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The cube represents a four-dimensional space. The vertical coordinate is time t. We perform an adiabatic twist in t. The depth corresponds to the plane x, y where the vortex lies. The horizontal coordinate corresponds to z. On the left, in the space (x, y, t, z = −∞), we have a double-twisted vortex. On the right, at (x, y, t, z = +∞), we have a vortex with no twist. In the middle an instanton of SO(N f ) is generated. This explains why the integral over the Chern-Simons term gets continuously changed from 2 to 0 for N f > 3 and from 4 to 0 for N f = 3.
Source publication
Dynamics of SU(N_c) Yang--Mills theories with N_f adjoint Weyl fermions is quite different from that of SU(N_c) gauge theories with fundamental quarks. The symmetry breaking pattern is SU(N_f) --> SO(N_f). The corresponding sigma model supports Skyrmions whose microscopic identification is not immediately clear. We address this issue as well as the...
Contexts in source publication
Context 1
... are different ways of performing lift- ing from SU(k)/SO(k) to SU(k). They differ from each other by an ele- ment of π 3 (SO(k)). These are indeed the topological configurations rele- vant for SO(k) instantons. They will enter when extending the space from SU(k)/SO(k) to SU(N f ) with the subgroup SO(N f ) gauged. An intuitive picture is shown in Fig. 1. Let us consider for simplicity N f ≥ 4. In this case π 3 (M N f ) = Z 2 . In the sigma model with the target space M N f topological vortices are present. They are associated with the second homotopy group π 2 , which is Z for N f = 2 and Z 2 for N f ≥ 3. As discussed in Ref. [21], the Hopf Skyrmion can be interpreted in term of an ...
Context 2
... is it possible? The answer, as explained in Figure 1, is that an instanton of SO(N f ) is generated when we perform the "unwinding" of the Hopf Skyrmion of topological charge 2. (Remember that in order to define a Hopf charge we need to extend the space to SU(N f ) with a subgroup SO(N f ) gauged.) For N f = 3 the situation is very similar but now the minimal instanton is doubled (see [22]). ...
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Citations
... Higher homotopy groups are π n (G QCD /H dd ) = π n (M 3 ) for n > 1 and the latter can be found in refs. [89,90]: π 2 (M 3 ) = Z 2 , π 3 (M 3 ) = Z 4 and so on. ...
A bstract
We find a novel confinement mechanism in the two-flavor dense quark matter proposed recently, that consists of the 2SC condensates and the P -wave diquark condensates of d -quarks. This quark matter exhibiting color superconductivity as well as superfluidity is classified into two phases; confined and deconfined phases of vortices. We establish that the criterion of the confinement is color neutrality of Aharonov-Bohm (AB) phases: vortices exhibiting color non-singlet AB phases are confined by the so-called AB defects to form color-singlet bound states. In the deconfined phase, the most stable vortices are non-Abelian Alice strings, which are superfluid vortices with fractional circulation and non-Abelian color magnetic fluxes therein, exhibiting color non-singlet AB phases. On the other hand, in the confined phase, these non-Abelian vortices are confined to either a baryonic or mesonic bound state in which constituent vortices are connected by AB defects. The baryonic bound state consists of three non-Abelian Alice strings with different color magnetic fluxes with the total flux canceled out connected by a domain wall junction, while the mesonic bound state consists of two non-Abelian Alice strings with the same color magnetic fluxes connected by a single domain wall. Interestingly, the latter contains a color magnetic flux in its core, but this can exist because of color neutrality of its AB phase.
... Higher homotopy groups are π n (G QCD /H dd ) = π n (M 3 ) for n > 1 and the latter can be found in Refs. [89,90]: π 2 (M 3 ) = Z 2 , π 3 (M 3 ) = Z 4 and so on. ...
We find a novel confinement mechanism in the two-flavor dense quark matter proposed recently, that consists of the 2SC condensates and the $P$-wave diquark condensates of $d$-quarks. This quark matter exhibiting color superconductivity as well as superfluidity is classified into two phases; confined and deconfined phases of vortices. We establish that the criterion of the confinement is color neutrality of Aharonov-Bohm (AB) phases: vortices exhibiting color non-singlet AB phases are confined by the so-called AB defects to form color-singlet bound states. In the deconfined phase, the most stable vortices are non-Abelian Alice strings, which are superfluid vortices with fractional circulation and non-Abelian color magnetic fluxes therein, exhibiting color non-singlet AB phases. On the other hand, in the confined phase, these non-Abelian vortices are confined to either a baryonic or mesonic bound state in which constituent vortices are connected by AB defects. The baryonic bound state consists of three non-Abelian Alice strings with different color magnetic fluxes with the total flux canceled out connected by a domain wall junction, while the mesonic bound state consists of two non-Abelian Alice strings with the same color magnetic fluxes connected by a single domain wall. Interestingly, the latter contains a color magnetic flux in its core, but this can exist because of color neutrality of its AB phase.
... The ground state allows topologically stable vortex configurations since π 1 ðG QCD =H dd Þ ¼ Z. See Refs. [59,60] for homotopy groups of M 3 . The broken generators of the coset space M 3 belong to 5 representation (traceless symmetric 2 × 2 tensor) of SOð3Þ C . ...
Quark-hadron continuity with two-flavor quarks that was proposed recently connects hadronic matter with neutron P23 superfluidity and two-flavor dense quark matter. This two-flavor dense quark phase consists of the coexistence of the 2SC condensates and the P-wave diquark condensates of d-quarks, which gives rise to color superconductivity as well as superfluidity. We classify vortices in this phase. The most stable vortices are what we call the non-Abelian Alice strings, which are superfluid vortices with non-Abelian color magnetic fluxes therein, exhibiting so-called topological obstruction, or a non-Abelian generalization of the Alice property. We show that a single Abelian superfluid vortex is unstable against decay into three non-Abelian Alice strings. We discover that a non-Abelian Alice string carries orientational moduli of the real projective space RP2 corresponding to the color flux therein in the presence of the P-wave condensates alone. We calculate Aharanov-Bohm (AB) phases around the non-Abelian Alice string, and find that the 2SC condensates and string’s orientational moduli must be aligned with each other because of single-valuedness of the AB phases of the 2SC condensates.
... 6 5 The existence of stable baryons in theories with adjoint fermions was investigated in refs. [30,31], where stable skyrmion solutions were identified and conjectured to correspond to composite states with mass of OpN 2 DC q, interpolated from the vacuum by non-local operators. We will not include these hypothetical states in our analysis. ...
A bstract
We introduce the gluequark Dark Matter candidate, an accidentally stable bound state made of adjoint fermions and gluons from a new confining gauge force. Such scenario displays an unusual cosmological history where perturbative freeze-out is followed by a non-perturbative re-annihilation period with possible entropy injection. When the gluequark has electroweak quantum numbers, the critical density is obtained for masses as large as PeV. Independently of its mass, the size of the gluequark is determined by the confinement scale of the theory, leading at low energies to annihilation rates and elastic cross sections which are large for particle physics standards and potentially observable in indirect detection experiments.
... To summarize our discussion on models, Table 1 reports the minimal blocks which have a potentially viable DM candidate and a sufficiently high cut-off, above 10 15 GeV, as re- 4 The existence of stable baryons in theories with adjoint fermions was investigated in Refs. [27,28], where stable skyrmion solutions were identified and conjectured to correspond to composite states with mass of OpN 2 DC q, interpolated from the vacuum by non-local operators. We will not include these hypothetical states in our analysis. ...
We introduce the gluequark Dark Matter candidate, an accidentally stable bound state made of adjoint fermions and gluons from a new confining gauge force. Such scenario displays an unusual cosmological history where perturbative freeze-out is followed by a non-perturbative re-annihilation period with possible entropy injection. When the gluequark has electroweak quantum numbers, the critical density is obtained for masses as large as PeV. Independently of its mass, the size of the gluequark is determined by the confinement scale of the theory, leading at low energies to annihilation rates and elastic cross sections which are large for particle physics standards and potentially observable in indirect detection experiments.
... A similar discussion is found in Ref.[70]. ...
We propose a new realization of strongly interacting massive particles (SIMP) as self-interacting dark matter, where SIMPs couple to the Standard Model sector through an axion-like particle. Our model gets over major obstacles accompanying the original SIMP model, such as a missing mechanism of kinetically equilibrating SIMPs with the SM plasma as well as marginal perturbativity of the chiral Lagrangian density. Remarkably, the parameter region realizing $\sigma_{\rm self}/m_{\rm DM} \simeq 0.1 \textrm{--} 1 \, {\rm cm}^{2}/{\rm g}$ is within the reach of future beam dump experiments such as the Search for Hidden Particles (SHiP) experiment.
... Such a symmetry breaking pattern can be achieved by considering technifermions transforming in the fundamental representation of SU(N). The mechanism guaranteeing the quantum stability of the skyrmion in SU(N)/SO(N) cosets is also completely different from the previous two cases, since there is no equivalent of a topological charge here43444546. The generators of the weak gauge group can be taken as in (39) as a subgroup of SO(4), but there is a unique implementation of the hypercharge — up to global transformations — yielding a quadruplet of SO(4) which can be identified with the Higgs, namely ...
... We see that the technifermions all have necessarily integer electric charge in this model, which is consistent with the integer charge of the skyrmion. Note that in contrary to the previous cases, the skyrmion in the SU(N)/SO(N) coset is not made of an antisymmetric bound state of N c technifermions, but appears as a bound state of N c (N c + 1) /2 of them43444546. A skyrmion of charge N c can for example be constructed with N c positively charged technifermions and N c (N c − 1) /2 neutral ones, although other combinations are possible. ...
Skyrmions are present in many models of electroweak symmetry breaking where
the Higgs is a pseudo-Goldstone boson of some strongly interacting sector. They
are stable, composite objects whose mass lies in the range 10-100 TeV and can
be naturally abundant in the universe due to their small annihilation
cross-section. They represent therefore good dark matter candidates. We show
however in this work that the lightest skyrmion states are electrically charged
in most of the popular little Higgs models, and hence should have been directly
or indirectly observed in nature already. The charge of the skyrmion under the
electroweak gauge group is computed in a model-independent way and is related
to the presence of anomalies in the underlying theory via the
Wess-Zumino-Witten term.
... Another interesting investigation would be to study chiral symmetry breaking in this system and its relation to the non-conservation of baryon number. See [12] for a recent investigation of chiral symmetry breaking and stability of topological solitons in SU(N) and SO(N) Yang-Mills theories. ...
We construct a classical dynamical system whose phase space is a certain infinite-dimensional Grassmannian manifold, and propose that it is equivalent to the large N limit of two-dimensional QCD with an O(2N+1) gauge group. In this theory, we find that baryon number is a topological quantity that is conserved only modulo 2. We also relate this theory to the master field approach to matrix models.
... Furthermore, a trace such as T rMu is rewritten as dx dy M(x, y)u(y, x). Now we can rewrite equation (12) in terms of integral kernels: ...
... Another interesting investigation would be to study chiral symmetry breaking in this system and its relation to the non-conservation of baryon number. See [12] for a recent investigation of chiral symmetry breaking and stability of topological solitons in SU(N) and SO(N) Yang-Mills theories. ...
... This contribution is a review of recent developments regarding the Skyrmion sector of higher representation QCD. This review is mostly based on the results of [1,2,3]. ...
... The purpose of the present section is to complete the study for generic values of n f , giving a brief review of the results of [3]. We shall skip a lot of technical, but important, details that can be found by the reader in the given reference. ...
... The second part of the paper has been carried out in Minneapolis at FTPI. I am grateful to R. Auzzi for useful discussions and the collaboration [3] on which the last part of this contribution is based. I want to thank especially M. Shifman for many useful discussions and collaborations [2,3]. ...
This is a review of recent developments regarding the Skyrmion sector of higher representation QCD. Ordinary QCD is a SU(n) gauge theory with n_f Dirac quarks in the fundamental representation. Changing the representation of quarks leads to different and interesting theories, which are not as well studied as ordinary QCD. In order to be able to have a consistent asymptotically free large n limit, we must limit ourselves to three cases: two-index representation (symmetric or anti-symmetric) and adjoint representation. Skyrmions of the low-energy effective Lagrangian shall be the main subject of this review. There are puzzling aspects, both in orientifold and adjoint QCD, regarding the identification of the Skyrmion and its quantum stability, that have not yet been understood. We shall explain these problems and the solution we proposed for them. The first part is dedicated to the two-index representation. Here the challenge is to identify the correct particle in the spectrum that has to be identified with the Skyrmion. It turns out not to be the simplest baryon (as in ordinary QCD) but a baryonic state with higher charge, precisely composed by n(n\pm 1)/2 quarks. Although not the simplest among the baryons, it is the one that minimizes the mass per unit of baryonic charge and thus is the most stable among them. The second part is devoted to the quarks in the adjoint representation. The task here assume a different perspective. We do not have a baryon charge, like in ordinary QCD. An important role is now played by a massive fermion that must be considered in the low-energy effective Lagrangian. Through this fermion, the Skyrmion acquires an anomalous fermionic number (-1)^F and, as a consequence, an odd relationship between the latter and its spin/statistic. This implies a Z_2 stability of the Skyrmion.
