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Present status of the pion-nucleon sigma term (and strange content of nucleon) 2018/11/12 QNP 2018 satellite workshop J-PARC Nodoka Yamanaka (IPN Orsay)

Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

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Page 1: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Present status of the pion-nucleon sigma term

(and strange content of nucleon)

2018/11/12 QNP 2018 satellite workshop

J-PARC

Nodoka Yamanaka(IPN Orsay)

Page 2: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Introduction

Applications:

Fundamental property of QCD:

Definition:

�⇡N ⌘ mu +md

2hN |uu+ dd|Ni

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�s ⌘ mshN |ss|Ni<latexit sha1_base64="VNKntUmvoRVeeWFx9yDFNhEaW04=">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</latexit><latexit sha1_base64="VNKntUmvoRVeeWFx9yDFNhEaW04=">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</latexit><latexit sha1_base64="VNKntUmvoRVeeWFx9yDFNhEaW04=">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</latexit><latexit sha1_base64="VNKntUmvoRVeeWFx9yDFNhEaW04=">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</latexit>

Pion-nucleon sigma term Strange content of nucleon

Current quark mass contribution to the nucleon mass (trace anomaly relation)

Quark scalar matrix element :

Gives the variation of chiral condensate in low density nuclear matter

Useful input in BSM search (especially in dark matter direct detection)

Important input in the baryon chiral perturbation theory

⇒ Shift from 3 is contribution from relativistic quarks inside nucleon

⇒ Probe of quark-gluon interaction and confinement in nucleons

mN =�QCD

2gshN |Gµ⌫

a Gµ⌫,a|Ni+X

q

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hN |qq|Ni<latexit sha1_base64="g4m/Nj7ymfEBngChEpd9WJj0y38=">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</latexit><latexit sha1_base64="g4m/Nj7ymfEBngChEpd9WJj0y38=">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</latexit><latexit sha1_base64="g4m/Nj7ymfEBngChEpd9WJj0y38=">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</latexit><latexit sha1_base64="g4m/Nj7ymfEBngChEpd9WJj0y38=">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</latexit>

Page 3: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

How to calculate?

Lattice QCD:

Chiral perturbation theory (ChPT) based extractions:Start from Cheng-Dashen (CD) low energy theorem (LET), relate σπN and πN scattering by using analyticity

Fit σπN by calculating πN scattering phase shift in ChPT

M. Hoferichter et al., Phys. Rev. Lett. 115, 092301 (2015) J. Ruiz de Elvira et al., J. Phys. G 45, 024001 (2017)

J. M. Alarcon et al., , Phys. Rev. D 85, 051503(R) (2012)

Direct calculation of the nucleon 3-point correlator

Derivative of the nucleon mass (Feynman-Hellmann theorem)

Perturbative QCD and fit from experimental data :

Other model approaches:

Integrate twist-3 parton distribution function eq(x) (uncertainty is large)

QCD sum rules, Dyson-Schwinger, etc.

P. Gubler and K. Ohtani, Phys. Rev. D 90, 094002 (2014)

RQCD, chiQCD, ETMC, JLQCD, …

BMW, Lutz et al., Ling et al., …

Page 4: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Present status of the calculations of πN sigma term

JLQCD(2018)ETM(2016)

chiQCD(2016)RQCD(2016)

Ling et al.(2017)BMW(2016)

Lutz et al.(2014)QCDSF-UKQCD(2012)

Ruiz de Elvira et al.(2017)Yao et al.(2016)

Hoferichter et al.(2015)Alarcon et al.(2012)

0 10 20 30 40 50 60 70σπN (MeV)

⇒ Visible disagreement between lattice QCD results

and phenomenological extractions

Phenomenological extractions based on ChPT

Application of Feynman-Hellmann theorem to lattice

Direct lattice QCD calculations

Phenomenological extractions : σπN ~ 60 MeV

Lattice QCD calculations : σπN ~ 30 MeV

Page 5: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Present status of the calculations of strange content

Phenomenological extractions based on ChPT

Application of Feynman-Hellmann theorem to lattice

Direct lattice QCD calculations

Other approaches have large uncertainties

Lattice QCD calculations : σs ~ 40 MeV

JLQCD(2018)ETM(2016)

chiQCD(2016)RQCD(2016)BMW(2016)

Lutz et al.(2014)Ren et al.(2014)

QCDSF-UKQCD(2012)Yao et al.(Fit-B)(2016)

Alarcon et al.(2014)

0 50 100 150 200σs (MeV)

Page 6: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Scope of this talk

In this talk, I will review the two approaches,

and let the audience consider it in the hope to resolve it one day

The ChPT based phenomenological and lattice QCD predictions of the pion-nucleon sigma term differ significantly

⇒ A puzzle in QCD which has not yet convincing answer

Unfortunately, this is one of the argument used by one community (either ChPT or lattice) to claim that the other approach does not work

(´∀`;)

Page 7: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Chiral perturbation based approaches

7

Page 8: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Chiral perturbation based approaches

Extraction from πN scattering experimental data (Cheng-Dashen low-energy theorem approach)

Full calculation in chiral perturbation theory

Page 9: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Principle of σπN extraction from Cheng-Dashen LET

Low-energy theorem (LET):

Why subtract Born amplitude?

Consider the πN scattering : ⇡a(q) +N(p) ! ⇡b(q0) +N(p0)<latexit sha1_base64="1TTv/bUCK6Dei+zURsAMspi0m0s=">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</latexit><latexit sha1_base64="1TTv/bUCK6Dei+zURsAMspi0m0s=">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</latexit><latexit sha1_base64="1TTv/bUCK6Dei+zURsAMspi0m0s=">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</latexit><latexit sha1_base64="1TTv/bUCK6Dei+zURsAMspi0m0s=">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</latexit>

Isoscalar, momentum-symmetric πN amplitudeD+ :

ΔR : Higher order SU(2) ChPT correction to D+

⇒ Remove the massless QCD contribution hN(p0)|LQCD(mud = 0)|N(p)i<latexit sha1_base64="A0FgwUxwzBDP1Hh9OVV82HmXne0=">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</latexit><latexit sha1_base64="A0FgwUxwzBDP1Hh9OVV82HmXne0=">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</latexit><latexit sha1_base64="A0FgwUxwzBDP1Hh9OVV82HmXne0=">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</latexit><latexit sha1_base64="A0FgwUxwzBDP1Hh9OVV82HmXne0=">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</latexit>

+

Estimated to be small, O(�⇡Nm2⇡/m

2N )

<latexit sha1_base64="fl/I60phShnGW4a+eRErj2V2xnQ=">AAAChnichVFNLwNBGH66vqo+WlwkLktDuNS0IcSp4eJEfRSJstld05p0v7K7bULjLPEHHJxIRERc+QEu/oCDnyCOJC4O3m43EQTvZGaeeeZ93nlmRnMM4fmMPUakpuaW1rZoe6yjs6s7nujpXfPsiqvzvG4btruhqR43hMXzvvANvuG4XDU1g69r5bn6/nqVu56wrVV/z+FbplqyRFHoqk+UkhhcHC14omSqSq3gCHnhQDYVAtsZeZzQwnZmTEkkWYoFIf8E6RAkEUbOTlyggB3Y0FGBCQ4LPmEDKjxqm0iDwSFuCzXiXEIi2Oc4QIy0FcrilKESW6axRKvNkLVoXa/pBWqdTjGou6SUMcwe2CV7Yffsij2x919r1YIadS97NGsNLXeU+FH/ytu/KpNmH7ufqj89+yhiOvAqyLsTMPVb6A19df/4ZWVmebg2ws7YM/k/ZY/sjm5gVV/18yW+fIIYfUD6+3P/BGuZVJrw0kQyOxt+RRQDGMIovfcUsphHDnk69xDXuMGtFJVS0qQ01UiVIqGmD19Cyn4AdJeUww==</latexit><latexit sha1_base64="fl/I60phShnGW4a+eRErj2V2xnQ=">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</latexit><latexit sha1_base64="fl/I60phShnGW4a+eRErj2V2xnQ=">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</latexit><latexit sha1_base64="fl/I60phShnGW4a+eRErj2V2xnQ=">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</latexit>

V. Bernard et al., Phys. Lett. B 389, 144 (1996)

Important point :πN sigma term is �(t = 0)

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�(t = 2m2⇡)

<latexit sha1_base64="FsoEuEuG7VsVpjYRc1b/wRE9XJ4=">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</latexit><latexit sha1_base64="FsoEuEuG7VsVpjYRc1b/wRE9XJ4=">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</latexit><latexit sha1_base64="FsoEuEuG7VsVpjYRc1b/wRE9XJ4=">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</latexit><latexit sha1_base64="FsoEuEuG7VsVpjYRc1b/wRE9XJ4=">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</latexit>

, not !!

corresponds to an off-shell, unphysical point!

We must extrapolate the amplitude to t = 0 !

f2⇡D

+(⌫ = 0, t = 2m2⇡)� (Born term)��R = �(t = 2m2

⇡)<latexit sha1_base64="x/rcqOXG/O2RKDU1Vf8Nwf/8Pw0=">AAACunichVHLThRBFD20D6B9MMKGxE3HCWaIQO5MSCTGSQiycAmDAyQ0dLrbmrEy/Up1zSQ44Qf4ARasIDHG+Aku3fgDLli6VJaYuGHB7Z5OBIl6K1V16tQ9t05VeUkgU010MmTcuHnr9vDIqHnn7r37Y6UH4+tp3FW+aPpxEKtNz01FICPR1FIHYjNRwg29QGx4nRfZ/kZPqFTG0Su9m4jt0G1HsiV9VzPllJotx07kTs1a3nliVeyoW6cZXa+FA3baMmcrfVuF1lKsInvG0kKFe9PmrL0sAu06DbNu2qlsh65VuaQynVKZ5igP6zqoFqCMIlbi0nvYeI0YProIIRBBMw7gIuW2hSoICXPb6DOnGMl8X2APJmu7nCU4w2W2w2ObV1sFG/E6q5nmap9PCbgrVlqYoq/0gc7oC32k73T+11r9vEbmZZdnb6AViTO2P7n267+qkGeNN79V//Ss0cJC7lWy9yRnslv4A33v7cHZ2rPGVP8xHdMp+z+iE/rMN4h6P/13q6JxiOwDqn8+93WwXpurMl6dLy8uFV8xgod4hAq/91Ms4iVW0ORzP+EbfuDUeG54hjQ6g1RjqNBM4EoY+gIMt6X1</latexit><latexit sha1_base64="x/rcqOXG/O2RKDU1Vf8Nwf/8Pw0=">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</latexit><latexit sha1_base64="x/rcqOXG/O2RKDU1Vf8Nwf/8Pw0=">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</latexit><latexit sha1_base64="x/rcqOXG/O2RKDU1Vf8Nwf/8Pw0=">AAACunichVHLThRBFD20D6B9MMKGxE3HCWaIQO5MSCTGSQiycAmDAyQ0dLrbmrEy/Up1zSQ44Qf4ARasIDHG+Aku3fgDLli6VJaYuGHB7Z5OBIl6K1V16tQ9t05VeUkgU010MmTcuHnr9vDIqHnn7r37Y6UH4+tp3FW+aPpxEKtNz01FICPR1FIHYjNRwg29QGx4nRfZ/kZPqFTG0Su9m4jt0G1HsiV9VzPllJotx07kTs1a3nliVeyoW6cZXa+FA3baMmcrfVuF1lKsInvG0kKFe9PmrL0sAu06DbNu2qlsh65VuaQynVKZ5igP6zqoFqCMIlbi0nvYeI0YProIIRBBMw7gIuW2hSoICXPb6DOnGMl8X2APJmu7nCU4w2W2w2ObV1sFG/E6q5nmap9PCbgrVlqYoq/0gc7oC32k73T+11r9vEbmZZdnb6AViTO2P7n267+qkGeNN79V//Ss0cJC7lWy9yRnslv4A33v7cHZ2rPGVP8xHdMp+z+iE/rMN4h6P/13q6JxiOwDqn8+93WwXpurMl6dLy8uFV8xgod4hAq/91Ms4iVW0ORzP+EbfuDUeG54hjQ6g1RjqNBM4EoY+gIMt6X1</latexit>

D+(⌫ = 0, t = 0)<latexit sha1_base64="5MyONN2W7OzhSguG9DnftBs9F1Y=">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</latexit><latexit sha1_base64="5MyONN2W7OzhSguG9DnftBs9F1Y=">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</latexit><latexit sha1_base64="5MyONN2W7OzhSguG9DnftBs9F1Y=">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</latexit><latexit sha1_base64="5MyONN2W7OzhSguG9DnftBs9F1Y=">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</latexit>

T. P. Cheng and R. F. Dashen, Phys. Rev. Lett. 26, 594 (1971)

⌫ ⌘ s� u

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�(t) ⌘ hN(p0)|mud(uu+ dd)|N(p)i<latexit sha1_base64="4giqWi2axAE05fEnZuXvd322Ox8=">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</latexit><latexit sha1_base64="4giqWi2axAE05fEnZuXvd322Ox8=">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</latexit><latexit sha1_base64="4giqWi2axAE05fEnZuXvd322Ox8=">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</latexit><latexit sha1_base64="4giqWi2axAE05fEnZuXvd322Ox8=">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</latexit>

Page 10: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Roy-Steiner equation

J. Gasser et al., PLB 253, 252 (1991); M. Hoferichter et al., Phys. Rep. 625, 1 (2016).

We can analytically extrapolate σπN to off-shell region

We consider a system of “subtracted“ dispersion relations (DRs)with channel couplings : πN→πN, ππ→KK, ππ→NN,…

+ …

DRs respects analyticity, unitarity, crossing symmetry,

(e.g. use scattering length for s-wave D+ at t=0)

⇒ To make converge the integral (get rid of the contour),

we set additional polynomial in the denominator (“subtraction”)

⇒ Due to Cauchy’s theorem, additional terms (c(t)) appear

⇒ If we can identify them with known quantities,

we can constrain the analytic form of D+

⇒ The analytic continuation has limits, but the error is quantifiable

by partial wave decomposition, changing subtraction, considering resonances, etc

DD

D

Page 11: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Overall : result of CD LET approach

�⇡N ⌘ �(0) = ⌃d +�D ��� ��R<latexit sha1_base64="Q3hCWEtrEh6dNCWP3LhcJtu35MI=">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</latexit><latexit sha1_base64="Q3hCWEtrEh6dNCWP3LhcJtu35MI=">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</latexit><latexit sha1_base64="Q3hCWEtrEh6dNCWP3LhcJtu35MI=">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</latexit><latexit sha1_base64="Q3hCWEtrEh6dNCWP3LhcJtu35MI=">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</latexit>

Synopsis of the pion-nucleon sigma term:

�D ⌘ D+(0, 2m2⇡)� ⌃d

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�� ⌘ �(2m2⇡)� �⇡N

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�R<latexit sha1_base64="SAClqlvM56EUwXvs40EW30bjoEE=">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</latexit><latexit sha1_base64="SAClqlvM56EUwXvs40EW30bjoEE=">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</latexit><latexit sha1_base64="SAClqlvM56EUwXvs40EW30bjoEE=">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</latexit><latexit sha1_base64="SAClqlvM56EUwXvs40EW30bjoEE=">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</latexit>

J. Gasser et al., PLB 253, 260 (1991)

Can be determined from DR analysis as well

Determined by DR analysis (Roy-Steiner eq.)

< 2 MeVV. Bernard et al., Phys. Lett. B 389, 144 (1996)

�D ��� = (�1.8± 0.2)MeV<latexit sha1_base64="R5bAYP7cdSxKBQJ7rP1Dxb69SDE=">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</latexit><latexit sha1_base64="R5bAYP7cdSxKBQJ7rP1Dxb69SDE=">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</latexit><latexit sha1_base64="R5bAYP7cdSxKBQJ7rP1Dxb69SDE=">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</latexit><latexit sha1_base64="R5bAYP7cdSxKBQJ7rP1Dxb69SDE=">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</latexit>

(Uncertainty of ChPT)

Small due to the cancellation (both are ~15 MeV):

⌃d ⌘ f2⇡(d

+00 + 2m2

⇡d+01)

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D+(⌫, t) =1X

n,m=0

d+mn⌫2mtn

<latexit sha1_base64="ufQwWB76LzNjdbDwcc3POPoW2/s=">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</latexit><latexit sha1_base64="ufQwWB76LzNjdbDwcc3POPoW2/s=">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</latexit><latexit sha1_base64="ufQwWB76LzNjdbDwcc3POPoW2/s=">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</latexit><latexit sha1_base64="ufQwWB76LzNjdbDwcc3POPoW2/s=">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</latexit>

Lowest terms the Taylor and partial wave expansion

Determined from Roy-Steiner analysis

Page 12: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Overall : result of CD LET approach

�⇡N ⌘ �(0) = ⌃d +�D ��� ��R<latexit sha1_base64="Q3hCWEtrEh6dNCWP3LhcJtu35MI=">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</latexit><latexit sha1_base64="Q3hCWEtrEh6dNCWP3LhcJtu35MI=">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</latexit><latexit sha1_base64="Q3hCWEtrEh6dNCWP3LhcJtu35MI=">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</latexit><latexit sha1_base64="Q3hCWEtrEh6dNCWP3LhcJtu35MI=">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</latexit>

Synopsis of the pion-nucleon sigma term:

�D ⌘ D+(0, 2m2⇡)� ⌃d

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�� ⌘ �(2m2⇡)� �⇡N

<latexit sha1_base64="95hFbqBMyOD8sbtsFDvMdEyI9yE=">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</latexit><latexit sha1_base64="95hFbqBMyOD8sbtsFDvMdEyI9yE=">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</latexit><latexit sha1_base64="95hFbqBMyOD8sbtsFDvMdEyI9yE=">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</latexit><latexit sha1_base64="95hFbqBMyOD8sbtsFDvMdEyI9yE=">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</latexit>

�R<latexit sha1_base64="SAClqlvM56EUwXvs40EW30bjoEE=">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</latexit><latexit sha1_base64="SAClqlvM56EUwXvs40EW30bjoEE=">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</latexit><latexit sha1_base64="SAClqlvM56EUwXvs40EW30bjoEE=">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</latexit><latexit sha1_base64="SAClqlvM56EUwXvs40EW30bjoEE=">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</latexit>

J. Gasser et al., PLB 253, 260 (1991)

Can be determined from DR analysis as well

Determined by DR analysis (Roy-Steiner eq.)

< 2 MeVV. Bernard et al., Phys. Lett. B 389, 144 (1996)

�D ��� = (�1.8± 0.2)MeV<latexit sha1_base64="R5bAYP7cdSxKBQJ7rP1Dxb69SDE=">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</latexit><latexit sha1_base64="R5bAYP7cdSxKBQJ7rP1Dxb69SDE=">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</latexit><latexit sha1_base64="R5bAYP7cdSxKBQJ7rP1Dxb69SDE=">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</latexit><latexit sha1_base64="R5bAYP7cdSxKBQJ7rP1Dxb69SDE=">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</latexit>

(Uncertainty of ChPT)

Small due to the cancellation (both are ~15 MeV):

⌃d ⌘ f2⇡(d

+00 + 2m2

⇡d+01)

<latexit sha1_base64="x2qGrDUC5lz0XZPiXHWWovGvVWM=">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</latexit><latexit sha1_base64="x2qGrDUC5lz0XZPiXHWWovGvVWM=">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</latexit><latexit sha1_base64="x2qGrDUC5lz0XZPiXHWWovGvVWM=">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</latexit><latexit sha1_base64="x2qGrDUC5lz0XZPiXHWWovGvVWM=">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</latexit>

D+(⌫, t) =1X

n,m=0

d+mn⌫2mtn

<latexit sha1_base64="ufQwWB76LzNjdbDwcc3POPoW2/s=">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</latexit><latexit sha1_base64="ufQwWB76LzNjdbDwcc3POPoW2/s=">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</latexit><latexit sha1_base64="ufQwWB76LzNjdbDwcc3POPoW2/s=">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</latexit><latexit sha1_base64="ufQwWB76LzNjdbDwcc3POPoW2/s=">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</latexit>

Lowest terms the Taylor and partial wave expansion

Determined from Roy-Steiner analysis

Total : �⇡N = (59.1± 3.5)MeV<latexit sha1_base64="Z2Slbx8OJQDMeOGzQk5wzvT+eZw=">AAACkXichVFNSxtBGH5cW41rq1EvBS+LQYkQwmyraIRC0EuhKBqbD3BD2N2O6eB+sbsJ2CV/oEcvHjwpSCn9A15bL/4BD/4E8RjBSw++2SyIStt3mJlnnnmfd56ZMTxLBCFjVwPS4IuXQ8OpEXn01eux8fTEZCVwW77Jy6ZruX7N0ANuCYeXQxFavOb5XLcNi1eNvbXefrXN/UC4zqdw3+N1W286YleYekhUI53TAtG09UakeULZ6Mjv5exiIa8qmmcr7/KL84qWUyLNt5V1Xuk00hmWZ3Eoz4GagAyS2HTT36HhM1yYaMEGh4OQsAUdAbUdqGDwiKsjIs4nJOJ9jg5k0rYoi1OGTuwejU1a7SSsQ+tezSBWm3SKRd0npYJZdsl+sC67YD/ZNfvz11pRXKPnZZ9mo6/lXmP825vtu/+qbJpDfHlQ/dNziF0sx14FefdipncLs69vfz3sbq+UZqM5dsJuyP8xu2LndAOnfWuebvHSEWT6APXpcz8Hlbd5lfDWQqa4mnxFCtOYQZbeewlFfMAmynTuAc7wC7+lKakgFaUkVxpINFN4FNLHe+yzl2g=</latexit><latexit sha1_base64="Z2Slbx8OJQDMeOGzQk5wzvT+eZw=">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</latexit><latexit sha1_base64="Z2Slbx8OJQDMeOGzQk5wzvT+eZw=">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</latexit><latexit sha1_base64="Z2Slbx8OJQDMeOGzQk5wzvT+eZw=">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</latexit>

M. Hoferichter et al., PRL 115, 092301 (2015).

Page 13: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Full calculation in chiral perturbation theory

J. M. Alarcon et al., Phys. Rev. D 85, 051503 (2012)

-14-12-10

-8-6-4-2 0

1.08 1.1 1.12 1.14 1.16 1.18 1.2 2 3 4 5 6 7 8 9

10 11

1.08 1.1 1.12 1.14 1.16 1.18 1.2

-4-3.5

-3-2.5

-2-1.5

-1-0.5

0 0.5

1.08 1.1 1.12 1.14 1.16 1.18 1.2-1.2

-1-0.8-0.6-0.4-0.2

0 0.2

1.08 1.1 1.12 1.14 1.16 1.18 1.2

0 10 20 30 40 50 60

1.08 1.1 1.12 1.14 1.16 1.18 1.2-2

-1.5-1

-0.5 0

0.5

1.08 1.1 1.12 1.14 1.16 1.18 1.2W (GeV)W (GeV)

W (GeV)W (GeV)

W (GeV)W (GeV)

S11

S31

P11

P13

P31

P33

�⇡N = �4c1m2⇡ � 3g2Am

3⇡

16⇡2f2⇡MN

3M2

N �m2⇡p

4M2N �m2

arccosm⇡

2MN+m⇡ log

m⇡

MN

!

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σπN at O(p3) in covariant ChPT at in extended on-shell mass scheme :

Extract low energy constants (c1, c2, c3, c4) from O(p2) Lorentz covariant ChPT

Fits for several πN scattering data sets :c1 c2 c3 c4

KH �0.80(6) 1.12(13) �2.96(15) 2.00(7)

GW �1.00(4) 1.01(4) �3.04(2) 2.02(1)

EM �1.00(1) 0.58(3) �2.51(4) 1.77(2)<latexit sha1_base64="1aOPbLGE41vI9qqrxQcVCeull6Q=">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</latexit><latexit sha1_base64="1aOPbLGE41vI9qqrxQcVCeull6Q=">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</latexit><latexit sha1_base64="1aOPbLGE41vI9qqrxQcVCeull6Q=">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</latexit><latexit sha1_base64="1aOPbLGE41vI9qqrxQcVCeull6Q=">AAADWnichVHLbtNQEB0nPFoXaHgskNhEJLWcBdZcJ2kCqwqEqISQ+iBNpbiK7NvbxKpjW7YTqURZsC0fwIIVSAghPoMNP8CiEj+AWBbEhgVz7RAeFTCWPTNn5ozP3OuEnhsniEdKLn/q9Jmzc/PqwrnzFxYLFy9txcEw4qLFAy+Ith07Fp7ri1biJp7YDiNhDxxPtJ39O7LeHokodgP/YXIQip2B3fPdPZfbCUHdwhfVckTP9ceJTZRJp7+jWlz4iYhcv/dLbejZ0WTMpU1Urcy7rFyUzsxcNXO1smWpVl+KUe+vauUbZTSaqC9XNGYwU2fVisRM4+ayzuoVzTQQ9UaFOPfassBkXpPNyKQnqGpgTTfTVuLL1rsPZq2soqFRb+o/xtZZxm40iKJawt+dSZ9ltKSqdgslpAHSiicDNg1KMLW1oPAKLNiFADgMYQACfEgo9sCGmJ4OMEAICduBMWERRW5aFzABlbhD6hLUYRO6T98eZZ0p6lMuZ8Ypm9NfPHojYhZhCd/jazzGd/gGP+K3v84apzOklgPyTsYVYXfx8Orm1/+yBuQT6P9k/VNzAnvQTLW6pD1MEbkFz/ijR0+PN29tLI01fIGfSP9zPMK3tIE/+sxfrouNZyAvgP153CeDLdNgFK+bpZXb06uYg2twHXQ67waswCqsQQu40lEeK4fKk9yHfC4/n1/IWnPKlHMZfrP8le/qLMkh</latexit>

Data sets of G. Washington Group :

σπN = 59(7)MeV

(unit: GeV-1)

Consistent with the result of CD LET approach!

G. Washington Group data set and O(p3) ChPT fit of πN scattering phase shift

Page 14: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Lattice QCD approaches

13

Page 15: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Lattice QCD approaches

Direct calculation of πN sigma term

Feynman-Hellmann theorem calculation

Page 16: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Formulation:

|tsnk-tsrc|, |tvtx-tsrc|

N N

Γ

N N

Γ3-point function on lattice:

Connected diagrams Disconnected diagram(Leading contribution of σπN) (strange content)

< N | qq | N > = < 0 | N (y,tsnk) (qq)(x,tvtx) N†(0,tsrc) | 0 >

< 0 | N (y,tsnk) N†(0,tsrc) | 0 >lim 2mN

⇒ Ratio of 3pt and 2pt functions

ZS : nonperturbative renormalization (µ = 2 GeV)

! 1

Lattice QCD : direct calculation

ZS

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16

gS

s

∆t/a

∆t’=11∆t’=12∆t’=13∆t’=14∆t’=15∆t’=16

fit-0.4

-0.2

0

0.2

0.4

0.6

0 2 4 6 8 10 12 14 16S

s∆t/a

∆t’=11∆t’=12∆t’=13∆t’=14∆t’=15∆t’=16

fit

Mandatory techniques:

Noise method

Truncated solver method

(tvtx — tsrc)/a (tvtx — tsrc)/a

Operator improvement

(smearing, etc)

(for Dirac high modes)

(effectively ↑ statistics)

(need careful treatment, see next page)

Page 17: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Renormalization of qq and operator mixing

(qq)phys = Z0(qq)<latexit sha1_base64="XSDcutCuaHc4RIuQ7iwzsQ4YFAA=">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</latexit><latexit sha1_base64="XSDcutCuaHc4RIuQ7iwzsQ4YFAA=">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</latexit><latexit sha1_base64="XSDcutCuaHc4RIuQ7iwzsQ4YFAA=">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</latexit><latexit sha1_base64="XSDcutCuaHc4RIuQ7iwzsQ4YFAA=">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</latexit>

(uu+ dd� 2ss)phys = Z8(uu+ dd� 2ss)<latexit sha1_base64="A6gPLWIjCJKutU1mLMYSd5XTGs0=">AAACs3ichVHLLgRBFD3au70GG4lNxYQQMakRQiQSYWMlXoOYkUl3T6GjX+nqnmRM/IAfsLAiERFbf2DjByzYshFLEhsLd3o6EQS3UlWnTt1z61SV7lmmDDi/q1Fq6+obGpua1ZbWtvaORGfXmnRD3xAZw7Vcf0PXpLBMR2QCM7DEhucLzdYtsa7vzVX214vCl6brrAYlT2zZ2o5jbpuGFhCVTywM5nTNZ2HIhlmECgU2MhohKdlQvpzzbebtluSBOq1u5ifZP/mJJE/xKNhPkI5BEnEsuolz5FCACwMhbAg4CAhb0CCpZZEGh0fcFsrE+YTMaF/gACppQ8oSlKERu0fjDq2yMevQulJTRmqDTrGo+6Rk6Oe3/IK/8Bt+yZ/4+6+1ylGNipcSzXpVK7x8x2HPytu/KpvmALufqj89B9jGZOTVJO9exFRuYVT1xf2jl5Wp5f7yAD/lz+T/hN/xa7qBU3w1zpbE8jFU+oD09+f+CdZGU2nCS2PJmdn4K5rQiz4M0ntPYAbzWESGzr3CPR7wqIwrWUVXCtVUpSbWdONLKPYHMNqjmA==</latexit><latexit sha1_base64="A6gPLWIjCJKutU1mLMYSd5XTGs0=">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</latexit><latexit sha1_base64="A6gPLWIjCJKutU1mLMYSd5XTGs0=">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</latexit><latexit sha1_base64="A6gPLWIjCJKutU1mLMYSd5XTGs0=">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</latexit>

Renormalizations of flavor singlet and nonsinglet are in general different:

Renormalizations of ss:

Divergence due to quark loop (mixing with identity operator)

(ss)phys =1

3

"(Z0 + 2Z8)(ss) + (Z0 � Z8)(uu+ dd) +

b0a3

+ · · ·#

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Connected diagram (Large)

Disconnected diagram (Small)

ss operator mixes with the large connected diagram and identity operator!

In Wilson-type fermion, Z0—Z8 ≠ 0 :

⇒ (ss)phys affected by the increase of statistical error due to subtraction

In chiral fermion (DW,overlap) with massless renormalization scheme, the trace of quark loop (disconnected diagram) vanishes ⇒ Z0 = Z8

⇒ No need of subtraction for qq operators!

JLQCD Collaboration, PRD 83, 114506 (2011); PRD 98, 054516 (2018)

Renormalizations of uu+dd: (uu+ dd)phys =2

3

"⇣Z0 +

1

2Z8

⌘(uu+ dd) + (Z0 � Z8)(ss) +

b0a3

+ · · ·#

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uu+dd operator is less affected since connected diagram is the largest

Page 18: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Lattice QCD : Feynman-Hellmann theorem approachOn lattice, one can artificially change the current quark mass (mq)

Feynman-Hellmann theorem:@mN

@mq= hN |qq|Ni

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M

[M

eV]

N

m [MeV]RGIud

M (Φ)N

β = 3.31β = 3.5

β = 3.61β = 3.7β = 3.8β = ∞

0 5 10 15 20 25 30 35 40

1200

800

900

1000

1100

N

RGIs

M (Φ)N

β = 3.31β = 3.5

β = 3.61β = 3.7β = 3.8β = ∞

m [MeV]

M

[M

eV]

100 110 120 130 140 150 160 170 180 190

1200

1100

1000

900

800

mud [MeV] ms [MeV]

mN [

MeV

]

mN [

MeV

]

BMW Collaboration, PRL 116, 172001 (2016)

Correct renormalization needed for Wilson-type quark (see next page)

Feynman-Hellmann theorem may be combined with ChPT :

Fit mud dependence of mN in ChPT with lattice data and take derivative

(see Lutz (2014), Ren (2015), Ling (2017))

Page 19: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Issue of renormalization in Feynman-Hellmann approach

C. Michael et al. (UKQCD Collaboration), NPB Proc. Suppl. 106, 293 (2002).

@(amN )

@mbval

= lim|tsnk�tsrc||txyz�tsrc|

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From Feynman-Hellmann theorem, we have

N N

Γ

N N

Γ

@(amN )

@mbsea

= �Nf lim|tsnk�tsrc||txyz�tsrc|

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and

mbsea = mA + amsea

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mbval = mA + amval

<latexit sha1_base64="p4Vwe6oDLWv9dK4bgNQlslEFNzM=">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</latexit><latexit sha1_base64="p4Vwe6oDLWv9dK4bgNQlslEFNzM=">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</latexit><latexit sha1_base64="p4Vwe6oDLWv9dK4bgNQlslEFNzM=">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</latexit><latexit sha1_base64="p4Vwe6oDLWv9dK4bgNQlslEFNzM=">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</latexit>

N N N N

@mN

@mval=

@(amN )

@mbval

<latexit sha1_base64="LXAnRwVUjw0Fs/E+3N/6GmXDP5s=">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</latexit><latexit sha1_base64="LXAnRwVUjw0Fs/E+3N/6GmXDP5s=">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</latexit><latexit sha1_base64="LXAnRwVUjw0Fs/E+3N/6GmXDP5s=">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</latexit><latexit sha1_base64="LXAnRwVUjw0Fs/E+3N/6GmXDP5s=">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</latexit>

Connected : simple derivative of valence q mass

Disconnected : derivative of sea q mass have several terms!

Renormalization of quark mass:

@mN

@msea(1� amseaB �X) =

@(amN )

@mbsea

+ (amseaB +X)@(amN )

@mbval

�mNB<latexit sha1_base64="AZRFGJffklY78fyG+Z+pb2zNSMU=">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</latexit><latexit sha1_base64="AZRFGJffklY78fyG+Z+pb2zNSMU=">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</latexit><latexit sha1_base64="AZRFGJffklY78fyG+Z+pb2zNSMU=">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</latexit><latexit sha1_base64="AZRFGJffklY78fyG+Z+pb2zNSMU=">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</latexit>

B ⌘ @(ln a)

@mbsea

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X ⌘ @mA

@mbsea

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mA : additive mass renormalization (in Wilson-type fermion) (must be determined for each gauge ensemble)

(Note that lattice spacing depends also on )mbsea

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⇒ Mixing with valence quark contribution

(bare mass)

(consequence of additive mass renormalization)

(connected) (disconnected)

Page 20: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Both approaches seem robust…

Then what to do??

19

Page 21: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Road to the resolution of the puzzleCalculation of πN (ππ, NN, …) scattering phase shift(s) in lattice QCD

Input of Cheng-Dashen LET approach with Roy-Steiner analysis

Check the consistency with the direct calculation

⇒ HAL QCD method (see talks of T. Hatsuda, K. Sasaki)

What if consistent ?If yes :

If no :

Lattice QCD has trouble with exp. data

ChPT based extractions have problems

Page 22: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Road to the resolution of the puzzleCalculation of πN (ππ, NN, …) scattering phase shift(s) in lattice QCD

Input of Cheng-Dashen LET approach with Roy-Steiner analysis

Check the consistency with the direct calculation

⇒ HAL QCD method (see talks of T. Hatsuda, K. Sasaki)

What if consistent ?If yes :

If no :

Lattice QCD has trouble with exp. data

ChPT based extractions have problems

Issue with the convergence of ChPT?σπN has small correction in SU(3) ChPT

⇒ Did we overlook contribution beyond SU(3) ChPT?V. Bernard et al., Phys. Lett. B 389, 144 (1996)

Page 23: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

Road to the resolution of the puzzleCalculation of πN (ππ, NN, …) scattering phase shift(s) in lattice QCD

Input of Cheng-Dashen LET approach with Roy-Steiner analysis

Check the consistency with the direct calculation

⇒ HAL QCD method (see talks of T. Hatsuda, K. Sasaki)

Direct measurement in high-energy QCD experiments?

The integration of the twist-3 quark distribution eq(x) gives the sigma term

⇒ Measure eu,d(x) in experiments to determine σπN

⇒ Measurement of es(x) may provide the first experimental <N|ss|N>

(Compete with the QCD sum rules based extraction from J-PARC exp., ask P. Gubler for detail)

What if consistent ?If yes :

If no :

Lattice QCD has trouble with exp. data

ChPT based extractions have problems

Issue with the convergence of ChPT?σπN has small correction in SU(3) ChPT

⇒ Did we overlook contribution beyond SU(3) ChPT?V. Bernard et al., Phys. Lett. B 389, 144 (1996)

Page 24: Present status of the pion-nucleon sigma term (and strange ......Pion-nucleon sigma term Strange content of nucleon Current quark mass contribution to the nucleon mass (trace anomaly

• A review of πN sigma term and strange content. • An important puzzle in the calculations of σπN :

ChPT based phenomenological approach vs lattice QCD⇒ Do not agree, but both seem OK…

• The evaluation of <N|ss|N> on lattice requires a careful renormalization : consistent among modern lattice QCD works, with relatively small absolute value.

Summary

Future prospects:• Lattice calculation of ππ, πN, NN, πK, … phase shifts

will give a strong hint whether ChPT approach is OK or not.

• The high-energy experimental measurement of the twist-3 PDF eq(x) will unveil the true value of σπN.

• Could you give any other ideas?