P. Wang, D. B. Leinweber, A. W. Thomas, and R. Young

Transcription

1 Chiral extrapolation of nucleon form factors from lattice data P. Wang, D. B. Leinweber, A. W. Thomas, and R. Young

2 1. Introduction CHPT Finite-Range- Regularization 2. Magnetic form factors 3. Extrapolation results Magnetic moments Magnetic form factors Strange form factor Octet Charge radii 4. Summary

3 Introduction M N = c 0 + c 2 m π2 + c 4 m π

4 Chiral perturbation theory: Chiral perturbation theory provides a systematic method for discussing the physics at low energy by means of an effective field theory. The general Lagrangian can be written with an increasing number of derivatives and quark(meson) mass terms. m = , MB = MB (1) + MB (2) +...

5 The lowest order Lagrangian including baryons: The Feynman diagrams can be arranged as power counting scheme (q D or m πd ). D = 4N L 2I M I B + Σ (2n N M 2n ) + Σ (n N B n ) D = = 3

6 Treating the nucleons as relativistic Dirac fields does not satisfy the power counting. Heavy baryon chiral perturbation theory: Separate the original baryon field into light and heavy components. Consider baryons as extremely heavy static source. The light component satisfies the massless Dirac equation. The heavy component is suppressed by powers of 1/m.

7 is the massless field which has the following properties: is the spin operator:

8 The Lagrangian in chiral perturbation theory is changed to be: The propagators for octet and decuplet baryons:,,

9 M N = a 0 + a 2 m π 2 + a 4 m π χ π I π (m π ) Leading order one loop Feynman diagram: With the dimensional regularization:

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13 Chiral perturbation theory with dimensional regularization fails to fit the lattice data. The chiral expansion is not convergent. High order terms are important at large pion mass. How to solve this problem? How to build the high order terms into the one loop contribution and make the chiral expansion be convergent quickly. Finite Range Regularization

14 DR: Large contributions to the integral from k portion of integral. Baryons are hard point particle. Short distance physics is highly overestimated. FRR: Remove the incorrect short distance contribution associated with the suppression of loop integral at ultraviolet region. Baryons are soft particle with structure which results in a vertex function in the loop integral.

15 The loop integral in FRR: u (k) is the regulator, for example for dipole: Expand in m π,

16 In FRR, the nucleon mass is expressed as: To any finite order, FRR is mathematically equivalent to dimensional regularization. For small pion mass, FRR and DR give almost same results. For large pion mass, can FRR fit lattice data?

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19 D. Leinweber, A. Thomas and R. Young, PRL 92 (2004) ; R. Young, D. Leinweber, A. Thomas, S. Wright, PRD 66 (2002)

20 For other regulators:

21 D. Leinweber, A. Thomas and R. Young, PRL 92 (2004)

22 D. Leinweber, A. Thomas and R. Young, PRL 92 (2004)

23 D. Leinweber, A. Thomas and R. Young, PRL 92 (2004)

24 D. Leinweber, A. Thomas and R. Young, PRL 92 (2004)

25 The FRR reproduces the lattice results very well. The high order terms are really automatically included in the one loop contribution in FRR. The residual analytic terms have a good convergent behavior. At any finite order, FRR and DR are equivalent. In this sense, DR is an approximation of FRR at low pion mass.

26 Magnetic Form factors The lowest order interaction: In the heavy baryon formalism: The compact notation:

27 Meson matrix: Octet baryons: Decuplet baryons:

28 Baryon octet magnetic moment Lagrangian: At the lowest order: Baryon decuplet magnetic moment Lagrangian: The transition magnetic operator:

29 The baryon magnetic moments can also be expressed as quark magnetic moments. For particular choice: Nucleon magnetic moments in one loop level:

30 The Pauli and Dirac form factors: In terms of charge and magnetic form factors: Equivalent definition of charge and magnetic form factors:

31 The one loop diagrams for magnetic form factors:

32 The contribution of diagram a: For dimensional regularization, I 1π NN m π.

33 The contribution of diagram b: O(m π lnm π ) The contribution of diagram c: O(m π 2 lnm π )

34 The contribution of diagram d: O(m π 2 lnm π ) The contribution of diagram e: O(m π 2 lnm π )

35 The contribution of diagram g: O(m π 2 lnm π )

36 The integrals are defined as:

37 Extrapolation results The extrapolation of magnetic moments: The mass relationships between mesons:

38 P. Wang, D. Leinweber, A. Thomas and R. Yong, PRD 75 (2007)

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40 Determination of Optimal Λ: 1. Have the best convergence, a Have the best fit of lattice data, χ 2 = (µ fit µ lat ) 2 is small. 3. Produce reasonable nucleon magnetic moments.

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44 The extrapolation of magnetic form factors: At finite momentum, we do not expand the form factors in terms of Q 2 and Q 4 as we did for the m π dependence. G MN (m π2 ) = a 0N + a 2N m π2 + a 4N m π4 + G MN (Loop) a N N 0 = b 1N + b N 2 Q 2 + b N 3 Q a N 2 = b 4N + b N 5 Q 2 + b N 6 Q a N 4 = b 7N + b N 8 Q 2 + b N 9 Q a 0, a 2 and a 4 are determined from the lattice data at finite momentum and Q 2 dependence is included in the parameters.

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50 Big error bars on lattice data. Meson form factors.

51 R. Baldini et al., EPJC 11 (1999) 709; NPA 666 (2000) 3.

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54 P. Wang, D. Leinweber, A. Thomas and R. Yong, PRD 75 (2007)

55 Quenched chiral perturbation theory

56 D. B. Leinweber, PRD 69 (2004)

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60 Strange magnetic form factor

61 u-sea and d-sea quark have same contribution,

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63 Leinweber et al., PRL94(2005)212001

64 Baryon octet charge radii

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70 Summary The nucleon magnetic moments and form factors are extrapolated from the lattice data with FRR in chiral perturbation theory. The optimal Lambda is discussed in different methods. To any finite order, FRR is mathematically equivalent to dimensional regularization. At low pion mass, both FRR and DR give reasonable results. At large pion mass, DR fails while FRR works well. High order terms in the chiral expansion are important which can be built in the one loop contribution in FRR. The residual analytic terms have a good convergent behavior. The extrapolated magnetic moments, form factors and charge radii at physical pion mass are reasonable compared with the experimental values. The same method can be applied to the extrapolation of many other lattice data in full QCD or quenched QCD.