The aim of this project is to build your own Effective field theory (EFT) interaction model that approximates the phase shifts you extracted with the simple interaction model from project 1. We will thus use as input the phase shift analysis from the previous project and use these as our theoretical benchmark data. We will follow closely Lepage's article *How to renormalize the Schroedinger equation*, see especially pages 8-17.
We will start by building a pionless EFT potential to describe nucleon-nucleon scattering low energies where even the longest range piece of the potential is unresolved. Therefore, the EFT potential is just a series of contact interactions. Recall that this is the domain of the effective range expansion, which is valid up to an onshell momentum \( k\sim m_{\pi}/2 \), which translates to about 10 MeV lab energy. If we were considering the true NN system with non-central terms (and full spin- and isospin-dependent terms, we drop isospin here), then the contact interactions take the form: $$ \begin{align*} V^\mathrm{LO}(\vec{q},\vec{k})=&C_S+C_T\vec{\sigma}_1\cdot\vec{\sigma}_2,\\ V^\mathrm{NLO}(\vec{q},\vec{k})=&C_1\vec{q}^{\,2}+C_2\vec{k}^2+ \vec{\sigma}_1\cdot\vec{\sigma}_2\left(C_3\vec{q}^2+C_4\vec{k}^2\right)\\ &+iC_5\frac{\vec{\sigma}_1+\vec{\sigma}_2}{2}\cdot\vec{q}\times\vec{k}+C_6\vec{q}\cdot\vec{\sigma}_1\vec{q}\cdot\vec{\sigma}_2 +C_7\vec{k}\cdot\vec{\sigma}_1\vec{k}\cdot\vec{\sigma}_2, \end{align*} $$ where \( \vec{q}=\vec{p}-\vec{p'} \) is the momentum transfer, \( \vec{k}=\frac{\vec{p}+\vec{p'}}{2} \) the average momentum, and \( \vec{p},\vec{p'} \) the relative momenta. However, since our "underlying theory" in Project 1 is spin- and isospin-independent, the above simplifies substantially, just strike out any terms with Pauli spin/isospin matrices!
Note that the coupling constants appearing in the \( ^1S_0 \) expressions are linear combinations of the coupling constants in the general expressions shown above. Their exact form will not concern us here, though you can find the relevant details on pp. 32-33 of Machleidt and Entem.
As it stands, this potential would give UV divergences if you naively used it in the LS equation of project 1. Therefore, we need to regularize the potential to cutoff the problematic high-momentum modes in the loop integrals. In class, we used a sharp cutoff so that the above would by multiplied by \( \theta(\Lambda - p)\theta(\Lambda-p') \). While conceptually simple, sharp cutoffs can be tricky, can you guess how you would need to modify the trick used to handle the principal value integral in the LS equation? (This is actually the least of our worries. If we try to use a sharp cutoff in some manybody method that relies on expanding the Hamiltonian in a harmonic oscillator basis, the expansion is very slowly converging in analogy with the Gibbs overshoot phenomena in Fourier analysis.) To avoid these subtleties, we will instead use a smooth UV regulator as $$ V(p',p)\Rightarrow f_{\Lambda}(p')V(p',p)f_{\Lambda}(p)\, $$ where we choose $$ f_{\Lambda}(p) = \exp{[-p^4/\Lambda^4]}\,. $$
Choose an appropriate value for \( \Lambda \) and fit \( C_0 \) to low energy phase shifts. Hint: Following Lepage, always use the most infrared data possible when fitting the couplings of contact terms. Also, remember Lepage's discussion on choosing the value for \( \Lambda \).
and fit the four parameters \( C_0,C_2,C_4,C'_4 \) to the low-energy phase shifts. Make a plot of your calculated LO, NLO, and NNLO phase shifts and compare them to the results from the Project 1 potential. Also make a plot a-la Fig. 3 in Lepage to show the power-law improvement with each additional order, and from this pick off the breakdown scale \( \Lambda_b \) where the EFT stops improving with additional orders. These log-log error plots are commonly known as Lepage plots, and they help show that an EFT is working as expected.
The pionless EFT basically reproduces the physics of the Effective Range Expansion, so we expect \( \Lambda_b\sim m_{\pi} \), which sets the scale of the longest-ranged Yukawa term in the toy potential of Project 1. We now want to build a toy EFT which works at higher energies. To do this, we now include the longest-ranged Yukawa potential as an explicit degree of freedom in our EFT. (We should put quotation marks around the one-pion exchange moniker since our underlying theory is a toy model, without the full spin/isospin dependence that would come with the true OPE potential.)
We will repeat parts 2a)-2b) by including the simple one-pion exchange contribution and refitting the contact term couplings. Therefore, at LO our EFT potential takes the form $$ V^\mathrm{LO}(p,p')= V_{\pi}(p,p') + C_0\, $$ while at NLO our potential takes the form $$ V^\mathrm{NLO}(p,p')= V_{\pi}(p,p') + C_0 + C_2(p^2+p'^2)\,, $$ with the analogous form for the NNLO potential.
Redo the analysis of parts 2a-2b, taking care to modify your choice of \( \Lambda \) to an appropriate value. Your analysis should be very similar as for the pionless theory, except that the value of \( \Lambda_b \) should be at a higher value, roughly equal to the mass of the next heaviest Yukawa term.
Redo the analysis of part 2c, and comment on any similarities/differences.
Here follows a brief recipe and recommendation on how to write a report for each project.
The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008 or Python. The following prescription should be followed when preparing the report: