We examine the contribution of small instantons to the axion mass in various UV completions of QCD. We show that the reason behind the potential dominance of such contributions is the non-trivial embedding of QCD into the UV theory. The effects from instantons in the partially broken gauge group appear as “fractional instanton” corrections in the effective theory. These will exhibit unusual dependences on the various scales in the problem whenever the index of embedding is non-trivial. We present a full one-instanton calculation of the axion mass in the simplest product group models, carefully keeping track of numerical prefactors. Rather than using a ’t Hooft operator approximation we directly evaluate the contributions to the vacuum bubble, automatically capturing the effects of closing up external fermion lines with Higgs loops. This approach is manifestly finite and removes the uncertainty associated with introducing a cutoff scale for the Higgs loops. We verify that the small instantons may dominate over the QCD contribution for very high breaking scales and at least three group factors.

Objective

To compare cross-sectional and longitudinal side-differences in thigh muscle anatomical cross-sectional areas (ACSAs), muscle strength, and specific strength (strength/ACSA), between knees with early radiographic change vs knees without radiographic knee osteoarthritis (RKOA), in the same person.

Design

55 (of 4796) Osteoarthritis Initiative (OAI) participants fulfilled the inclusion criteria of early RKOA in one limb (definite tibiofemoral osteophytes; no radiographic joint space narrowing [JSN]) vs no RKOA (no osteophyte; no JSN) in the contralateral limb. ACSAs of the thigh muscles and quadriceps heads were determined using axial MRIs at 33%/30% femoral length (distal to proximal). Isometric extensor and flexor muscle strength were measured (Good Strength Chair). Baseline quadriceps ACSA and extensor (specific) strength represented the primary analytic focus, and 2-year changes of quadriceps ACSAs the secondary focus.

Results

No statistically significant side-differences in quadriceps (or other thigh muscle) ACSAs, muscle strength, or specific strength were observed between early RKOA vs contralateral limbs without RKOA (P ≥ 0.44), neither in men nor in women. The 2-year reduction in quadriceps ACSA in limbs with early RKOA was -0.9 ± 6% (mean ± standard deviation) vs -0.5 ± 6% in limbs without RKOA (statistical difference P = 0.85).

Conclusion

Our results do not provide evidence that early unilateral radiographic changes, i.e., presence of osteophytes, are associated with cross-sectional or longitudinal differences in quadriceps muscle status compared with contralateral knees without RKOA. At the stage of early unilateral RKOA there thus appears to be no clinical need for countervailing a potential dys-balance in quadriceps ACSAs and strength between both knees.