Gunion Fest, March 2014
Naturalness and the Higgs Boson
James Wells March 29, 2014 Great to be back at UC Davis
Was faculty here from 1999-2002. I was very fortunate to have such a tremendous role model at that stage.
(Somehow Jack and others had pa ence with me.)
Brash young man I had certain preconceived no ons of Jack before arriving….
But over me I saw a different Jack
Over me I also saw his incredible dedica on to his physics research
Nothing got in the way.
During a lunch conversa on somebody was talking about Kobe Bryant’s amazing performance the previous day.
And Jack said … Who’s Kobe Bryant? Kobe Bryant: Forward for Los Angeles Lakers. NBA MVP and frequent All-Star. One of greatest scorers of all me. I looked up to Jack and decided to re-dedicated myself. No more watching basketball and Kobe Bryant. No more excessive a en on to poli cs. No more obsessing over “Buffy the Vampire Slayer”
(never mind) Re-dedica on peak “Who’s Kobe Bryant?” Jack’s precision calcula ons prowess went well beyond par cle physics. I was on a faculty commi ee charged to give an award to a student.
There were three nomina ons.
Each one looked great, and indis nguishable to me. How can we decide who is best?
My strategy, as usual, was … … WWJD = What would Jack Do?
So, I asked, “Jack, who’s best?” Without hesita on Jack said,
1st 2nd 3rd That ended the discussion for me…
Other commi ee members needed more convincing
Somebody had the brilliant idea to… … look at the students’ GPAs!!
We shuffled through the folders and found the data 3.82 3.81 3.77
1 part in 103 precision calcula on ability, not just in par cle physics! We all admire Jack’s incredible Energy Dedica on Consistency Technical Skill Crea ve Insight Go-to exper se in Higgs Physics etc
Jack’s contribu ons to Higgs physics in par cular has been second to none in the world, and con nues to have great influence. Naturalness and the Higgs Boson
Audience cau on: you may experience slight dizziness, minor irrita ons at mes, and in rare circumstances loss of consciousness. 24 HISTORICAL SLIDE – 3 YEARS OLD Should we believe in the Higgs boson? The Higgs boson is a specula ve par cle explana on for elementary par cle masses.
Cons: 1. One par cle carries all burdens of mass genera on? 2. Fundamental scalar not known in nature. 3. Hasn’t been found yet. 4. Too simplis c -- dynamics for vev not built in. 5. Idea not stable to quantum correc ons. Fixing that New Physics
Pros: S ll consistent with experimental facts!
25 New Physics Ideas and Higgs boson viability
Trying to fix and understand Higgs physics leads to new ideas that have new states/dynamics (susy, extra dimensions, t’, etc.) and a scalar state that look very similar to the Standard Model Higgs boson.
What have we found so far?
26 mass = 126 GeV New York Times 27 But nothing else has been found so far…. $WODV6HDUFKHVBH[RWLFVBKFSSQJ ð
28
KWWSV WZLNLFHUQFKWZLNLSXE$WODV3XEOLF&RPELQHG6XPPDU\3ORWV$WODV6HDUFKHVBH[RWLFVBKFSSQJ Losing the Naturalness Religion
Star ng to hear many more comments like:
“Quadra c divergence Naturalness problem is just philosophical – not really a data-driven concern.”
“Dimensional regulariza on has no quadra c divergence Naturalness problem, so maybe it doesn’t exist”
H H W
2 (Note, there is no Λ cutoff funny business – only 1/(4-n)) 29 Sensi vity to higher physical scales persists
However, all it takes is for any massive par cle to interact with 2 2 1 mW the Higgs and there is a real physical quantum correc on to m γE +ln4π +1 ln + (1) W 4 n − − µ2 ··· contend with. −
∆ = λ H 2 Φ 2 (2) L Φ| | | | H H Φ
It is inconceivable to me that there is nothing else between 2 2 2 ∆m λ m ln m18 (3) “here” (10 GeV) and the Planck scale (10H ∝ Φ Φ Φ GeV). And if there is another scalar (even if exo cally charged!) there is no simple symmetry to forbid it from coupling to the Higgs boson. 30
1 Implicit Postulate of Absolute Naturalness: A large hierarchy in QFT (even "technically natural") requires further explana on by further dynamics or an addi onal principle. Either way: “new physics”.
Higgs without yet discovering suppor ng entourage (susy, Xdim, composite states, etc.) is disheartening to some.
Anguished query from the youngsters: Should I s ll take this “philosophical concept” of Naturalness seriously as a guiding principle? 31 Showing that Naturalness Principle is Effec ve
Naturalness has been the oxygen of “Beyond the Standard Model Physics”.
We can show the concept’s effec veness (if it is) by wai ng for new dynamics to arise at LHC that stabilizes the Higgs boson to these Naturalness- voiding quadra c divergences.
Or we can try to put it on trial now.
Naturalness defense a orney 32 QED Applica on
Let's test the principle of Naturalness as a guide by applying it to the past.
Example: The early days of Quantum Electrodynamics.
Specifically, why is the electron so light??
(I wonder: Why didn’t “they” ask that ques on more earnestly?)
33 2
TECHNICAL NATURALNESS perturbative gauge coupling in the ultraviolet flows to strong value at the low scale and confinement happens at Λ 1 GeV. This gives to characteristic scale of There is another notion of Naturalness that has been QCD ∼ articulated from as least the early 1980’s [], which is the hadrons in the theory, and the proton and neutron called Technical Naturalness, or sometimes ’t Hooft Nat- obtain mass approximately equal to this scale. However, uralness. A theory has Technical Naturalness even if the pion masses are much lower, and can be understand as the Goldstone bosons of SU(2) SU(2) SU(2) some parameters are small, if an enhanced symmetry de- L × R → V velops when the small parameter is taken to zero. For flavor symmetry breaking. The mass is exactly zero when example, a very light fermion mass is Technically Natu- there are no explicit quark masses in the theory, and this ral since an enhanced chiral symmetry emergences when “hierarchy” is very well understood. Furthermore, no- the mass is taken to zero. In quantum field theory this body is concerned that the proton mass mp 1 GeV 19∼ protects the small parameter from any large quantum is much less than the Planck mass MPl 10 GeV = 1/2 ∼ (G )− . The reason is that an (1) number, namely correction, and the small value is technically stable. N O Agreeing that theories with small parameters need to the QCD gauge coupling, is an input at some high scale possess Technical Naturalness is less philosophically tax- that through renormalization group flow generates an ex- ing than demanding nature must be described by a theory ponentially suppressed scale through dimensional trans- with Absolute Naturalness. In other words, more funda- mutation. This is Natural because no very big or very mental theories with Absolute Naturalness always lead to small numbers were needed as input. low-energy effective theories with Technical Naturalness. However, there is a Naturalness problem in QCD. However, it is not as clear that all Technically Natural Namely Goldstone bosons are not exactly massless, but low-energy theories must arise from theories that have gain small mass due to explicit breaking from quark Absolute Naturalness. Thus, Absolute Naturalness is a masses. These quark masses are neither MPl nor ΛQCD. stronger form of Naturalness. They are very small compared to both. The up and down Requiring theories possess Technical Naturalness is quark masses are roughly mq 10 MeV. This is about a 2 ∼ generally assumed a non-negotiable requirement in the- 10− suppression with respect to ΛQCD, which is border- ory model building, whereas strides toward improving the line acceptable from a very generic point of view (even Absolute Naturalness of our theories are generally con- though there is no obvious connection between ΛQCD and sidered as further steps toward more fundamental law. It quark masses). The light quark masses are more than a 20 is this working hypothesis, either explicitly or more often factor of 10− suppression compared to MPl, which of implicitly assumed in the field, that more fundamental course looks much worse. The theory is very obviously theories should possess Absolute Naturalness that is sub- not Natural. ject to controversy and refutation upon further scrutiny. What we have seen here is that Technical Natural- In the subsequent discussion I will use the word Natural- ness helped explain the pion masses, and we were able ness to mean the stronger form of Absolute Naturalness, to see that Absolute Naturalness is satisfied when ex- and will use Natural to mean possessing the qualities of plaining the proton and neutron mass. However, if we Absolute Naturalness. want Absolute Naturalness to explain the small non-zero pion masses, or equivalently the light quark masses, we must go to a deeper theory. Writing down small, explicit JUSTIFYING NATURALNESS masses is not allowed in a theory with Absolute Natural- ness. Assuming Naturalness as a law of nature imposes very strong constraints on model building. In the case of the Higgs boson, it leads not only to ideas like supersymme- QUANTUM ELECTRODYNAMICS AND try, which protect the Higgs boson from having a large NATURALNESS mass, but the devotion one has to strict naturalness leads to radically different superpartner spectra. Compare the We could proceed further with a discussion about ap- spectrum of heavy superpartners in PeV scale supersym- plying Absolute Naturalness demands on the quarks of metry [4? , 5], where naturalness is not so strictly bowed QCD, but quantum electrodynamics (QED) provides ex- to, versus the spectrum of supersymmetry that requires actly the same conundrum, except that it is simpler to no parameters and finetunings more than one percent [7]. discuss and the problem to overcome is numerically more In the past, Technical Naturalness has been used to severe. understand experimental results that have already been Let us consider the QED theory as it was known in its measured. For example, the masses of the pions, pro- early days.Quantum Electrodynamics The lagrangian is quite simple ton and neutron are understood well from symmetries and Technical Naturalness. We know from asymptotic 1 = F F µν + iψγ¯ µ(∂ ieA )ψ + m ψψ¯ (1) freedom of quantum chromodynamics (QCD) that the L 4 µν µ − µ e -4 A,F contain the photon and ψ is the electron (me=5x10 GeV). Extraordinary theory:
1. Rela vis c invariant 2. Dirac equa on built in 3. Massless photon – electromagne c radia on 4. Electromagne c gauge invariance 5. Renormalizable! (infini es easily handled) 6. Fits the low-energy data very well
Subsequent theories had to live up to QED (e.g.,
renormalizability of the massive weak interac ons, etc.) 34 The electron mass
Hard to imagine QED being cri cized from the perspec ve of the 1940s. But what if they took naturalness seriously?
Naturalist: Why is the electron mass so small?
Skep c: Small compared to what?
-1/2 18 Naturalist: Newton’s gravity scale (GN) =10 GeV
Skep c: What’s gravity got to do with a li le par cle’s mass?
Naturalist: GN is a dimensionful scale parameter in the ac on of natural law just like the electron mass is. How can we have such a large hierarchy between dimensionful numbers?
35 Skep c: Nobody understands gravity. It’s not renormalizable. It’s too remote. We don’t understand it.
Naturalist: Precisely! We don’t understand it, so let’s try.
Skep c: Maybe Dirac was right with his Large Number Hypothesis. The universe spots us one very large number and we just have to live with it. I’m willing to live with that, especially if it involves mysterious and remote gravity!
36 Naturalist: Ok. I’ll give you that for now. Who am I to argue with
Dirac. But what about the electron to proton mass ra o mp/me = 103.
Skep c: 103 is no big deal. That can just be an accident. People do get struck by lightening you know, and that’s much more rare.
Naturalist: Ok, then what about Fermi’s new theory of β decay. His constant is orders of magnitude larger yet than the proton!
Skep c: Hmm. (Stalling) Remind me about Fermi’s Theory….
37 161 E. Fermi, Zeitschri für Physik, 88 (1934) 161. Versuch einer Theorie der p-Strahlen. I1). Von E. Fermi in Rom. M.it 3 Abbildungen.Versueh einer Theorie(Eingegangen der fl-Strahlen. am 16.I. Januar 1934.)165 Eine quantitative Theorie des fl-Zerfalls wird vorgesehlagen, in weleher man diewo Existenzcs, ~ und c*,~ des GrSgen Neutrinos darstellen, annimmt, die yon undden Koordina~en,die Emission I_mpulsen der Elektronen usw. und Neutrinosdes schweren aus einemTeilchens Kern abh~ngen beim ~-Zeffall kSnnen. mit einer ~hnliehen Methode behandelt, wie dieZur Emission n/iheren einesBestimmung Lichtquants yon Haus is~ einemman aufangeregten Einfachheitskriterien Atom in der Strah- lungstheorie.angewiesen. EineFormeln wesentliche fiir die EinschriinkungLebensdauer undin derfiir Freiheitdie Form der des Wahl emittierten kontinuierlichen/~-Strahlenspektrumsyon H ist durch die Erhaltung des Impulses werden sowie abgeleitet durch unddie mitBedingung der Effahrung 176 E. Fermi,verglichen. gesetzt, dais bei einer Drehang oder einer Translation der Raumkoordi- 1. Grundannahmen der Theorie. diena~en folgenden (9) invariant Daten: ~bleiben --= 0,87; muli Uo~- 1,24; F (Y/o) =- 0,102; vF (~7o) = 0,09, also einBeiSehen TP-Wert dem wir Versuch, mmitehstetwa zehnmal eine yon Theorie den kleiner Relativitgtskorrektionen der als Kernelektronendie der ersten Gruppe. andsowie der der/~-Emission Spin-Ffir l~aDaufzubauen,wirkung hat man ab, sobegegnet~ =ist 320000;wohl dieman Uo eintaehst =bekanntlieh 0,38 mSgliche(sehr unsieher); zwei Wahl Sehwierigkeiten. yon F (Uo) (9) ~die 0,00011; folgende: Die ersteTheory ~Fist (Uo)dutch =: 35. das RaD kontinuierlicheH liegt---- g also{Qv ungefiihr(x) fl-Strahlenspektrum~0 (x) +in Q*~o*der Mitte (x) q0* zwisehen (x)}, bedingt. den beidenFalls(10) der Er-innova on Gruppen.haltungssatz Ich habe der kei~eEnergie Daten giiltig fiber bleiben die anderen sell, fl-strahlendenmu~ man annehmen, Elemente dab ein wo g eine Konstan~e mi~ den Dimensionen LSMT -~ darstellt; x repr~- MsTh 1, UY, Ae, AcC, UZ, RaC" gefunden. Brnehteilsentiert die der Koordinaten beim /%Zeffall des sehweren ffei werdendenTeilehens; ~o, Energie ~o, ~o*, ~*unseren sind dureh bisherigen Aus den Daten der Tabelle 2 kann man eine, wenn aueh sehr grobe, BeobachtungsmSglichkeiten(2) und (4) gegeben und sind anentgeht. dent Orte Naeh x, y, demz des Vorschlagsehweren Teilchens von W. Pauli Absch~i~zung der Konstante g gewirmen. Nimmt man etwa an, dal~ in kannzu nehmen. man z.B. annehmen, dab beim /~-Zerfalt nieht nut ein Elektron, den F~llen wo (50) gleieh Eins wird, man ~:F (Uo) ~- 1 hat (d. h., in Se- Experimental sondern(10) auch stellt einkeineswegs neues Teilchen,die einzig dasmSgliche sogenannte Wahl yon ,,Neutrino" H dar. Jeder(Masse yon kunden, ---- 8600), so bekommt man aus (45): savvy derskalare GrSBenordnung Ausdruek, wie oder etwa kleiner als die Elektronenmasse; keine elektrisehe g = 4.10-5~ 3. erg. Ladung) emittiertL (p) wird.~o (x) MIn (p) der ~o (x)vorliegenden N (p) + kompl, Theorie konjug., werden wir die Hypo- Dieser Weft gibt natfirlich nut die Grsl~enordnm~g yon g. 38 thesewo Ldes (p), NeutrinosM (p), N (p) zugrunde passende legen.Funktionen des Impulses des schweren Zusammenfassend kann man sagen, dal~ dieser Vergleieh yon Theorie TeilchensEine weitere darsr Schwierigkeitwiirde ebensogut fi~r mSglichdie Theorie sein. derDa Kernelektronenjedoch die Folge- besteht and Erfahrung eine so gute Ubereinstimmung gibt, wie man nur erwarten rungen aus (10) bisher mit der Erfahrung in Einklang zu sein scheinen, darin, dab die jetzigen relativistischenkonnte. Theorien Die bei derden alsleiehten experi- Teilehen ist es wohl besser, sieh vorl~ufig auf die einfachste Wahl zu beschri~nken. (Elektronen oder Neutrinos) niehf imstandementell unsieherenslnd, in einwandfreier Elementen Weise zu erkl~ren,Wesentlieh wie ist solche es jedoch, Teilehen den Ausdruck in BahnenRaD (10) yonund derart KerndimensionenAeB zu verallgemeinern,festgestellten gebunden dab man mindestens die leichten Teilehen relativistisoh behandeln kann. werden kSnnen. Abweichungen kSnnenwohl teil- Aueh bei dieser Verallgemeinerung ist natiirlich eine gewisse Willkiir nieht Es seheint deswegen zweckm~Biger,weise mit dutch Heisenberg Ungenauigkeit ~) anzunehmen, der auszuschlielSen. Die einfachste LSsung des Problems diirf~e die folgende sein: dab ein Kern nut aus schweren Teilchen,Messungen t'rotonen erkl~rt werden,und Neutronen, teil- be- Relativis~iseh treten an Stelle yon Vweise and aueh~0 je Vierdurch Diraesche etwas abnorm Fank- steht. Um trotzdemFig. 2. die M5gliehkeit der/~-Emission zu verst~hen, wollen tionen V1 ~o~ V3 YJa und ~01 q% ~oa ~o4. Wir be~rachtengroi~e abet nan gar die" 16 nicht unabh~ngigen unplau- wit versuchen, eine Theorie der Emission leiehter Teilehen aus einem Kern siblebilinearen Schwankungen Kombinationen des l~Iatrixelements aus ~o1 ~o2 (50). V~ ~P4 Man und hat ~o 1~02 weiter ~o8 epa.zu bemerken,Bei einer in Lorentz-TransformationAnalogie zur Theorie der der Koordinaten Emission erfahreneines Liehtquants diese 16 trSlSen aus eineeinem an- da] man aus der den fl-Zerfall begleitenden 7-Strahlung sehliel]en kann, lineare Transformation, eine Darstellung der Ordnung 16 der Lorentz- dal~geregten die meisten Atom fl-Zerf~llebeim 'gewShnlichen zu verschiedenen StrahlungsprozeB Endzust~inden aufzubauen.des Protons In der Gruppe. Diese Darstellung sparer sich in versehiedene einfaehere Dar- fflhrenStrahlungst.heorie kSnnen, wodureh ist die wieder totale Schwankungen Anzahl der Lichtquantenin dem TF (~o)-~Vertkeine Konstante: stellungen; ira besonderen transformieren sich die vier bilinearen Kombi- erkl~irtLichtquanten werden kSnnen.entstehen, wenn sie von einem Atom emittiert werden, nationen : undWit versehwinden, wenden uns jetzt wenn zur sie Frage absorbiert nach der werden. Form der InGeschwindigkeits- Analogie hierzu wollen A o :~ v21qo 2 + ~oi + y%~04-- y~4~03, ] verteilungskurvewir der fl-Strahlentheorie der emittierten/3-Strahlen. folgende Annahmen FOr den zugrunde Fall der erlaubtenlegen: 2tl ~--- ~01 ~P3-- V2~04 - Vs~O1 -'}- V4~2, I (11) Uberg~nge ist die Verteilungskurve als Funktion yon U (d. h. his auf den 1) Vgl. die vorl~ufige Mitteilung : La Ricerca Scientifica 2, Heft 12, 1933. -- Faktore) W. Heisenberg,1700, von H~o) ZS. dutch f. Phys.(44) gegeben. 77, 1, 1932.Verteilungskurven ! flit ver- schiedene Werte yon ~o sind in der Fig. 9. zusammengestellt, wobei ffir die 11" Bequemlichkeit der Zeichnung die Ordinateneinheit in den versehiedenen F~llen passend gew~hlt worden ist. Diese Karven zeigen eine befriedigende ~hnliehkeit etwa zu den yon Sargent 1) zusammengestellten Verteilungs- kurven. Nur in dem Tell der Kurve kleiner Energie liegen die Kurven von Sargent etwas tiefer als die theoretischen. Dies ist deutlieher in der
1) B. W. Sargent, Proe. Cambridge Phil. Soc. 28, 538, 1932. Fermi’s Constant in the day… 2 2 1 mW mW γE +ln4π +1 ln + (1) 4 -50 n −3 -6− -2 µ2 ··· g = 4 x 10 − cm erg = 3.25 x 10 GeV -1/2 This translates into scale MF = g = 555 GeV. 2 2 6 ∆ = λΦ H Φ (2) MF/me = 10 Now that’s star ng to be concerning!L | | | |
Today, we quote Fermi’s constant as ∆m2 λ m2 ln m (3) H ∝ Φ Φ Φ -5 -2 GF = 1.166 x 10 GeV where the normaliza on is set by
GF µ (1 γ5) (1 γ5) u¯γ − d e¯γµ − ν (4) √2 2 2 39
GF µ (¯uLγ dL e¯Lγµν + h.c.) (5) √2
1 Agreed: smal me not Natural Now what do we do?
We cogitate… We wait… We ponder…
And then a Realis c Intellectual Leap (RIL) happens.
RIL #1: me is closer zero than it is to MF.
Skep c: What does that do for us?
Answer: Let’s try to start with a theory that forbids electron mass and see if we have an idea to let in a li le bit of mass later.
40 Forbidding the electron mass
It’s obvious that the problem is that ψψ is gauge invariant. How can we make it non-gauge invariant.
We cogitate… We wait… We ponder…
RIL #2: The representa on structure of the Lorentz group allows us to write QED in two component spinors:
41 Laporte & Uhlenbeck (1931):
Staring at this we see a qualita ve difference between the mass term and the photon interac on term. Mass requires both right and le components, whereas photon int. does not.
This is the introduc on of chirality into the theory.
42 3
3 where ψ is the spin-1/2 electron field, Aµ is the spin- lagrangian in the constituent two-component formalism: 1 photon field, e 0.31 is the gauge coupling constant where ψ is the spin-1/2 electron field, Aµ is the spin- lagrangian1 in the constituentµν µ two-component formalism: and m =0.511 MeV is the electron mass. The coupling = F F + iψ† γ (∂ ieA )ψ (3) 1e photon field, e 0.31 is the gauge coupling constant L 4 µν L µ − µ L constant appears to satisfy the demands of Absolute Nat- 1 µν µ and me =0.511 MeV is the electron mass. The coupling = Fµν F µ + iψ† γ (∂µ ieAµ)ψL (3) L 4+ψ† γ (∂µ LieAµ)ψ−R + me(ψ† ψR + ψ† ψL) uralness,constant but the appears electron to satisfy mass the does demands not. of Absolute Nat- R L R µ − +ψ† γ (∂ ieA )ψ + m (ψ† ψ + ψ† ψ ) Oneuralness, violation but of the the electron electron mass mass does with not. respect to Ab- We can accomplishR µ − ourµ firstR taske byL R somehowR L treating 23 solute NaturalnessOne violation is of that the electronme/MPl mass with10− respect. However, to Ab- ψWeL di canfferently accomplish than ourψR first. The task electric by somehow charge treating for both we 23 one mightsolute object Naturalness that this is that involvesme/M thePl mysteries10− . of However, gravi- ψ differently than ψ . The electric charge for both we mustL keep at 1, butR we must assign different charges ties whichone might are too object diffi thatcult this to sort involves out and the mysteries that perhaps of gravi- all must keep at −1, but we must assign different charges for each under− the new symmetry G. A simple concrete of theties Naturalness which are too considerations difficult to sort of out the and Standard that perhaps Model all startfor each to underthis would the new be tosymmetry let G beG some. A simple new abelian concrete group can beof satisfied the Naturalness at mass considerations scales far removed of the Standard from M Model.I start to this would be to let G be some new abelian group can be satisfied at mass scales far removed from MPl .I U(1) and assign ψR double the charge of ψL. In other do not agree with this statement, and I believe a theoryPl U(1) and assign ψR double the charge of ψL. In other do not agree with this statement, and I believe a theory words, our our spectrum spectrum is is that trulythat truly possesses possesses Absolute Absolute Naturalness Naturalness will will have have no no unexplained large hierarchies, including with respect to UnderUnderU(1)U(1)EMU(1)U (1): : unexplained large hierarchies, including with respect to EM × × MPl.M However,Pl. However, the problemthe problem with with the the electron electron mass mass is is ψψL=(=(1, 1,1),1)ψ, ψ=(R =(1, 2)1, 2) (4) (4) L −−− − R − − − alreadyalready transparent transparent without without invoking invokingMMPlPl. . With these these charge charge assignments assignments the theψψ¯ ψψterm¯ term is no is longer no longer In 1933In Fermi1933 Fermi proposed proposed his his theory theory of of nuclear nuclear beta beta de- de- allowed. cay, whichcay, which in modern in modern language language is governed is governed by by the the four four µ TheThe next next step step is is to to somehow somehow regain regain the electron the electron mass mass fermion operator GF u¯µLγ dLe¯γµν,whereGF =1.17 43 fermion operator5 2 GF u¯Lγ dLe¯γµν,whereGF =1.17 × through some other means. Everything would be tried, 5 10− GeV2 − is the Fermi constant. This constant defines× through some other means. Everything would be tried, 10− GeV− is the Fermi constant.1/ This2 constant defines and in time it would be recognized that one could form a new mass scale MF =(GF1)/−2 = 293 GeV. The vio- and in time it would be recognized that one could form a new mass scale MF =(GF )− = 293 GeV. The vio- a renormalizable operator with ψL and ψR if a scalar is lations of Absolute Naturalness can now be phrased as a a renormalizable operator with ψL and ψR if a scalar is lations of Absolute Naturalness can now be phrased as a added. The operator is yeψL† ΦψR where ye is a dimen- ratio that is unnaturally too small: added. The operator is y ψ† Φψ where y is a dimen- ratio that is unnaturally too small: sionless coupling constant ande L the quantumR numberse of sionlessΦ are (0, coupling1). I believe constant it would and be inevitable the quantum to write numbers this of me 6 =1.7 10− (Absolute Naturalness problem) (2) Φdownare because (0, 1). I there believe would it would be an attempt be inevitable to get toψL writeand this me MF ×6 =1.7 10− (Absolute Naturalness problem) (2) downψR together because in an there operator would somehow be an attempt since a mass to get termψL and MF × ψhasR together to connect in the an two operator together somehow by some since means. a mass The term If pursuing Absolute Naturalness is a valid guide to con- easiest way to do that, which surely would be found, is structing more fundamental theories, then we should be has to connect the two together by some means. The If pursuing Absolute Naturalness is a valid guide to con- easiestto add this wayΨ toscalar. do that, which surely would be found, is able to apply the principle to this problem and see that The next inevitable step is to enhance the Lagrangian structing more fundamental theories, then we should be to add this Ψ scalar. it could have guided one to deeper insights if adhered to to include kinetic terms and potential for the scalar field able touncompromisingly. apply the principle to this problem and see that The next inevitable step is to enhance the Lagrangian it could have guided one to deeper insights if adhered to 2 2 2 to include∆ kinetic= ∂µ termsΦ andµ Φ† potentialΦ λ(Φ†Φ for) the scalar(5) field We should reiterate that small me is Technical Natu- L | | − − uncompromisingly.ral, as a chiral symmetry enhancement develops in the I believe that once this step2 occurs2 the Higgs mechanism2 Welimit should of m reiterate0. The that small small massm valueis Technical is of course Natu- sta- ∆ = ∂µΦ µ Φ†Φ λ(Φ†Φ) (5) e e follows almostL immediately.| | − A systematic− study of the ble to quantum→ corrections of the theory. But Absolute ral, as a chiral symmetry enhancement develops in the parameters of this potential would easily recognize that Naturalness says that m ψψ¯ really should be M ψψ¯ . I believe that once this step occurs the Higgs mechanism limit of me 0. The smalle mass value is of courseF sta- when µ2 < 0 there is a vacuum expectation value of the → ¯ follows almost immediately. A systematic study of the ble to quantumThe QED corrections gauge invariance of the of theory. the operator But Absoluteψψ is the Higgs boson and stability of the potential requires λ > 0 ¯ ¯ parameters of this potential would easily recognize that Naturalnessproblem. says This that operatormeψψ really is dimension should three be M andF ψψ so. to at the renormalizable2 level. This leads to a minimum of round out the dimensions to four one needs a massive when µ < 0 there is a vacuum expectation value of the The QED gauge invariance of the operator ψψ¯ is the the potential where the Φ field has a vacuum expectation coupling, which Absolute Naturalness demands should be Higgsvalue of bosonΦ = andµ2 stability/λ. of the potential requires λ > 0 problem. This operator is dimension three and so to − similar to MF . A reasonable starting path is to somehow atThe the reader renormalizable might initially level. recoil This at leads such a to nonchalant a minimum of ¯ roundaugment out the the dimensions theory to make to fourψψ onenot an needs invariant a massive in the theassumption potential that where the Higgs the Φ mechanismfield has a would vacuum be expectation an in- 2 coupling,spirit which that Absoluteme is “closer Naturalness to zero than demands it is to shouldMF .” One be valueevitable of breakthrough,Φ = µ /λ. given that it is so highly cele- − similarcan to thenMF . look A reasonable to set the starting mass to path zero and is to then somehow find a bratedThe today,reader but might recall initially that many recoil people at such simultane- a nonchalant ¯ augmentnew the means theory by which to make to recoverψψ not a correction an invariant to that in the and assumptionously understood that it the in di Higgsfferent contextsmechanism historically. would be The an in- a give a finite value. spirit that me is “closer to zero than it is to MF .” One evitablebiggest leap breakthrough, is not the Higgs given mechanism that it but is so coming highly to cele- can thenIf look adherence to set to the Absolute mass Naturalness to zero and is then religious, find no a bratedthe realization today, that but it recall is important that many to study people a scalar simultane- po- new meansugliness by or which complexity to recover should a stop correction us from findingto that a and way tential. Thus, in our problem at hand, the biggest leap ¯ ously understood it in different contexts historically. The a giveto a finitebanish value. this offending operator ψψ. In time it would in all of our discussion in my view is first agreeing to be inevitable that theorists would recognize that there biggestAbsolute leap Naturalness, is not the and Higgs then second, mechanism recognizing but coming that to If adherenceis something to special Absolute about Naturalness that operator is religious, compared no to theψψ¯ can realization be banished that by it treating is important the two to components study a scalar of po- uglinessothers: or complexity it mixes the should right stop and leftus from handed finding components a way tential.the spinor Thus, differently. in our problem at hand, the biggest leap to banishof the this spinor. offending The kinetic operator termψψ¯ and. In gauge time interactions it would inOnce all of the our Higgs discussion mechanism in my is view understood is first progress agreeing to be inevitabledo not. that It is theorists more transparent would recognize if one write that the there QED Absolutewould be Naturalness,very fast. It and would then be second, recognized recognizing that the that is something special about that operator compared to ψψ¯ can be banished by treating the two components of others: it mixes the right and left handed components the spinor differently. of the spinor. The kinetic term and gauge interactions Once the Higgs mechanism is understood progress do not. It is more transparent if one write the QED would be very fast. It would be recognized that the The next step is to regain the electron mass through some other means.
A er me I think it is inevitable that people would think of
This is basically the start of the Higgs discussion, but certainly by 1950 with Ginzburg-Landau theory it was in the air to have a complex scalar func on order parameter interact with QED!
RIL #4 might look to be the least plausible RIL to some, but I think it is might be the most plausible. 44 Building on Landau's theory of phase transi ons (1937), Ginzburg- Landau theory of superconduc vity (1950) looks like the Higgs boson.
45 At this point it's all chug and crank. RIL#5 below would be obvious, and hardly an "intellectual leap" given RIL#1-#4:
This is the most obvious RIL, and easiest to achieve, but its consequences are huge. We started with a non-empirical philosophical criterion (Naturalness) and are led to our first "real" physics predic on. 46 4
fermion mass is obtained by m = y Φ .Thus,the a Higgs boson, but there will also be a massless pseu- e e electron mass is generated and for suitable choice of ye doscalar particle in the spectrum from the spontaneous and Φ the correct mass is obtained. breaking of the global symmetry. I believe it is also in- The reader may object at this point and say that we evitable, and perhaps even more likely from the start, have introduced several problems by doing all the steps that the choice would be made to promote U(1) to a above. First, we have introduced more parameters, an- gauge symmetry, in which case it is readily seen that other field, and more complexity in the theory, all in the massless pseudoscalar can be gauged away and said service of a debatable philosophical devotion. However, to be “eaten” by the photon of U(1) giving it mass of the objections are also based on debatable philosophical M 2 = 1 g 2 Φ 2. A 2 notions so we should carry on. More damaging is that we have replaced one problem with Absolute Naturalness (the electron mass) for another problem. If the Φ mass CONCLUSIONS 2 6 2 2 is µ2 mF then ye 10− , or if µ me then −2 2∼ 6 ∼ − ∼ µ /M 10− . Either way there is a problem with − F ∼ One path to justifying Absolute Naturalness as a guide respect to the Absolute Naturalness. But all this means to theory model building is to consider how science could is that we have to go further for a more fundamental have progressed in the past if we firmly devoted ourselves RIL #6: Researchers trying to renormalize this theory theory, and working harder would inevitably lead to con- to this principle. Would it have led us astray or would cepts like are found in today’s theories of flavor [1]. A would discover that U(1)’ has chiral anomalies, and would it have led to more fundamental theories? Although it is fully Natural theory can in principle be built and from be forced to add cancelling exo cs. conceivable that it could lead us astray at times, I have the point of view of Absolute Naturalness we have made presented here evidence of its salutary influence. tremendous progress. In this article I have considered QED as a theory in Now, as it stands our theory is sick, because there 3 gross violation of Absolute Naturalness, and followed are anomalies. The U(1)-graviton-graviton and U(1) plausible steps scientists could have taken if they were anomaly cancelation conditions are not met among the wholly devoted to recasting the theory in a way com- fermions. This necessitates additional fermions in the patible with Absolute Naturalness. The inferences that spectrum that are charged under U(1). There are many results are 1) the existence of an extra scalar Higgs bo- possibilities that could be written down. In an exhaustive son field that couples according to the mass of the elec- table among these possibilities would be the following ex- tron, 2) an exotic gauge symmetry with a massive pho- otic fermions: ton, 3) parity violation in the fermion interactions with the gauge boson, 4) the necessity of additional exotic Exotics : 6Q1 /3 +3Q4 /3 +3Q 2/3 +1Q 1 − − fermions to cancel anomalies. These inferences are ne- where nQq means n copies of fermions with charge q. cessitated by the approach we took. What’s also possible These add to our original fermions 1Q 1 +1Q 2.These is 5) the prospect of multiple copies (generations) of ex- − − charge assignments and multiplicities are exactly those otic fermions, and 6) the prospect of non-abelian gauge of the SM fermions under twice hypercharge (see, e.g., symmetries chirally protecting the fermion masses. 47 table 1 of [8]). The steps along the way required would have required There was infinite freedom in how I chose U(1) charge creativity and dedication to discovery, and also would assignments for ψL and ψR, and the magic of recovering have required strength in the face of criticisms regarding the SM spectrum was that I chose the simple case of the lack of simplicity and the complexity invoked. Losing ψR charge being twice that of ψL. We did not have to faith in Absolute Naturalness would have been easy, but choose those charges, of course. Furthermore, we could unwavering devotion to the concept would have paid off have even made G a non-abelian group, as that would do very handsomely. We may be finding ourselves in a sim- just as well in protecting the fermion masses. For exam- ilar situation today with respect to the continuing but ple, ψL could transform as a fundamental under SU(N), pressure-strained research to recast particle physics in a ψR as a singlet, and Φ as an anti-fundamental. Neverthe- theory with more Absolute Naturalness. less, a key qualitative point remains in all choices, and that is that exotics are expected in the spectrum. Fur- Acknowledgments: Acks. thermore, whatever set of exotics satisfies the anomaly cancelation conditions can be repeated many times and still satisfy all the constraints. This we can call the num- ber of generations. There is another complication in this theory that we [1] K. S. Babu, “TASI Lectures on Flavor Physics,” arXiv:0910.2948 [hep-ph]. have yet to discuss. We have treated U(1) as a global [2] G. F. Giudice, “Naturally Speaking: The Naturalness Cri- symmetry, and once Φ obtains a vacuum expectation terion and Physics at the LHC,” In *Kane, Gordon (ed.), 2 value there will be scalar in the spectrum of mass mϕ = Pierce, Aaron (ed.): Perspectives on LHC physics* 155- 2λ Φ , which is the analog of what we know today to be 178 [arXiv:0801.2562 [hep-ph]]. 4 fermion mass is obtained by m = y Φ .Thus,the a Higgs boson, but there will also be a massless pseu- e e electron mass is generated and for suitable choice of ye doscalar particle in the spectrum from the spontaneous and Φ the correct mass is obtained. breaking of the global symmetry. I believe it is also in- The reader may object at this point and say that we evitable, and perhaps even more likely from the start, have introduced several problems by doing all the steps that the choice would be made to promote U(1) to a above. First, we have introduced more parameters, an- gauge symmetry, in which case it is readily seen that other field, and more complexity in the theory, all in the massless pseudoscalar can be gauged away and said service of a debatable philosophical devotion. However, to be “eaten” by the photon of U(1) giving it mass of the objections are also based on debatable philosophical M 2 = 1 g 2 Φ 2. A 2 notions so we should carry on. More damaging is that we have replaced one problem with Absolute Naturalness (the electron mass) for another problem. If the Φ mass CONCLUSIONS 2 6 2 2 is µ2 mF then ye 10− , or if µ me then −2 2∼ 6 ∼ − ∼ µ /M 10− . Either way there is a problem with − F ∼ One path to justifying Absolute Naturalness as a guide respect to the Absolute Naturalness. But all this means to theory model building is to consider how science could is that we have to go further for a more fundamental have progressed in the past if we firmly devoted ourselves theory, and working harder would inevitably lead to con- to this principle. Would it have led us astray or would cepts like are found in today’s theories of flavor [1]. A it have led to more fundamental theories? Although it is fully Natural theory can in principle be built and from conceivable that it could lead us astray at times, I have the point of view of Absolute Naturalness we have made presented here evidence of its salutary influence. tremendous progress. In this article I have considered QED as a theory in Now, as it stands our theory is sick, because there 3 gross violation of Absolute Naturalness, and followed are anomalies. The U(1)-graviton-graviton and U(1) plausible steps scientists could have taken if they were anomaly cancelation conditions are not met among the wholly devoted to recasting the theory in a way com- fermions. This necessitates additional fermions in the patible with Absolute Naturalness. The inferences that spectrum that are charged under U(1). There are many results are 1) the existence of an extra scalar Higgs bo- possibilities that could be written down. In an exhaustive son field that couples according to the mass of the elec- table among these possibilities would be the following ex- tron, 2) an exotic gauge symmetry with a massive pho- otic fermions: ton, 3) parity violation in the fermion interactions with the gauge boson, 4) the necessity of additional exotic Exotics : 6Q1 /3 +3Q4 /3 +3Q 2/3 +1Q 1 − − fermions to cancel anomalies. These inferences are ne- where nQq means n copies of fermions with charge q. cessitated by the approach we took. What’s also possible These add to our original fermions 1Q 1 +1Q 2.These is 5) the prospect of multiple copies (generations) of ex- − − charge assignments and multiplicities are exactly those otic fermions, and 6) the prospect of non-abelian gauge of the SM fermions under twice hypercharge (see, e.g., symmetries chirally protecting the fermion masses. table 1 of [8]). The steps along the way required would have required There was infinite freedom in how I chose U(1) charge 3 Y 3 Y creativity and dedication to discovery, and also would Field SU(3) SU(2)L T 2 Q = T + 2 assignmentsa for ψL and ψR, and the magic of recovering gµ (gluons) 81 00 0 have required strength in the face of criticisms regarding the SM spectrum0 was that I chose the simple case of the (W ±,W ) 13( 1, 0) 0 ( 1, 0) lack of simplicity and the complexity invoked. Losing µ µ ± ± ψR charge0 being twice that of ψL. We did not have to Bµ 11 00 0 faith in Absolute Naturalness would have been easy, but choose those charges, of course. Furthermore,1 we2 could uL 2 1 3 unwavering devotion to the concept would have paid off QL = 32 have even maded G a non-abelian group,1 as6 that would1 do ! L " ! − 2 " ! − 3 " very handsomely. We may be finding ourselves in a sim- u 31 0 2 2 just as wellR in protecting the fermion masses.3 For3 exam- ilar situation today with respect to the continuing but d 31 0 1 1 ple, ψL couldR transform as a fundamental3 under SU3 (N), 1 − − pressure-strained research to recast particle physics in a νL 2 1 0 ψR as aE singlet,L = and Φ12as an anti-fundamental. Neverthe- e 1 − 2 1 theory with more Absolute Naturalness. less, a key! qualitativeL " point remains! − 2 " in all choices,! − " and e 11 0 1 1 that is thatR exotics are expected in the− spectrum.− Fur- Acknowledgments: Acks. φ+ 1 1 thermore,Φ = whatever set12 of exotics satisfies2 1 the anomaly φ0 1 2 0 cancelation! conditions" can be repeated! − 2 " many! times" and φ0 1 0 Φc = 12 2 1 still satisfy all the constraints. This1 we can2 call the num- ! φ− " ! 2 " − ! 1 " ber of generations. − − There is another complication in this theory that we [1] K. S.48 Babu, “TASI Lectures on Flavor Physics,” Table 1: Charges of Standard Model fields. arXiv:0910.2948 [hep-ph]. have yet to discuss. We have treated U(1) as a global [2] G. F. Giudice, “Naturally Speaking: The Naturalness Cri- interaction:symmetry, and once Φ obtains a vacuum expectation terion and Physics at the LHC,” In *Kane, Gordon (ed.), 2 value there will be scalar in the spectrum of mass mϕ = Pierce, Aaron (ed.): Perspectives on LHC physics* 155- c yt v + h 2λ Φ , which∆ = isy thetQL† Φ analogtR + c.c. of= what(tL† bL† we) know todaytR + c.c. to be (48) 178 [arXiv:0801.2562 [hep-ph]]. L √2 ! 0 " h h = m (t† t + t† t ) 1+ = m tt¯ 1+ (49) t R L L R v t v # $ # $ where mt = ytv/√2isthemassofthet quark.
The mass of the charged leptons follows in the same manner, yeEL† ΦeR + c.c.,and interactions with the Higgs boson result. In all cased the Feynman diagram for Higgs boson interactions with the fermions at leading order is m hff¯ : i f . (50) v
We see from this discussion several important points. First,thesingleHiggs boson of the Standard Model can give mass to all Standard Modelstates,evento the neutrinos as we will see in the next lecture. It did not havetobethatway.It could have been that quantum numbers of the fermions did not enable just one Higgs
10 But Naturalness is not yet fully solved! (never ending pursuit...)
Responses: (1) Working to solve Naturalness (not necessarily solving it) frui ul. (2) Theory space of where to go next is infinite. Naturalness provides discrete path(s) and gateway(s) through which to inves gate. (3) We are not done in our Naturalness quest: this is the next door. 49 Conclusion 1/2
Thus, it is plausible that if Naturalness were religiously held to by researchers in the 1940’s/50’s, we would have been led to
1. Exo c gauge symmetry and exo c massive gauge boson 2. Higgs boson and Higgs mechanism 3. Exo c charged fermion scenarios, including iden fying full SM content as one viable predic on
These are correct and extraordinarily frui ul results from merely taking Naturalness seriously.
But it would nevertheless take many decades to experimentally find all these implica ons.
(Depressed unimagina ve skep cs would gloat during that me.) 50 Conclusion 2/2
Same situa on today? Naturalness for Higgs leads to many exo c implica ons.
Analogies: Supersymmetry & Composite Higgs & Etc
1. Solves Naturalness problem 2. Exo cs are required for self-consistency 3. Cri cism that exchanges one problem (quadra c instability) for another problem (SUSY: µ term problem; Xdim: size of compact dimensions) 4. No theorem for how long it will take to discover it (could be decades, and decades more)
Historical considera ons suggest taking Naturalness seriously has value. Present circumstances may not warrant its abandonment. 51