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The Strong CP Problem

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The Strong CP Problem The Strong CP Problem Sourav Sen Duke University October 2017 Abstract The QCD part of the SM lagrangian has no reason to not include a CP violating term, akin to its electro-weak counterpart. However, no evidence of strong CP violation experimentally thus far has lead to suppressing the CP violating term coefficient a very small number. The reason why nature chose the CP violating term very small is \the Strong CP problem" and is yet to be understood. The most popular and elegant solution put forth is by Peccei and Quinn, who assume that is a dynamical field, instead of a constant value, forming the functional form of a U(1) global symmetry of a complex scalar field. This scalar field has a vacuum expectation value when = 0, where this U(1) symmetry is spontaneously broken giving rise to a Goldstone boson - axions. In this essay, I also discuss the approaches towards experimentally detecting the axions. 1 Introduction A physical theory has CP symmetry as a natural consequence, if the complex phases of the coupling constants have no relative difference for the theory, such as in pure quantum electrodynamics (QED). The quantum chromodynamics (QCD), however, does not possess such obvious characteristics to make it a CP symmetric theory naturally. This fact became clearer after the resolution of a problem called the "U(1)A problem" in QCD. 2 U(1)A problem and QCD vacuum The QCD lagrangian has a large global symmetry: U(N)V × U(N)A for N flavors in the limit of vanishing quark masses mf 0. Therefore, at least for up and down quark flavors (as mu, md ΛQCD), the strong interaction should approximately have U(2)V × U(2)A symmetry. ! The U(1)V vector symmetry is the symmetry corresponding to isospin times baryon number [U(2)V = SU(2)I × U(1)B], and is indeed a good approximate symmetry of the strong interactions. This can be evident by the appearance of nucleon and pion multiplets in the spectrum of hadrons. 1 The U(2)A axial symmetry, however, appears to be spontaneously broken, by the formation of quark condensates < uu¯ >=< dd¯ >6= 0. Therefore, one does expect to find four Nambu-Goldstone bosons of the broken U(2)A axial symmetry. But except for mπ, no other light state in the hadronic spectrum has been found. As no Nambu-Goldstone boson is found for the U(1)A axial symmetry, Steven Weinberg suggested that the strong interaction doesn't have a U(1)A symmetry somehow, and termed it as the "U(1)A problem" [1]. This problem was resolved by Gerard 't Hooft [2], by explaining that the QCD vacuum has a more complicated structure. This complex structure of the QCD vacuum, leads to chiral anomaly [7] for the U(1)A axial currents. Thus, although the LQCD appears to have the U(1)A symmetry, its corresponding action does not. This resolves the U(1)A problem by making U(1)A not a true symmetry of QCD. The resolution of the U(1)A problem, by recognizing the complicated nature of the QCD vacuum, effectively adds an extra term involving a free dimensionless parameter θ [12] to the QCD Lagrangian: g2 L = θ Fµν~Fa ; (1) θ 32π2 a µν µν ~a 1 µναβ a where Fa is a gluon field and Fµν = 2 Fαβ. Since all the quarks are massive, this term violates parity (P-) and time reversal (T-) and conserves the charge-conjugaton (C-) symmetry. Thus, this term violates CP, unless θ 0. 3 The Strong CP problem ! Apart from the θ term, diagonalizing the quark mass matrix Mq introduces another FF~ CP violating with phase parameter of -arg detMq. However, this term has no natural reason to be equal to θ. As a result, the effective CP violating parameter θeff becomes: θ¯ = θ - arg detMq (2) The phase θ¯ is experimentally measured from the neutron electric dipole mo- ment (EDM) dn [3] [4], since there is a leading contribution (for light quarks) 3 to the neutron EDM via chiral loop (dn ≈ 2 × 10 θ¯ efm). The existing mea- -13 surement of neutron EDM [5] (jdnj < 3:6 × 10 e fm), constraining the CP parameter θ¯ < 10-9. But why is the CP violating parameter θ¯ so small, even though there is no natural reason for it to happen? This is the "strong CP problem". 4 Peccei Quinn mechanism and Axions The solution proposed by Roberto Peccei and Helen Quinn [6], is still the most elegant and popular solution to the strong CP problem. In this essay, we will 2 confine our discussion to this solution only. In the Peccei Quinn solution, they introduced an additional global chiral U(1) symmetry, now known as U(1)PQ, to the QCD lagrangian. To break this U(1)PQ symmetry, they included a complex a(x)=f scalar field ΦPQ = jΦje PQ with a potential: λ V(Φ ) = (jΦj2 - f2 )2: (3) PQ 4 PQ Here, fPQ is the order parameter for the U(1)PQ spontaneous symmetry breaking (see Fig.1). Hence, a(x) field becomes the Nambu-Goldstone boson of this broken U(1)PQ, christened the "axion" [8]. This gives the axion a(x) field a shift symmetry: a(x) a(x) + αfPQ; (4) with 0 ≤ a ≤ 2π. ! Figure 1: The potential of the QP field. The axion is the Nambu-Goldstone boson after spontaneous breaking of U(1)PQ symmetry. The U(1)PQ chiral current is not conserved, due to the Adler-Bell-Jackiw (AJB) axial-anomaly [7] and the low-energy instanton effects: g2 @ Jµ = FµνF~a : (5) µ PQ 32π2 a µν The axion field therefore acquires anomalous coupling to gluons (see Fig.2) The total lagrangian, including the axion field, becomes: 2 1 µ µ ¯ a g µν~a Ltot = LSM - @µa@ a + Lint[@ a=fPQ; Ψ] + θ + ξ 2 Fa Fµν; (6) 2 fPQ 32π where Ψ fields are the quark fields. 3 Figure 2: Triangle loop diagram corresponding to the anomalous axion-gluon- gluon coupling. This figure has been taken from [9] Figure 3: The periodic effective potential of axion breaks the shift symmetry of fPQ ¯ axion making < a >= - ξ θ. fPQ ¯ Hence, the effective potential of the axion has a minimum at a = - ξ θ [10]: 2 @Veff ξ g µν a = - F F~ j f = 0: (7) 2 a µν <a>=- PQ θ¯ @a fPQ 32π ξ This makes the CP violating F~F term in LQCD zero, and thus solves the strong CP problem dynamically. a In other words, the Veff ∼ cos θ¯ - ξ , which is obtained by redefining fPQ the quark fields such that there is no F~F term in the action [11]. This explicitly breaks the axion shift symmetry in eq. 4 (see Fig.3). As a result, the axion ac- quires a mass by the QCD-instanton (and other, more general, non-perturbative) effects, and becomes a pseudo Nambu-Goldstone boson. The axion mass (ma) is [10]: 2 2 @ Veff ξ g @ µν a m = = - F F~ j f (8) a 2 2 a µν <a>=- PQ θ¯ @a fPQ 32π @a ξ To prove this elegant Peccei-Quin theory, one has to discover these axions in nature. 4 5 Experimental searches of Axions Since, the axions have no charge and spin, they do not interact very much with the particles in standard model. However, when axions are passed through strong magnetic field, they decay into photons via Primakoff effect [13]. Early searches for axions ruled out 'standard axions' (fPQ is related to electroweak scale fEW ). Models were constructed for 'invisible axion' (where fPQ fEW ), which are good dark matter candidates (such as [14]). Some of the major axion Figure 4: Exclusion ranges (in grey) on axion mass mA and scale of Peccei- Quinn symmetry breaking fPQ (referred in this figure as fA) from the axion search experiments. The green bands represent the projected reach of these experiments. This figure is taken from [17] search experiments, which exploits the Primakoff effect are: • the Axion Dark Matter eXperiment (ADMX) [15] which searches for ax- ions by converting into microwave photons using an 8 Tesla magnet in an RF cavity. This experiment is trying to detect the axions from the galactic dark matter halo. • the CERN Axion Solar Telescope (CAST) [16] is helioscope with a 9 Tesla magnet trying to detect solar axions. These experiments have not yet been able to discover the axion, so limits have been set on axion mass and PQ symmetry breaking scale (see Fig. 4). Apart from the current experiments, which are exploiting the axion-photon coupling, proposed future experiments such as the Cosmic Axion Spin Precession Exper- iment (CASPEr) are going to exploit the nuclear axion couplings (for nucleon 5 EDM or spin) using Nuclear Magnetic Resonance (NMR). These experiments would search for lower mass axions. 6 Conclusion We discussed how the QCD vacuum gives rise to a CP violating term in the LQCD through chiral anomaly from the U(1)A axial current. But the experimen- tal evidence suggests that strong interactions have CP symmerty. The elegance of Peccei Quinn solution lets QCD preserves its CP symmetry naturally, with a caveat of introducing a yet undiscovered particle - the axion. These axions are chargeless and massless, and are very weakly interacting with the SM particles. This makes them a good candidate for dark matter. The experimental searches for axion have not yet discovered the particle. Hence, the strong CP problem still remains an open and interesting problem in particle physics.
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