## Author(s)

Baeza-Ballesteros, Jorge## Abstract

This doctoral thesis focuses on the application of lattice techniques to study two separate topics: hadron interactions within quantum chromodynamics (QCD), and the emission of particles and gravitational waves (GWs) from cosmic string loops. The first part of the dissertation deals with the study of two- an three-particle interactions using lattice QCD simulations, complemented by the limit of large number of colors and chiral perturbation theory. In particular, this part presents results on the study of meson interactions as a function of the number of colors, three-pion interactions in chiral perturbation theory, and two- and three-particle scattering in the (1+1)-dimensional O(3) non-linear sigma model. The second part of the thesis is devoted to the study of the dynamics and decay into particles and GWs of cosmic string loops, using classical-field-theory lattice computations. Results are presented for both global and local string loops.

## Figures

Schematic representation of the gluon propagator and self-interacting vertices in the standard (top row) and double-line (bottom row) notation.

Schematic representation of the two-particle correlation function. Circles represent the Bethe-Slapeter kernels and squares are the operator insertions, while $C_{2,0}$ contains all diagrams with no two-particle $s$-channel loop.

: $\tilde{F}$ factor

: $G$ factor

: Disconnected ($D$) diagram, $\cO(\Nc^2)$

: Connected ($C$) diagram, $\cO(\Nc)$

: $\cO(\Nc^0)$

: $\cO(\Nf\Nc^0)$

Additional one-loop Feynman diagrams needed to study two-pion scattering in the $SS$ and $AA$ channels in large $\Nc$ ChPT. Solid lines depict pions, while dotted ones represent the $\eta'$.

Additional one-loop Feynman diagrams needed to study two-pion scattering in the $SS$ and $AA$ channels in large $\Nc$ ChPT. Solid lines depict pions, while dotted ones represent the $\eta'$.

Additional one-loop Feynman diagrams needed to study two-pion scattering in the $SS$ and $AA$ channels in large $\Nc$ ChPT. Solid lines depict pions, while dotted ones represent the $\eta'$.

: $SS$ channel

: $AA$ channel

: Results for the $s$-wave scattering length obtained using the threshold expansion to $\cO(L^{-5})$, together with LO ChPT predictions. Both axes are multiplied by a factor that eliminates leading $\Nc$ dependencies. The physical point is indicated with a vertical dashed line.

: $SS$ channel, 3A10 ensemble

: $SS$ channel

: $AA$ channel

: Linear interpolation of $q\cot\delta_0$ to the reference value $\xi_\text{ref}=0.14$ (dashed line) for the ``3A'' ensembles at $q_\text{ref}^2=-0.08$. We show the results for the unitary (magenta squares) and the mixed-action (green dots) setups, and depict the result as an empty point. Only the three closest points are used for the interpolation.

: Constrained continuum extrapolation of $q\cot\delta_0$ computed from a unitary (magenta squares) and a mixed-action (green dots) setups at ($q_\text{ref}^2, \xi_\text{ref}=(-0.08,0.14)$. The continuum results is shown as a black diamond.

: $L_R$ from SU(4) ChPT

: $L_R$ from U(4) ChPT

: $K_R$ from U(4) ChPT

Results of a simultaneous chiral and $\Nc$ fit to U(4) ChPT predictions of both the $SS$ (top) and $AA$ (bottom) channels. Points represent our lattice results multiplied by a factor that eliminates leading chiral and $\Nc$ dependencies, and horizontal lines are the best fit predictions from U(4) ChPT, as summarized in \cref{tab:largeNpions:bothchannelglobalfits}. Empty points in the $SS$ channel are not fitted. Within each color, $\xi$ decreases from left to right.

Unitarized result for $q\cot\delta_0$ for the $AA$ channel obtained applying the inverse amplitude method to the U($\Nf$) ChPT amplitude in \cref{eq:largeNpions:AAUamplitudeChPT}, with values of the parameters as defined in the text. We observe a change of sign, indicating the possible presence of a tetraquark resonance.

Relative overlaps of the lowest-lying finite-volume states, $|n\rangle$, with the operators used to solve the GEVP---see \cref{eq:largeNmesons:overlaps}, for the rest-frame $A_1^+$ irrep for the $AA$ channel with $\Nc=3$. Different colors correspond to different types of operators: $\pi\pi$ (red), $\rho\rho$ (blue) and local tetraquarks (green). Numbers in parenthesis indicate the magnitude squared of the single-particle momentum, in units of $(2\pi/L)^2$.

Best-fit results to \cref{eq:largeNmesons:ratiodefinition} for the ground-state energy in the $\bm{P}=[0,0,1]$ frame of the $AA$ channel with $\Nc=3$, for different values of $t_\text{min}$ and two choices of $t_\text{max}$. The final result (gray band) is obtained by averaging the results using the weights from \cref{eq:QCD:weightsplateaux}, in the bottom panel. Empty points are manually excluded from the average.

Finite-volume spectra for the $SS$ channel, extracted using the full set of operators. Each panel corresponds to a different $\Nc$ and each column represents a different irrep of the cubic group and momentum frame, with $|\bm{P}|^2$ indicated in parenthesis. Horizontal solid and dashed segments indicate non-interacting $\pi\pi$ and $\rho\rho$ energies, respectively, while gray dashes lines are relevant inelastic thresholds---see \cref{fig:largeNmesons:threshold} for a full list.

Same as \cref{fig:largeNmesons:energiesSS} for the $AA$ channel.

Same as \cref{fig:largeNmesons:energiesSS} for the $AS$ channel.

: Relevant inelastic threshold in our ensembles, computed using the meson masses in \cref{tab:largeNmesons:mesonmasses}. We indicate in which channels these threshold are present in the figure legend.

: Results for the finite-volume spectrum of the $AA$ channel for $\Nc=3$ extracted using different choices of the operator set, as indicated in the figure legend.

: Results of the $s$-wave phase shift for the $SS$ channel, determined for different $\Nc$ and cubic group irreps, as indicated in the figure legends. Shaded regions indicate the best-fit results to a modified ERE---see \cref{eq:largeNmesons:mERE}.

: Same as \cref{fig:largeNmesons:PSSS} for the $AA$ channel.

Results of the $p$-wave phase shift for the $AS$ channel determined for different $\Nc$ and cubic group irreps, as indicated in the figure legends.

Results for the scattering length divided by LO ChPT predictions (left)---see \cref{eq:largeNpions:LOChPTscatteringalength}---and for the scattering range (right), for both the $SS$ and $AA$ channels. These are determined from the results of a fit to a modified effective range expansion, see \cref{eq:largeNmesons:mEREandERErelation}. Results for $\Nc=4-6$ are fitted to a linear relation (dashed lines) to extrapolate to the large $\Nc$ limit. We also indicate the result of a constrained large $\Nc$ extrapolation for all $\Nc=3-6$ results and including up to $\cO(\Nc^{-2})$ corrections (solid lines).

Results for the scattering length divided by LO ChPT predictions (left)---see \cref{eq:largeNpions:LOChPTscatteringalength}---and for the scattering range (right), for both the $SS$ and $AA$ channels. These are determined from the results of a fit to a modified effective range expansion, see \cref{eq:largeNmesons:mEREandERErelation}. Results for $\Nc=4-6$ are fitted to a linear relation (dashed lines) to extrapolate to the large $\Nc$ limit. We also indicate the result of a constrained large $\Nc$ extrapolation for all $\Nc=3-6$ results and including up to $\cO(\Nc^{-2})$ corrections (solid lines).

Scattering phase shift for the $AA$ channel with $\Nc=3$, together with the best-fit results to a modified ERE---see \cref{eq:largeNmesons:mERE}. We observe the presence of a virtual bound state, given by the solution to \cref{eq:largeNmesons:virtualstatesolution}, indicated by a star.

LO (dashed black line) and LO+NLO (grey line and band) ChPT predictions for $\Kiso$ (top) and $\Kisoone$ (bottom) as functions of $(\Mpi/\Fpi)^4$, using LECs from \rrcite{Colangelo:2001df,FLAG:2021npn} [see \cref{eq:pipipiKmatrix:LECref}]. These predictions are compared to lattice results from \rcite{Blanton:2021llb} (orange points). We also present the best fit to the lattice data (dotted orange line and band).

LO (dashed black line) and LO+NLO (grey line and band) ChPT predictions for $\Kiso$ (top) and $\Kisoone$ (bottom) as functions of $(\Mpi/\Fpi)^4$, using LECs from \rrcite{Colangelo:2001df,FLAG:2021npn} [see \cref{eq:pipipiKmatrix:LECref}]. These predictions are compared to lattice results from \rcite{Blanton:2021llb} (orange points). We also present the best fit to the lattice data (dotted orange line and band).

NLO ChPT predictions for $\Kisotwo$ and $\KA$ (left), and $\KB$ (right) as functions of $(\Mpi/\Fpi)^6$, using LECs from \rrcite{Colangelo:2001df,FLAG:2021npn} [see \cref{eq:pipipiKmatrix:LECref}]. In the case of $\KB$, we compare to lattice results from \rcite{Blanton:2021llb} (blue points).

NLO ChPT predictions for $\Kisotwo$ and $\KA$ (left), and $\KB$ (right) as functions of $(\Mpi/\Fpi)^6$, using LECs from \rrcite{Colangelo:2001df,FLAG:2021npn} [see \cref{eq:pipipiKmatrix:LECref}]. In the case of $\KB$, we compare to lattice results from \rcite{Blanton:2021llb} (blue points).

: Comparison between numerical results and the threshold expansion for $\Kdf$, evaluated for the first momentum family in \cref{tab:pipipiKmatrix:momentafamilies}. The comparison is presented for two pion masses, with $\Mpi=340$ MeV corresponding to the heaviest pion mass used in \rcite{Blanton:2021llb}. The dashed vertical line indicates the inelastic threshold at $E^*=5\Mpi$.

: Comparison of the numerical and the threshold results for the different contributions to $\Kdf$---see \cref{eq:pipipiKmatrix:schematicequation}---evaluated for the first momentum family in \cref{tab:pipipiKmatrix:momentafamilies} for $\Mpi=340$ MeV. The panels correspond to the OPE part (top left), non-OPE part (top right) and BH subtraction (bottom). The dashed vertical line indicates the inelastic threshold at $E^*=5\Mpi$.

: Caption not extracted

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Comparison of contributions to $\Kdf^{\NLO,\OPE}$ from different partial waves of the NLO dimer amplitude, numerically evaluated for the first momentum family in \cref{tab:pipipiKmatrix:momentafamilies} with $\Mpi=340$ MeV. The black line is the full result including all partial waves, and the vertical dashed line is the inelastic threshold. Contributions from $\ell\geq 4$ are negligible.

LO (thin lines) and LO+NLO (thick lines and bands) ChPT predictions for the threshold-expansion coefficients of $\Kdf$ as a function of $(\Mpi/\Fpi)^4$, divided by a factor of $10^3$. Bands represent the uncertainties arising from the LECs---see \cref{eq:pipipiKmatrix:comparisonlattice}. Each panel corresponds to a different isospin and irrep of the permutation group, for the $\Ippp=1$ channel. Some coefficients are rescaled for legibility. We note $\cK^\text{AS}_{1}$, $\cK^\text{AS}_{3}$ and $\cK^\text{AS}_{4}$ have no error, as they are independent of the LECs.

Convergence of the threshold expansion at $\Mpi=340$ MeV, for various components of $\mKdf^\NLO$ (LO is omitted) in the symmetric basis using the kinematic \mbox{configuration} in \cref{eq:isospinKmatrix:kinematicconfiguration}. Results are represented for the total $K$-matrix as well as for the separate contributions according to \cref{eq:isospinKmatrix:NLOseparation}, where ``BH'' refers to $\mK^\BH=-\mD^\BH$. Lines represent the threshold expansion, and the the width of the bands is the difference to the exact results, which thus correspond to the other end of the band. All values are divided by a \mbox{factor $10^3$}. Vertical lines represent the five-pion threshold.

Comparison of contributions to $\mKdf^{\NLO,\OPE}$ from different dimer partial waves in $\mM_2^\NLO$, evaluated numerically at $\Mpi=340$ MeV using the kinematic configuration in \cref{eq:isospinKmatrix:kinematicconfiguration}. All values are divided by a \mbox{factor $10^3$}. Partial waves that are identically zero, as well as the negligibly small $\ell>3$, are omitted.

Diagramatic representation of the two equivalent ways of computing the three-particle $S$-matrix in a factorizable theory. This equivalence leads to the Yang-Baxter equations, \cref{eq:O3model:factorization}. Note we have imposed that the final set of momenta is identical to the initial one, which is not true for the flavor indices. We use $S_{ab}=S(\theta_{ab})$, where $\theta_{ab}$ is the relative rapidity of particles $a$ and $b$, and leave the flavor indices implicit.

Diagramatic representation of the two equivalent ways of computing the three-particle $S$-matrix in a factorizable theory. This equivalence leads to the Yang-Baxter equations, \cref{eq:O3model:factorization}. Note we have imposed that the final set of momenta is identical to the initial one, which is not true for the flavor indices. We use $S_{ab}=S(\theta_{ab})$, where $\theta_{ab}$ is the relative rapidity of particles $a$ and $b$, and leave the flavor indices implicit.

Analytic results for the two-particle scattering phase shift in all isospin channels. We focus on the $\Iss=2$ and $\Iss=1$ channels in this work.

: Two-particle contractions

: Three-particle contractions

: Diagramatic representation of the Wick contractions needed for the computation of two- and three-particle correlation functions in this work. Initial and final states are represented at the right and left of the diagrams, respectively, and the corresponding momenta are $\{k_i\}$ and $\{p_i\}$, assigned to the vertices in order from top to bottom.

: Caption not extracted

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Summary of the simulations used in this work. All have $mT\approx 20.4$ to avoid thermal effects.

: First excited state in the $\Iss=2$ channel

: Ground state in the $\Isss=3$ channel

: Best-fit results to a single exponential of one of the generalized eigenvalues, $\lambda_n$, of the correlation matrix of the two-particle isospin-two channel (left) and three-particle isospin-three channel (right). Both cases correspond to the rest-frame and the B1 ensemble. Fits are performed with varying $\tmin$ and fixed $\tmax$ and the final result is obtained by averaging the results using weights based on the Akaike Information Criterion---see \cref{eq:QCD:weightsplateaux}---presented in the lower pannels.

Comparison of the finite-volume energies for the $\Isss=2$ channel in the B1 ensemble determined using different sets of operators to compute the matrix of correlators.

: First excited state for $\bm{P}=2\pi/L$ in the $\Iss=1$ channel.

: Ground state for $\bm{P}=0$ with positive parity in the $\Isss=3$ channel.

: $\Iss=2$ channel

: $\Iss=1$ channel

: Results for the finite-volume energies assuming $\Kdf=0$ for the $\Isss=3$ channel in the rest frame with positive parity, obtained using two different cutoff functions. We observe the presence of unphysical solutions (red dashed lines) with the standard cutoff choice, \cref{eq:hadrons:standardcutoff}, which decay faster than expected and do not converge to threshold. Physical results (blue solid lines and dots), on the other hand, show negligible differences between the two cutoff choices.

: Eigenvalues of $\tilde{K}_2^{-1}-\tilde{F}-G$ for the $\Isss=3$ channel rest frame with positive parity, for $mL=10$, presented for the standard (blue solid lines) and the hard cutoff (blue dots). Vertical dashed lines indicate the energies at which the size of the matrix is increased.

Results for the continuum-extrapolated finite-volume energies in the $\Isss=3$ channel. Lattice results (red dots) are represented together with analytical predictions made assuming $\Kdf=0$ and using the hard cutoff in \cref{eq:O3model:hardcutoff} (solid blue lines), and non-interacting energies (dashed black lines). Each panel corresponds to a different momentum frame, with $\bm{P}=2\pi n/L$. We also present the two parity sectors, $p$, for the $n=0$ case.

Same as \cref{fig:O3model:energiesthreeparticlesI3} for the $\Isss=2$ channel.

Same as \cref{fig:O3model:energiesthreeparticlesI3} for the $\Isss=0$ channel.

Results for the isotropic $K$-matrix in the $\Isss=3$ channel, computed using the standard cutoff for those states with $E^*<4m$.

Radial profile of the global NO vortex with $k=1$. We represent the short-range (dashed line, left panel) and the long-range (dotted line, left panel, in logarithmic scale) approximations.

Radial profile of the global NO vortex with $k=1$. We represent the short-range (dashed line, left panel) and the long-range (dotted line, left panel, in logarithmic scale) approximations.

Representation of the algorithm used to join the segments of strings determined from the pierced plaquettes, in the case of a four-segment loop.

: End of diffusion.

: End of extra-fattening.

: Moment when an isolated loop is found.

Evolution of the scale factor (top), the $\lambda$ parameter (center) and the core width of the string (bottom) during the extra-fattening phase and the subsequent evolution in a RD background, separated by a vertical dashed line. Both $\lambda$ and $w_\text{c}$ are normalized by their initial values at the end of diffusion.

Time-evolution of the comoving mean string separation (left) and the mean-squared velocity (right) of a network of global strings averaged over 10 independent realizations. The networks are generated with $\tilde{\ell}_\str=15$ and simulations are performed with $\tilde{L}=256$ and $\delta\tilde{x}=0.25$. The vertical dotted line indicates the end of extra-fattening and the shaded region corresponds to one standard deviation.

Time-evolution of the comoving mean string separation (left) and the mean-squared velocity (right) of a network of global strings averaged over 10 independent realizations. The networks are generated with $\tilde{\ell}_\str=15$ and simulations are performed with $\tilde{L}=256$ and $\delta\tilde{x}=0.25$. The vertical dotted line indicates the end of extra-fattening and the shaded region corresponds to one standard deviation.

Schematic representation of the relevant variables used for the creation of a pair of parallel boosted strings. We have exagerated $b$ compared to the actual simulations, which use $\rc\lesssim b\ll a =\Lonefourth/2$.

: Beginning of the simulation.

: Instant before isolating the inner loop.

: Instatn after isolating the inner loop.

: Network loops

: Artificial loops

Lifetime of artificial loops with different initial velocities as a function of their initial angular momentum, measured using \cref{eq:global:stringJweighteddefinition}. The line is the result of a power-law fit to all families.

: Massive modes

Lifetime of artificial loops with different initial velocities as a function of their initial angular momentum, measured using \cref{eq:global:stringJweighteddefinition}. The vertical dashed line indicates the scale of the core radius, $\tilde{k}_\text{c}=2\pi/\tilde{r}_\text{c}$, and the dot-dashed line is the result of a power-law fit to the $r_\text{ini}=r_\text{c}$ result.

: Network loop

: Artificial loop

: Evolution of the GW energy density power spectrum for artificial loops generated with $v_1=0.6$, $v_2=0.7$ and $\sin\alpha=0.5$, with fixed $\delta\tilde{x}=0.25$ $\tilde{L}_{1/4}=64$ and varying $L$. The vertical dotted line indicates the scale of the initial length of the string, $\tilde{k}_0=2\pi/\tilde{L}_0$. Spectra go from early to late times from bottom to top, with a separation of four units of program time between consecutive lines.

: Evolution of the GW energy density power spectrum of an artificial loop with $\delta\tilde{x}=0.25$, $\tilde{L}=512$ and $\tilde{L}_{1/4}=64$, generated with $v_1=0.6$, $v_2=0.7$ and $\sin\alpha=0.5$ Each line corresponds to a different time, going from purple to red, with separation of two units of program time. The vertical dotted line indicates the scale of the initial length of the string, $\tilde{k}_0=2\pi/\tilde{L}_0$, and the dot-dashed line is a fit to the high-frequency tail of the final-time spectrum.

Rolling-averaged GW power emitted by network loops of several lengths (left) and artificial loops with different boost velocities (right), computed using \cref{eq:global:rollingaverage}. In the case of network loops, each color corresponds to a different initial length and the linestyle indicates the lattice parameters, as indicated in the plot legends. For artificial loops, each color refers to a different pair of boost velocities. For comparison, the typical NG result (for $\mu=\pi v^2$ and $\Gamma=50$) is shown as a horizontal dashed line. Finally, The grey bands represent the average value of the emission power in the range $\tilde{t}=30-100$ for each type of loop.

Radial profile of the NO vortex for $\beta=1$ and $k=1$, for the scalar and the gauge fields.

Time evolution of the mean string separation and the mean-squared velocity of a network of local strings averaged over 20 independent realizations. Networks are generated with $\tilde{\ell}_\str=15$ and simulations are performed with $\tilde{L}=256$ and $\delta\tilde{x}=0.25$. The vertical lines corresponding to the end of the extra-fattening, RD evolution and the transient phases, as indicated. The time coordinate corresponds to comoving time during the first two phases, while it is the standard time coordinate in Minkowski spacetime. Bands indicate one standard deviation.

Time evolution of the mean string separation and the mean-squared velocity of a network of local strings averaged over 20 independent realizations. Networks are generated with $\tilde{\ell}_\str=15$ and simulations are performed with $\tilde{L}=256$ and $\delta\tilde{x}=0.25$. The vertical lines corresponding to the end of the extra-fattening, RD evolution and the transient phases, as indicated. The time coordinate corresponds to comoving time during the first two phases, while it is the standard time coordinate in Minkowski spacetime. Bands indicate one standard deviation.

Representation of a single plaquette in the $(x,y)$-plane pierced by a string going in the $z$ diraction (out of the paper). We indicate the initial values to which the different links are set. The scalar field is fixed to $\varphi=v$ everywhere.

: End of diffusion.

: Instant after the outer loop disappears.

: Network loop.

: Artificial loop of type \RNum{1} for $v_1=v_2=0.6$.

: Artificial loop of type \RNum{1} for $v_1=0.3$ and $v_2=0.6$.

: Artificial loop of type \RNum{2}.

: Power-law fit

: Linear fit

Three-dimensional snapshots of $|\varphi|^2=0.2v^2$ surfaces of different moments of the evolution of an artificial loop of type \RNum{1}, simulated with $\tilde{L}=64$, $\delta\tilde{x}=0.25$, $v_1=v_2=0.25$ and $\sin\alpha=0.4$. The secondary loop has already vanished at $\tilde{t}=50$, while the other one oscillates several times before decaying. The field has been periodically shifted by $L/2$ in the $x$ and $z$ directions so that the outer loop is centered in the figures.

Three-dimensional snapshots of $|\varphi|^2=0.2v^2$ surfaces of different moments of the evolution of an artificial loop of type \RNum{1}, simulated with $\tilde{L}=64$, $\delta\tilde{x}=0.25$, $v_1=v_2=0.25$ and $\sin\alpha=0.4$. The secondary loop has already vanished at $\tilde{t}=50$, while the other one oscillates several times before decaying. The field has been periodically shifted by $L/2$ in the $x$ and $z$ directions so that the outer loop is centered in the figures.

Three-dimensional snapshots of $|\varphi|^2=0.2v^2$ surfaces of different moments of the evolution of an artificial loop of type \RNum{1}, simulated with $\tilde{L}=64$, $\delta\tilde{x}=0.25$, $v_1=v_2=0.25$ and $\sin\alpha=0.4$. The secondary loop has already vanished at $\tilde{t}=50$, while the other one oscillates several times before decaying. The field has been periodically shifted by $L/2$ in the $x$ and $z$ directions so that the outer loop is centered in the figures.

: Estimated lifetime without double-line collapse

: Measured lifetime with double-line collapse

Time evolution of the length of an artificial loop of type \RNum{1}, generated with $v_1=0.3,v_2=0.6$ and $\sin\alpha=0.5$, in a lattice of size $\tilde{L}=56$, for varying UV resolution. The length is measured from the number of pierced plaquettes.

GW power spectra produced by an artificial loop of type \RNum{1}, generated with $v_1=0.3,v_2=0.6$ and $\sin\alpha=0.5$, in a lattice of size $\tilde{L}=56$ for varying UV resolution. The vertical dashed line indicates the scale of the string core, $\tilde{k}=2\pi/\tilde{r}_\text{c}$. Spectra are represented every unit of program time, going from early (purple) to late (red) times.

GW emission power for both network (left) and artificial loops of \mbox{type \RNum{1}} (right), computed using \cref{eq:global:rollingaverage} and \cref{eq:local:PowerAverageArtificial}, respectively. The grey band and line represents an average of the emission power, and we compare the NG predictions (horizontal dashed line), estimated with $\Gamma=50$ and $\mu=\pi v^2$. The power is represented as a function of the time since the emission of GWs started, normalized by the total decay time of each loop. Network and artificial loops are simulated with $\delta\tilde{x}=0.25$ and $\delta\tilde{x}=0.1875$, respectively.

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