of renormalization constants and mixing coefficients for DIS structure function moments, utilizing improved actions and automated symbolic computation to enhance accuracy and efficiency.

Contribution

It introduces a novel automated computational framework for calculating operator renormalization in lattice QCD with improved actions.

Findings

Calculated renormalization constants for DIS moments

Developed automated symbolic computation codes

Reduced systematic errors with improved lattice actions

Abstract

We present the computation, in lattice QCD, of the renormalization constants and mixing coefficients of operators that measure the first two moments of DIS Structure Functions. These calculations have been performed using the Sheikholeslami-Wohlert O(a) improved ``clover'' action, which is known to reduce the systematic error associated with the finiteness of the lattice spacing a. Due to the complexities of the computations, we have developed, using the computer languages Schoonschip and Form, general codes that are able to automatically carry out all the analytic lattice manipulations.

Equations

$$W_{\mu \nu }=\frac{1}{2\pi }\int d^{4}x{e}^{iqx}<p|J_{\mu }(x)J_{\nu }(0)|p>.$$

$$<p|O_{{\mu }_1\cdots {\mu }_N}^{(N)}|p>=A_N(\mu )p_{{\mu }_1}\cdots p_{{\mu }_N}+\text{mbox}traceterms,$$

$$\int d\text{mbox}x\text{mbox}{x}^{N-1}{\mathcal{F}}_k(q^2,\text{mbox}x)=C_N(q^2/{\mu }^2)A_N(\mu ),$$

$$O_{\mu \nu }^q=\frac{1}{4}\overline{\psi }{\gamma }_{\{\mu }{\overleftrightarrow{D}}_{\nu \}}\psi ⟶<\text{mbox}x{>}_q$$

$$O_{\mu \nu }^g=\sum _{\rho }\text{mbox}Tr\left[F_{\{\mu \rho }{F}_{\rho \nu \}}\right]⟶<\text{mbox}x{>}_g$$

$$O_{\mu \nu \tau }^q=\frac{1}{8}\overline{\psi }{\gamma }_{\{\mu }{\overleftrightarrow{D}}_{\nu }{\overleftrightarrow{D}}_{\tau \}}\psi ⟶<\text{mbox}{x}^2{>}_q.$$

$$S_W^f$$

$$\Delta S_I^f=-i{g}_0{a}^4\sum _{n,\mu \nu }\frac{r}{4a}{\overline{\psi }}_n{\sigma }_{\mu \nu }{F}_{n,\mu \nu }{\psi }_n,$$

$$F_{n,\mu \nu }=\frac{1}{4}\sum _{\mu \nu =\pm }{P}_{n,\mu \nu }=\frac{1}{8i{g}_0{a}^2}\sum _{\mu \nu =\pm }(U_{n,\mu \nu }-U_{n,\mu \nu }^{+}),$$

$$(V{)}_{\rho }^{bc}=-g_0(t^A{)}_{bc}\left[r\sin \frac{a(k+k^{\prime }{)}_{\rho }}{2}+i{\gamma }_{\rho }\cos \frac{a(k+k^{\prime }{)}_{\rho }}{2}\right],$$

$$(V^I{)}_{\rho }^{bc}=-g_0\frac{r}{2}(t^A{)}_{bc}\cos \frac{a(k-k^{\prime }{)}_{\rho }}{2}\sum _{\lambda }{\sigma }_{\rho \lambda }\sin a(k-k^{\prime }{)}_{\lambda }.$$

$$\psi ⟶\left(1-\frac{ar}{2}\overrightarrow{/D}\right)\psi \text{mbox},\overline{\psi }⟶\overline{\psi }\left(1+\frac{ar}{2}\overleftarrow{/D}\right).$$

$${⟨p\left|{\hat{\mathcal{O}}}_L\right|p^{\prime }⟩}_{MonteCarlo}=a^d\left[{⟨p\left|\hat{\mathcal{O}}\right|p^{\prime }⟩}_{phys}+O(a/\log a)\right].$$

$${\hat{O}}^l(\mu )=Z_{lk}(\mu a)O^k(a).$$

$$\begin{array}[]{c}\widehat{O}^{q}=Z_{qq}O^{q}+Z_{qg}O^{g}\\ \widehat{O}^{g}=Z_{gq}O^{q}+Z_{gg}O^{g},\end{array}$$

$$\left(\begin{array}[]{cc}<q|O^{q}|q>&<g,\sigma|O^{q}|g,\sigma>\\ <q|O^{g}|q>&<g,\sigma|O^{g}|g,\sigma>\end{array}\right)$$

$$\begin{array}[]{rll}\widehat{O}^{q}_{\{12\}}=&1.027~{}O^{q}_{\{12\}}&\mbox{Wilson}\\ (\widehat{O}^{q}_{\{12\}})^{\rm{I}}=&1.134~{}(O^{q}_{\{12\}})^{\rm{I}}&\mbox{Improved}\\ &&\\ \widehat{O}^{q}_{\{123\}}=&1.160~{}O^{q}_{\{123\}}&\mbox{Wilson}\\ (\widehat{O}^{q}_{\{123\}})^{\rm{I}}=&1.252~{}(O^{q}_{\{123\}})^{\rm{I}}&\mbox{Improved}\\ &&\\ \widehat{O}^{q}_{\rm{DIS}}=&\frac{1}{3}\left[1.184~{}O_{A}+1.156~{}O_{B}\right]&\mbox{Wilson}\\ (\widehat{O}^{q}_{\rm{DIS}})^{\rm{I}}=&\frac{1}{3}\left[1.331~{}(O_{A})^{I}+1.187~{}(O_{B})^{\rm{I}}\right]&\mbox{Improved},\end{array}$$

Full text

SHEP 96/16 July 4, 1996

LATTICE DIS STRUCTURE FUNCTIONS

aaaTalk presented at the DIS96 International Workshop on “Deep Inelastic Scattering and Related Phenomena”, Rome, April 15-19, 1996

STEFANO CAPITANI bbbFrom October 1996 at the Desy Theory Group

We present the computation, in lattice QCD, of the renormalization constants and mixing coefficients of operators that measure the first two moments of DIS Structure Functions. These calculations have been performed using the Sheikholeslami-Wohlert $O(a)$ improved “clover” action, which is known to reduce the systematic error associated with the finiteness of the lattice spacing $a$. Due to the complexities of the computations, we have developed, using the computer languages Schoonschip and Form, general codes that are able to automatically carry out all the analytic lattice manipulations.

1 Introduction

The computation on the lattice of renormalization constants is a necessary ingredient for the connection of lattice operators and matrix elements to their continuum counterparts and the extraction of physical quantities from the numbers obtained in Monte Carlo simulations. We have computed, in lattice 1-loop perturbation theory using the Sheikholeslami-Wohlert $O(a)$ improved action, the renormalization constants and mixing coefficients of the quark and gluon operators of rank two \!{}^{{\bf?}}$${}^{\!-\,}$$\!{}^{{\bf?}} and of the quark operators of rank three \!{}^{{\bf?}}$${}^{\!-\,}$$\!{}^{{\bf?}} that measure the first two moments of DIS Structure Functions. The use of improvement \!{}^{{\bf?}}$${}^{\!-\,}$$\!{}^{{\bf?}} allows us to reduce the systematic error associated with the finiteness of the lattice spacing $a$. In the particular formulation we have used this error is lowered, for on-shell quantities, from $O(a)$ to $O(a/\log a)$.

2 Moments of Structure Functions

The hadronic tensor $W_{\mu\nu}$, from which the Structure Functions can be defined in a well-known way, is written in terms of the hadronic currents as

[TABLE]

By means of a Wilson OPE expansion near the light-like region of the kind $A(x)B(0)\sim\sum_{N,i}c_{N,i}(x^{2})\,x^{\mu_{1}}\cdots x^{\mu_{N}}O^{(N,i)}_{\mu_{1}\cdots\mu_{N}}(0)$ , it can be expressed in terms of a set of symmetric and traceless operators $O^{(N,i)}_{\mu_{1}\cdots\mu_{N}}$ with vanishing vacuum expectation values, of which the dominant ones have twist two. The matrix elements of these operators have the general form

[TABLE]

and are related to the moments of the Structure Functions by the formula

[TABLE]

where ${\cal F}_{1}=2F_{1},{\cal F}_{2}=F_{2}/\mbox{x}$ and ${\cal F}_{3}=F_{3}$. The coefficients $C_{N}$ are known from continuum P.T., and thus we can extract a given moment $<x^{N}>$ once we know the corresponding matrix element $<p|O^{(N)}_{\mu_{1}\cdots\mu_{N}}|p>$. Such matrix elements contain long distance (non-perturbative) physics, thus the only viable way to compute the moments of Structure Functions is with the use of lattice methods.

We have considered in our calculations the unpolarized Structure Functions, and in particular we have computed the operators

[TABLE]

[TABLE]

[TABLE]

3 Improved Lattice QCD

Lattice QCD allows the evaluation from first principles of the hadronic matrix elements needed for the computation of the moments of the Structure Functions. Once discretization is introduced, the quark fields, $\psi_{n}$, exist on the lattice sites, and the gauge fields, $U_{n,\mu}=e^{ig_{0}at^{A}A_{n,\mu}^{A}}$, exist as links between these sites. The Wilson action $\!{}^{{\bf?}}$

[TABLE]

is widely used as discretization of the (Euclidean) QCD action. The corresponding regularization scheme is gauge-invariant, but the terms proportional to $r$ ($0<r\leq 1$), introduced to get rid of lattice spurious fermions, break chiral symmetry even in the case of a vanishing quark mass $m_{f}$.

A matrix element like $<p|O^{(N)}_{\mu_{1}\cdots\mu_{N}}|p>$ (i.e. a given moment of the Structure Functions) can be determined from the computation of two- and three-point correlation functions. However, its determination from Monte Carlo simulations is affected by both statistical and systematic errors. The statistical errors come from the finite number of configurations used, while the systematic errors are of various nature: finite lattice spacing $a$, finite volume $V$, quenched approximation (that is, dropping the contribution of the internal quark loops), and from the necessary extrapolation to recover chiral symmetry.

We are interested here in the systematic error associated to the finiteness of the lattice spacing, and to reduce this error improvement procedures have been proposed.\!{}^{{\bf?}}$${}^{\!-\,}$$\!{}^{{\bf?}} The formulation that we use consists of adding an irrelevant operator to the standard Wilson action in such a way to cancel, in on-shell matrix elements,$\!{}^{{\bf?}}$ all terms that in the continuum limit are effectively of order “$a$”. This “clover-leaf” Sheikholeslami-Wohlert term is $\!{}^{{\bf?}}$

[TABLE]

where $F_{n,\mu\nu}$ is the average of the four plaquettes lying in the plane $\mu\nu$, stemming from the point $n$:

[TABLE]

with $U_{n,\mu\nu}=U_{n,\mu}U_{n+\mu,\nu}U_{n+\nu,\mu}^{\dagger}U_{n,\nu}^{\dagger}$. This new term in the action means that we have to add to the Wilson quark-quark-gluon interaction vertex,

[TABLE]

the improved quark-quark-gluon interaction vertex

[TABLE]

The fermion propagator is not modified by this improved action, and neither is the gluon propagator as the first corrections to the pure gauge term of the Wilson action are already of order $a^{2}$.

As well as adding the term (8), in the calculation of a fermionic Green function the fermion fields undergo the rotation

[TABLE]

This rotation combined with the use of the Sheikholeslami-Wohlert action has been shown to remove, from on-shell hadronic matrix elements, all terms that in the continuum limit are effectively of order $a$.\!{}^{{\bf?}}$${}^{\!,\,}$$\!{}^{{\bf?}} Using this recipe, the systematic error related to the lattice discretization drops from order $a$ to order $a/\log a$:

[TABLE]

The magnitude of the order $a$ terms is about 20–30 %, while the magnitude of the order $a/\log a$ terms is about 5–10 %. Therefore, with the use of Sheikholeslami-Wohlert improvement one can achieve a remarkable decrease of the systematic error coming from the finiteness of the lattice spacing.

4 Renormalization constants

The renormalization constants connect the bare lattice operators, $O(a)$, to finite operators, $\widehat{O}(\mu)$, renormalized at a scale $\mu$:

[TABLE]

These constants are fixed in perturbation theory by the same renormalization conditions used in the continuum. In the flavor Singlet case there is a mixing between quark and gluon operators of the same rank that have the same conserved quantum numbers. We can then write:

[TABLE]

and in this case all elements of the mixing matrix

[TABLE]

have to be computed.

To this mixing (already present in continuum QCD) the lattice formulation adds additional mixings, induced by the breaking of (Euclidean) Lorentz invariance.$\!{}^{{\bf?}}$ In some cases it is possible, by a careful choice of the Lorentz indices, to choose operators that are multiplicatively renormalizable on the lattice, but the higher the moment the more complicated the mixing pattern is.\!{}^{{\bf?}}$${}^{\!,\,}$$\!{}^{{\bf?}} The breaking of the Lorentz invariance has also bound us to develop special computer routines to correctly perform the Dirac algebra on the lattice. They are a necessary ingredient in our general codes that are able to automatically carry out all the stages of the algebraic manipulations from the elementary building blocks of each Feynman diagram.

5 Some results

Some simulations have been performed in the past with the unimproved Wilson action.\!{}^{{\bf?}}$${}^{\!-\,}$$\!{}^{{\bf?}} Within errors, the results of these simulations are consistent with experiment. However, the values of the renormalization constants change non-trivially when they are computed in the improved theory. As an example we give here a selection of the results for quark operators. At $\beta=6/g_{0}^{2}=6.0$ (the general dependence is $Z=1+\mbox{const}/\beta$), and for $r=1$, we have:

[TABLE]

where $O^{q}_{\rm{DIS}}\equiv O^{q}_{\{411\}}-\frac{1}{2}(O^{q}_{\{422\}}+O^{q}_{\{433\}})$ (which can be writtten as one-third of the sum of the non-symmetric operators $O_{A}\equiv O^{q}_{411}-\frac{1}{2}(O^{q}_{422}+O^{q}_{433})$ and $O_{B}\equiv O^{q}_{141}+O^{q}_{114}-\frac{1}{2}(O^{q}_{242}+O^{q}_{224}+O^{q}_{343}+O^{q}_{334})$) is not multiplicatively renormalizable on the lattice. We see there is a great difference between the Wilson and the improved results; in particular, the improved renormalization constants are somewhat higher. For this reason, improved simulations using the new values of the renormalization constants will give better insight into the agreement with the experimental values of the moments.

Finally, we want to mention that some results are now available also for the polarized Structure Functions,$\!{}^{{\bf?}}$ although so far limited to the Wilson case.

Acknowledgments

This work has been supported by the EC Contract ERBCHBGCT940665. I would like to thank Darren Burford for checking the manuscript.