Hyperon Physics from Lattice QCD Huey-Wen Lina∗ Athomas Jeﬀerson National Accelerator Facility, Newport News, VA 23606

Hyperon Physics from Lattice QCD Huey-Wen Lina∗ aThomas Jeﬀerson National Accelerator Facility, Newport News, VA 23606

I review recent lattice calculations of hyperon physics, including hyperon spectroscopy, axial coupling constants, form factors and semileptonic decays.

1. Introduction The gauge fields are hypercubic-smeared and the source field is Gaussian-smeared to improve the The hyperons are extremely interesting because signal. Details on the configurations can be found they provide an ideal system in which to study in Ref. [6]. For three-point functions, the source- SU(3) flavor symmetry breaking by replacement sink separation is fixed at 10 time units. of up or down quarks in nucleons by strange ones. Hyperon semileptonic decays provide an addi- tional way of extracting the CKM matrix element 2. Spectroscopy Vus and offer unique opportunities to understand On the lattice, continuum SO(3) rotational baryon structure and decay mechanisms. How- symmetry is broken to the more restricted sym- ever, since hyperons decay in less than a nanosec- metry of the cubic group (also known as the oc- ond under weak interactions, their experimental tahedral group). Hadronic states at rest are clas- study is not as easy as for the nucleons, and thus sified according to irreducible representations (ir- hyperon properties are not as well determined. reps) of the cubic group. Since cubic symmetry Recent successes of lattice QCD in computing nu- is respected by the lattice action, an operator be- cleon structure[1–3] provide assurance that lat- longing to a particular irrep will not mix with tice QCD can reliably predict the properties of states in other irreps. The most general baryon hyperons as well. Knowledge of these properties spectrum from a lattice calculation is given by can be very valuable in understanding hypernu- a small number of irreps of the double-cover of clear physics, the physics of neutron stars and the the cubic group: G1g/u, Hg/u and G2g/u, where g structure of nucleons. (German: gerade) and u (ungerade) denote posi- In this proceeding, I review the latest progress tive and negative parity, respectively. on the hyperon spectroscopy, axial coupling con- We follow the technique introduced in Ref. [7] stants, electric charge radii, magnetic moments to construct all the possible baryon interpolating and semileptonic decay form factors using a operators that can be formed from local or quasi- mixed action, in which the sea and valence local u/d and s quark fields; the G2g/u irreps are fermions use different lattice actions. Lattice cal- excluded, since they require non-(quasi-)local op- culations of hyperon scattering lengths are not in- erators. In this paper, we present the calcula- cluded in this proceeding; for more details, please tions of the masses of the lowest-lying states in see Ref. [4]. In our case, the sea fermions are the G1g/u and Hg/u representations. 2+1 flavors of staggered fermions (in configura- The non-locality in four dimensions of our va- tion ensembles generated by the MILC collabo- lence DWF action manifests in oscillations of the ration[5]), and the valence fermions are domain- two-point effective mass close to the source. As wall fermions (DWF). The pion mass ranges from a result, a phenomenological form for fitting such 300 to 700 MeV in a lattice box of size 2.6 fm. data was proposed in Ref. [8], employing both an ∗Speaker oscillating contribution describing the non-local

1 2 H.-W. Lin lattice artifacts and two positive-deﬁnite contri- 2.5

butions: L

2.0 GeV

−M (t−t ) −M (t−t ) H C(t) = A e 0 src + A e 1 src 0 1 L

t −Mosc(t−tsrc) M 1.5 + Aosc(−1) e . (1)

0 0.2 0.4 0.6 The ﬁrst excited-state mass M1 is included to ex- 2 2 MΠ HGeV L tract a better ground-state mass M0; Mosc is a L non-physical oscillating term due to lattice ar- 2.5 GeV H

tifacts. We use two-point correlators with the W same smearing and interpolating operators at M 2.0 both source and sink and select fitted results with varying fit ranges, optimized for quality of fit. A 0 0.2 0.4 0.6 2 2 standard jackknife analysis is employed here. MΠ HGeV L Here we will focus on orbitally excited hyperon resonances. The ground states of the octet and Figure 1. The squared pion-mass dependence of decuplet and the SU(3) Gell-Mann–Okubo mass Λ and Ω-flavor baryons with their extrapolated relation can be found in Ref. [6]. There is in values. The bar at the left indicates the experi- this reference an extensive description of baryon- mental values for the corresponding spin. mass extrapolation (to the physical pion mass) using continuum and mixed-action heavy-baryon chiral perturbation theory for two and three dynamical flavors. However, no chiral perturbation Although our naive extrapolation neglects con- theory results have been published for orbitally tributions at next-to-leading-order chiral pertur- excited hyperon resonances; therefore, we apply bation theory, several interesting patterns are 2 naive linear extrapolations in terms of Mπ. Fig- seen. The better-known Λ (with Hg being spin- ure 1 summarizes our results for the lowest-lying 5/2) and Σ spectra match up with our calcula- Λ and Ω G1g/u (upward/downward-pointing tri- tions well. G1u Ξ lines up well with the Ξ(1690), angles) and Hg/u (diamonds and squares) (since indicating that the spin-parity of this resonance these have less overlap with the data in in Ref. [6]) could be 1/2−; this agrees with SLAC’s recent and their chiral extrapolations. The leftmost spin measurement. The spin assignments for the points are extrapolated masses at the physical Ω channel are the least known; from our mass pion mass, and the horizontal bars are the ex- pattern, we predict that Ω(2250) is likely to be perimental masses (if they are known). 3/2− (although we cannot rule out the possibil- We summarize the lattice hyperon-mass calcu- ity of 5/2−), and Ω(2380) and Ω(2470) are likely lations for the Σ, Λ, Ξ and Ω in Figure 2, divided spin 1/2− and 1/2+ respectively. into vertical columns according to their discrete To successfully extract any reliable radially ex- lattice spin-parity irreps (G1g/u and Hg/u), along cited hyperon states, we would need finer lattice with experimental results by subduction of con- spacing, which would require massive computa- tinuum J P quantum numbers onto lattice irreps. tional resources to achieve. In Euclidean space G1 ground states only overlap with spin-1/2. The the excited signals exponentially decay faster spin identification for H can be a bit trickier, than the ground state. One possible solution is to since this irrep could match either to spin-3/2 or use an anisotropic lattice, where the temporal lat- 5/2 ground states. We simply select the lowest- tice spacing is made finer than the spatial ones to lying of 3/2 or 5/2 indicated in the PDG, so it reduce overall costs. A full-QCD calculation us- could be either depending on which one is the ing anisotropic lattices is underway, and we would ground state for a particular baryon flavor. (We expect new results on these excited states within note below if the lowest H is not spin-3/2.) the next couple of years. Hyperon Physics from Lattice QCD 3

2 as a function of (Mπ/fπ) with the correspond- W ing chiral extrapolation; the band shows the 2.5 W L jackknife uncertainty. We conclude that gA = 1.18(4)stat(6)syst, gΣΣ = 0.450(21)stat(27)syst

L W X and gΞΞ = −0.277(15)stat(19)syst. In addi- tion, the SU(3) axial coupling constants are es- 2.0 S S L L timated to be D = 0.715(6)stat(29)syst and F = GeV X X H W 0.453(5)stat(19)syst. The axial charge couplings of B LW X Σ and Ξ baryons are predicted with signiﬁcantly S S

M S 1.5 X L smaller errors than estimated in the past. XL S We also study the SU(3) symmetry breaking in X S the axial couplings through the quantity δ , S L SU(3) L X n 1.0 δSU(3) = gA − 2.0 × gΣΣ + gΞΞ = cnx , (2) G1 g G1 u Hg Hu n

2 2 2 Figure 2. A summary of our hyperon spectrum where x is (MK − Mπ)/(4πfπ). Figure 3 shows compared to experiment. Our data are the short δSU(3) as a function of x. Note that the value bars with large labels on the right. The experi- increases monotonically as we go to lighter pion mental states are long bars with small labels on masses. Our lattice data suggest that a δSU(3) ∼ the left. x2 dependence is strongly preferred, as the plot of 2 δSU(3)/x versus x in Figure 3 also demonstrates. A quadratic extrapolation to the physical point gives 0.227(38), telling us that SU(3) breaking is 3. Axial Coupling Constants roughly 20% at the physical point, where x = The hyperon axial couplings are important pa- 0.332 using the PDG values[10] for Mπ+ , MK+ rameters entering the low-energy effective field and fπ+ . We compare the result of heavy-baryon theory description of the octet baryons. At the SU(3) chiral perturbation theory[11] for δSU(3) as leading order of SU(3) heavy-baryon chiral per- a function of x, and we find that the coefficient turbation theory, these coupling constants are lin- of the linear term in Eq. 2 does not vanish. This ear combinations of the universal coupling con- implies that an accidental cancellation of the low- stants D and F , which enter the chiral expansion energy constants is responsible for this behavior. of every baryonic quantity, including masses and scattering lengths. These coupling constants are 4. Form Factors needed in the effective field theory description of The study of the hadron electromagnetic form both the non-leptonic decays of hyperons, and the factors reveals information important to our un- hyperon-nucleon and hyperon-hyperon scattering derstanding of hadronic structure. The electro- phase shifts[9]. Hyperon-nucleon and hyperon- magnetic form factors of an octet baryon B can hyperon interactions are essential in understand- be written as ing the physics of neutron stars, where hyperon · 2 ¸ and kaon production may soften the equation of 2 F2(q ) state of dense hadronic matter. hB |Vµ| Bi = uB γµF1(q ) + σµν qν uB 2MB We have calculated the axial coupling constants for Σ and Ξ strange baryons using lat- from Lorentz symmetry and vector-current con- tice QCD for the first time. We have done servation. F1 and F2 are the Dirac and Pauli the calculation using 2+1-flavor staggered dy- form factors. Another common form-factor defi- namical configurations with pion mass as light nition, widely used in experiments, are the Sachs as 350 MeV. Figure 4 shows our lattice data form factors; these can be related to the Dirac 4 H.-W. Lin

0.4 1.2

0.2 A g -----0.9

0 0.6 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 2 Π2 2 4 mΠ 4 fΠ -----

SS 0.4 2 g

0 0.2 0 0.1 0.2 0.3 0.4 2 2 2 2 0 0.1 0.2 0.3 0.4 0.5 - Π 2 2 2 HmK mΠ L4 fΠ mΠ 4Π fΠ -0.23 XX g - Figure 3. (Top) The SU(3) symmetry breaking -----0.28 measure δSU(3). The circles are the measured val- -0.33 ues at each pion mass, the square is the extrapo- 0 0.1 0.2 0.3 0.4 0.5 2 Π2 2 lated value at the physical point, and the shaded mΠ 4 fΠ region is the quadratic extrapolation and its error band. (Bottom) δ /x2 plot. Symbols as SU(3) Figure 4. Lattice data (circles) for g , g and above, but the band is a constant ﬁt. A ΣΣ gΞΞ and chiral extrapolation (lines and bands). The square is the extrapolated value at the physical point. and Pauli form factors through

2 2 2 q 2 GE(q ) = F1(q ) − 2 F2(q ) (3) be able to extract the form factors from Eq. 3. 4MB 2 2 2 We solve for GM,E using singular value decom- GM (q ) = F1(q ) + F2(q ). (4) position (SVD) at each time slice from source to In this work, we will concentrate on Sachs form sink with data from all momenta with the same 2 factors. Note that here we only calculate the q and all µ. “connected” diagram, which means the inserted We constrain the fit form to go asymptotically 4 2 2 quark current is contracted with the valence to 1/Q at large Q and to have GE(Q = 0) = 1: quarks in the baryon interpolating fields. AQ2 + 1 1 On the lattice, we calculate the quark- GE = 2 2 2 2 . (6) CQ + 1 (1 + Q /Me ) component inserted current, Vµ = qγµq, with q = u, d for the light-quark current and q = s for By trying various combinations of fit constraints the strange-quark vector current. A single inter- on our data with squared momentum transfer polating field for the nucleon, Sigma and cascade < 2 GeV2, we find C is always consistent with octet baryons has the general form 0; therefore, we set C = 0. The mean-squared

B abc aT b c electric charge radii can be extracted from the χ (x) = ² [q1 (x)Cγ5q2(x)]q1(x), (5) electric form factor GE via µ ¶ where C is the charge conjugation matrix, and q1 2 ¯ 2 d GE(Q ) ¯ and q2 are any of the quarks {u, d, s}. For ex- hrEi = (−6) 2 ¯ . (7) dQ GE(0) Q2=0 ample, to create a proton, we want q1 = u and − q2 = d; for the Ξ , q1 = s and q2 = d. By cal- In Figure 5 we plot the electric charge radii with culating two-point and three-point correlators on the neutron and Ξ0 omitted, since the vector con- 2 the lattice with the same baryon operator, we will served current gives GE,{n,Ξ0}(Q ) ≈ 0 for these. Hyperon Physics from Lattice QCD 5

We see that there is small SU(3) symmetry break- 0.6 ing between the SU(3) partners p and Σ+ (or Σ− and Ξ−); their charge radii are consistent within 0.4 statistical errors. Overall, the SU(3) symmetry 0.2 breaking in the charge radii is much smaller than 2 2 2 X \ p Xr \ S Xr \ X what we observed in our study of the axial cou- rE u E l E s pling constants; for charge radii, the effect is neg- 0.60 0.2 0.4 0.6 2 2 ligible. mΠ HGeV L 0.4 We can take a ratio of the electric radii of the baryons which coincide in the SU(3) limit; for 0.2 + − − 2 2 2 example, p and Σ and Σ and Ξ . The domi- Xr \ p XrE \ S XrE \ X 0 E d s l nant meson-loop contribution is suppressed; thus, 0 0.2 0.4 0.6 a naive linear fit to a ratio could be a better de- 2 2 mΠ HGeV L scription than fitting individual channels. Follow- ing Sec. 3, we use the SU(3) symmetry measure x Figure 5. The electric mean-squared radii in units 2 2 2 to parametrize the deviation of the ratio from 1 of fm as functions of Mπ (in GeV ) from each PN n due to symmetry breaking: 1+ n=1 cnx , where quark contribution the next-order corrections contribute at the order of xN+1; taking n = 1, we expect the remain- ing effect should be less than 1% in the expansion. The fit works fairly well for the extrapo- GE,n and GE,Ξ0 . Figure 6 shows the magnetic 2 2 hrE ip hrE iΣ− lation of 2 and 2 . By using the ex- moments of each baryon compared with its SU(3) hrE iΣ+ hrE iΞ− + 0 − − 2 2 partner: {p, Σ }, {n, Ξ }, {Σ , Ξ }. We find perimental value of hrEip and hrEiΣ− , we can 2 2 that as seen in experiment, the SU(3) breakings make predictions for hr i + and hr i − : 0.93(3) E Σ E Ξ of the magnetic moments are rather small. As and 0.501(10) fm2 respectively. (Using n = 2 we go to larger pion masses (that is, as the light in for the fit yields c zero within error and thus 2 mass goes to the strange mass), the discrepancy gives results consistent with the extrapolation us- gradually goes to zero as SU(3) is restored. But ing n = 1.) even at our lightest pion mass, around 350 MeV, Studying the momentum-transfer dependence the effects of SU(3) symmetry breaking effect can of magnetic form factors gives us the magnetic be ignored. The fitted results are consistent with moment via what we obtained from the dipole extrapolations. B 2 We examine the radii differences from the quark µB = GM (Q = 0) (8) contributions and observe less than 10% discrep- e with natural units , where MB are the baryon ancy. The ratio approach also benefits from can- 2MB (B ∈ {N, Σ, Ξ}) masses. To compare among dif- cellation of noise due to the gauge fields, and thus ferent baryons, we convert these natural units it has smaller statistical error. Therefore, we will e into nuclear magneton units µN = ; there- concentrate on the results from this approach for 2MN fore, we convert the magnetic moments with fac- the rest of this work. tors of MN . We extrapolate the baryon magnetic moments MB We can obtain the magnetic moments and radii using the SU(3) ratios with the SU(3) symme- from polynomial fitting to the ratio of magnetic try breaking measure x. Again, the ratio has and electric form factors, GM /GE. From the def- cleaner statistical signal due to the cancellation of inition of the electric and magnetic radii, we ex- fluctuations within gauge configurations, and the 2 pect that GM /GE ≈ A + CQ , where the mag- linear extrapolation is not a bad approximation, netic moment is µ = AGE(0), and C is propor- since potential log terms are suppressed. With 2 2 0 tional to hrM i − hrEi. In the case of n and Ξ , the help of experimental µΣ+ , µΞ− and µΞ0 [10], we use GE,p and GE,Ξ− in the ratio instead of we obtain µp = 2.56(7), µn = −1.55(8) and 6 H.-W. Lin

4 about 2σ away from 1. This symmetry is softly broken, possibly due to finite lattice-spacing ef- 3 fects. Finer lattice-spacing calculations would be 2 needed to confirm this. ΜP 1 ΜS+ -0.6 5. Semileptonic Decays -0.80 0.2 0.4 0.6 2 2 The hyperons differ from the nucleons by their mΠ HGeV L -1.2 strangeness, and although hyperon decays via the weak interaction have been known for more than -1.6 Μn half a century, interest in their study has not de- Μ 0 -2.0 X cayed with time. They provide an ideal systems -0.40 0.2 0.4 0.6 in which to study SU(3) flavor symmetry break- 2 2 mΠ HGeV L -0.8 ing and offer unique opportunities to understand baryon structure and decay mechanisms. The -1.2 low-energy contribution to the transition matrix Μ - S − ΜX- elements for hyperon beta decay, B1 → B2e ν -1.6 0 0.2 0.4 0.6 can be written in general form as 2 2 mΠ HGeV L Gs V A α M = √ uB2 (Oα + Oα )uB1 ueγ (1 + γ5)vν . (9) Figure 6. Baryon magnetic moments in units of 2 2 2 µN as functions of Mπ (in GeV ). The leftmost From Lorentz symmetry, we expect the matrix points are the experimental numbers. element composed of any two spin-1/2 nucleon states, B1 and B2, to have the form

2 2 2 V 2 α f2(q ) β f3(q ) µΣ− = −1.00(3). (The fit using up to x terms Oα = f1(q )γ + σαβq + qα MB MB results in a zero-consistent fit parameter c2 and 1 1 yields numbers consistent with the above.) µ 2 2 ¶ A 2 α g2(q ) β g3(q ) SU(6) symmetry predicts the ratio µ p /µ p O = g (q )γ + σ q + q γ d u α 1 M αβ M α 5 should be around −1/2. Compared with what B1 B1 we obtain in this work, the ratio agrees within with transfer momentum q = pB2 − pB1 and V,A 2σ for all the pion-mass points. The heaviest two indicating the vector and axial currents respec- pion points have roughly the same magnitude as tively. f1 and g1 are the vector and axial form in the quenched calculation[12]. However, at the factors, and f2 and g2 are the weak magnetic and lightest two pion masses, they are consistent with induced-pseudoscalar form factors. They are non- the −1/2 value. The difference could be due to zero even with B1 = B2. SU(3) flavor symmetry sea-quark effects, which become larger as the pion breaking accounts for the non-vanishing induced mass becomes smaller. A naive linear extrapola- scalar and weak electric form factors, f3 and g2 tion through all the points gives −0.50(10). SU(6) respectively. Due to the difficulty in disentangling symmetry is preserved in the lattice calculations. experimental form factor contributions, they tend We also check the sum of the magnetic mo- to be set to zero. We will be able to determine ments of the proton and neutron, µp + µn, which these form factors using theoretical technique and should be about 1 from isospin symmetry. Again, will find them to be non-negligible. the values from different pion masses are consis- So far, there are only two “quenched” (where tent with each other within 2 standard deviations the fermion masses in the sea sector are infinitely and differ from 1 by about the same amount. heavy) lattice calculations of hyperon beta de- A naive linear extrapolation suggests the sum is cay, and they are in different channels, Σ → n 0.78(13), which is consistent with experiment but and Ξ0 → Σ+. Guadagnoli et al.[13] calculated Hyperon Physics from Lattice QCD 7 the matrix element Σ → n with all of the pion masses larger than 700 MeV. Sasaki et al.[14] used lighter pion masses in the range 530–650 MeV and DWF to look at the Ξ0 decay channel. They extrapolate the vector form factor f1 using the parameter δ = (MB2 − MB1 )/MB2 . In this work, we calculate both decay channels and remove the quenched approximation, which often causes no- toriously large systematic error. Our fermion sea sector contains degenerate up and down quarks plus the strange. We use pion masses as light as 350 MeV, which ameliorates some of the uncertainty in the extrapolation to the physical pion mass. To condense our work for this proceeding, we will concentrate on the results from Σ → n. To obtain Vus, we need to extrapolate f1 to zero momentum-transfer (we cannot calculate Figure 7. Simultaneous extrapolation in q2 and this point directly due to the discrete values of mass momentum accessible in a finite volume) and the physical pion and kaon masses. Fortunately, f1 is protected by the Ademollo-Gatto (AG) theorem such that there is no first-order SU(3) breaking. The axial form factor g1 is not protected by the Therefore, the quantity deviates from its SU(3) AG theorem. Experimentally, one is interested in value by the order of the symmetry breaking its ratio with the vector form factor, g1/f1, at zero term of O(H02), where H0 is the SU(3) symmetry momentum transfer. Here we adopt naive linear 2 2 2 2 breaking Hamiltonian; the natural candidate an mass combinations (MK + Mπ) and (MK − Mπ) observable to track this breaking is the mass split- and momentum dependence to extrapolate and ting between the kaon and pion. Combining with find g1(0)/f1(0) = −0.336(52), which is consis- momentum extrapolation (using a dipole form in tent with the experimental value of −0.340(17). this case), we use a single simultaneous fit: Similar extrapolations are applied to other form factor ratios, such as f2(0)/f1(0) = −1.28(19), ¡ ¢ ¡ ¡ ¢¢ 2 2 2 2 2 which is consistent with Cabibbo model value of 2 1 + MK − Mπ A1 + A2 MK + Mπ f1(q ) = µ ¶2 . −1.297[15]. q2 Finally, we turn our discussion toward the 1 − 2 2 M0+M1(M +M ) K π SU(3)-vanishing weak-electric g2 and induced- (10) scalar f3 form factors. Figure 8 presents the mo- 2 2 mentum dependence of the ratios g2(q )/f1(q ) 2 2 Figure 7 shows the result from simultaneously and f3(q )/f1(q ) for sea-pion masses rang- fitting over all q2 and mass combinations for ing 350–700 MeV. The band indicates our the Σ− → n decay. The z-direction indicates mass extrapolation in terms of a naive lin- 2 2 2 f1, while the x- and y-axes indicate mass and ear dependence of MK − Mπ and MK + 2 transfer momentum. The surface is the fit us- Mπ. We find f3(0)/f1(0) = −0.17(11) ing Eq. 10 with color to indicate different masses. and g2(0)/f1(0) = −0.29(20), which are 1.5 The columns are the data and the momentum standard deviations from zero. Experimen- points from different pion masses line up in bands. tally, only a combination of axial form factors Our preliminary result for f1 is −0.95(3) (which |g1(0)/f1(0) − 0.133g2(0)/f1(0)| is determined. is consistent with the quenched result[13]: f1 = They find 0.327(7)(19), which is consistent with −0.988(29)stat.) our result, 0.297(60). 8 H.-W. Lin

Acknowledgements 0.2 HWL thanks collaborators Kostas Orginos,

0.0 David Richards and Saul Cohen for useful dis- 1 f cussions. These calculations were performed us-

3 f ing the Chroma software suite[16] on clusters at -0.2 Jeﬀerson Laboratory using time awarded under the SciDAC Initiative. Authored by Jeﬀerson Sci- - 0.4 ence Associates, LLC under U.S. DOE Contract No. DE-AC05-06OR23177. The U.S. Govern- 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ment retains a non-exclusive, paid-up, irrevoca- Q2 HGeV2L 0.0 ble, world-wide license to publish or reproduce this manuscript for U.S. Government purposes.

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