Nucleon charges and form factors using clover and HISQ ensembles
Sungwoo Park, Tanmoy Bhattacharya, Rajan Gupta, Yong-Chull Jang, Balint Joo, Huey-Wen Lin, Boram Yoon
NNucleon charges and form factors using clover andHISQ ensembles
Sungwoo Park ∗ , Tanmoy Bhattacharya, Rajan Gupta T-2, Los Alamos National Laboratory, Los Alamos, NM 87545, USAE-mail: [email protected] , [email protected] , [email protected] Yong-Chull Jang
Phsics Department, Brookhaven National Laboratory, Upton, NY 11973, USAE-mail: [email protected]
Balint Joo
Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USAE-mail: [email protected]
Huey-Wen Lin
Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USAE-mail: [email protected]
Boram Yoon
CCS-7, Los Alamos National Laboratory, Los Alamos, NM 87545, USAE-mail: [email protected]
We present high statistics ( O ( × ) measurements) preliminary results on (i) the isovectorcharges, g u − dA , S , T , and form factors, G u − dE ( Q ) , G u − dM ( Q ) , G u − dA ( Q ) , (cid:101) G u − dP ( Q ) , G u − dP ( Q ) , onsix 2 + N π and N ππ ) states is in the axial channel. (ii)Flavor diagonal axial, tensor and scalar charges, g u , d , sA , S , T , are calculated with the clover-on-HISQformulation using nine 2+1+1-flavor HISQ ensembles generated by the MILC collaboration [2]with lattice parameters given in Table 2. Once finished, the calculations of g u , d , sA , T will update theresults given in Refs. [3, 4]. The estimates for g u , d , sS and σ N π are new. Overall, a large part of thefocus is on understanding the excited state contamination (ESC), and the results discussed providea partial status report on developing defensible analyses strategies that include contributions ofpossible low-lying excited states to individual nucleon matrix elements. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] F e b ucleon charges and form factors using clover and HISQ ensembles Sungwoo Park
1. Isovector charges with 2+1-flavor clover fermions
Examples of ESC in the vector charge, g u − dV , and form factors G E and G M are illustrated inFig. 1. g u − dV does not vary monotonically with source-sink separation τ , however it is constant towithin 1–2%. So we take the average of the central points with the largest τ (the plateau method).With this choice the identity Z V g u − dV = Z V calculated in Ref. [5].Data for the charges, g u − dA , S , T , show significant ESC as discussed in [5, 6]. As described inRef. [1], the key parameter controlling ESC is the energy, E , of the first excited state. Its value,obtained from a 4-state fit to the 2-point function using emperical Bayesian priors [7], is muchlarger than that of non-interating N π or N ππ -states, especially for physical M π ensembles. Usingthe energy of the N π state as a prior for E in a 3-state fit gives a much lower output value for E but with an equally good χ / DOF, indicating a flat direction in the parameter space. Note thatwith a small E , even E is slightly smaller. We have, therefore, analyzed the ESC using multiplestrategies and, here, compare two for g u − dA , S , T based on 3 ∗ -state fits (3-state truncation of the spectraldecomposition of the 3-point functions with (cid:104) (cid:48) | O | (cid:105) = { , ∗ } and { N π , ∗ } . In { , ∗ } , the spectrum is taken from the standard 4-state fit [8]. In { N π , ∗ } , the energy E N π of thelowest possible state, N ( ) π ( − ) , is used as a prior for E in a 3-state fit and the resulting outputs,ground-state amplitude A and energies E , E and E , are used as inputs in fits to the 3-pointfunctions. The data and fits for { , ∗ } and { N π , ∗ } are compared in Fig. 2 for the a m L ensemble, where one expects the largest effect as it has the smallest M π ∼
170 MeV and Q , with Q = (cid:126) p − ( E N − M N ) being the Euclidean 4-momentum squared transferred.The value of g u − dA is sensitive to input E used in the ESC fits, however, different fits are not dis-tinguished by χ / DOF, again indicating a flat direction. Renormalized charges in the MS scheme at2 GeV, g u − dA , S , T | R = Z u − dA , S , T g u − dA , S , T , are obtained using Z u − dA , S , T from Ref. [5]. Their chiral-continuum (CC)extrapolation is done using the ansatz f ( a , M π ) = c + c a + c M π (see Fig. 3), and the results at M π =
135 MeV and a = g u − dA is a measure of the systematicuncertainty associated with ESC fits. Data for g u − dS , T from the two strategies, shown in Fig. 3 and theextrapolated values in Table 3, are consistent within 1 σ and the ESC fits do not prefer the low E N π .
2. Form factors
We pointed out in Ref. [1] that the large violation of the PCAC relation between axial and
Ensemble ID a (fm) M π (MeV) L × T M π L N conf N HPmeas N LPmeas τ a m
285 0.127(2) 285(3) 32 ×
96 5.85 2002 8008 256,256 {8, 10, 12, 14} a m
270 0.094(1) 270(3) 32 ×
64 4.11 2469 7407 237,024 {8, 10, 12, 14, 16} a m L ×
128 6.16 1854 7416 237,312 {8, 10, 12, 14, 16, 18} a m
170 0.091(1) 170(2) 48 ×
96 3.7 2754 11016 352,512 {8, 10, 12, 14, 16} a m L ×
128 5.08 1825 9125 292,000 {8, 10, 12, 14, 16} a m
270 0.0728(8) 272(3) 48 ×
128 4.8 2454 9816 314,112 {11, 13, 15, 17, 19}
Table 1:
Lattice parameters of 2 + N LPmeas low-precision and N HPmeas high-precision measurements of 2- and 3-point functions aremade using the bias corrected truncated solver method (see Ref. [5] for details.). τ gives the source-sinkseparations studied. Statistics on a m L , a m
170 and a m L ensembles are being increased. ucleon charges and form factors using clover and HISQ ensembles Sungwoo Park t − τ/ τ : ∞
16 14 12 10 8 g V u-d t − τ/ τ : ∞
16 14 12 10 G E , n = u-d χ /19 =1.22 t − τ/ τ : ∞
16 14 12 10 G M , n = u-d χ /19 =1.38 Figure 1:
Example of ESC in unrenormalized isovector vector charge g u − dV and form factors G u − dE , M ( (cid:126) p ) at (cid:126) p = ( π / L ) (cid:126) n with (cid:126) n = a m L clover lattices. For g u − dV , the τ → ∞ value (grey band) is theaverage of the 5 middle data points with τ =
16. Fits to G u − dE , M use the { , ∗ } strategy. pseudoscalar form factors observed in [7] is due to lower energy N π excited states that are notexposed by the { , ∗ } analysis. Including them addressed the PCAC relation [1]. We, thereforeexplore 2 fit strategies here. The top 5 panels in Fig. 4 show renormalized G u − dE , G u − dM , axial ( G u − dA ),induced pseudoscalar ( (cid:101) G u − dP ) and pseudoscalar ( G u − dP ) form factors analyzed using the standard { , ∗ } strategy. The bottom 5 panels are with (i) 2-state simultaneous fit to all V µ channels for G E and G M with E left free, and (ii) the S A strategy defined in [1] for the axial channels.The G E and G M data show better collapse onto a single curve (indicating no significant a , M π , volume dependence) plotted versus Q / M N , and the agreement with the Kelly curve is bettercompared to the clover-on-HISQ data discussed in Ref. [8]. The main difference between the twostrategies is in the errors: the errors from the simultaneous fits are larger, especially at the larger Q .In the axial channels, data with S A satisfies PCAC with most of the change occuring in (cid:101) G u − dP and G u − dP as discussed in [1]. Note that data for G u − dA , (cid:101) G u − dP and G u − dP , shown in Fig. 4, will moveup or down depending on the value of g A , which, as shown in Tab. 3, has unresolved systematics.Thus, resolving the ESC in g A is essential before comparing/using G A ( Q ) in phenomenology. Ensemble ID a (fm) M π (MeV) M π L L × T N l conf N l src N s conf N s src N LP / N HP a m
310 0.1510(20) 320(5) 3.93 16 ×
48 1917 2000 1919 2000 50 a m
310 0.1207(11) 310(3) 4.55 24 ×
64 1013 5000 1013 1500 30 a m
220 0.1184(10) 228(2) 4.38 32 ×
64 958 11000 958 4000 30 a m
310 0.0888(8) 313(3) 4.51 32 ×
96 1081 4000 1081 2000 30 a m
220 0.0872(7) 226(2) 4.79 48 ×
96 712 8000 847 10000 30/50 a m
130 0.0871(6) 138(1) 3.90 64 ×
96 1270 10000 877 10000 50 a m
310 0.0582(4) 320(2) 3.90 48 ×
144 830 4000 956 10000 50 a m
220 0.0578(4) 235(2) 4.41 64 ×
144 593 10000 554 10000 50 a m
135 0.0570(1) 136(1) 3.7 96 ×
192 553 500 553 500 50
Table 2:
Parameters of the 2 + + N l , s conf gives the number of gauge configurations analyzed for light ( l )and strange ( s ) flavors. N l , s src the number of random sources used per configurations, and N LP / N HP the ratioof low- to high-precision meausurements. Results for the connected contributions are taken from Ref. [6]. Charge { , ∗ } { N π , ∗ } g u − dA | R g u − dS | R g u − dT | R Table 3: g u − dA , S , T | R in MS scheme at 2 GeV calculated in 2 ways to remove ESC, and [ χ / DOF] of CC fits. ucleon charges and form factors using clover and HISQ ensembles Sungwoo Park t − τ/ τ : ∞
16 14 12 10 g A u-d χ /22 =1.55 t − τ/ τ : ∞
16 14 12 10 g S u-d χ /22 = 0.80 t − τ/ τ : ∞
16 14 12 10 g T u -d χ /
22 = t − τ/ τ : ∞
16 14 12 10 g A u-d χ / 22 =1.5 t − τ/ τ : ∞
16 14 12 10 g S u-d χ / 22 = 0. t − τ/ τ : ∞
16 14 12 10 g T u - d χ /
22 = Figure 2:
Data and ESC fits for unrenormalized charges g u − dA , S , T on a m L clover lattices using the { , ∗ } fit (top 3 panels) and the { N π , ∗ } fit (bottom 3 panels). Values of τ and [ χ / DOF] are given in the legend. g A u - d a [fm] a127m285a094m270a094m270L a091m170a091m170La073m270 Extrap χ /dof = 0.26 g Su - d a [fm] a127m285a094m270a094m270L a091m170a091m170La073m270 Extrap χ /dof = 0.32 g T u - d a [fm] a127m285a094m270a094m270L a091m170a091m170La073m270 Extrap χ /dof = 0.05 g A u - d a [fm] a127m285a094m270a094m270L a091m170a091m170La073m270 Extrap χ /dof = 0.24 g Su - d a [fm] a127m285a094m270a094m270L a091m170a091m170La073m270 Extrap χ /dof = 0.49 g T u - d a [fm] a127m285a094m270a094m270L a091m170a091m170La073m270 Extrap χ /dof = 0.35 Figure 3:
Chiral-continuum extrapolation of the renormalized (in MS at 2 GeV) isovector charges using theansatz f ( a , M π ) = c + c a + c M π . Results with { , ∗ } ( { N π , ∗ } ) strategy are shown in the top (bottom)3 panels. In each pannel, the pink band shows the result of the simultaneous fit plotted versus the latticespacing a with M π set to 135 MeV. The value in the continuum limit, a =
0, is marked with a red star.
3. Flavor diagonal charges on + + -flavor HISQ lattices The flavor diagonal charges presented here are obtained using the same ESC strategy as dis-cussed in [3, 4]. Alternate analyses taking into account possible lower excited states are in progress.The connected and disconnected contributions shown in Fig. 5 are analyzed separately to constructthe renormalized charges g fA , S , T | R = Z f f (cid:48) A , S , T ( g f (cid:48) , conn A , S , T + g f (cid:48) , disc A , S , T ) , where f , f (cid:48) are quark flavors. Theconnected contribution, g f , conn A , S , T , are taken from Ref. [6]. Here, we update g f , disc Γ using the largerdata set shown in Table 2, and present new results on the connected and disconnected contributions(right two panels in Fig. 5) for the renormalization matrix Z f f (cid:48) A , S , T in the 3-flavor theory using theRI-sMOM scheme. The matching between the lattice RI-sMOM and continuum MS schemes, andthe running to 2 GeV are done using 2-loop perturbation theory. Additionally, we give our firstpreliminary data for the scalar charges. 3 ucleon charges and form factors using clover and HISQ ensembles Sungwoo Park . . . . . . G E / g V Q /M N a m a m a m La m a m La m . . . . . . G M / g V Q /M N a m a m a m La m a m La m . . . . G A / g A Q /M N a m a m a m La m a m La m M A = 1 . M A = 1 . e G P / g A Q /M N a m a m a m La m a m La m pion-pole, M A = 1 . M A = 1 . G P / g A Q /M N a m a m a m La m a m La m . . . . . . G E / g V Q /M N a m a m a m La m a m La m . . . . . . G M / g V Q /M N a m a m a m La m a m La m . . . . G A / g A Q /M N a m a m a m La m a m La m M A = 1 . M A = 1 . e G P / g A Q /M N a m a m a m La m a m La m pion-pole, M A = 1 . M A = 1 . G P / g A Q /M N a m a m a m La m a m La m Figure 4:
Renormalized isovector form factors ( G u − dE , G u − dM , G u − dA , (cid:101) G u − dP and G u − dP ) versus Q / M N . Top(bottom) 5 panels show data with standard { , ∗ } (new) strategy. The value of g A is taken from { , ∗ } fit. Figure 5:
Connected and disconnected diagrams for (i) the 3-point functions that give nucleon charges (left 2panels) and (ii) the renormalization of the flavor diagonal quark bilinear operators in 3-flavor theory (right 2).
The new data for Z factors in Table 4 show that the difference between the isovector ( u − d )and isoscalar ( u + d ) renormalization constants for the axial and tensor operators is small for all4 values of a . This validates the approximation g u + dA , T | R = Z u + d , u + dA , T g u + dA , T + Z u + d , sA , T g sA , T ≈ Z u − dA , T g u + dA , T made in Refs. [3, 4] for our clover-on-HISQ calculations. Also, the off-diagonal mixing Z u + d , sA , T g sA , T is tiny since Z u + d , sA , T (cid:46) . g sA ≈ . g sT ≈ . g l , s , disc A , T | R are carried out using the simple ansatz g ( a , M π ) = c + c a + c M π and thedata and fits are shown in Fig 6. Combining the disconnected contributions with the connected4 ucleon charges and form factors using clover and HISQ ensembles Sungwoo Park a Z u − dA / Z u − dV Z u + d , u + dA / Z u − dV Z u − dT / Z u − dV Z u + d , u + dT / Z u − dV Table 4:
Renormalization factors Z A and Z T for u − d (isovector) and u + d (isoscalar) operators in MS scheme at 2 GeV on HISQ lattices. Renormalizing using ratios with Z u − dV is intended to cancel some of thestatistical and systematic uncertainties as discussed in Ref. [6]. Errors quoted are the larger of the two: halfthe difference between RI-MOM and RI-sMOM results or the largest statistical error. contributions g l , conn A , T | R presented in Ref. [6], our preliminary updated flavor diagonal charges are g uA | R = . ( )( ) g dA | R = − . ( )( ) g sA | R = − . ( ) (3.1) g uT | R = . ( )( ) g dT | R = − . ( )( ) g sT | R = − . ( ) , (3.2)where the second is a systematic error assigned to the chiral-continuum extrapolation [6].There remain issues regarding the systematics in the calculation of the matrix element of thescalar operator that are still being investigated: the values for the the renormalization constants, Z S ,show significant differences between the RI-MOM and RI-sMOM schemes. For example, there are5 ∼
30% differences in Z u − dS , and 5 ∼
10% differences in Z u + d , u + dS with the differences increasingas the lattice spacing becomes larger. For the mixing matrix element Z s , u + dS , the RI-MOM schemegives − . ∼ − .
1, which is much larger than the RI-sMOM scheme result of − . ∼ − . g sS | R = Z s , sS g sS + Z s , u + dS g u + dS , the larger value of mixing Z s , u + dS in RI-MOM scheme gives a negative value for g sS | R ! For the time being, we use the RI-sMOM scheme, in which case the corresponding mixing term Z s , u + dS g u + dS gives about 6 ∼ Z s , sS g sS .The renormalized strangeness g sS | R , from the clover-on-HISQ calculation, is plotted versus a and M π in Fig. 7, along with the nucleon sigma term σ π N = m l g u + dS that is independent of the renor-malization scheme. We have used am l = ( κ − − κ − ) for the definition of the bare quark massand g u + dS = g u + d , conn S + g u + d , disc S for the unrenormalized isoscalar scalar charge. The data show asignificant a dependence in g sS | R while the large linear dependence of σ π N on M π comes from thequark mass in the definition of σ π N . The dependence of g u + dS | R on a and of g sS | R on M π is not clear. Conclusions:
We have presented the status of ongoing calculations of nucleon matrix elementsand are performing a more detailed analysis of the excited state contamination in the extraction of all nucleon matrix elements. The analysis of flavor diagonal scalar charges, g u , d , sS is new, however,a complete understanding of all the systematics is still under investigation. Acknowledgments:
We thank the MILC collaboration for sharing their 2 + + ucleon charges and form factors using clover and HISQ ensembles Sungwoo Park -0.14-0.12-0.10-0.08-0.06-0.04-0.020.00 0 0.03 0.06 0.09 0.12 0.15 g A l , d i s c a [fm] a15m310a12m310a12m220 a09m310a09m220a09m130 a06m310a06m220Extrap -0.14-0.12-0.10-0.08-0.06-0.04-0.020.00 0 0.03 0.06 0.09 0.12 0.15 χ /dof = 1.33 -0.14-0.12-0.10-0.08-0.06-0.04-0.020.00 0 0.03 0.06 0.09 0.12 g A l , d i s c M π [GeV ] a15m310a12m310a12m220 a09m310a09m220a09m130 a06m310a06m220 M π -Extrap -0.14-0.12-0.10-0.08-0.06-0.04-0.020.00 0 0.03 0.06 0.09 0.12 -0.08-0.06-0.04-0.020.000.02 0 0.03 0.06 0.09 0.12 0.15 g A s , d i s c a [fm] a15m310a12m310a12m220 a09m310a09m220a09m130 a06m310a06m220Extrap -0.08-0.06-0.04-0.020.000.02 0 0.03 0.06 0.09 0.12 0.15 χ /dof = 0.13 -0.08-0.06-0.04-0.020.000.02 0 0.03 0.06 0.09 0.12 g A s , d i s c M π [GeV ] a15m310a12m310a12m220 a09m310a09m220a09m130 a06m310a06m220 M π -Extrap -0.08-0.06-0.04-0.020.000.02 0 0.03 0.06 0.09 0.12 -0.012-0.010-0.008-0.006-0.004-0.0020.0000.002 0 0.03 0.06 0.09 0.12 0.15 g T l , d i s c a [fm] a15m310a12m310a12m220 a09m310a09m220a09m130 a06m310a06m220Extrap -0.012-0.010-0.008-0.006-0.004-0.0020.0000.002 0 0.03 0.06 0.09 0.12 0.15 χ /dof = 0.77 -0.012-0.010-0.008-0.006-0.004-0.0020.0000.002 0 0.03 0.06 0.09 0.12 g T l , d i s c M π [GeV ] a15m310a12m310a12m220 a09m310a09m220a09m130 a06m310a06m220 M π -Extrap -0.012-0.010-0.008-0.006-0.004-0.0020.0000.002 0 0.03 0.06 0.09 0.12 -0.006-0.004-0.0020.0000.0020.004 0 0.03 0.06 0.09 0.12 0.15 g T s , d i s c a [fm] a15m310a12m310a12m220 a09m310a09m220a09m130 a06m310a06m220Extrap -0.006-0.004-0.0020.0000.0020.004 0 0.03 0.06 0.09 0.12 0.15 χ /dof = 0.14 -0.006-0.004-0.0020.0000.0020.004 0 0.03 0.06 0.09 0.12 g T s , d i s c M π [GeV ] a15m310a12m310a12m220 a09m310a09m220a09m130 a06m310a06m220 M π -Extrap -0.006-0.004-0.0020.0000.0020.004 0 0.03 0.06 0.09 0.12 Figure 6:
The extrapolation of the disconnected contributions of the renormalized (in MS at 2 GeV) fla-vor diagonal charges g l ( s ) , disc A | R (top row) and g l ( s ) , disc T | R (bottom row) using the chiral-continuum fit ansatz g ( a , M π ) = c + c a + c M π . The parameters for the eight clover-on-HISQ ensembles are given in Table 2. σ π N [ M e V ] a [ fm ] a15m310a12m310a12m220a09m310a09m220a09m130a06m310a06m220a06m130 04080120160 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 σ π N [ M e V ] M π [ GeV ] a15m310a12m310a12m220a09m310a09m220a09m130a06m310a06m220a06m1300.20.30.40.50.6 0 0.05 0.1 0.15 0.2 0.25 g s S | R a [ fm ] a15m310a12m310a12m220a09m310a09m220a09m130a06m310a06m220a06m130 0.20.30.40.50.6 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 g s S | R M π [ GeV ] a15m310a12m310a12m220a09m310a09m220a09m130a06m310a06m220a06m130 Figure 7: (Top) The nucleon sigma term σ N π plotted versus a and M π . (Bottom) The nucleon strangeness g ss | R renormalized in MS scheme at 2 GeV versus a and M π . References [1] Y.-C. Jang, R. Gupta, B. Yoon, and T. Bhattacharya .[2]
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