Long strings emerge in many Quantum Field Theories, for example as vortices in
Abelian Higgs theories, or flux tubes in Yang-Mills theories. The actions of such
objects can be expanded in the number of derivatives, around a long straight string
solution. This corresponds to the expansion of energy levels in powers of

Article funded by SCOAP3

3$ classical Lorentz invariance allows a six-derivative term, but its presence modifies the form of the generators (while higher-derivative allowed terms do not); and then quantum considerations show that its value is actually fixed. \item The orthogonal (``conformal'') gauge in which diffeomorphism is fixed up to conformal transformations and Lorentz symmetry is maintained. In this formalism, the action is constrained by conformal invariance. \end{enumerate} This work aims at generalizing the results of Aharony and Komargodski to the case of Supersymmetry (SUSY), specifically $D=4$, $N=1$ SUSY. In a supersymmetric theory, a string may break $D=4$, $N=1$ SUSY either completely, or partially into $D=2$, $N=\left(2,0\right)$, as was shown by Hughes and Polchinski~\cite{key-11}. The breaking of SUSY generators adds massless fermionic modes of excitation, known as Goldstinos. The action can then be written as a functional of the NGBs and Goldstinos, and expanded as in the fully bosonic case by the number of derivatives. For the two cases of complete and partial breaking of SUSY, a complete classification of action terms has yet to be made. In the scope of this work we will only explore the case of complete SUSY breaking, which is relevant in particular for confining strings in supersymmetric Yang-Mills theory, and it is the main goal of this work to classify action terms for this case. As a final step, we will calculate the form of the energy level correction for a closed string on a circle, arising from the lowest order new term we find, so that our results can be verified by lattice simulations at some later point. The outline of this paper is as follows. In the next section we review well established results, as well as notations and definitions we will use, and eventually a graphical approach, originally presented by Gliozzi and Meineri~\cite{key-2}, to find invariant actions for bosonic effective strings. In section~\ref{sec:3} we extend this approach to include Goldstinos, and in section~\ref{sec:4} we use the extended approach to find invariant actions for SUSY breaking effective strings, including a new term at order $1/L^{5}$. In section~\ref{sec:5} we formulate prohibition rules which show that our list of invariant actions is indeed exhaustive, and in section~\ref{sec:6} we derive the energy corrections that follow from our new term. Finally we discuss our results and draw some conclusions. ]]>

1\right)$ \begin{align} \delta\left(\d_{a_{1}\cdots a_{n}}^{n}X^{i}\right) & =-\epsilon^{bj}\paren{\d_{b}X^{i}\d_{a_{1}\cdots a_{n}}^{n}X^{j}+\sum_{k}\d_{a_{k}}X^{j}\d_{ba_{1}\cdots a_{k-1}a_{k+1}\cdots a_{n}}^{n}X^{i}}+\nonumber \\ & \qquad\ \thesis{+\sum_{k,l}\d_{a_{k}a_{l}}^{2}X^{j}\d_{ba_{1}\cdots a_{k-1}a_{k+1}\cdots a_{l-1}a_{l+1}\cdots\cdots a_{n}}^{n-1}X^{i}+\dots} . \end{align} Where the first two terms add a scale zero vertex on each on the legs, and are canceled by the moves $\eta^{ab}\r g^{ab},\delta^{ij}\r t^{ij}$ as we have seen in the previous section. The third term has a scale $n-2$ vertex connected to a scale 1 vertex so it can only be canceled by terms containing such vertices, the fourth has a scale $n-3$ vertex connected to a scale 3 vertex and so on. We can cancel these terms by defining a sort of covariant derivative. GM define this for the scale 2 term \begin{equation} \d_{abc}^{3}X^{i}\r\del_{abc}^{3}X^{i}=\d_{abc}^{3}X^{i}-\left(\d_{ab}^{2}X^{j}\d_{d}X^{j}\d_{ec}^{2}X^{i}g^{de}+\text{cyclic permutations of }abc\right) \end{equation} so that \begin{equation} \delta\left(\del_{abc}^{3}X^{i}\right)=-\epsilon^{bj}\left(\d_{b}X^{i}\d_{a_{1}\cdots a_{n}}^{n}X^{j}+\sum_{k}\d_{a_{k}}X^{j}\d_{ba_{1}\cdots a_{k-1}a_{k+1}\cdots a_{n}}^{n}X^{i}\right) \end{equation} which can be generalized to the $n$-th derivative with \begingroup \allowdisplaybreaks \medmuskip=0mu \begin{align} \del_{a_{1}\cdots a_{n}}^{n}X^{i} & =\d_{a_{1}\cdots a_{n}}^{n}X^{i}-\left(\d_{a_{1}\cdots a_{n-1}}^{n-1}X^{j}\d_{b}X^{j}\d_{ca_{n}}^{2}X^{i}g^{bc}+\text{cyclic permutations of }a_{1}\dots a_{n}\right)+\nonumber \\ &\phantom{{}={}} -\left(\d_{a_{1}\cdots a_{n-2}}^{n-2}X^{j}\d_{b}X^{j}\d_{ca_{n-1}a_{n}}^{3}X^{i}g^{bc}+\text{cyclic permutations of }a_{1}\dots a_{n}\right)+\dots \end{align} \endgroupWe can generalize this for the supersymmetric case by noting that \begin{align} \delta\d_{a_{1}\cdots a_{n}}^{n}X^{i} & =-\epsilon^{bj}\paren{\d_{b}X^{i}\d_{a_{1}\cdots a_{n}}^{n}X^{j}+\sum_{k}\d_{a_{k}}X^{j}\d_{ba_{1}\cdots a_{k-1}a_{k+1}\cdots a_{n}}^{n}X^{i}}\nonumber \\ &\quad \qquad\ \thesis{+\sum_{k,l}\d_{a_{k}a_{l}}^{2}X^{j}\d_{ba_{1}\cdots a_{k-1}a_{k+1}\cdots a_{l-1}a_{l+1}\cdots\cdots a_{n}}^{n-1}X^{i}+\dots}\nonumber \\ &\quad +i\overline{\theta}\paren{\gamma^{i}\d_{a_{1}\cdots a_{n}}^{n}\psi+\gamma^{b}\sum_{k}\d_{a_{k}}\psi\d_{ba_{1}\cdots a_{k-1}a_{k+1}\cdots a_{n}}^{n}X^{i}}+\label{eq:d1}\\ &\quad \qquad\thesis{+\gamma^{b}\sum_{k,l}\d_{a_{k}a_{l}}^{2}\psi\d_{ba_{1}\cdots a_{k-1}a_{k+1}\cdots a_{l-1}a_{l+1}\cdots\cdots a_{n}}^{n-1}X^{i}+\dots}\nonumber \end{align} where the first two terms of the $\theta$ variation are canceled by the move $\d_{a_{1}\cdots a_{n}}^{n}X^{i}\r C_{a_{1}\cdots a_{n}}^{i}$, and the following terms can be canceled by generalizing the above covariant derivative to the supersymmetric case such that \begingroup \allowdisplaybreaks \medmuskip=0mu \begin{align} \del_{a_{1}\cdots a_{n}}^{n}X^{i} & =\d_{a_{1}\cdots a_{n}}^{n}X^{i}-\left(\d_{a_{1}\cdots a_{n-1}}^{n-1}X^{j}\d_{b}X^{j}\d_{ca_{n}}^{2}X^{i}g^{bc}+\text{cyclic permutations of }a_{1}\dots a_{n}\right)+\nonumber \\ &\phantom{{}={}} -\left(\d_{a_{1}\cdots a_{n-2}}^{n-2}X^{j}\d_{b}X^{j}\d_{ca_{n-1}a_{n}}^{3}X^{i}g^{bc}+\text{cyclic permutations of }a_{1}\dots a_{n}\right)+\dots\nonumber \\ &\phantom{{}={}} -\left(i\overline{\psi}\gamma_{b}\d_{ca_{n}}^{2}\psi\d_{a_{1}\cdots a_{n-2}}^{n-2}X^{i}g^{bc}+\text{cyclic permutations of }a_{1}\dots a_{n}\right)+\\ &\phantom{{}={}} -\left(i\overline{\psi}\gamma_{b}\d_{ca_{n-1}a_{n}}^{3}\psi\d_{a_{1}\cdots a_{n-3}}^{n-3}X^{i}g^{bc}+\text{cyclic permutations of }a_{1}\dots a_{n}\right)+\dots\nonumber \end{align} \endgroupand similarly for derivatives acting on fermions where \begin{align} \delta\left(i\overline{\psi}\gamma^{i}\d_{a_{1}\cdots a_{n}}^{n}\psi\right) & =-i\epsilon^{bj}\paren{\overline{\psi}\gamma^{i}\d_{b}\psi\d_{a_{1}\cdots a_{n}}^{n}X^{j}+\sum_{k}\d_{a_{k}}X^{j}\overline{\psi}\gamma^{i}\d_{ba_{1}\cdots a_{k-1}a_{k+1}\cdots a_{n}}^{n}\psi}+\nonumber \\ &\quad \qquad\ \thesis{+\sum_{k,l}\d_{a_{k}a_{l}}^{2}X^{j}\overline{\psi}\gamma^{i}\d_{ba_{1}\cdots a_{k-1}a_{k+1}\cdots a_{l-1}a_{l+1}\cdots\cdots a_{n}}^{n-1}\psi+\dots}+\nonumber \\ &\quad +i\overline{\theta}\paren{\gamma^{i}\d_{a_{1}\cdots a_{n}}^{n}\psi+\gamma^{b}\sum_{k}\d_{a_{k}}\psi i\overline{\psi}\gamma^{i}\d_{ba_{1}\cdots a_{k-1}a_{k+1}\cdots a_{n}}^{n}\psi}+\nonumber \\ &\quad \qquad\thesis{+\gamma^{b}\sum_{k,l}\d_{a_{k}a_{l}}^{2}\psi i\overline{\psi}\gamma^{i}\d_{ba_{1}\cdots a_{k-1}a_{k+1}\cdots a_{l-1}a_{l+1}\cdots\cdots a_{n}}^{n-1}\psi+\dots} \end{align} and a similar expression when replacing the transverse index $i$ with a worldsheet index $c$. We get \begingroup \medmuskip=0mu \begin{align} \del_{a_{1}\cdots a_{n}}^{n}\psi & =\d_{a_{1}\cdots a_{n}}^{n}\psi-\left(\d_{ba_{1}\cdots a_{n-2}}^{n-1}\psi\d_{c}X^{i}\d_{a_{n-1}a_{n}}^{2}X^{i}g^{bc}+\text{cyclic permutations of }a_{1}\dots a_{n}\right)+\nonumber \\ &\quad -\left(\d_{ba_{1}\cdots a_{n-3}}^{n-2}\psi\d_{c}X^{i}\d_{a_{n-2}a_{n-1}a_{n}}^{3}X^{i}g^{bc}+\text{cyclic permutations of }a_{1}\dots a_{n}\right)+\dots\nonumber \\ &\quad -\left(\d_{ba_{1}\cdots a_{n-2}}^{n-1}\psi i\overline{\psi}\gamma_{c}\d_{a_{n-1}a_{n}}^{2}\psi g^{bc}+\text{cyclic permutations of }a_{1}\dots a_{n}\right)+\label{eq:d2}\\ &\quad -\left(\d_{ba_{1}\cdots a_{n-3}}^{n-2}\psi i\overline{\psi}\gamma_{c}\d_{a_{n-2}a_{n-1}a_{n}}^{3}\psi g^{bc}+\text{cyclic permutations of }a_{1}\dots a_{n}\right)+\dots\nonumber \end{align} \endgroup So we can get invariants by taking any bosonic seed graph, and acting on it with the following moves \begin{enumerate} \item Replacing $\eta^{ab}\r g^{ab}$ on worldsheet edges, \item Replacing the bosonic vertices with combination vertices for $n\geq2$, $\d_{a_{1}\cdots a_{n}}^{n}X^{i}\r D_{a_{1}\cdots a_{n}}^{i}$, \item Replacing $\delta^{ij}\r t^{ij}$ on transverse edges, \end{enumerate} where $D_{a_{1}\cdots a_{n}}^{i}$ is a combination vertex with higher derivatives replaced with covariant derivatives \begin{equation} D_{a_{1}\cdots a_{n}}^{i}=\del_{a_{1}\cdots a_{n}}^{n}X^{i}-i\overline{\psi}\gamma^{i}\del_{a_{1}\cdots a_{n}}^{n}\psi-\left(\d_{c}X^{i}-i\overline{\psi}\gamma^{i}\d_{c}\psi\right)\left(\delta_{\,d}^{c}+i\overline{\psi}\gamma^{c}\d_{d}\psi\right)^{-1}i\overline{\psi}\gamma^{d}\del_{a_{1}\cdots a_{n}}^{n}\psi . \end{equation} GM generate two scale 4 invariants \begin{align} I_{3} & =\sqrt{-g}t^{ij}t^{kl}\d_{ab}^{2}X^{i}\d_{cd}^{2}X^{j}\d_{ef}^{2}X^{k}\d_{gh}^{2}X^{l}g^{ha}g^{bc}g^{de}g^{fg}\\ I_{4} & =\sqrt{-g}\del_{abc}^{3}X^{i}\del_{efg}^{3}X^{j}t^{ij}g^{ae}g^{bf}g^{cg} \end{align} which we can use to generate the supersymmetric invariants \begin{align} I_{3} & =\sqrt{-g}t^{ij}t^{kl}D_{ab}^{i}D_{cd}^{j}D_{ef}^{k}D_{gh}^{l}g^{ha}g^{bc}g^{de}g^{fg}\\ I_{4} & =\sqrt{-g}D_{abc}^{i}D_{efg}^{j}t^{ij}g^{ae}g^{bf}g^{cg} \,. \end{align} We are left with the term $\d^{m}\overline{\psi}\d^{n}\psi$, which is the only vertex which gives us non-trivial supersymmetric invariants. First we note that it is antisymmetric, which means any invariant we generate must have an even number of these vertices. Second, we note that we can use the above argument for the variation of $\d^{n}\psi$ to show that given a seed graph which contains such vertices, the above moves are sufficient to generate an invariant, along with replacing $\d^{m}\overline{\psi}\d^{n}\psi\r\del^{m}\overline{\psi}\del^{n}\psi$. We will define a seed graph at scaling higher than zero as a graph containing only boson and $\d^{m}\overline{\psi}\d^{n}\psi$ vertices, which does not contain scale zero vertices. The procedure for generating invariants at scale $n$ will then be \begin{enumerate} \item Draw all seed graphs at this scale \item Perform the above moves to generate an invariant \end{enumerate} We can then generate two non-trivial scale 4 invariants \begin{align} I_{5} & =\sqrt{-g}\d_{a}\overline{\psi}\d_{bc}^{2}\psi\d_{d}\overline{\psi}\d_{ef}^{2}\psi g^{ad}g^{be}g^{cf}\\ I_{6} & =\sqrt{-g}\d_{a}\overline{\psi}\d_{b}\psi\d_{cd}^{2}\overline{\psi}\d_{ef}^{2}\psi g^{ad}g^{be}g^{cf} \end{align} as well as many higher scaling invariants. As in the scale zero case, this method is exhaustive since up to the overall multiplicative constant it fixes the coefficients of all possible terms. ]]>

}\\ \delta_{J}\mathcal{L}_{d} & =\mathcal{L}_{d-1}^{<}+\mathcal{L}_{d,d+1}^{>} \end{align} where $f$ is the number of fermions and $d$ is the number of derivatives in the term. In the case of supercharges the lowering part $\mathcal{L}_{f-1}^{<}$ is obtained by removal of one bare fermion (fermion without any derivatives acting on it); in the Lorentz transformation case to obtain the lowering part $\mathcal{L}_{d-1}^{<}$ we need to erase one $\partial X$ term from the initial $\mathcal{L}_{d}$. Let's call the bare fermion $\psi$ and $\partial X$ --- the lowering factor. If and only if a seed term contains at least one lowering factor, its lowering variation isn't zero. ]]>

1$. ]]>