Bosón de Higgs: historia, hallazgo y perspectivas

Alejandro Jenkins, U. de Costa Rica

Academia Nacional de Ciencias26 de febrero del 2014

Motivación

Dilbert, 4 oct. 2012

Ver: AJ, “Noticias del mundo subatómico”, La Nación, 12 jul. 2012

Aclaración

Dilbert, 21 feb. 2012

Aclaración, bis

✤ “Partícula de Dios” se debe a L. Lederman (1993)

✤ El bosón de Higgs no juega ningún papel en biología, química, o física nuclear

✤ Su importancia está en lo que pueda decir sobre la estructura de las interacciones a muy altas energías

Nobel 2013: Englert y Higgs

“por el descubrimiento teórico de un mecanismo que contribuye a nuestra comprensión del origen de la masa de las partículas subatómicas y que fue recientemente confirmado gracias al descubrimiento de la partícula fundamental predicha, por parte de los experimentos ATLAS y CMS en el Large Hadron Collider del CERN.”

Ley de Stigler

✤ “Ningún descubrimiento científico lleva el nombre de su descubridor original”

✤ (Stigler dice que esta ley se debe a Robert K. Merton)

✤ David Goodstein: “le damos el crédito del descubrimiento a quien descubre algo tan bien que no sea necesario volverlo a descubrir”.

Cronología

✤ London y London ’35: explican efecto Meissner en superconductores

✤ Nambu ’60 / Anderson ’63: sugieren aplicabilidad del mismo principio en física de altas energías

✤ Brout y Englert / Higgs / Guralnik, Hagen y Kibble ’64: modelos relativistas con ruptura espontánea de simetría de calibre

✤ Weinberg ’67: usa campo de Higgs para dar masas a partículas elementales

Migdal y Polyakov

✤ Dos físicos soviéticos llegaron independientemente a conclusiones similares

✤ ¡Tenían 19 años!

✤ Trabajo encontró resistencia de las autoridades

✤ No fue publicado hasta 1966

SOVIET PHYSICS JETP VOLUME 24, NUMBER 1 JANUARY, 1967

SPONTANEOUS BREAKDOWN OF STRONG INTERACTION SYMMETRY AND THE

ABSENCE OF MASSLESS PARTICLES

A. A. MIGDAL and A. M. POL YAKOV

Submitted to JETP editor November 30, 1965; resubmitted February 16, 1966

J. Exptl. Theoret. Physics (U.S.S.R.) 51, 135-146 (July, 1966)

The occurrence of massless particles in the presence of spontaneous symmetry breakdown is discussed. By summing all Feynman diagrams, one obtains for the difference of the mass operators Ma(P) - Mb(P) of particles a and b belonging to a supermultiplet an equation which is identical to the Bethe-Salpeter equation for the wave function of a scalar bound state of van-ishing mass (a "zeron") in the annihilation channel ab of the corresponding particles. It is shown that if symmetry is spontaneously violated in a Yang-Mills type theory involving vector mesons, the zerons interact only with virtual particles and therefore unobservable. On the other hand, the vector mesons acquire a mass in spite of the generalized gauge invariance. It is shown in Appendices A and B that the asymmetrical solution corresponds to a minimal en-ergy of the vacuum and that C-invariance of the solution implies strangeness conservation for it.

1. INTRODUCTION

SPONTANEOUS symmetry breakdown related to an instability of the system under consideration with respect to an infinitesimally weak asymmetric perturbation is often encountered in quantum sta-tistical mechanics (ferromagnetism, superconduc-tivity, etc.). A large number of such examples has been analyzed in Bogolyubov's review. [ 1l

In the theory of elementary particles the possi-bility of spontaneous asymmetry has been discussed by Nambu and Jona-Lasinio[ 2J and by Vaks and Larkin, (SJ on the example of violated '}'5-invariance. In their calculations these authors ran into the ex-istence of massless particles (zerons). Later, Goldstone has proved the theorem [ 4• sJ which states that bound states of zero mass must appear in systems with unstable symmetries. An example are the spin waves which occur in a ferromagnetic body and the acoustic oscillations in a boson gas. On the other hand, there are no acoustic oscilla-tions in a superconductor. Lange(SJ has shown that the cause for the inapplicability of the Gold-stone theorem in this case are the long-range Coulomb forces.

In elementary particle theory, forces analogous to the Coulomb forces appear as a result of the exchange of massless Yang-Mills vector mesons[ 7 J which guarantee SU(3) symmetry and generalized gauge invariance (cf. e.g., the paper of Glashow and Gell-Mann[ 8 J). It is to be expected that no zerons would appear with spontaneous symmetry

91

breakdown in such a theory, and that the vector mesons acquire a physical mass in analogy with the screening of Coulomb forces in a superconduc-tor.

We show that the vector mesons do indeed ac-quire a mass and that the zerons do not interact with real particles. Such a "phantom" zeron man-ifests itself as an off-mass-shell pole in the scat-tering amplitudes. The scattering amplitudes for real particles do not have such poles and there-fore the zerons are unobservable. We note that in a Yang-Mills theory with stable symmetry the vector mesons would not have a physical mass; the instability of the symmetry removes this dif-ficulty, which has been repeatedly discussed in the literature. [ 9]

Our paper is organized in the following manner: In Sec. 2 the equations of N am bu [ 2 J and Vaks and Larkin, [SJ which were introduced for weak point interactions, are extended to the case of an arbi-trary interaction. The Goldstone theorem [ 4] is confirmed for this case: a condition for the solva-bility of the equations so derived is the existence of a bound state of zero mass (zeron). In Sec. 3 we prove the unobservability of zerons in a theory with vector mesons. In Appendix A the necessity of choosing (from energy considerations) the asym-metric solution, whenever it exists, is justified. Appendix B contains a proof of the connection be-tween the conservation of strangeness and C-invariance in the presence of spontaneous symme-try breaking.

Landau: Transición de fase de 2do orden

f

m

T>Tc

T=Tc

T<Tc

Figure 1: Free energy density f as a function of (uniform)m. Below Tc new minima at nonzero±m develop(the dashed curve denotes m(T ). If the magnetization of the system is varied (rather than the field) the dottedline is the tie line giving the variation of the free energy as a function of the total magnetization densitym = M/V .

andm ≃

!a

b

"1/2(Tc − T )1/2 for T < Tc. (7)

Evaluating f at m gives

f = f0 − a2(T − Tc)2

2b(8)

showing the lowering of the free energy by the ordering. Note that f (T ) deviates from f0 quadratically,as we found for the mean field theory of the Ising ferromagnet. This will yield a jump discontinuity in thespecific heat

c =#

c0 T ≥ Tc

c0 + abT T < Tc

, (9)

with c0 the smooth contribution coming from f0(T ).

We can gain useful insight into the transition by plotting f (m) for a uniform m for various temperatures, asin Fig. 1. For T > Tc the free energy has a single minimum at m = 0. Below Tc two new minima at ±m

develop. Right at Tc the curve is very flat at the minimum (varying as m): we might expect fluctuations tobe particularly important here.

It is easy to add the coupling to a magnetic field

f (m, T , B) = f0(T ) + a(T − Tc)m2 + 1

2bm4 + γ (∇m)2 − mB. (10)

Themagnetic field couples directly to the order parameter and is a symmetry breaking field: with themagneticfield the full Hamiltonian is not invariant under spin inversion. Now the free energy is minimized by a nonzero m. Minimizing f again with respect to m, we find above Tc the diverging susceptibility

χ = m

B

$$$$B=0

= 12a

(T − Tc)−1, (11)

3

Espín

✤ Naturaleza de la partícula está identificada por números cuánticos (carga, “sabor”, “color”, etc.)

✤ Lleva energía y moméntum

✤ Puede tener moméntum angular intrínseco (espín)

✤ Espín cuenta número de polarizaciones

Fermiones

✤ En honor a Enrico Fermi (1901-1954)

✤ Partículas con espín j = n/2, para n entero impar

✤ Obedecen principio de exclusión de Pauli: dos fermiones no pueden ocupar mismo estado cuántico

✤ Electrones son fermiones

✤ Esto explica mucha de la estructura de la tabla periódica

Bosones

✤ Honra a Satyendra Nath Bose (1894-1974)

✤ Partículas con espín j entero

✤ No obedecen principio de exclusión

✤ Fotones son bosones (j = 1)

✤ Permiten estados cuánticos macroscópicos (e.g., superconductores y superfluidos)

Condensado de Bose-Einstein

Física teórica

mecánicaclásica

mecánicacuántica

mecánicarelativista

teoría cuánticadel campo

pequeño

rápido

Teoría cuántica del campo

✤ Las partículas son perturbaciones en un campo

✤ A diferencia de las ondas clásicas, su energía está cuantizada

✤ Espín nos dice si las partículas correspondientes son bosones o fermiones

✤ Para partículas con masa y con espín j, la partícula tiene 2j+1 polarizaciones

E = cp

p2 + m2c2 � =h

|p|

Partículas sin masa

✤ Partículas con m = 0 son perturbaciones con E → 0 para λ → ∞

✤ No tienen marco de reposo

✤ Todos los observadores las ven viajar a la velocidad de la luz

✤ Fotón tiene j=1, pero solo 2 polarizaciones: carece de modo longitudinal

E = h⇥ =hc

�= c|p|

Potenciales

1

V (r) = � g2

4�

e�µr

rYukawa:

V (r) =14�

e2

rCoulomb:

~ ⌘ h/2� = 1; c = 1En adelante usaré unidades en que

Ruptura espontánea de simetría

Teorema de Goldstone

✤ Para simetrías continuas espontáneamente rotas, existen perturbaciones del estado base que corresponden a partículas sin masa

✤ “Bosones de Nambu-Goldstone”

✤ Sin espín (j = 0)

✤ Ejemplo: sonido en un sólidolim

�!1E = 0

Quiralidades

✤ Para una partícula sin masa, espín debe ser paralelo o antiparalelo a la dirección de propagación

✤ Solo puede tener 2 polarizaciones, llamadas “quiralidades”

✤ Para el fotón, son las polarizaciones circulares de la luz

Birrefringencia

Electromagnetismo

Aµ = (V,A)Potencial cuadrivectorial:

B = r⇥ACampo magnético:

E = �rV � �A�t

Campo eléctrico:

Aµ ! Aµ + �µ�Transformación de calibre:

Aharonov-Bohm

Teorías de calibre (gauge)

✤ Descripción relativista de partículas con m = 0 y j > 1/2 introduce redundancia matemática

✤ Llamada “simetría local de calibre”

✤ Electromagnetismo (j = 1) es teoría de calibre abeliana

✤ Fotón no interactúa con sí mismo ⇒ linealidad de ecs. de Maxwell

✤ Interacciones nucleares requieren teorías de calibre no abelianas (Yang-Mills), con ecs. no lineales

Analogías

x

µ � x

0µ({x�})

✤ Gravedad puede interpretarse como teoría de calibre para j = 2

✤ Ver: AJ, arXiv:hep-th/0607239; 0904.0453 [gr-qc]

U({xi})� f(U({xi})); f

0(x) > 0

✤ Invariancia general de coordenadas en relatividad general:

✤ Transfomación monotónica de “función de utilidad” en microeconomía:

Simetría global de calibre

✤ Campo de calibre Aμ debe acoplarse a corriente conservada

✤ Transformación global de calibre ( Ω = const.) actúa solo sobre las cargas ψ:

✤ Verdadera simetría física

✤ Implica conservación de carga

� ! eie⌦�

Brout-Englert-Higgs (BEH)

V (⇥) = ��|⇥|2 � v2

�2

BEH, bis

+ + (1)

1

✤ Adquiere marco de reposo y polarización longitudinal (dada por el bosón de Goldstone)

✤ Evita divergencias en dispersión del modo longitudinal

✤ Viola conservación de carga: medio puede absorber o ceder carga

✤ Campo de calibre se propaga como si tuviera masa

Unificación electrodébil ✤ Comenzamos con 4 bosones

vectoriales y un doblete de Higgs

✤ Doblete tiene 4 grados de libertad

✤ 3 bosones de Goldstone son comidos, dando masas a W± y Z0

✤ queda 1 bosón vectorial sin masa (el fotón)

✤ y 1 escalar masivo (el bosón de Higgs)

1

rango

⇠Mweak =

1

2

gv

Nobel 1979: Glashow, Weinberg y Salam (GSW)

Masas de fermiones

✤ Weinberg ‘67 usa campo de Higgs para dar masas a leptones y quarks

✤ Masa de átomo de hidrógeno ~ 1 GeV

✤ El 99% de la masa de la materia ordinaria se debe a interacciones entre quarks y gluones en el núcleo

QCDΛ

1 GeV 10 GeV100 MeV10 MeV1 MeV0.1 MeV 100 GeV

u d bs c te μ τ

Mweak

h

Perversiones cuánticasTendril Perversion in Intrinsically Curved Rods 243

Fig. 2. An actual perversion in a telephone cord.

seashells in a species of overwhelming right-handed individuals (for instance, there areonly six known left-handed specimens out of the million known Cerion, a West Indianland snail [12], [13]).A qualitative explanation of the creation of helices with opposite handedness (see

Fig. 2) can readily be given. Consider a filament with given nonvanishing intrinsiccurvature, that is, the unstressed configuration is coiled on itself (a multicovered ring,if ones disregards self-contact). Now, take this filament and completely untwist it andstraighten it out. By applying sufficient tension (and proper end moments), one cancompletely straighten out the filament. Note that the total twist of the straight filament iszero. Now, as one reduces the tension, there is a critical value of the tension below whichthe straight filament becomes unstable. The optimal solution (the solution with lowestenergy) is a helix whose torsion and curvature are functions of the tension. However, inorder to create a helix, one of the ends must rotate. As we do not allow the ends to rotate,this solution cannot be obtained. Nevertheless, another solution with zero twist can beobtained by smoothly pasting two optimal helices together with a small inversion. Thisis the perversion solution. It can be obtained by reducing the tension of an intrinsicallycurved twistless straight filament whose total twist is fixed. One of the most fascinatingnatural manifestations of perversion can be found in the growth of some climbing plants.Among the many different mechanisms climbing plants use to climb and grow alongsupports, the so-called tendril-bearers constitute an important class (e.g., the grape-vine,the hop, the bean, the melon). In the first stage of their development, tendrils are tender,soft, curly, and flexible organs originating from the stem. As they grow, the tendrilscircumnutate [15], [16]. That is, the tip of the tendril describes large loops in space bycompletely rotating on itself until it touches a support, such as a trellis, a pole or a branch.If the circumnutation does not result in a contact, the tendril eventually dries and fallsoff the stem. The tendrils which are in contact with a support enter another phase of theirdevelopment, and their tissues develop in such a way that they start to curl and tighten up,eventually becoming woody, robust, and tough. This curling provides the plant with anelastic springlike connection to the support that enables it to resist high winds and loads.Since neither the stem nor the support can rotate, the total twist in the tendril cannotchange. Thus the conditions are ripe for obtaining a perversion, and as the tendril curlson itself, the coils of the helix are reversed at some point so that the tendril goes froma left-handed helix to a right-handed one, the two being separated by a small inversion,the perversion (see Fig. 3).The phenomenon of perversion in climbing plants has a long, interesting scientific

history. Inversion of helicity in tendrils already appears in the illustration (see Fig. 4) ofLinnaei in Philosophia Botanica [17]. However, according to de Candolle [18], [19], thefirst record of a scientific observation of perversion goes back to a letter of the Frenchscientist Ampere to the French Academy of Sciences. From then on, almost all majorbotanists in the nineteenth century, such as Dutrochet in 1844 [20], von Mohl in 1852[21], and Leon in 1858 [22], [23] describe the perversion found in tendrils.

Fuente: T. McMillen and A. Goriely, “Tendril Perversion in Intrinsically Curved Rods”, J. Nonlinear Sci. 12, 241 (2002)

✤ En un fermión con masa, quiralidad oscila constantemente

✤ Pero los fermiones zurdos tienen carga débil, mientras que los diestros no

✤ Requiere acoplamiento del fermión con el campo de Higgs

Large Hadron Collider (LHC)

8.6 km

175 m

Colisiones protón-protón

LHC ring: 27 km circumference

Instalaciones

Dentro del túnel, 2007

Instalación del tubo de haz que entra al detector de ATLAS

Instalación del “tracker” en el centro del detector de CMS

Detector de ATLAS jul. 2007

Centro de cómputo del CERN, durante puesta en operación de los servidores principales

Producción y desintegración

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En el LHC se hacen chocar protones a velocidades muy elevadas y se estudian las partículas producidas en las colisiones. Si se genera un bosón de Higgs, se sabe que este puede desintegrarse en dos foto- nes. Sin embargo, la misma señal puede provenir de un gran número de procesos «de fondo» que nada tienen que ver con el bosón de Higgs. Esta gráfica muestra los sucesos con dos fotones registrados por el experimento CMS frente a la señal de fondo esperada. El pico indica la exis- tencia de fotones «de más»: una clara señal de que se han originado en la desin- tegración de una nueva partícula cuya masa ronda los 125 gigaelectronvoltios.

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Interrogantes

✤ Glashow llamó al sector de Higgs “el excusado del modelo estándar”

✤ v, λ, y’s son parámetros libres

✤ no explica masa del Higgs

V (⇥) = ��|⇥|2 � v2

�2

mf = yfv

Problema de la jerarquía

✤ μ2 = 4 λ v2 sufre grandes correcciones radiativas

✤ μ2 ~ (102 GeV)2, mientras que ΛUV2 ~ (1018 GeV)2

✤ La fuerza nuclear débil no parece ser suficientemente débil, comparada con la gravedad

�µ2 = � |yf |2

8�2⇥2

UV + . . .

Problema de la jerarquía, bis

✤ En un superconductor, correspondería a finísimo ajuste de (TC - T)/T

✤ No es inconsistente, pero parece requerir explicación

✤ Wilson motiva 40 años de trabajo en posibles soluciones dinámicas

✤ (tecnicolor, supersimetría, grandes dimensiones extra, etc.)

f

m

T>Tc

T=Tc

T<Tc

Figure 1: Free energy density f as a function of (uniform)m. Below Tc new minima at nonzero±m develop(the dashed curve denotes m(T ). If the magnetization of the system is varied (rather than the field) the dottedline is the tie line giving the variation of the free energy as a function of the total magnetization densitym = M/V .

andm ≃

!a

b

"1/2(Tc − T )1/2 for T < Tc. (7)

Evaluating f at m gives

f = f0 − a2(T − Tc)2

2b(8)

showing the lowering of the free energy by the ordering. Note that f (T ) deviates from f0 quadratically,as we found for the mean field theory of the Ising ferromagnet. This will yield a jump discontinuity in thespecific heat

c =#

c0 T ≥ Tc

c0 + abT T < Tc

, (9)

with c0 the smooth contribution coming from f0(T ).

We can gain useful insight into the transition by plotting f (m) for a uniform m for various temperatures, asin Fig. 1. For T > Tc the free energy has a single minimum at m = 0. Below Tc two new minima at ±m

develop. Right at Tc the curve is very flat at the minimum (varying as m): we might expect fluctuations tobe particularly important here.

It is easy to add the coupling to a magnetic field

f (m, T , B) = f0(T ) + a(T − Tc)m2 + 1

2bm4 + γ (∇m)2 − mB. (10)

Themagnetic field couples directly to the order parameter and is a symmetry breaking field: with themagneticfield the full Hamiltonian is not invariant under spin inversion. Now the free energy is minimized by a nonzero m. Minimizing f again with respect to m, we find above Tc the diverging susceptibility

χ = m

B

$$$$B=0

= 12a

(T − Tc)−1, (11)

3

Jerarquía en el multiverso

✤ ¿Puede la solución ser antrópica?

✤ Ver: AJ y G. Perez, Sci. Am. 302(1), 42 (2010)

✤ O. Gedalia, AJ y G. Perez, arXiv:1010.2626

V (⇥) = ��|⇥|2 � v2

�2

mf = yfv

Mweak =12gv

Cuestiones oscuras

Perspectivas

✤ Descubrimiento del bosón de Higgs es un triunfo del modelo estándar

✤ Esperamos que haya algo más

✤ Tal vez materia oscura y supersimetría

✤ ¿Contacto con cosmología y gravedad cuántica?

✤ LHC vuelve a operar en el 2015...