QCD Mass Inequalities in the Heavy Quark Limit

QCD inequalities are derived for the masses of mesons and baryons containing a single heavy quark using the heavy quark effective field theory. A rigorous lower bound is obtained for the Λ parameters of the heavy quark effective theory that parameterize 1/m corrections, Λ ≥ 237 MeV for mesons, and Λ ≥ 657 MeV for baryons. The inequalities on Λ imply the inequalities mc < 1627 MeV and mb < 5068 MeV for the mass parameters of the heavy quark effective field theory. CERN-TH.6766/92 UCSD/PTH 92-45 hep-ph/9212289 December 1992 * On leave from the University of California at San Diego. 1 Rigorous inequalities between hadron masses in QCD can be derived using the Euclidean functional integral formulation of the theory [1][2][3][4][5]. The basic idea is that the functional integral measure for a vector-like theory such as QCD is real and positive, so that the Cauchy-Schwarz inequality can be used to derive inequalities on Euclidean correlation functions. The inequalities among correlation functions imply inequalities among hadron masses. The usual QCD inequalities hold for arbitrary quark masses, and so apply also to the heavy quark case e.g., m(ρ) ≥ m(π) [1] has the heavy quark analog m(B) ≥ m(B), etc. These will not be discussed further here. QCD inequalities are derived in this paper for the heavy quark effective field theory [6][7][8], which is a systematic expansion about the infinite quark mass limit. The quark propagator in the infinite mass limit is the path ordered integral of the exponential of the vector potential, which is a unitary matrix. This implies certain inequalities which would not necessarily hold for light quarks. In this paper, mass inequalities will be derived for hadrons containing only a single heavy quark Q. The heavy quark effective theory has the leading order fermion Lagrangian Lv = iQv (v ·D)Qv, (1) where Qv is a Dirac spinor field that annihilates a heavy quark with velocity v, and D is the gauge covariant derivative. The total Lagrangian is the sum of the heavy quark Lagrangian and the usual QCD Lagrangian for the light quarks and gluons. (In this paper, light quarks will refer to quarks with mass mq that is finite, but not necessarily small compared to ΛQCD.) The field Qv satisfies the constraint Qv = 1 + v/ 2 Qv. (2) The masses of hadrons in the heavy quark effective theory containing a single heavy quark Q are m−mQ, where m is the mass of the hadron, and mQ is the mass of the heavy quark. The heavy quark mass mQ, which is defined so that the effective Lagrangian eq. (1) has no residual mass term, is a well defined quantity which is a parameter of the heavy quark effective theory. A detailed discussion of this issue can be found in Ref. [9]. 1 A good discussion of QCD inequalities can be found in the Caltech lecture notes for Ph230 of J. Preskill (unpublished). 2 The mass inequalities can be easily derived using the continuum formulation of the theory, provided one uses certain “obvious” properties of the products of Dirac delta functions. The same results can also be obtained using a lattice regulated version of the theory which avoids “problems” with Dirac delta functions, and is the method used here. The Lorentz frame for the heavy quark theory can be chosen so that the velocity vector v is (1, 0, 0, 0), before analytically continuing the theory to Euclidean space. The hypercubic Euclidean lattice is chosen with edges that are parallel or perpendicular to v, for simplicity. The Euclidean coordinate parallel to v will be denoted by t, and the transverse coordinates will be denoted by x. The lattice spacing is a, with t ≡ n0a and x ≡ na. The Euclidean propagator for the heavy quark theory is then [10] W (x, t;y, s) = a 1 + γ 2 U(x, t;x, t− a)U(x, t− a;x, t− 2a) . . . . . . U(x, s+ 2a;x, s+ a)U(x, s+ a;x, s) if x = y, t > s = 0 otherwise. (3) Here U is a unitary matrix in color space which is defined on the links of lattice, and can be thought of as the path-ordered exponential of the gluon field, U ∼ P exp ig ∫ Aμdx . W satisfies the discretized version of the Green’s function equation (v ·D)W (x, t;y, s) = 1 + γ 2 δ(x − y)δ(t− s), 1 a [W (x, t;y, s)− U(x, t;x, t− a)W (x, t− a;y, s)] = a 1 + γ 2 δxy δts, (4) where δab is a Kronecker delta. To avoid complicating the notation in the correlation functions to be computed below, (x, t) will be denoted by x. Consider the meson correlation function (at y = x, y > x)