LOG#239. Higgspecial.

The 2012 found Higgs particle is special. The next generations of physicists and scientists, will likely build larger machines or colliders to study it precisely. The question is, of course, where is new physics, i.e., where is the new energy physics scale? Is it the Planck scale? Is it lower than $10^{28}eV$ ?

What is a particle?

Particles are field excitations. Fields satisfy wave equations. Thus particles, as representations of fields, also verify field or wave equations. Fields and particles have also symmetries. They are invariant under certain group transformations. There are several types of symmetry transformations:

Spacetime symmetries or spacetime invariance. They include: translations, rotations, boosts (pure Lorentz transformations) and the full three type combination. The homogeneous Lorentz group does not include translations. The inhomogeneous Lorentz group includes translations and it is called Poincaré group. Generally speaking, spacetime symmetries are local spacetime transformations only.
Internal (gauge) symmetries. These transformations are transformations of the fields up to a phase factor at the same spacetime-point. They can be global and local.
Supersymmetry. Transformations relating different statistics particles, i.e., relating bosons and fermions. It can be extended to higher spin under the names of hypersymmetry and hypersupersymmetry. It can also be extended to N-graded supermanifolds.

We say a transformation is global when the group parameter does not depend on the base space (generally spacetime). We say a transformation is local when it depends of functions defined on the base space.

Quantum mechanics is just a theory relating “numbers” to each other. Particles or fields are defined as functions on the spacetime momentum (continuum in general) and certain discrete set of numbers (quantum numbers all of them)

$\ket{p^\mu,\sigma}$

and thus

$U(\Lambda)\ket{p,\sigma}=D_{\sigma\sigma'}\ket{\Lambda p,\sigma'}$

represents quantum particles/waves as certain unitary representations of the Poincaré group (spacetime)! Superfields generalize this thing. Diffferent particles or fields are certain unitary representations of the superPoincaré group (superspacetime)! Equivalently, particles are invariant states under group or supergroup transformations. Particle physics is the study of fundamental laws of Nature governed by (yet hidden and mysterious) the fusion of quantum mechanics rules and spacetime rules.

From the 17th century to 20th century: we had a march of reductionism and symmetries. Whatever the ultimate theory is, relativity plus QM (Quantum Mechanics) are got as approximations at low or relatively low (100GeV) energies. Reductionism works: massless particles interact as an Greek Y (upsilon) vertices.

$\chemfig{A-B(-[:-30]C)-[:30]D}$

Massless particles can be easily described by twistors, certain bispinors (couples of spinors):

$p_{\alpha\dot{\alpha}}=\begin{pmatrix} p_0+p_3 & p_1-ip_2\\ p_1+ip_2 & p_0-p_3\end{pmatrix}=\lambda_{\alpha}\overline{\lambda}_{\dot{\alpha}}$

Indeed, interactions are believed to be effectively described by parallel twistor-like variables $\lambda_A\propto \lambda_B\propto \lambda_C$ and $\overline{\lambda_A}\propto\overline{ \lambda_B}\propto \overline{\lambda_C}$ . The Poincaré group completely fixes the way in which particles interact between each other. For instante, the 4-particle scattering constraints

$(\langle 1 2 \rangle \left[3 4 \right])^{2S}F(s,t,u)$

where $s$ is the spin of the particle. Be aware of not confusing the spin with the Mandelstam variable $s$ . Locality implies the factorization of the 4-particle amplitude into two $Y$ pieces, such as

$F(s,t,u)=\begin{cases}\dfrac{g^2}{s t^S}\\ \dfrac{g^2}{t u^S}\\ \dfrac{ g^2}{u s^S}\end{cases}$

Two special cases are $S=0$ (the Higss!) and $S=2$ (the graviton!):

$F(s,t,u)=g^2\left(\dfrac{1}{s}+\dfrac{1}{t}+\dfrac{1}{u}\right)$

$F(s,t,u)=g^2\dfrac{1}{stu}$

where the latter represents the 2×2 graviton scattering. For spin S=1 you have

$Y\propto gf^{abc}\dfrac{\langle 1 2 \rangle^3}{\langle{13}\rangle\langle 23\rangle}$

Interactions between both, massive and massless spin one particles must contain spin zero particles in order to unitarize the scattering amplitudes! Scalar bosons are Higgs bosons. Of course, at very high energies, the Higgs and the chiral components of the massive gauge bosons (spin one) are all unified into a single electroweak interaction. A belief in these principles has a paid-off: particles have only spin 0,1/2,1,3/2,2,… The 21st century revelations must include some additional pieces of information about this stuff:

The doom or end of spacetime. Is the end of reductionism at sight?
Why the Universe is big?
New ideas required beyond spacetime and internal symmetries. The missing link is usually called supersymmetry (SUSY), certain odd symmetry transformations relating boson and fermions. New dogma.
UV/IR entanglement/link/connection. At energies bigger than Planck energy, it seems physics classicalize. We have black holes with large sizes, and thus energies (in rest) larger than Planck energy. High energy is short distance UV physics. Low energy is large distance IR physics.
Reductionism plus wilsonian effective field theory approaches plus paradigmatic model is false. Fundamental theories or laws of Nature nothing like condensed matter physics (even when condensed matter systems are useful analogues!). Far deeper and more radical ideas are necessary. Only at Planck scale?

Photons must stay massless for consistent Quantum Electrodynamics, so they are Higgs transparent. $2\neq 3$ is the Nima statement on this thing. massless helicities are not the same of massive helicities. This fact is essential to gauge fields and chiral fermions. So they can be easily engineered in condensed matter physics. However, Higgs fields are strange to condensed matter systems. Higgs is special because it does NOT naturally arise in superconductor physics and other condensed matter fields. Why the Higgs mass is low compared to the Planck mass? That is the riddle. The enigma. Higgs particles naturally receive quantum corrections to mass from boson and fermion particles. The cosmological constant problem is beyond a Higgs-like explanation because the Higgs field energy is too-large to handle it. Of course, there are some ideas about how to fix it, but they are cumbersome and very complicated. We need to go beyond standard symmetries. And even so, puzzles appear. Flat spacetimes, de Sitter (dS) spacetimes or Anti de Sitter (AdS) spacetimes? They have some amount of extra symmetries: $SO(5,1)\rightarrow \mbox{Poincaré}\rightarrow SO(4,2)$ . The cases of $\Lambda>0$ (dS), $\Lambda=0$ (flat spacetime), $\Lambda<0$ (AdS). Recently, we found (not easily) dS vacua in string and superstring theories. But CFT/AdS correspondences are better understood yet. We are missing something huge about QM of the relativistic vacuum in order to understand the macroscopic Universe we observe/live in.

Why is the Higgs discovery so important? Our relativistic vacuum is qualitatively different than anything we are seen (dark matter, dark energy,…) in ordinary physics. Not just at Planck scale! Already at GeV and TeV scale we face problems! The Higgs plus nothing else at low energies means that something is wrong or at least not completely correct. The Higgs is the most important character in this dramatic story of dark stuff. We can put it under the most incisive and precise experimental testing! So, we need either better colliders, or better dark matter/dark energy observations. The Higgs is new physics from this viewpoint:

We have never seen scalar (fundamental and structureless?) fields before.
Harbinger of deep and hidden new principles/sectors at work at the quantum realm.
We must study it closely.

It could arise that Higgs particles are composite particles. How pointlike are Higgs particles? Higgs particles could be really composite of some superstrong pion-like stuff. But also, they could be truly fundamental. A Higgs factory working at $125 GeV$ (pole mass) of the Higgs should serve to see if the Higgs is point-like (fundamental). Furthermore, we have never seen self-interacting scalar fields before. A 100 TeV collider or larger could measure Higgs self-coupling up to 5%. The Higgs is similar to gravity there: the Higgs self-interacts much like gravitons!

Yang-Mills fields (YM) plus gravity changes helicity of particles AND color. 100 TeV colliders blast interactions and push High Energy Physics. New particles masses up to 10 times the masses accessible to the LHC would ve available. They would probe vacuum fluctuations with 100 times the LHC power. The challenge is hard for experimentalists. Meanwhile, the theorists must go far from known theories and go into theory of the mysterious cosmological constant and the Higgs scalar field. The macrouniverse versus the microuniverse is at hand. On-shell lorentzian couplings rival off-shell euclidean couplings of the Higgs? Standard local QFT in euclidean spacetimes are related to lorentzian fields. UV/IR changes this view! Something must be changed or challenged!

Toy example

Suppose $F=1/t$ . By analytic continuation,

$\dfrac{1}{2\pi i}\oint dt F(t)=1$

Is Effective Field Theory implying 0 Higgs and that unnatural value? Wrong! Take for instance

$F(t)=\dfrac{1-10^{-120t/t_{uv}}}{t}$

Then

$\oint dt F(t)=0!$

This mechanism for removing bulk signs works in AdS/CFT correspondences and alike theories. For $\Lambda=0$ we need something similar to remove singularities and test it! For instance, UV-IR tuning could provide sensitivities to loop processes arising in the EFT of the Higgs potential

$V(1-loop)=\lambda^4h^4\log( \lambda^2+k^2)+(M\pm h)^4\log (M\pm k)^2=\sum M^4\log M^2(k)$

However, why should we tree cancel the 1-loop correction? It contains UV states and $\lambda^2 M^2 h^2$ terms. Tree amplitudes are rational amplitudes. Loop amplitudes are trascendental amplitudes! But long known funny things in QFT computations do happen. For instance,

$\Gamma(positronium)=\mbox{something}\times (M^2-9)$

Well, this not happens. There is a hidden mechanism here, known from Feynman’s books. Rational approximations to trascendental numbers are aslo known from old mathematics! A High School student knows

$\ln 2=\int_0^1 \dfrac{dx}{1+x}$

This is trascendental because of the single pole at $x=-1$ . If you take instead

$I(x)=\int \dfrac{dx P(x)}{1+x}=P(-1)\log 2+\mbox{Rational part}$

you get an apparen tuning of rational to trascendental numbers

$\int_0^1\dfrac{dx}{1+x}\left(\dfrac{x(1-x)}{2}\right)^N=\pm \log 2+\mbox{Rational part}$

and thus, e.g., if N=5, you get a tiny difference to $\log 2$ by a factor of $10^{-5}$ (the difference up to this precision is $2329/3360$ ). The same idea works if you take

$4\int_0^1\dfrac{dx}{1+x^2}\left(\dfrac{x(1-x)}{4})\right)^{4N}=\pi +\mbox{Rational number}$

You get for N=1 $\pi-22/7\sim 10^{-3}$ and $\pi-47171/15015\sim 10^{-6}$ if N=2. Thus, we could conjecture a fantasy: there is a dual formulation of standard physics that represents physical amplitudes and observeables in terms of integral over abstract geometries (motives? schemes? a generalized amplituhedron seen in super YM?). In this formulation, the discrepancy between the cosmological constant scale and the Higgs mass is solved and it is obviously small. But it can not be obviously local physics. Another formulation separates different number theoretical parts plus it looks like local physics though! However, it will be fine-tuned like the integrals above! In the end, something could look like

$V(h)=\int dk^2 F(k^2)=\mbox{Logs+rational}=\mbox{exponentially small}$

Fine-tuning could, thus, be only an apparent artifact of local field theory!

A final concrete example:

$F(h)=\int_k^1 \dfrac{dx (x-h)^4}{1+x}\left(\dfrac{x(1-x)}{2}\right)^N$

Take $V(h)=F(h)+F(-h)$ . Then,

$V(h)=\sum_{\pm}(1\pm h)^4\log (1\pm h)+\mbox{Rational parts}$

And it guarantees to be fine-tuning! This should have critical tests in a Higgs factory or very large LHC and/or 100TeV colliders of above. In the example above, if $N=5$

$V(h)=\ldots+10^{-6}+10^{-5}h^2+\ldots+h^{10}$

with no sixth power or eight power terms. Precision circular electron-positron colliders could handle with this physics. Signals from tunning mechanisms could be searched. It is not just $m_h^2$ terms only. High dimensional operators and corrections to the Higgs potential (the vacuum structure itself!) could be studied. But also, we could search for new fields or tiny effects we can not do within the LHC.

Summary: scientific issues today are deeper than those of 1930s or even 1900s. Questions raised by the accelerated universe and Higgs discovery go at the heart of the Nature of the spacetime and the vacuum structure of our Universe!

What about symmetries? In the lagrangian (action) approach, you get a symmetry variation (quasiinvariance) of the lagrangian as follows

$\delta_s L=\partial_\mu \left[\left(\dfrac{\partial L}{\partial(\partial_\mu\phi)}\right)\delta_s \phi\right]+\left[\dfrac{\partial L}{\partial\phi}-\partial_\mu\left(\dfrac{\partial L}{\partial(\partial_\mu\phi)}\right)\right]\delta_s\phi$

Then, by the first Noether theorem, imposing that the action is extremal (generally minimal), and the Euler-Lagrange equations (equations of motion), you would get

$E(L)=0, \mbox{Plus quasiinvariance}\;\; \delta_s L=\partial_\mu K^{\mu}$

a conserved current (and a charged after integration on the suitable measure):

$\partial_\mu J^\mu=0=\partial_\mu\left[\dfrac{\partial L}{\partial(\partial_\mu \phi)}\delta_s \phi -K^\mu\right]=\varepsilon \partial_\mu J^\mu$

such as

$J^\mu=\dfrac{\partial L}{\partial( \partial_\mu \phi)}\Delta_s\phi-\dfrac{K^\mu}{\varepsilon}$

where $\Delta_s\phi= \delta_s\phi/\varepsilon$ .

This theorem can be generealized for higher order lagrangians, in any number of dimension, and even for fractional and differentigral operators. Furthermore, a second Noether theorem handles ambiguities in this theorem, stating that gauge local transformations imply certain relations or identities between the field equations (sometimes referred as Bianchi identities but go further these classical identities). You can go further with differential forms, exterior calculus or even with Clifford geometric calculus. A p-form

$A_p=\dfrac{1}{p!}A_{\mu_1\cdots\mu_p} dx^{\mu_1}\wedge\cdots \wedge dx^{\mu_p}\equiv A_{\mu_1\cdots\mu_p} dx^{\mu_1}\wedge\cdots \wedge dx^{\mu_p}$

defines p-dimensional objects that can be naturally integrated out. For a p-tube in D-dimensions

$\tau_{\mu_{p+1}\cdots \mu_D}=\dfrac{1}{p!}\int_C\varepsilon_{\mu_1\cdots\mu_p\mu_{p+1}\cdots\mu_D}\delta(x-y) dy^{\mu_1}\wedge\cdots dy^{\mu_p}$

On p-forms, the Hodge star operator for a p-form $A_p$ in D-dimensions turn it into a $(D-p)$ -form

$\star A=\dfrac{\sqrt{\vert g\vert}}{p!(D-p)!}A_{\mu_1\cdots\mu_p}\varepsilon^{\mu_1\cdots\mu_p} dx^{\nu_{p+1}}\wedge\cdots\wedge dx^{\nu_D}$

As $D=Dim(X)$ , then we have $\star^2=\star\star=(-1)^{p(D-p)+q}$ , where $q=1$ if the metric is Lorentzian, $q=0$ for euclidean metrics and $q=T$ , the number of time-like dimensions, if the metric is ultrahyperbolic. Moreover,

$vol=\star 1=\dfrac{\sqrt{\vert g\vert}}{D!}\varepsilon_{\mu_1\cdots \mu_{D}}dx^{\mu_1}\wedge\cdots\wedge dx^{\mu_D}=\sqrt{\vert g\vert} dx^{\mu_1}\wedge\cdots \wedge dx^{\mu_D}$

For $\star:\Omega^p\rightarrow \Omega^{D-p}$ maps, you can also write

$\langle A,B\rangle=\int A\wedge \star B=\int B\wedge \star A$

$\int A\wedge B=\int \star A\wedge \star B$

where the latter is generally valid up to a sign. The Hodge laplacian reads

$\Delta=(d^++d)^2$

and you also gets

$\langle A, dB\rangle=\langle d^+A,B\rangle$

If $\partial X$ is not zero (the boundary is not null), then it implies essentially Dirichlet or Neumann boundary conditions for $d, d^+$ . When you apply the adjoint operator $d^+$ on p-forms you get

$d^{+}=(-1)^{Dp+D+1}\star d\star$

in general but you pick up an extra $-$ sign in euclidean signatures.

To end this eclectic post, first a new twist on Weak Gravity Conjectures(WGC). Why the electron charge is so small in certain units? That is, $e<m_e$ . Take Coulomb and Newton laws

$F_C=K_C\dfrac{Qq}{r^2}=\dfrac{1}{4\pi\varepsilon_0}\dfrac{Qq}{r^2}$

$F_N=G_N\dfrac{Mm}{r^2}$

Then,

$\dfrac{F_N}{F_C}\sim 10^{-42}$

Planck mass is

$M_P=\sqrt{\dfrac{\hbar b}{G_N}}$

and then

$Q_P=\sqrt{4\pi\varepsilon_0\hbar c}=\left(\dfrac{\hbar c}{K_C}\right)^{1/2}$

Planckian entities satisfy instead $F_G/F_C=1$ ! then, the enigma is why

$\dfrac{m_e}{M_p}<10^{-22}<<\dfrac{q_e}{q_P}\sim 0.1=10^{-1}$

In other words, $q_e/m_e\approx 10^{21}$ in relativistic natural units with $c=\hbar=4\pi\varepsilon_0=1$ with $\hbar\neq 1$ . The WGC states that the lightest charge particle with $m,q$ in ANY U(1) (abelian gauge) theory admits UV embedding into a consistent quantum gravity theory only and only if

$\dfrac{qg}{\sqrt{\hbar}}\geq \dfrac{m}{M_p}$

where $g$ is the gauge coupling and QED satisfies WGC since $ge/\sqrt{\hbar}\sim 10^{-3}>>10^{-22}$ . WGC ensures that extremal black holes are unstable and decay into non-extremal black holes rapidly (if even formed!) via processes (and avoid to be extremal too) that are QG consistent. Furhtermore, WGC could imply the 3rd thermodynamical law easily. For Reissner-Nordström black holes

$T=\dfrac{\hbar \sqrt{M^2-Q^2}}{2\pi\left(M+\sqrt{M^2-Q^2}\right)^2}$

and a grey body correction to black hole arises from this too

$\langle N_{j\omega l p}\rangle=\dfrac{\Gamma(j\omega l p)}{e^{(\omega-e\phi)/T}\pm 1}$

Generalised Uncertainty principles (GUP) plus Chandrasekhar limits enjoy similarities with the WGC:

$M_C\sim \dfrac{1}{m_e^2}\left(\dfrac{\hbar c}{G}\right)^{3/2}\simeq 1.4M_\odots$

The S-matrix

$\braket{\Psi(+\infty) |\Psi(+\infty)}=1=\bra{\Psi(- \infty)} S^+S\ket{\Psi(-\infty)}=\braket{\Psi(-\infty) |\Psi(-\infty)}$

and by time reversal, the principle of deteailed balance holds so

$\braket{\Psi(-\infty) |\Psi(+\infty)}=1=\bra{\Psi(+ \infty)} S^+S\ket{\Psi(-\infty)}=\braket{\Psi(+\infty) |\Psi(-\infty)}$

Quantum determinism implies via unitarity $\Psi'=U\Psi$ . However, Nature could surprise us. And that would affect Chandrasekhar masses or TOV limits. Stellar evolucion implies luminosities $L=4\pi R^2 \sigma T_e^4$ , where $T_e$ is the effective temperature for black bodies (Planck law), and $\sigma$ is the Stefan-Boltzmann constant

$\sigma=5.67\cdot 10^{-5}\dfrac{erg}{cm^2 K^4 s}$

Maximal energy for a set of baryons under gravity is

$E_G=-\dfrac{GMm_B}{R}=-G\dfrac{Nm_B^2}{R}=\hbar c\dfrac{N^{1/3}}{R}-G\dfrac{Nm_B^2}{R}$

as the baryon number for a star is

$N_B=\left(\dfrac{\hbar c}{Gm_B}\right)^{3/2}\simeq 2\cdot 10^{57}$

Using the Wien law

$\lambda_{max} T_e\simeq 2.9\cdot 10^6 mm\cdot K$

the stars locally have an equation for baryon density about

$\left(\dfrac{V}{N}\right)^{1/3}=n^{1/3}=\dfrac{n^{1/3}}{M_P^{1/3}}\sim \left(\dfrac{4}{3} \pi R_\odot^3\dfrac{m_P}{M_{ \odot}}\right)^{1/3}\sim 10^{-8}$

Stars are sustained by gas and radiation against gravitational collapse. The star pressure

$P_\star=P(gas)+P(radiation)=\dfrac{K}{\mu}\rho T+\dfrac{1}{3}n T^4$

Thus, the maximal mass for a white dwarf star made by barions is about $1.5M_\odot$ . Ideal gas law implies the HR diagram!!!! Luminosity scales as the cube of mass. The Eddington limit maximal luminosity for any star reads off as

$L_E=\dfrac{AMc Gm(r)}{K}$

and a Buchdahl limit arises from this and the TOV limit as follows

$TOV\rightarrow \dfrac{GM}{c^2R}<\dfrac{4}{9}$

and then

$M>\dfrac{4}{9}\dfrac{Rc^2}{G}$

implies a BH as inevitable consequence iff $M>3M_\odot$ approximately!

Epilogue: heterodynes or superheterodynes? Jansky? dB scales? Photoelectric effect is compatible with multiphoton processes and special relativity too. SR has formulae for Compton effect, inverse Compton effect, pair creations, pair annihilations, and strong field effects!

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The Spectrum Of Riemannium

The Art Of Science and Physmatics. This is the geeknerd way!

LOG#239. Higgspecial.

What is a particle?

Toy example

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What is a particle?

Toy example

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