# Nucleon form factors in dispersively improved chiral effective field theory: Scalar form factor

## Abstract

We propose a method for calculating the nucleon form factors (FFs) of $G$-parity-even operators by combining Chiral Effective Field Theory ($$\chi$$EFT) and dispersion analysis. The FFs are expressed as dispersive integrals over the two-pion cut at $$t > 4 M_\pi^2$$. The spectral functions are obtained from the elastic unitarity condition and expressed as products of the complex $$\pi\pi \rightarrow N\bar N$$ partial-wave amplitudes and the timelike pion FF. $$\chi$$EFT is used to calculate the ratio of the partial-wave amplitudes and the pion FF, which is real and free of $$\pi\pi$$ rescattering in the $t$-channel ($N/D$ method). The rescattering effects are then incorporated by multiplying with the squared modulus of the empirical pion FF. The procedure results in a marked improvement compared to conventional $$\chi$$EFT calculations of the spectral functions. We apply the method to the nucleon scalar FF and compute the scalar spectral function, the scalar radius, the $t$-dependent FF, and the Cheng-Dashen discrepancy. Higher-order chiral corrections are estimated through the $$\pi N$$ low-energy constants. Results are in excellent agreement with dispersion-theoretical calculations. We elaborate several other interesting aspects of our method. The results show proper scaling behavior in the large-$$N_c$$ limit of QCD because the $$\chi$$EFT includes $N$ and $$\Delta$$ intermediate states. The squared modulus of the timelike pion FF required by our method can be extracted from Lattice QCD calculations of vacuum correlation functions of the operator at large Euclidean distances. Our method can be applied to the nucleon FFs of other operators of interest, such as the isovector-vector current, the energy-momentum tensor, and twist-2 QCD operators (moments of generalized parton distributions).

- Authors:

- Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)

- Publication Date:

- Research Org.:
- Thomas Jefferson National Accelerator Facility, Newport News, VA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26)

- OSTI Identifier:
- 1410288

- Alternate Identifier(s):
- OSTI ID: 1409461

- Report Number(s):
- JLAB-THY-17-2525; DOE/OR/-23177-4275; arXiv:1707.07682

Journal ID: ISSN 2469-9985; PRVCAN

- Grant/Contract Number:
- AC05-06OR23177; FPA2016-77313-P

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physical Review C

- Additional Journal Information:
- Journal Volume: 96; Journal Issue: 5; Journal ID: ISSN 2469-9985

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; Form factors; dispersion relations; chiral effective field theory; scalar operators; 1=Nc expansion

### Citation Formats

```
Alarcon Soriano, Jose Manuel, and Weiss, Christian.
```*Nucleon form factors in dispersively improved chiral effective field theory: Scalar form factor*. United States: N. p., 2017.
Web. doi:10.1103/PhysRevC.96.055206.

```
Alarcon Soriano, Jose Manuel, & Weiss, Christian.
```*Nucleon form factors in dispersively improved chiral effective field theory: Scalar form factor*. United States. doi:10.1103/PhysRevC.96.055206.

```
Alarcon Soriano, Jose Manuel, and Weiss, Christian. Mon .
"Nucleon form factors in dispersively improved chiral effective field theory: Scalar form factor". United States.
doi:10.1103/PhysRevC.96.055206. https://www.osti.gov/servlets/purl/1410288.
```

```
@article{osti_1410288,
```

title = {Nucleon form factors in dispersively improved chiral effective field theory: Scalar form factor},

author = {Alarcon Soriano, Jose Manuel and Weiss, Christian},

abstractNote = {We propose a method for calculating the nucleon form factors (FFs) of $G$-parity-even operators by combining Chiral Effective Field Theory ($\chi$EFT) and dispersion analysis. The FFs are expressed as dispersive integrals over the two-pion cut at $t > 4 M_\pi^2$. The spectral functions are obtained from the elastic unitarity condition and expressed as products of the complex $\pi\pi \rightarrow N\bar N$ partial-wave amplitudes and the timelike pion FF. $\chi$EFT is used to calculate the ratio of the partial-wave amplitudes and the pion FF, which is real and free of $\pi\pi$ rescattering in the $t$-channel ($N/D$ method). The rescattering effects are then incorporated by multiplying with the squared modulus of the empirical pion FF. The procedure results in a marked improvement compared to conventional $\chi$EFT calculations of the spectral functions. We apply the method to the nucleon scalar FF and compute the scalar spectral function, the scalar radius, the $t$-dependent FF, and the Cheng-Dashen discrepancy. Higher-order chiral corrections are estimated through the $\pi N$ low-energy constants. Results are in excellent agreement with dispersion-theoretical calculations. We elaborate several other interesting aspects of our method. The results show proper scaling behavior in the large-$N_c$ limit of QCD because the $\chi$EFT includes $N$ and $\Delta$ intermediate states. The squared modulus of the timelike pion FF required by our method can be extracted from Lattice QCD calculations of vacuum correlation functions of the operator at large Euclidean distances. Our method can be applied to the nucleon FFs of other operators of interest, such as the isovector-vector current, the energy-momentum tensor, and twist-2 QCD operators (moments of generalized parton distributions).},

doi = {10.1103/PhysRevC.96.055206},

journal = {Physical Review C},

number = 5,

volume = 96,

place = {United States},

year = {Mon Nov 20 00:00:00 EST 2017},

month = {Mon Nov 20 00:00:00 EST 2017}

}

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