Hyperon couplings from $N_f = 2 + 1$ lattice QCD

TL;DR

This paper calculates isovector charges and parton distribution functions of octet baryons using $N_f=2+1$ lattice QCD, revealing flavor symmetry breaking effects and providing insights into baryon structure.

Contribution

It presents new lattice QCD results for baryon isovector charges and PDFs, including flavor symmetry breaking and low energy constants, with simulations at multiple lattice spacings.

Findings

Significant SU(3) flavor symmetry breaking for $g_A$.

Small flavor symmetry violations for $g_T$ and $ angle_{u^+-d^+}$.

First lattice results on low energy constants $F$ and $D$.

Abstract

We compute various (generalized) isovector charges of the octet baryons. These include $g_A$, $g_T$ and $g_S$ as well as the unpolarized, polarized and transversity parton distribution function (PDF) momentum fractions $\langle x\rangle_{u^+-d^+}$, $\langle x\rangle_{\Delta u^--\Delta d^-}$ and $\langle x\rangle_{\delta u^+-\delta ^+}$. The simulations are carried out on a subset of the (isospin symmetric) $N_f=2+1$ flavour Coordinated Lattice Simulations (CLS) gauge ensembles with lattice spacings ranging from $a\approx 0.086\,$fm down to $a\approx 0.050\,$fm. First results on the breaking of flavour symmetry and the low energy constants $F$ and $D$ are presented. While SU(3) flavour symmetry violations are found to be sizeable for $g_A=\langle 1\rangle_{\Delta u^+-\Delta d^+}$, these are quite small for $g_T=\langle 1\rangle_{\delta u^--\delta d^-}$ and $\langle x\rangle_{u^+-d^+}$.

Tables

Table 1. Table 1: CLS gauge ensembles analysed here. t 𝑡 t denotes the source-sink separations and bracketed digits indicate the number of measurements carried out for each distance on each configuration.

Ensemble	$\beta$	$a\left[\text{fm}\right]$	$N_t\cdot N_s^3$	$M_{\pi }\left[\text{MeV}\right]$	$L{M}_{\pi }$	$t/a$	$N_{\text{configs}}$
H102	3.4	$0.0856$	$96\cdot {32}^3$	355	4.9	[8(2), 10(2), 12(2), 14(2)]	2004
N202	3.55	$0.0642$	$128\cdot {48}^3$	412	6.4	[11, 14(2), 16(2), 19(4)]	1768
X250			$64\cdot {48}^3$	348	5.4	[11, 14(2), 16(4), 19(4)]	345
X251			$64\cdot {48}^3$	269	4.2	[11, 14(2), 16(4), 19(4)]	436
N203			$128\cdot {48}^3$	346	5.4	[11(2), 14(2), 16(2), 19(2)]	1543
N200			$128\cdot {48}^3$	284	4.4	[11, 14, 16, 19]	1712
D201			$128\cdot {64}^3$	199	4.1	[11(2), 14(2), 16(2), 19(2)]	1078
N302	3.7	$0.0497$	$128\cdot {48}^3$	347	4.1	[14, 17, 21, 24]	1383

Equations

$$g_J^B$$

$$m_B⟨x{⟩}_J^B$$

$$N_{\alpha }=p_{\alpha }={ϵ}^{ijk}{u}_{\alpha }^i\left(u^{j⊺}C{\gamma }_5{d}^k\right),{\Sigma }_{\alpha }^{+}={ϵ}^{ijk}{u}_{\alpha }^i\left(u^{j⊺}C{\gamma }_5{s}^k\right),{\Xi }_{\alpha }^0={ϵ}^{ijk}{s}_{\alpha }^i\left(s^{j⊺}C{\gamma }_5{u}^k\right),$$

$$R_J^B(t,\tau )$$

$$Z_J(g^2,a\mu )\left(1+a{m}_{\ell }{b}_J(g^2)+3a\overline{m}{\tilde{b}}_J(g^2)\right),$$

$$\displaystyle C(\mathbf{p}^{\prime},\mathbf{q},x^{\prime}_{4},y_{4},x_{4})^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha^{\prime}\,\alpha\,\beta^{\prime}\,\beta\,\delta^{\prime}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\delta\,\gamma^{\prime}\gamma}}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}U\,D\,}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{U\,U\,}}=\frac{1}{N_{\mathrm{sto}}}\sum\limits_{n=1}^{N_{\mathrm{sto}}}\!\sum\limits_{k=1}^{3}\left(S_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{UD}}(\mathbf{p}^{\prime},x^{\prime}_{4},x_{4})^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha^{\prime}\,\alpha\,\beta^{\prime}\,\beta\,\delta^{\prime}}}_{nk}\cdot\,I_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}UU}(\mathbf{q},y_{4},x_{4})^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\delta\,\gamma^{\prime}\gamma}}_{nk}\right).$$

$$C_{2pt,B}(t)=A_0{e}^{-m_{B}t}\left(1+A_1{e}^{-\Delta m_{B}t}\right),R_J^B(t,\tau )=B_0+B_1{e}^{-\Delta m_{B}t/2}\cosh (\Delta m_B(\tau -t/2))+B_2{e}^{-\Delta m_{B}t}$$

$$g_A^N=F+D,g_A^{\Sigma }=2F,g_A^{\Xi }=F-D.$$

$${\delta }_{\text{SU(3)}}^J=\left(g_J^N+g_J^{\Xi }\right)/g_J^{\Sigma }-1.$$

Full text

1 Introduction

In recent years quite a few lattice calculations of the nucleon axial charge $g_{A}^{N}$ have been carried out, assuming isospin symmetry, most of which are compatible with the experimental ratio $g_{A}^{N}/g_{V}^{N}=1.2732(23)$ [1], as measured from neutron $\beta$ decay. On the lattice $g_{A}^{N}$ can be accessed, e.g., via the proton matrix element $\langle p|\bar{u}\gamma_{5}\gamma_{\mu}u-\bar{d}\gamma_{5}\gamma_{\mu}d|p\rangle$. The hyperon axial charges $g_{A}^{B}$ for $B\neq N$ are less well known since a direct measurement, e.g., of $\Sigma^{-}\rightarrow\Sigma^{0}e^{-}\bar{\nu}_{e}$, is unrealistic. So far these can only be inferred, assuming an approximate flavour symmetry, from $\beta$ decays such as $\Xi^{-}\rightarrow\Lambda\ell\bar{\nu}_{\ell}$ or $\Sigma^{-}\rightarrow n\ell\bar{\nu}_{\ell}$, where an $s$-quark is converted into a $u$-quark. In terms of lattice simulations only few direct results [2, 3, 4, 5, 6, 7] of these charges exist. Such determinations, however, are interesting both in terms of testing the extent of SU(3) flavour symmetry and to determine the low energy constants (LECs) $F$ and $D$ from first principles, which also appear in other SU(3) BChPT expansions, e.g., of octet baryon self-energies.

In addition to the axial charges we also compute the hyperon isovector charges in other channels, in particular the scalar charges $g_{S}^{B}$, and the tensor charges $g_{T}^{B}$ as well as the unpolarized, polarized and transversity Bjorken momentum fractions $\langle x\rangle^{B}_{u^{+}-d^{+}}$, $\langle x\rangle^{B}_{\Delta u^{-}-\Delta d^{-}}$ and $\langle x\rangle^{B}_{\delta u^{+}-\delta d^{+}}$, respectively, where $q^{\pm}=q\pm\bar{q}$. Much less is known for these (generalized) charges since these are even more elusive to experimental measurement than $g_{A}^{B}$.

2 CLS gauge ensembles used

For our analysis we employ gauge ensembles generated by the Coordinated Lattice Simulations (CLS) [8] effort, that combine the $N_{f}=2+1$ non-perturbatively $\mathcal{O}(a)$ improved Wilson fermionic action with the tree-level Symanzik improved gauge action. To avoid freezing of the topological charge most of these ensembles have open boundary conditions in time [9]. The ensembles are generated along three different trajectories, namely:

• •

$\text{tr}\;M=\text{const}$: keeping the trace of the quark mass matrix constant near its physical value [8], thereby increasing the strange quark mass while the light quark mass is decreased.

• •

$\widehat{m}_{s}\approx\widehat{m}_{s}^{\text{ph}}$: setting the renormalized strange quark mass to its physical value [10].

• •

$m_{\ell}=m_{s}$: the symmetric line.

This enables us to extrapolate to the physical point along two different quark mass trajectories ($\text{tr}\;M\approx\text{tr}\;M^{\text{ph}}$ and $\widehat{m}_{s}\approx\widehat{m}_{s}^{\text{ph}}$), while along the symmetric line an extrapolation to the $N_{f}=3$ chiral limit can be carried out.

The parameters of the ensembles investigated so far are listed in table 1. These cover a range of pion masses from $\sim 410\,\text{MeV}$ down to $\sim 200\,\text{MeV}$, three different lattice spacings $a$ and volumes with $LM_{\pi}$ between 4.1 and 6.4. D201 is on the $\widehat{m}_{s}\approx\widehat{m}_{s}^{\text{ph}}$ trajectory, which at this light pion mass is very close to the $\text{tr}\;M=\text{const}$ line. H102, N202, N203, N200 and N302 are on the $\text{tr}\;M=\text{const}$ line and X250 and X251 (as well as N202) are on the SU(3) symmetric line.

3 Definitions, numerical methods and the fit procedure

The isovector charges are defined as matrix elements of local operators at zero momentum transfer. Here we consider two kinds of (generalized) charges:

[TABLE]

where the $\Gamma$-structures of the latter currents contain one derivative and in both cases we use isovector combinations $O(\Gamma_{J})=\bar{u}\Gamma_{J}u-\bar{d}\Gamma_{J}d$. We destroy the nucleon, the $\Sigma$ and the $\Xi$ octet baryon components with Dirac index $\alpha$, using the interpolators

[TABLE]

where $q\in\{u,d,s\}$ are Wuppertal smeared quark fields with $m_{s}\geq m_{\ell}=m_{u}=m_{d}$. When constructing zero-momentum two-point functions $C_{2pt,B}$, we project with $\mathbb{P}_{+}=\tfrac{1}{2}(\mathds{1}+\gamma_{4})$ onto positive parity, while for the three-point functions, we also project on helicity differences if this is required.

The matrix elements are obtained from ratios of three-point over two-point functions

[TABLE]

where $t$ is the temporal source-sink separation and $\tau$ the distance of the current from the source. In our case the discretization effects are of orders $a^{2}$ and $a$ for matrix elements without and with derivatives, respectively. To quote results in the continuum $\overline{\text{MS}}$ scheme at the scale $\mu=2\,\text{GeV}$, we renormalize the lattice matrix elements, multiplying these by

[TABLE]

where $3\overline{m}=\text{tr}\;M$. We non-perturbatively determine the renormalization with respect to the intermediate RI’-SMOM scheme and employ the improvement coefficients $b_{J}(g^{2})$ of [11], where first estimates of $\tilde{b}_{J}$ were found to be compatible with zero.

With the exception of the $m_{\ell}=m_{s}$ ensembles the three-point correlation functions are computed using the stochastic method described in [12] (see also [13, 14, 15, 16]), estimating a timeslice-to-all propagator (depicted as a wiggly line in figure 1). This allows us to factorize the three-point correlation function into two parts, the spectator part $S$ and the insertion part $I$, which can be evaluated separately, where we leave all Dirac indices (Greek letters) as well as the stochastic and colour indices ($n$ and $k$) open:

[TABLE]

The baryon source time $x_{4}$ and the current insertion time $y_{4}$, where the spatial momentum $\mathbf{q}$ is injected, are varied, while the sink time $x_{4}^{\prime}$ of the baryon with momentum $\mathbf{p}^{\prime}$, where the stochastic source is located, is fixed. We have a similar backward sink (not shown) to enable forward-backward time averaging. For the baryons other than the proton, we substitute thequark flavours accordingly. We then compute the contractions for the matrix ele-ments of interest. In this analysis we restrict ourselves to non-flavour changing currents and $\mathbf{p}^{\prime}=\mathbf{q}=\mathbf{0}$.

Note that for different baryon sinks or momenta no additional inversions are needed. However, stochastic noise is added to the gauge noise. This is eventually offset by averaging over three-point functions in the forward and backward directions as well as over equivalent polarizations and momentum combinations at very little computational overhead. Any baryonic three-point function with currents containing (in our case) up to one derivative can be computed at the analysis stage from the building blocks $S$ and $I$ that we store, making this method extremely versatile.

In addition to statistical errors there will be systematic uncertainties related to the precision of the renormalization constants and improvement coefficients, the continuum limit extrapolation, finite volume effects and the use of unphysical quark masses. Not all of these systematics can be explored on the presently analysed subset of CLS ensembles. With all lattice extents $L>4.1/M_{\pi}$, finite size effects should be negligible. Before discussing the quark mass dependence, we will study the effect of excited state contributions.

For all ensembles we use a set of four different, approximately matched source-sink separations t/\text{fm}=(x_{4}^{\prime}-x_{4})/\text{fm}\mathrel{\mathop{\in}\limits^{\vbox to-0.90417pt{\kern-2.0pt\hbox{\scriptstyle\sim}\vss}}}\{0.7,0.8,1.0,1.2\}. In figure 2 we show the ratio $R_{T}^{N}$ (see eq. (3)) for the tensor channel, in which excited state contributions are clearly visible. To obtain the final result, we carry out simultaneous fits to the two-point function and the ratios using all four (or three) source-sink separations, according to the ansatz

[TABLE]

with the mass difference $\Delta m_{B}=m_{B}^{\prime}-m_{B}$. In these fits the position of the current $\tau=y_{4}-x_{4}$ is allowed to run within the interval $\tau\in[\delta t,t-\delta t]$, where $\delta t\in\{2a,3a,4a\}$. Overall, we find stable results for all six fits as can be seen in the bottom right panel of the figure, where the error band indicates our final result.

4 Results and Outlook

In figure 3 we show results for the examples of the axial and tensor charges and the unpolarized isovector PDF Mellin moment $\langle x\rangle^{B}_{u^{+}-d^{+}}$. The dashed red lines that are drawn to guide the eye connect the $m_{\ell}=m_{s}$ points that will approach the SU(3) chiral limit while the blue lines connect the $\text{tr}\;M\approx\text{tr}\;M^{\text{ph}}$ points. The left-most point corresponds to D201, which is on the $\widehat{m}_{s}\approx\widehat{m}_{s}^{\text{ph}}$ line, but also near the $\text{tr}\;M=\text{tr}\;M^{\text{ph}}$ trajectory, due to the small pion mass. The first row shows the nucleon charges which are subject to visible cut-off effects. When normalizing hyperon charges with respect to the nucleon charges (second and third rows) the renormalization and improvement factors cancel and the data collapse onto two curves for the two quark mass trajectories, indicating that lattice spacing effects cancel to a large extent.

Assuming SU(3) flavour symmetry, the axial vector charges can be expressed as combinations of the two LECs $F$ and $D$:

[TABLE]

We can directly extract these LECs, extrapolating the combinations $F=(g_{A}^{N}+g_{A}^{\Xi})/2$ and $D=(g_{A}^{N}-g_{A}^{\Xi})/2$ along the $m_{\ell}=m_{s}$ trajectory as linear functions in $M_{\pi}^{2}$, see figure 4. Our values are consistent with the recent lattice results obtained in [7], albeit the latter correspond to the SU(2) and not to the SU(3) chiral limit.

Finally, we investigate SU(3) flavour symmetry breaking effects. Note that according to eq. (7) $(g^{N}_{J}+g^{\Xi}_{J})/g^{\Sigma}_{J}=(2F_{J})/(2F_{J})=1$ holds for $m_{\ell}=m_{s}$. Thus the breaking effect for a current $J$ can, e.g., be quantified in terms of the parameter

[TABLE]

These effects are small for $g_{T}$ and $\langle x\rangle_{u^{+}-d^{+}}$ as is also evident from figure 3, where the blue and red data sets do not significantly deviate from each other. However, for the axial moments $g_{A}=\langle\mathds{1}\rangle_{\Delta u^{+}-\Delta d^{+}}$ and $\langle x\rangle_{\Delta u^{-}-\Delta d^{-}}$ we find symmetry breaking effects of about 10% at physical quark masses, see figure 5. This will be investigated in detail, employing the systematic approach used by QCDSF [5].

We are in the process of analysing additional ensembles that should enable us to carry out controlled physical point and continuum limit extrapolations. We also plan to determine the $\Delta S=1$ (generalized) form factors to complete the picture of flavour symmetry violation and to determine the relevant baryon ChPT LECs.

Acknowledgements: The work of GB and SW is funded by the German BMBF grant 05P18WRFP1. Partial supported was also granted by the EU ITN EuroPLEx (grant 813942) and by the German DFG (SFB/TRR 55). We thank Benjamin Gläßle, Simon Heybrock and Marius Löffler for co-developing some of the software used here. We gratefully acknowledge computing time granted by the John von Neumann Institute for Computing (NIC), provided on the Booster partition of the supercomputer JURECA [17] at Jülich Supercomputing Centre (JSC). Additional simulations were carried out at the QPACE 3 Xeon Phi cluster of SFB/TRR 55.