The deconstructed Standard Model equation _ - symmetry magazine.pdf

La ecuación más larga en física
El modelo Lagrangiano es una expresión matemática que resume el Modelo Estándar de física de partículas, que es la teoría más exitosa de las interacciones fundamentales entre partículas elementales.
Se compone de cuatro partes diferentes, cada una de las cuales describe un aspecto diferente del Modelo Estándar.
El modelo lagrangiano está escrito en una notación compacta que utiliza símbolos y operadores de la teoría cuántica de campos, como derivadas covariantes, tensores de intensidad de campo, matrices de Dirac y generadores de grupos de calibre.
También utiliza varias constantes y parámetros que se determinan mediante experimentos, como constantes de acoplamiento, masas y ángulos de mezcla. Es una de las ecuaciones más largas de la física porque contiene muchos términos y factores que explican todas las posibles interacciones y simetrías del modelo estándar.
Fue transcrito por Thomas Gutiérrez, quien lo derivó de Diagrammatica: The Path to Feynman Diagrams de Martinus Veltman.

1. Yvonne Tang. SLAC National Accelerator Laboratory

The deconstructed Standard Model
equation

(07/28/16 | 8y Rashmi Shivni

The Standard Model is far more than elementary particles arranged in a table.

The Standard Model of particle physics is often visualized as a table, similar to
the periodic table of elements, and used to describe particle properties, such as

2. mass, charge and spin. The table is also organized to represent how these teeny,
tiny bits of matter interact with the fundamental forces of nature.

But it didn’t begin as a table. The grand theory of almost everything actually
represents a collection of several mathematical models that proved to be
timeless interpretations of the laws of physics.

Here is a brief tour of the topics covered in this gargantuan equation.

The whole thing

This version of the Standard Model is written in the Lagrangian form. The
Lagrangian is a fancy way of writing an equation to determine the state of a
changing system and explain the maximum possible energy the system can
maintain.

Technically, the Standard Model can be written in several different formulations,
but, despite appearances, the Lagrangian is one of the easiest and most
‘compact ways of presenting the theory.

OI — ISO EI — TILL gi g guide +
ara) + G20°G* + gs f°O,G°G y O WOW
MW,FW,, — £0,200, 25 — E M 2021 —10,A,0,A, —10,H0,H —
Im} H? — 8,6*0,6° — M26*d- 10,0%, MO — p [PE +
2H + L(H? + 6090 + 26*6°)] + May — igew[9,ZUW,FW, -

We) - ZUW+&W; — WW) + ZUW+8,W, —
WOW] - igsuld A (WW — WW;) — AW+9,W; —
WWE) + A,(Wi0,W,, - Wa W+)] — 192WW,W+W, +

MI + e Zu ZW, — Zi ZW, Wir) +
gs (A WE AW — A A WW) + 9 sucio Ay Zo (W EW
WW) = 24, Z0W SW, ] — galH® + HS +2H6*6"]
¿Pal HR (6)! +4(6+07)?+4(0°)?6*6° + 4H?76* O- +2(6°)? A] —
gMW;W;H — Lg1Z0Z20H — LigW (6%, 07 — 670,0") —

3. POS — 9 + 3907, (10,0 —O 9,4) —W, (HO —

o Ha, ga (ADS —¿%0,H) ¡gi MZW¿0"—W0%)+
igsyMA,(Wi6" —W¿6*) — ig

ig8wAy(o*O.6~ — 7 dpt) — 4 ‘a LH? + (9°)? + 26767] —

ZIZUR + (9)? + 2(282, — 126* 07] — IA ZU (WE +
Wot) — Lig ZOH(W+ o> — Wr dt) + hp su AW o> +

Wot) + Hig?swAuH (WT —W,,6*) — 922" (262, -1)Z0A,6*0" —

(6* 8,0" — 670,0*) +

Is ApAót9 | 200 + me = Po — WAV + mud —
Æ 300+m08 + ig. A) + 039) — a] +
1 (1 + Pr) + (ey (482, — 1 — Per) + (Bre (der
Y) + (dt (1 — $83, — PA) + WP a“ + Je) +
(a +0, ad) y+ Ve (Pa. +) + Behr +
Pw) HBR A) AAN

ava |
ame HPA) NAPA] + o om (Ol Pd) +
CUE a ad [m3(@CI, (1+78)ut) me (BCL, (1

Pur] — LH) — SADA) + PRI) —

ig g(a) + [KO — M2)X* + X (0? — MYX + KP —
a MONEY PY Hg Wi ee OpX*X°) +95 WE (D Ÿ X=
— O,X°X*) + igsuW (0, X Y —
= RX”) + igswAy(O,.X
tH+ X-X-H+ OH] +
XX+G-] +

ae

Lograngian standard mode!
Courtesy of 0. Gutierrez

Section 1
These three lines in the Standard Model are ultraspecific to the gluon, the boson

4. that carries the strong force. Gluons come in eight types, interact among
themselves and have what's called a color charge.

Section 2
Almost half of this equation is dedicated to explaining interactions between
bosons, particularly W and Z bosons.

Bosons are force-carrying particles, and there are four species of bosons that
interact with other particles using three fundamental forces. Photons carry
electromagnetism, gluons carry the strong force and W and Z bosons carry the
weak force. The most recently discovered boson, the Higgs boson, is a bit
different its interactions appear in the next part of the equation.

0,20 = sie M2207, 1,0,A, —10,HO,H -

Zn 2

Zar 049,0 MO

+P + PO +2 +

, AE
TAF

+07 +4H?6* & +2(0°)? H°
9MW¿W, — 298 Zi ZRH — sig[W,:(9%0,07 — 970,0") —

5. Wr (6°,.6* 079,0) + sa jf (119,07 — 67 OH) WW.
0°, H)) +492 ZUMO, — 699, H) -igE MZUW 6 —W, 0
¡gs MA, (W¿ó" - W,, 6*) A)
igsuA OOO” — 70,97) — F9 WW, [H? + (6°) + 26767]
AI
b Je zig? EHE $7 — Wet) + 5928 A O (WE
—W,, 07) - PE (2, —NRAO TO —

Section 3

This part of the equation describes how elementary matter particles interact
with the weak force. According to this formulation, matter particles come in three
generations, each with different masses. The weak force helps massive matter
particles decay into less massive matter particles.

This section also includes basic interactions with the Hi
some elementary particles receive their mas

Intriguingly, this part of the equation makes an assumption that contradicts
discoveries made by physicists in recent years. It incorrectly assumes that
ed neutrinos have no mass

die 0 + mad; + igswAy[—(@y"e>) + Ace M)
i 71 CL + 7° )v*) + (Ar (452 — 1 — 7 5e) +
DE Br
EPICA WE +

ri

6. Section 4

In quantum mechanics, there is no single path or trajectory a particle can take,
which means that sometimes redundancies appear in this type of mathematical
formulation. To clean up these redundancies, theorists use virtual particles they
call ghosts.

This part of the equation describes how matter particles interact with Higgs
ghosts, virtual artifacts from the Higgs field.

Jus) mi (DC,

in u) —

Section 5

This last part of the equation includes more ghosts. These ones are called
Faddeev-Popov ghosts, and they cancel out redundancies that occur in
interactions through the weak force.

X+(? — M?)X+ + X~(@ — M?)X- + X°(
AUOT + ige W (OX 0, XX) + 1950 W(0, Y X=
XV) + igeuW, (OX TX — ,X°X*) + igsu We (0, X Y —
AKT + ia RUN ENT AN N) Has AA N EXT —

7. YX) MIX EX TA + X-X-A + EXCH] +
igM[X*+ XG — X-X%
XOX",

Note: Thomas Gutierrez, an assistant professor of Physics at California
Polytechnic State University, transcribed the Standard Model Lagrangian for the
web. He derived it from Diagrammatica, a theoretical physics reference written
by Nobel Laureate Martinus Veltman. In Gutierrez's dissemination of the
transcript, he noted a sign error he made somewhere in the equation. Good luck
finding it!

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