Supersymmetry, Axial Anomaly and Holomorphy: a problem and a solution

Reason, or the ratio of all we have already known, is not the same that it shall be when we know more.~ William Blake!

The essential aspect of supersymmetry (SuSy), its bosonic-fermionic symmetry, also poses the gravest problem for the theory: since bose-fermi symmetry is not observed in ‘nature’, it must be broken if SuSy is to make contact with reality. A promising possibility is spontaneous SuSy-breaking that occurs at tree-approximation level induced by scalar multiplet interaction: this leads to zero-mass Goldstone fermions; that however cannot be identified with the neutrino: a solution naturally presents itself, namely, involving the Ward identities of the theory, which connect the Green’s functions of the supercurrent to other matrix elements: but the validity of the Ward identities require that no anomalies of the supercurrent exist. The reason this is so important is that axial currents create via creation-annihilation Hilbert Space operators, in the context of SuSy, the graviton with all the properties needed for unification to occur between General Relativity and Quantum Field Theory. Such gravitonic axial-current creation is a consequence of the convolution of the SuSy group generators and the generators of the Poincaré group: it is in this sense, along many others, that SuperString Theory is essentially a quantum theory of gravity. However, one must show that axial anomaly cancellation can be achieved, for two reasons: 1) axionic (super)-current gravitonic creation is one of the central reasons why SuperString/M-Theory is finite (it needs no renormalization) and thus is a quantum theory of gravity, and 2) is the only theory that ‘bypasses‘ (avoids the loopholes of) the severely limitative Coleman–Mandula theorem! The question now arises, in the crucial Yang-Mills context: is axionic anomally cancellation possible and is it paradoxes-free? I will show that one can solve both problems. But first: why are General Relativity (GR) and Quantum field Theory (QFT) so fundamentally incompatible? Here are just a few reasons:

GR is in an essential sense a theory of the gravitational field represented by the pseudo-Riemannian metric of an ‘inseparable’ spacetime. Since QFT quantizes all fields in spacetime, the gravitational field also must be quantized. Let us look at the field equation of GR:

${R_{\mu \nu }} - \frac{1}{2}\,{g_{\mu \nu }}\,R + {g_{\mu \nu }}\Lambda = \frac{{8\pi G}}{{{c^4}}}{T_{\mu \nu }}$

The side of the GR field equation that describes matter sources are under the descriptive domain of QFT, while the other side describes gravitation as a classical field: since a quantization of gravity implies that spacetime is not ‘smooth’ and since integration is a smoothing process, the GR-action principle would have no solution; and, since the action contains the content of a physical theory, that means that GR does not even ‘describe’ spacetime. So, if the right hand side of the GR-field equation represent quantized matter, GR can be shown inconsistent since consistency presupposes the existence of solutions.

Moreover, the gravitational field is represented by the spacetime metric, and a quantization of the gravitational field is a quantization of that metric: the problem now is that the quantum dynamical description of the gravitational field of GR implies a dynamical quantum-spacetime, but QFT presupposes a fixed non-dynamical background spacetime for the description of all fields. Thus, a quantum theory of the GR-gravitational field cannot be independent as a QFT because the ‘active’ diffeomorphism invariances of GR are ontologically incompatible with any fixed background spacetime!

Also and more serious, in the Hilbert space setting, energy and time are ‘entangled’ in the Heisenberg Uncertainty Principle: let

${H_{E,t}}$

stand in for that relation. If QFT treats time as it must, as a global background parameter, ${H_{E,t}}$ would imply that time is in a superpositional relation with energy. But, by Special Relativity’s $E = m{c^2}$ and the condition of solvability of the Klein–Gordon–Fock equation

$\frac{1}{{{c^2}}}\frac{{{\partial ^2}}}{{\partial {t^2}}}\psi - {\nabla ^2}\Psi + \frac{{{m^2}{c^2}}}{{{\hbar ^2}}}\Psi = 0$

time would cease to represent a physical observable represented by a Hilbert space operator that is compatible with the Dirac equation in natural units: $(i\hbar - m)\,\psi = 0$ . However, in GR, time cannot be, at any ‘TIME‘, in superpositionality since it is an inseparable part of spacetime, and that would mean that differentiating under the integral of GR’s action

$S = \frac{1}{{2\kappa }}\int {R\sqrt { - g} } \,{{\rm{d}}^4}x\,,{\rm{ }}k = 8\pi G{c^{ - 4}}$

with

$G$

being the Newtonian gravitational constant, becomes meaningless mathematically, and by Fredholm kernel analysis, spacetime would not only cease to be inseparable but would cease to causally interact with its own pseudo-Reimannian metric: this is clearly a consequence of the ${H_{E,t}}$ relation – this can be proven by using the Hilbert-Schmidt theorem, and a quantized gravitational field equation would, as mentioned, have no solution, also via ${H_{E,t}}$ , by a clever use of the The Cauchy–Kowalevski–Kashiwara theorem involving cohomological analysis in the language of modules. Again, proofs of all claims are forthcoming in subsequent blogs.

Now: SuperString/M-theory will be shown in this blog series, to not only solve all of the above problems, but also as to stand out as the only possible Grand Unification Theory (sorry Karl Popper): first, it is provably finite (it needs no renormalizability); it implies the Yukawa coupling of the sparticle ‘fields’ with the Higgs field; SuSy also cancels the different Planck scales corresponding to a quantum theory of gravity, with Planck scale $\sim {10^{ - 35}}$ and the Standard Model’s quark-leptonic scale $\sim {10^{ - 17}}$ . Also, sparticles are unstable, and decay into LSPs (lightest super partners), thus offering a prediction that dark matter is composed of LSPs, and that such LSPs explain the quantum cosmological nature of dark matter; and, by the AdS/CFT correspondence, a compactificational dimensional analysis would allow for phenomenological/experimental ‘testing’. Furthermore, if the Standard Model of physics is extended to a supersymmetric theory, it would provide a complete explanation of ‘Higgs physics‘; lastly for this post, SuSy resolves the hierarchy problem. More on the ‘Witten-miracles’ of supersymmetry/M-Theory in subsequent posts. Now, back to some possible paradoxes and a lurking inconsistency within supersymmetry: and some resolutions.

By Holomorphic group analysis, one gets a representation of the SuSy-Group on the Fréchet topological vector space definable by H being the space of entire (everywhere holomorphic) functions of exponential type $\tau$ via the norm

${\left\| f \right\|_n} = \mathop {\sup }\limits_{z \in \mathbb{C}} \exp \left[ { - \left( {\tau + \frac{1}{n}} \right)\left| z \right|} \right]\left| {f(z)} \right|$

$\rho :{G_{SuSy}}^Q \to GL({{\rm{F}}_{Hol,f}})$

with

${G_{SuSy}}^Q$

being the SuSy group considered as a Quantum group: the importance of such a homomorphic representation is that the holomorphic Fréchet vector space has properties that solve the spacetime ‘separability‘ implied by quantization, given ${H_{E,t}}$ and the validity of Special relativity: namely its local convexity; that its topology is induced by a translation-invariant metric; and a holomorphic function’s being a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain. The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series. With such weapons at our disposal, one can confront what could seem like paradoxes and inconsistencies in Supersymmetric Yang-Mills theory.

One source of confusion comes from holomorphic analysis of the gauge coupling: at tree level, such a gauge is induced from a term in the action that takes the form:

$\int {{d^2}} \theta S\,W_\alpha ^2$

This action in perturbation theory has a symmetry

$S \to S + i\alpha$

This is just a symmetric axion shift. Combined with holomorphy, this almost prevents us from defining the form of the effective action. It seems to me that the only terms are

${L_{eff}} = \int {{d^2}} \theta (S + {\rm{ constant}})W_\alpha ^2$

The constant term is associated with one-loop-corrections: but – higher loop ones are forbidden via consistency demands. On the other side of the story, there are two-loop corrections to the beta function in supersymmetric Yang-Mills theories! Does that constitute an inconsistency? Let me rephrase the puzzle. The axial anomaly, which must me cancelled, is in supermultiplet with conformal anomaly, which is the anomaly in the trace of the stress tensor: but, axial anomaly is not renormalizable; however, the trace anomaly is proportionate to the beta function. The simplest resolution is described in a $U(1){\rm{ - gauge - theory}}$ with charged superfields ${\phi ^ \pm }$ . If those fields have no masses, the irreducible effective action

${L_{eff}} = \int {{d^2}} \theta (S + {\rm{ constant}})W_\alpha ^2$

has infrared singularities and we also must regulate ultraviolet divergences. One such regulation is carried out by Paulli-Villard fields, while infrared divergences will be regulated by incorporating masses for the superfields ${\phi ^ \pm }$ . The ${\phi ^ \pm }$ and mass-regulator terms are provably holomorphic:

$\int {{d^2}} \theta M\Phi _{p\nu }^ + \Phi _{p\nu }^ -$

Now in the effective action, the gauge coupling term is a holomorphic function of $S$ and $M$ : however, the effective action

${L_{eff}} = \int {{d^2}} \theta (S + {\rm{ constant}})W_\alpha ^2$

also includes wave-functional renormalization for the regulating fields

$\int {{d^4}} \theta {Z^{ - 1}}\left( {{\Phi ^{ + * }}{\Phi ^ + } + {\Phi ^{ - * }}{\Phi ^ - }} \right)$

Parametrically, Z, the wave functional factorials are not holomorphic. So to determine the coupling renormalization constant of this scale, one needs to calculate and compute the Zs also. Let us start with the holomorphic term

$\frac{{8{\pi ^2}}}{{{g^2}(m)}} = \frac{{8{\pi ^2}}}{{{g^2}\Lambda }} + {b_0}{\rm{In}}\,{\rm{(}}\frac{m}{M})$

then one has in terms of mass,

$\frac{{8{\pi ^2}}}{{{g^2}(m)}} = \frac{{8{\pi ^2}}}{{{g^2}\Lambda }} + {b_0}{\rm{In}}\,{\rm{(}}\frac{m}{M}) - {\rm{In}}\left( {\frac{{Z(m)}}{{Z(M)}}} \right)$

Now one must construct the Beta function: just take

$\beta (g) = \frac{{ - {g^3}}}{{16{\pi ^2}}}\frac{\partial }{{\partial {\rm{In}}(m)}}{g^{ - 2}}(m) = - \frac{{{b_0}{g^3}}}{{16{\pi ^2}}} + \frac{{{b_0}{g^3}}}{{16{\pi ^2}}}\gamma$

where

$\gamma = \frac{d}{{d{\rm{In}}(M)}}{\rm{In}}(Z) = - \frac{{4{g^2}}}{{16{\pi ^2}}}$

Hence, there are higher than one-loop corrections to the Beta function: for the

$U(1){\rm{ - theory}}$

The two-loop order is

$- \beta (g) = \frac{{{g^3}}}{{16{\pi ^2}}} + \frac{{4{g^5}}}{{{{\left( {16{\pi ^2}} \right)}^2}}}$

Differentiating, one finally gets the Beta expression

$\beta (g) = \frac{{{g^3}}}{{8{\pi ^2}}}\frac{{3{C_A} - \sum {T_F^i(1 - {\gamma ^i})} }}{{1 - {{\left( {\frac{{{C_A}{g^2}}}{{8\pi }}} \right)}^2}}}$

which is the desired result.

I will conclude with a micro-summary: since axion currents in supersymmetric field theories ‘create’ the graviton and the fact that the supersymmetry generators and the Poincaré generators are in integral convolution analytically implies that supersymmetric field theories are theories of quantum gravity and of spacetime: a quantization of General Relativity is not hampered by axional anomaly cancelation as holomorphic analysis shows. To be continued. I appreciate your patience!

And, paraphrasing Lee Smolin, if indeed Supersymmetry and String/M-Theory turn out to be ‘false’, their ‘falsities’ will indeed teach us more than most theories can by being true!