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The subtracted (Equation presented) in Eq. (1) is defined as (Equation presented) [11], where the usual (perturbative) scale anomaly, (Equation presented) ((Equation presented)), is characterized by the intrinsic scale (Equation presented), a scale responsible for the asymptotically free (perturbative) running of the coupling in the ultraviolet region. Hence, (Equation presented) is nonzero only in the broken phase, (Equation presented), where (Equation presented), in sharp contrast to the usual QCD, where (Equation presented). For details see, e.g., the third reference in Ref. [11].
-
The subtracted (Equation presented) in Eq. (1) is defined as (Equation presented) [11], where the usual (perturbative) scale anomaly, (Equation presented) ((Equation presented)), is characterized by the intrinsic scale (Equation presented), a scale responsible for the asymptotically free (perturbative) running of the coupling in the ultraviolet region. Hence, (Equation presented) is nonzero only in the broken phase, (Equation presented), where (Equation presented), in sharp contrast to the usual QCD, where (Equation presented). For details see, e.g., the third reference in Ref. [11].
-
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22
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84907325410
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The Lagrangian is constructed uniquely (up to total derivatives) by the requirement that the action (Equation presented) be invariant under the scale (and also chiral) transformation (Equation presented), (Equation presented). This implies that (Equation presented) namely, the scale dimension of (Equation presented) must be 4, (Equation presented). Equation (2) is a unique chirally invariant Lagrangian having scale dimension four and a correct kinetic term of (Equation presented). Note that the dimensionless field (Equation presented) transforms as an operator with the scale dimension one, with (Equation presented). Similarly, the Lagrangians, Eqs. (2)-(4) are unique in a way to have the scale dimension four and to satisfy the anomalous WT identities as well as the stable vacuum. See also Ref. [19].
-
The Lagrangian is constructed uniquely (up to total derivatives) by the requirement that the action (Equation presented) be invariant under the scale (and also chiral) transformation (Equation presented), (Equation presented). This implies that (Equation presented) namely, the scale dimension of (Equation presented) must be 4, (Equation presented). Equation (2) is a unique chirally invariant Lagrangian having scale dimension four and a correct kinetic term of (Equation presented). Note that the dimensionless field (Equation presented) transforms as an operator with the scale dimension one, with (Equation presented). Similarly, the Lagrangians, Eqs. (2)-(4) are unique in a way to have the scale dimension four and to satisfy the anomalous WT identities as well as the stable vacuum. See also Ref. [19].
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23
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84907305428
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The scale-WT identity for the operator (Equation presented) reads (Equation presented), where (Equation presented), with (Equation presented), and (Equation presented) is the scale dimension of the operator (Equation presented). Note that the second term comes from the (Equation presented) pole contribution. Thus, we have (Equation presented).
-
The scale-WT identity for the operator (Equation presented) reads (Equation presented), where (Equation presented), with (Equation presented), and (Equation presented) is the scale dimension of the operator (Equation presented). Note that the second term comes from the (Equation presented) pole contribution. Thus, we have (Equation presented).
-
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27
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84865770259
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PRVDAQ 1550-7998 10.1103/PhysRevD.86.035025
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S. Matsuzaki and K. Yamawaki, Phys. Rev. D 86, 035025 (2012). PRVDAQ 1550-7998 10.1103/PhysRevD.86.035025
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84907365479
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The coefficient of the scale-invariant term (Equation presented) is uniquely determined as in Eq. (4) by the vacuum stability requirement to be proportional to the fermion mass (Equation presented), and hence does not exist in the chiral limit Lagrangian Eq. (2) with (Equation presented). See Ref. [18] for details. Thus, our Lagrangian, Eq. (5), as well as Eq. (2), is unique at order (Equation presented).
-
The coefficient of the scale-invariant term (Equation presented) is uniquely determined as in Eq. (4) by the vacuum stability requirement to be proportional to the fermion mass (Equation presented), and hence does not exist in the chiral limit Lagrangian Eq. (2) with (Equation presented). See Ref. [18] for details. Thus, our Lagrangian, Eq. (5), as well as Eq. (2), is unique at order (Equation presented).
-
-
-
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30
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0000498366
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A similar formula was also discussed in a completely different context, i.e., hadron physics, which we believe has no approximate scale symmetry and is irrelevant to our discussions. See, e.g., (); NUPBBO 0550-3213 10.1016/0550-3213(70)90422-0
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A similar formula was also discussed in a completely different context, i.e., hadron physics, which we believe has no approximate scale symmetry and is irrelevant to our discussions. See, e.g., J. Ellis, Nucl. Phys. B22, 478 (1970); NUPBBO 0550-3213 10.1016/0550-3213(70)90422-0
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|