Status of scalar quark matrix elements from Lattice QCD. André Walker-Loud

Transcription

1 Status of scalar quark matrix elements from Lattice QCD André Walker-Loud

2 Outline Nucleon matrix element calculations Direct method - 3 point function Indirect method - Feynman-Hellman Theorem Scalar Matrix Elements hn m q qq Ni q l q l ss q h q h Conclusions and questions

3 Nucleon Matrix Elements Z hn( ) O(t) N(0)i = 1 Z DUD D e S N( )O(t)N (0) e.g. axial current in the nucleon O = 1 2 q a µ 5q e.g. scalar matrix elements O = m q qq NOTE: these involve 3-point correlation functions. These are numerically difficult calculations.

4 Nucleon Matrix Elements Z hn( ) O(t) N(0)i = 1 Z DUD D e S N( )O(t)N (0) We also need to know the 2, 3, body matrix elements hnn( ) O(t) NN(0)i hnnn( ) O(t) NNN(0)i No reason (I know of) to expect these N-body matrix elements are significantly more important than in standard nuclear physics i.e. the precise nuclear matrix element will require knowledge of these but they are subcorrections O(10%) see Will Detmold s talk

5 Nucleon Matrix Elements Z hn( ) O(t) N(0)i = 1 Z DUD D e S N( )O(t)N (0) e.g. axial current in the nucleon O = 1 2 q a µ 5q So far, we (LQCD) have failed to convincingly compute the most basic nucleon 3-point function - ga

6 Nucleon Matrix Elements Z hn( ) O(t) N(0)i = 1 Z DUD D e S N( )O(t)N (0) e.g. axial current in the nucleon O = 1 2 q a µ 5q We have good ideas about the systematics that need to be controlled, but as yet, there has been no convincing calculation See Sergey Syritsyn s talk

7 Nucleon Matrix Elements Z hn( ) O(t) N(0)i = 1 Z DUD D e S N( )O(t)N (0) nucleon axial current is numerically easier than scalar matrix elements - no disconnected diagrams O = 1 2 q a µ 5q O(t) N(τ) N(0)

8 Nucleon Matrix Elements Z hn( ) O(t) N(0)i = 1 Z DUD D e S N( )O(t)N (0) scalar matrix elements: q = {u,d} has connected and disconnected q = {s,c} has only disconnected O(t) O = m q qq O(t) N(τ) N(0) N(τ) N(0)

9 Nucleon Matrix Elements Z hn( ) O(t) N(0)i = 1 Z DUD D e S N( )O(t)N (0) scalar matrix elements: Fortunately, there is an alternate means to compute these matrix elements Feynman-Hellman Theorem

10 Nucleon Matrix Elements Feynman-Hellman Theorem H = H 0 + H ni In our lattice QCD calculations, we can study the quark mass dependence of the nucleon and infer these matrix elements m q = hn m q qq Ni

11 Nucleon Matrix Elements Feynman-Hellman Theorem m q = hn m q qq Ni This is important: the thing we can do best with lattice QCD is spectroscopy. h0 N(t)N (0) 0i = X n Z n e E nt

12 Nucleon Matrix Elements Feynman-Hellman Theorem h0 N(t)N (0) 0i = X n Z n e E nt 0.92 /prot/m030m030 20x64/px0py0pz0 systematic eff (t, ) 1 C(t) 1 ln C(t+1) m ln t P(m)

13 Nucleon Matrix Elements Feynman-Hellman Theorem 0.92 /prot/m030m050 20x32/px0py0pz0 systematic /prot/m030m030 20x64/px0py0pz0 systematic eff (t, ) 1 ln C(t) C(t+1) m ln eff (t, ) 1 ln C(t) C(t+1) m ln t P(m) t P(m) We can perform the lattice calculations with different values of the quark mass (with all other parameters held fixed) and approximate the derivative m q hn qq Ni[MeV] = am(2) N am (1) N am (2) q am (1) q am q a 1 [MeV]

14 Nucleon Matrix Elements Feynman-Hellman Theorem 0.92 /prot/m030m050 20x32/px0py0pz0 systematic /prot/m030m030 20x64/px0py0pz0 systematic eff (t, ) 1 ln C(t) C(t+1) m ln eff (t, ) 1 ln C(t) C(t+1) m ln t P(m) t P(m) 1620 m ' 675 MeV mn [MeV] r 1 m s m q hn qq Ni[MeV] = am(2) N am (1) N am (2) q am (1) q am q a 1 [MeV]

15 Nucleon Matrix Elements Feynman-Hellman Theorem 0.92 /prot/m030m050 20x32/px0py0pz0 systematic /prot/m030m030 20x64/px0py0pz0 systematic eff (t, ) 1 ln C(t) C(t+1) m ln eff (t, ) 1 ln C(t) C(t+1) m ln t P(m) t P(m) MN HGeVL NNLO - m p 4, with g A =1.2H1L, g DN =1.5H3L Ë Or fit the light quark mass dependence and determine the derivative analytically m p ê H2 2 p f 0 L

16 Light quark matrix element N = m u + m d 2 hn ūu + dd Ni = m l (2m l = m u + m d ) ' m N [MeV] = 45 ± 5 How believable is this value? (30 80)

17 R. Young Lattice 2012 arxiv: N = m l hn ūu + dd Ni The only green value (from FLAG standards) Ú Ú Fukugita et al Dong et al SESAM 1998 Leinweber et al Leinweber et al Procura et al Procura et al ETM 2008 JLQCD 2008 QCDSF 2011 QCDSF 2012 Ê Young & Thomas 2009 Ê PACS-CS 2009 Ê Martin-Camalich et al Ê Dürr et al Ê QCDSF-UKQCD 2011 Ê Ê Shanahan et al Ren et al s l HMeVL These colors have nothing to do with FLAG [taken from R Young review]

18 Light quark matrix element MN GeV NNLO m Π 4, with g A 1.2 1, g N with m Π 2 2 Π f 0 M N = 941 ± 42 MeV N = 84 ± 17 MeV MN GeV cov. NNLO: g A 1.2 1, c 2 3.2, c without (statistical uncertainties only) m Π 2 2 Π f 0 M N = 956 ± 12 MeV more parameters fixed from phenomenology N = 42 ± 14 MeV

19 Light quark matrix element M N (m, )=M 0 + N (µ) m2 4 f 3 g 2 NN m 3 8 (4 f ) 2 3 g2 N F(m,,µ) (4 f ) 2 + O m 4 (4 f ) Chiral corrections to the nucleon mass m q LO for the nucleon mass, the delta loop 0.2 a little more than doubles the M N M 0 1 GeV m q m q m N loop 3 2 m loop m q 3 2 NLO nucleon loop. the nucleon masses rises uniformly as a function of the pion mass MN GeV m Π GeV NNLO m Π 4, with g A 1.2 1, g N including explicit delta dof forces the leading term to be larger to compensate the fit m Π 2 2 Π f 0

20 Baryons in lattice QCD Light quark mass dependence of MN NNLO - m p 4, with g A =1.2H1L, g DN =1.5H3L M N = a 0 N + a 1 N m p MN HGeVL MN HGeVL Ë Ë m p ê H2 2 p f 0 L m p ê H2 2 p f 0 L NNLO Heavy Baryon Fit M N = 941 ± 42 ± 17 MeV Ruler Approximation M N = 0 N + 1 N m = 938 ± 9 MeV I am not advocating this as a good model for QCD! LHP Collaboration arxiv:

21 Baryons in lattice QCD Light quark mass dependence of MN MN HGeVL Ë MILC: M N = a 0 N + a 1 N m p Ì Ì Ì Á Á NPLQCD Ì M N = a 0 N + a 1 N m p Ì Ì Ì MILC: coarse MILC: super fine LHP m p ê H2 2 p f 0 L Latt 2008, arxiv: MN HGeVL M N = a 0 N + a 1 N m p Ë M N = a 0 N + a 1 N m p RBC LHP m p ê H2 2 p f 0 L  MN HGeVL Ë PACS-CS LHP MN HGeVL Ë Â Â Â Â BMW: a~0.125 fm BMW: a~0.085 fm BMW: a~0.065 fm  m p ê H2 2 p f 0 L m p ê H2 2 p f 0 L

22 Baryons in lattice QCD Light quark mass dependence of MN Chiral Dynamics 2012 arxiv: What is the status now (2012)? M N = m = 938 ± 9 MeV Physical point NOT included in fit

23 Baryons in lattice QCD Light quark mass dependence of MN Chiral Dynamics 2012 arxiv: What is the status now (2012)? M N = m = 938 ± 9 MeV QCD Collaboration uses Overlap Valence fermions on Domain-Wall (RBC-UKQCD) sea fermions

24 Baryons in lattice QCD Light quark mass dependence of MN Chiral Dynamics 2012 arxiv: What is the status now (2012)? M N = m = 938 ± 9 MeV RBC-UKQCD Collaboration uses Domain-Wall valence and sea fermions

25 Baryons in lattice QCD Light quark mass dependence of MN Chiral Dynamics 2012 arxiv: What is the status now (2012)? M N = m = 938 ± 9 MeV m ' 758 MeV m ' 174 MeV Taking this seriously yields N = 67 ± 4 MeV I am not advocating this as a good model for QCD!

26 Baryons in lattice QCD Light quark mass dependence of MN Chiral Dynamics 2012 arxiv: What is the status now (2012)? M N ' m MeV m ' 758 MeV m ' 174 MeV Taking this seriously yields N = 67 ± 4 MeV I am not advocating this as a good model for QCD!

27 Light quark matrix element There is a larger uncertainty in the light-quark scalar matrix element in the nucleon than is often appreciated - at least the determination from lattice QCD 30. N. 70 MeV (conservatively) See talk by Martin Hoferichter for phenomenological determination

28 strange content of the nucleon P. Junnarkar and AWL PRD 87 (2013) mshn ss N i [MeV] fs b fm, m l = b fm, m l = b fm, m l = b fm, m l =0.010 b fm, m l =0.030 b fm, m l = m 2 [GeV2 ] m 2 [GeV2 ] Nucleon strange content seems to have very mild lightquark mass dependence

29 strange content of the nucleon P. Junnarkar and AWL PRD 87 (2013) Direct Excluded Feynman-Hellmann f s f s 0.063(11) [21] n f = (25) [14] n f = ( ) [16] n f = (06) [17] n f = (15) [18] n f = (22) [19] n f = (11) [20] n f = (09) [22] n f = (40) [19] n f = (17) [13] n f =2+1, SU(3) 0.036( ) [23] n f = (73) [24] n f = (22) [25] n f =2+1, SU(3) 0.022( ) [26] n f =2+1, SU(3) 0.134(63) [27] n f =2+1, SU(3) 0.053(19) present work 0.043(11) lattice average (see text) f s = m s N ss N /m N SU(3) baryon Chiral Perturbation Theory is NOT quantitatively reliable AWL with LHPC PRD79 (2009) PACS-CS PRD80 (2009) AWL with NPLQCD PRD81 (2010) colors inspired by FLAG good ok bad absence of many good results leads one to consider all results but we don t want to treat all results as equal in fact, there is a second layer of badness, reliance upon SU(3) baryon Chiral Perturbation Theory

30 strange content of the nucleon P. Junnarkar and AWL PRD 87 (2013) Direct Excluded Feynman-Hellmann f s f s 0.063(11) [21] n f = (25) [14] n f = ( ) [16] n f = (06) [17] n f = (15) [18] n f = (22) [19] n f = (11) [20] n f = (09) [22] n f = (40) [19] n f = (17) [13] n f =2+1, SU(3) 0.036( ) [23] n f = (73) [24] n f = (22) [25] n f =2+1, SU(3) 0.022( ) [26] n f =2+1, SU(3) 0.134(63) [27] n f =2+1, SU(3) 0.053(19) present work 0.043(11) lattice average (see text) f s = m s N ss N /m N all numbers included have been extrapolated to the physical pion mass excluded numbers were: - improved and included - not extrapolated to physical pion mass - only u,d dynamical quarks

31 strange content of the nucleon P. Junnarkar and AWL PRD 87 (2013) Direct Excluded Feynman-Hellmann f s f s 0.063(11) [21] n f = (25) [14] n f = ( ) [16] n f = (06) [17] n f = (15) [18] n f = (22) [19] n f = (11) [20] n f = (09) [22] n f = (40) [19] n f = (17) [13] n f =2+1, SU(3) 0.036( ) [23] n f = (73) [24] n f = (22) [25] n f =2+1, SU(3) 0.022( ) [26] n f =2+1, SU(3) 0.134(63) [27] n f =2+1, SU(3) 0.053(19) present work 0.043(11) lattice average (see text) f s = m s N ss N /m N simple weighted average produces unbelievably small uncertainty also - these results tend to be dominated by systematic uncertainties, which are not Gaussian i) For each lattice result, generate random, uniform distribution uniform(f i i,f i + + i,n dist = 10 4 ) ii) For each random sample, perform a weighted sum over the lattice results with weight w i = y i / i y i = {3, 2, 1} {1,.5} reliance upon SU(3) (generous) iii) quote 99% confidence interval of resulting distribution as uncertainty (easy way to chop off potential outliers in uniform distribution)

32 strange content of the nucleon P. Junnarkar and AWL PRD 87 (2013) Direct Excluded Feynman-Hellmann f s 0.063(11) [21] n f = (25) [14] n f = ( ) [16] n f = (06) [17] n f = (15) [18] n f = (22) [19] n f = (11) [20] n f = (09) [22] n f = (40) [19] n f = (17) [13] n f =2+1, SU(3) 0.036( ) [23] n f = (73) [24] n f = (22) [25] n f =2+1, SU(3) 0.022( ) [26] n f =2+1, SU(3) 0.134(63) [27] n f =2+1, SU(3) 0.053(19) present work f N + SU(3) f s µ N + SU(3) [old] 0.043(11) lattice average (see text) f s = m s N ss N /m N f 2 X f lattice average N + SU(3) µ N + SU(3) [old] f = q=u,d,s f q f s

33 Heavy quark matrix elements hn µ Ni µ = m N N N X = hn q=u,d,s,... f c = (1 x uds ) f b = (1 x uds ) f t = (1 x uds ) P Junarkar & AWL arxiv: (1 m)m q qq + (n f ) G 2 Ni 2 f q = hn m q qq Ni m N x uds = f u + f d + f s Shifman, Vainshtein and Zakharov PLB A Kryjevski PRD [hep-ph/ ] Solon and Hill, arxiv: m c hn cc Ni = 76( )MeV

34 Heavy quark matrix elements hn µ Ni µ = m N N N X = hn q=u,d,s,... f c = (1 x uds ) f b = (1 x uds ) f t = (1 x uds ) P Junarkar & AWL arxiv: (1 m)m q qq + (n f ) G 2 Ni 2 f q = hn m q qq Ni m N x uds = f u + f d + f s Shifman, Vainshtein and Zakharov PLB A Kryjevski PRD [hep-ph/ ] Solon and Hill, arxiv: m c hn cc Ni = 76( +1 2 )MeV (thanks Mikhail Solon for finding this mistake) I do not believe the updated precision - I am not sure what the pqcd uncertainty on this is

35 Heavy quark matrix elements SI(cm 2 ) had pert triplet Hill & Solon PRL 112 (2014) arxiv: doublet hn m c cc Ni [MeV] Factors of 2 normally do not seem to matter much in this business, but in some cases - there are large cancelations which are sensitive to the precise values

36 How much does each quark contribute to the mass of the nucleon? mqhn qq Ni[MeV] ((currently unknown uncertainty) pqcd: f Q = hn m Q QQ Ni m N =0.08xxx(1 x uds ) 10 LQCD: x uds = f u + f d + f s 0 u d s c b t

37 Conclusions Lattice QCD calculations of scalar matrix elements are nearing averageability, but still a bit premature SU(3) baryon Chiral Perturbation Theory should NOT be used for quantitative studies Other nucleon matrix element calculations lag significantly compared to scalar quark content lattice QCD calculations of scalar charm content beginning, but in preliminary stage: charm particularly interesting to compare LQCD to pqcd values We can also begin looking at but more challenging m q hn(p + q) qq N(p)i

38 Thank You