Exact Results for Nuclear Ground State Properties From QCD Inequalities
We thank Tom Schafer for pointing out difficulties in this paper.
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arXiv subjects
Thomas D. Cohen
Publications and source records attributed to Thomas D. Cohen.
Helium-4 Nuclear Ground State and QCD
We thank Tom Schafer for pointing out difficulties in this paper.
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Low Energy Theorems For Nucleon-Nucleon Scattering
Low energy theorems are derived for the coefficients of the effective range expansion in s-wave nucleon-nucleon scattering valid to leading order in an expansion in which both $m_π$ and $1/a$ (where $a$ is the scattering length) are treated as small mass scales. Comparisons with phase shift data, however, reveal a pattern of gross violations of the theorems for all coefficients in both the $^1S_0$ and $^3S_1$ channels. Analogous theorems are developed for the energy dependence $ε$ parameter which describes $^3S_1 - ^3D_1$ mixing. These theorems are also violated. These failures strongly suggest that the physical value of $m_π$ is too large for the chiral expansion to be valid in this context. Comparisons of $m_π$ with phenomenological scales known to arise in the two-nucleon problem support this conjecture.
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Systematic Power Counting in Cutoff Effective Field Theories for Nucleon-Nucleon Interactions and the Equivalence With PDS
An analytic expression for the ${}^1S_0$ phase shifts in nucleon-nucleon scattering is derived in the context of the Schrödinger equation in configuration space with a short distance cutoff and with a consistent power counting scheme including pionic effects. The scheme treats the pion mass and the inverse scattering length over the intrinsic short distance scale as small parameters. Working at next-to-leading order in this scheme, we show that the expression obtained is identical to one obtained using the recently introduced PDS approach which is based on dimensional regularization with a novel subtraction scheme. This strongly supports the conjecture that the schemes are equivalent provided one works to the same order in the power counting.
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The Spectral Density of the Dirac Operator above T_c
The importance of the spectral density of the Dirac operator in studying spontaneous chiral symmetry breaking and anomalous U(1) axial symmetry breaking are reviewed. It is shown that both types of symmetry breaking can be traced to effects of modes near zero virtuality. Above T_c, where chiral symmetry is restored, it is shown on general grounds that (in the massless quark limit), the density of states vanishes at zero virtuality faster than $λ$, where $λ$ is the virtuality-- $ρ(λ) \sim |λ|^α$ is not possible for $α\le 1$. Isospin invariance is used to show that $ρ(λ) \sim m_q^{1-α} |λ|^α$ is also not possible for $α\le 1$. State-of-the-art lattice calculations are reviewed in light of these constraints. In particular, it is argued that violations of these constraints by lattice calculations indicate possible large systematic errors; this raises questions about $U(1)_A$ violating effects seen on the lattice.It is also shown that above $T_c$, the Dirac spectrum has a gap near zero (in the $m_q \to 0$ limit) unless contributions from quark-line-connected and disconnected contributions conspire to cancel.
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Qunatum Number Exotic Mesons and Large N QCD
Exotic mesons with quantum numbers of a hybrid ($\bar{q} q g$) such as $J^{PC} = 1^{-+}$are shown to exist as narrow resonant states in the large $N_c$ limit of QCD. The width of these states is proportional to $1/N_c$. Possible phenomenological implications for the world of $N_c=3$ are discussed.
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Non-perturbative regularization and renormalization: simple examples from non-relativistic quantum mechanics
We examine several zero-range potentials in non-relativistic quantum mechanics. The study of such potentials requires regularization and renormalization. We contrast physical results obtained using dimensional regularization and cutoff schemes and show explicitly that in certain cases dimensional regularization fails to reproduce the results obtained using cutoff regularization. First we consider a delta-function potential in arbitrary space dimensions. Using cutoff regularization we show that for $d \ge 4$ the renormalized scattering amplitude is trivial. In contrast, dimensional regularization can yield a nontrivial scattering amplitude for odd dimensions greater than or equal to five. We also consider a potential consisting of a delta function plus the derivative-squared of a delta function in three dimensions. We show that the renormalized scattering amplitudes obtained using the two regularization schemes are different. Moreover we find that in the cutoff-regulated calculation the effective range is necessarily negative in the limit that the cutoff is taken to infinity. In contrast, in dimensional regularization the effective range is unconstrained. We discuss how these discrepancies arise from the dimensional regularization prescription that all power-law divergences vanish. We argue that these results demonstrate that dimensional regularization can fail in a non-perturbative setting.
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Fixed-Velocity Chiral Sum Rules for Nuclear Matter
Infinite sets of sum rules involving the excitations of infinite nuclear matter are derived using only completeness, the current algebra implicit in QCD, and relativistic covariance. The sum rules can be used for isospin-asymmetric nuclear matter, including neutron matter. They relate the chiral condensate and the isospin density to weighted sums over states with fixed velocity relative to the nuclear matter ground state.
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Relations Among Correlation Functions in the High Temperature Phase of QCD with Broken SU(3)
Group-theoretic arguments are used to determine the dependence of two-point correlators of quark bilinears on the current quark masses. The leading difference between $π$ and $δ$ correlators is found to be of order $m_s$ times a U(1)$_{\scriptscriptstyle A}$-violating correlator. These general arguments are consistent with Schaefer's observation that if U(1)$_{\scriptscriptstyle A}$ violation persists to high enough temperatures then the strange $η$ can be lighter than the non-strange one.
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Chiral Multiplets of Hadron Currents
Using group theoretical methods, we enumerate possible chiral representations in which hadron interpolating currents can be classified. We give simple examples of currents in each representation, some of which are well known. The classification enables one to find relations among current vacuum correlators in a chirally and/or axial U(1) symmetric phase of QCD in a straightforward way. Besides recovering many well-known relations among two-point correlation functions, a number of novel relations are found.
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How short is too short? Constraining zero-range interactions in nucleon-nucleon scattering
We discuss a number of constraints on the effects of zero-range potentials in quantum mechanics. We show that for such a potential $p \cot(δ)$, where $p$ is the momentum of the nucleon in the center of mass frame and $δ$ is the S-wave phase shift, must be a monotonically decreasing function of energy. This implies that the effective range of the potential is non-positive. We also examine scattering from the sum of two potentials, one of which is a short-range interaction. We find that if the short-range interaction is of zero-range then it must be attractive, and the logarithmic derivative of the radial wave function at the origin must be a monotonically decreasing function of energy. If the short-range interaction is not of zero range then a constraint which gives the minimum possible range for it to fit the phase shifts exists. The implications of these results for effective field theory treatments of nucleon-nucleon interactions are discussed.
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U(1) Axial Symmetry and Correlation Functions in the High Temperature Phase of QCD
Simple group-theoretical arguments are used to demonstrated that in the high temperature (chirally restored) phase of QCD with N massless flavours, all n-point correlation functions of quark bilinears are invariant under U(1) axial transformations provided n < N. In particular this implies that the two-point correlation function in the eta' channel is identical to that in the pion channel for N > 2. Unlike previous work, this result does not depend on the topological properties of QCD and can be formulated without explicit reference to functional integrals.
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Regularization, Renormalization and Range: The Nucleon-Nucleon Interaction from Effective Field Theory
Regularization and renormalization is discussed in the context of low-energy effective field theory treatments of two or more heavy particles (such as nucleons). It is desirable to regulate the contact interactions from the outset by treating them as having a finite range. The low energy physical observables should be insensitive to this range provided that the range is of a similar or greater scale than that of the interaction. Alternative schemes, such as dimensional regularization, lead to paradoxical conclusions such as the impossibility of repulsive interactions for truly low energy effective theories where all of the exchange particles are integrated out. This difficulty arises because a nonrelativistic field theory with repulsive contact interactions is trivial in the sense that the $S$ matrix is unity and the renormalized coupling constant zero. Possible consequences of low energy attraction are also discussed. It is argued that in the case of large or small scattering lengths, the region of validity of effective field theory expansion is much larger if the contact interactions are given a finite range from the beginning.
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Large-$N_c$ and Chiral Limits of QCD and Models of the Baryon
Two ideas have greatly contributed to our understanding of baryon structure in the framework of Quantum Chromodynamics (QCD). The first, chiral symmetry, received its fundamental justification from QCD and has been developed into the powerful technique of chiral perturbation theory. The other notion is the large-$N_c$ limit, namely the behavior of QCD in the limit where there are a large number of colors. Like chiral symmetry, the large-$N_c$ limit predicts many relations between baryon matrix elements. These so-called ``model-independent'' predictions should agree with each other, but paradoxically that is not always the case. The paradox is resolved by recognizing that the two limiting processes (large-$N_c$ and the chiral limit) do not commute. This noncommutivity can be traced to the special role played by the $Δ$ resonance in large-$N_c$ QCD. This gives rise to a new parameter in the effective theory of baryons.
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Isospin Violation and Possible Signatures of Disoriented Chiral Condensates in Ultrarelativistic Heavy Ion Collisions
It is possible that isospin violating dynamical effects are amplified due to coherence if a disoriented chiral condensate (or other source of a coherent pion state) is formed in a heavy ion collison. It is shown explicitly that altering the isospin of the coherent state by order of one unit can change the number of $π^0$'s in the condensate by some finite fraction of the total number. A possible signature of such amplified isospin violating effects is that the ratio of neutral to total low $p_T$ pions averaged over many events may differ markedly from 1/3. Such a signal would not be washed out if multiple domains of coherent pions form.
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Nucleon electric polarizability in soliton models and the role of the seagull terms
The full Hamiltonian of the soliton models contains no electric seagull terms. Here it is shown that if one restricts the fields to the collective subspace then electric seagull terms are induced in the effective Hamiltonian. These effective seagull contributions are consistent with gauge invariance. They also reproduce the leading nonanalytic behavior of a large $N_c$ chiral perturbation theory calculation of the electric polarizability.
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Rho-Omega Mixing Off Shell and Charge Symmetry Breaking In the N-N Potential
The role of the off-shell dependence of $ρ-ω$ mixing in the charge symmetry breaking nucleon-nucleon potential is discussed. It is shown that models describing the off-shell dependence of $ρ-ω$ mixing are not sufficient to determine the charge symmetry breaking nucleon-nucleon potential.
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Baryon Isovector Electric Properties and the Large N_c and Chiral Limits
A model independent calculation is given for the nucleon isovector electric charge radius which is valid in the limit $N_c \rightarrow \infty$, $m_π\rightarrow 0$, $N_c m_π$ fixed. This expression reduces to that of the Skyrme model in the limit $N_c m_π \rightarrow \infty$.
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