On mass spectrum in SQCD. Unequal quark masses
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UCLEI, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
On Mass Spectrum in SQCD. Unequal Quark Masses1 V. L. Chernyak Budker Institute of Nuclear Physics, Russian Academy of Sciences, Novosibirsk, 630090 Russia email: [email protected] Received December 12, 2009
Abstract—The ᏺ = 1 SQCD with Nc colors and two types of light quarks, Nl flavors with the smaller mass ml and Nh = NF – Nl flavors with the larger mass mh, Nc < NF < 3Nc, 0 < ml ≤ mh ΛQ, is considered within the dynamical scenario in which quarks can form a coherent colorless diquark condensate 〈 QQ〉 . There are sev eral phase states at different values of the parameters r = ml/mh. Nl, and NF. Properties of these phases and their mass spectra are described. DOI: 10.1134/S1063776110120071 1
1. INTRODUCTION We generalize the results obtained in [1] for equal quark masses to the case of unequal masses. We do not consider the most general case of arbitrary quark masses here. Only one specific (but sufficiently repre sentative) choice of unequal masses is considered: there are Nl ≠ Nc flavors with the smaller mass ml and Nh = NF – Nl flavors with the larger mass mh ≥ ml > 0, Nc < NF < 3Nc. Some abbreviations used below are fol lows: DC is the diquark condensate, HQ is a heavy quark, the lquarks are the quarks with the smaller mass ml, and the hquarks are those with the larger mass mh. The masses ml = ml(μ = ΛQ) and mh = mh(μ = ΛQ) are the running current quark masses normalized at μ = ΛQ, and ᏹ ch or ᏹ ch are the chiral diquark con densates of the l or hquarks, also normalized at μ = l l l h ΛQ, 〈 Q l Q ( μ = Λ Q )〉 = δ l ᏹ ch , 〈 Q h Q ( μ = Λ Q )〉 = l
h
δ h ᏹ ch , and ΛQ (independent of quark masses) is the scale parameter of the gauge coupling constant. All quark masses are small, but nonzero: 0 < ml ≤ mh ΛQ. The whole theory can therefore be regarded as being defined by the three numbers Nc, NF, and Nl and three dimensional parameters ΛQ, ml, and mh (i.e., all dimensional observables are expressed through these three). It is shown below that within the dynamical sce nario used, there are different phase states in this the ory at different values of the parameters r = ml/mh ≤ 1, Nl, and NF: h
h
pole
(a) the DCl – DCh phase appears for m h
(b) the DCl – HQh phase appears for ᏹ ch h
ᏹ ch m pole ΛQ at Nl > Nc only; h l
(c) another regime of the DCl – HQh phase appears for ᏹ ch m h ᏹ ch ΛQ in both cases Nl > Nc and Nl < Nc; (d) the Higgsl–DCh or Higgsl–HQh phases appear h
pole
( mh
l
at ᏹ ch ΛQ at Nl < Nc only. l
It is implied that the reader is familiar with the pre vious paper [1], because all the results in [1] are essen tially used in this paper. The paper is organized as follows. The properties of the DCl–DCh phase are considered in Section 2. The DCl–HQh phase (in two regimes) is considered in Sections 3 and 4. The Higgsl–DCh and Higgsl–HQh phases with Higgsed lquarks are considered in Sec tion 5. Section 6 contains a short conclusion. 2. THE DCl–DCh PHASE We first recall the effective Lagrangian for equ
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