The Loss of Naturalness? James Wells University of Michigan
The Meaning of LHC’s results: The Loss of Naturalness? James Wells University of Michigan
CTEQ, July 2015 1 2 Size of the ball is propor onal to its “elementary par cle mass”
Max Planck Ins tut
How do all of these get masses….? The Higgs Boson. 3 Peter Higgs and the Maths
University of Edinburgh 4 www.kip.uni-heidelberg.de 5 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.
Pros: S ll consistent with experimental facts!
6 Higgs boson unstable to QM
A quantum loop is quadra cally divergent. Higgs mass, connected to Higgs vev, is unstable to the highest mass scales in the theory.
18 Confusing: MPl is 10 mes more massive than Higgs boson.
7 Cures of the Naturalness Problem, and the Resul ng Higgs boson Entourage
1. Disallow all scalars in the theory (Technicolor).
2. Symmetry cancels quadra c divergences (supersymmetry)
3. Disallow higher mass scales (extra dimensions).
Implica on: “New Physics” needs to be found at LHC.
8 OLD SLIDE – 3 YEARS OLD
New Physics Ideas and Higgs boson viability
Trying to fix and understand Higgs physics leads to new ideas that have states that look very similar to the Standard Model Higgs boson.
Precision Electroweak Data almost demands this to be true.
What does the future hold for the Higgs boson?
DISCOVERY!!
9 mass = 126 GeV New York Times 10 But nothing else has been found…. 4/24/13 AtlasSearches_exotics_hcp12.png (1096×827)
11
https://twiki.cern.ch/twiki/pub/AtlasPublic/CombinedSummaryPlots/AtlasSearches_exotics_hcp12.png 1/1 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 à
2 (Note, there is no Λ cutoff funny business – only 1/(4-n)) 12 Sensi vity to higher physical scales persists 1 m2 m2 +ln4⇡ +1 ln W + (1) W 4 n E µ2 ··· However, all it takes is for any massive par cle to interact with ✓ ◆ the Higgs and there is a real physical quantum correc on to contend with. = H 2 2 (2) L | | | |
H H Φ m2 m2 ln m (3) H / It is inconceivable to me that there is nothing else between “here” (102 GeV) and the Planck scale (1018 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. 13
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? 14 Showing that Naturalness Principle is Useful
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 15 QED Applica on
In the mean me, we can 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?)
16 2
TECHNICAL NATURALNESS perturbative gauge coupling in the ultraviolet flows to strong value at the low scale and confinement happens at ⇤ 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 e↵ective 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 ⇤QCD. 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 ⇤QCD, 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 ⇤QCD 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 di↵erent 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 µ⌫ + i ¯ µ(@ ieA ) + m ¯ (1) freedom of quantum chromodynamics (QCD) that the L 4 µ⌫ 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.) 17 The electron mass
Hard to imagine QED being cri cized from the perspec ve of the 1940s. But what if they took naturalness seriously?
Naturalness: Why is the electron mass so small?
Skep cism: Small compared to what?
-1/2 18 Naturalness: Newton’s gravity scale (GN) =10 GeV
Skep cism: What’s gravity got to do with a li le par cle’s mass?
Naturalness: 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?
18 Skep cism: Nobody understands gravity. It’s not renormalizable. It’s too remote. We don’t understand it.
Naturalness: Precisely! We don’t understand it, so let’s try.
Skep cism: 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!
19 Naturalness: 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 3 mp/me = 10 .
Skep cism: 103 is no big deal. That can just be an accident. People do get struck by lightening you know, and that’s much less rare.
Naturalness: Ok, then what about Fermi’s new theory of β decay. His constant is orders of magnitude larger yet than the proton!
Skep cism: Hmm. (Stalling) Remind me about Fermi’s Theory….
20 161 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 Einewo cs,quantitative ~ und c*,~ GrSgen Theorie darstellen, des fl-Zerfalls die yon den wird Koordina~en, vorgesehlagen, I_mpulsen in weleherusw. man diedes Existenz schweren desTeilchens Neutrinos abh~ngen annimmt, kSnnen. und die Emission der Elektronen und Neutrinos aus einem Kern beim ~-Zeffall 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 sowieabgeleitet durch unddie mitBedingung der Effahrung 176 E. Fermi, gesetzt, dais bei einer Drehang oderverglichen. einer Translation der Raumkoordi- na~en (9) invariant bleiben muli die folgenden Daten: ~ --=1. 0,87; Grundannahmen Uo~- 1,24; F (Y/o) der =- Theorie.0,102; vF (~7o) = 0,09, Sehen wir mmitehst yon den Relativitgtskorrektionen and der Spin- also einBei TP-Wert dem Versuch, etwa zehnmal eine Theorie kleiner der als Kernelektronendie der ersten Gruppe. sowie der/~-EmissionFfir wirkung ab, so ist wohl die eintaehst mSgliche Wahl yon (9) die folgende: l~aDaufzubauen, hat man begegnet~ = 320000; man Uo =bekanntlieh 0,38 (sehr unsieher); zwei Sehwierigkeiten. F (Uo) ~ 0,00011; Die ersteTheory H ---- g {Qv (x) ~0 (x) + Q*~o* (x) q0* (x)}, (10) ~Fist (Uo)dutch =: 35. das RaD kontinuierliche liegt also ungefiihr fl-Strahlenspektrum in der Mitte zwisehen bedingt. den beidenFalls der Er-innova on Gruppen. Ich habe kei~e Daten fiber die anderen fl-strahlenden Elemente haltungssatzwo g eine Konstan~e der Energie mi~ dengiiltig Dimensionen bleiben sell, LSMT mu~ -~ mandarstellt; annehmen, x repr~- dab ein 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~zungzu nehmen. der Konstante g gewirmen. Nimmt man etwa an, dal~ in kann man z.B. annehmen, dab beim /~-Zerfalt nieht nut ein Elektron, den F~llen(10) stelltwo (50) keineswegs gleieh Eins die wird,einzig man mSgliche ~:F (Uo) Wahl ~- 1 yonhat H(d. dar.h., in JederSe- Experimental kunden,sondernskalare ---- Ausdruek,auch 8600), ein so neueswiebekommt etwa Teilchen, man aus das (45): sogenannte ,,Neutrino" (Masse yonsavvy der GrSBenordnung oder kleiner als die Elektronenmasse; keine elektrisehe L (p) ~o (x)g M= (p)4.10-5~ ~o (x) N (p)3. erg.+ kompl, konjug., Ladung) emittiert wird. In der vorliegenden Theorie werden wir die Hypo- Dieser Weft gibt natfirlich nut die Grsl~enordnm~g yon g. 21 wo L (p), M (p), N (p) passende Funktionen des Impulses des schweren theseZusammenfassend des Neutrinos kann zugrunde man sagen, legen. dal~ dieser Vergleieh yon Theorie Teilchens darsr wiirde ebensogut mSglich sein. Da jedoch die Folge- and ErfahrungEine weitere eine soSchwierigkeit gute Ubereinstimmung fi~r die Theorie gibt, wie der man Kernelektronen nur erwarten besteht 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 Wesentlieh ist es jedoch, den Ausdruck (10) derart zu verallgemeinern, RaD und AeB festgestellten zudab erkl~ren, man mindestens wie solche die Teilehen leichten inTeilehen Bahnen relativistisoh yon Kerndimensionen behandeln kann. gebunden Abweichungen kSnnenwohl teil- werdenAueh beikSnnen. dieser Verallgemeinerung ist natiirlich eine gewisse Willkiir nieht weise dutch Ungenauigkeit der auszuschlielSen.Es seheint Diedeswegen einfachste zweckm~Biger, LSsung des Problems mit Heisenberg diirf~e die folgende ~) anzunehmen, sein: Messungen erkl~rt werden, teil- dab einRelativis~iseh Kern nut tretenaus schwerenan Stelle yon Teilchen, V and ~0t'rotonen je Vier Diraesche und Neutronen, Fank- be- weise aueh durch etwas abnorm tionen V1 ~o~ V3 YJaFig. und 2. ~01 q% ~oa ~o4. Wir be~rachten nan die"16 unabh~ngigen steht. Um trotzdem die M5gliehkeit groi~eder/~-Emission abet gar nicht zu verst~hen, unplau- wollen bilinearen Kombinationen aus ~o1 ~o2 V~ ~P4 und ~o1~02 ~o8 epa. Bei einer siblewit Schwankungenversuchen, eine des Theorie l~Iatrixelements der Emission (50). Man leiehter hat weiter Teilehen zu bemerken, aus einem Kern Lorentz-Transformation der Koordinaten erfahren diese 16 trSlSen eine da]in Analogieman aus derzur den Theorie fl-Zerfall der begleitenden Emission 7-Strahlungeines Liehtquants sehliel]en auskann, einem an- 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- fflhren kSnnen, wodureh wieder Schwankungen in dem TF (~o)-~Vert Strahlungst.heoriestellungen; ira besonderen ist die totaletransformieren Anzahl sichder Lichtquantendie vier bilinearen keine Kombi- Konstante: erkl~irt werden kSnnen. Lichtquantennationen : entstehen, wenn sie von einem Atom emittiert werden, Wit wenden uns jetzt zur Frage nach der Form der Geschwindigkeits- und versehwinden,A o :~wenn v21qo sie 2 absorbiert+ ~oi + y%~04--werden. y~4~03, In Analogie ] hierzu wollen verteilungskurve der emittierten/3-Strahlen. FOr den Fall der erlaubten wir der fl-Strahlentheorie2tl ~--- ~01 ~P3-- folgende V2~04 - AnnahmenVs~O1 -'}- V4~2, zugrunde I legen: (11) Uberg~nge ist die Verteilungskurve als Funktion yon U (d. h. his auf den Faktor1) 1700,Vgl. dievon vorl~ufige H~o) dutch Mitteilung (44) gegeben. : La Ricerca Verteilungskurven Scientifica 2, flitHeft ver- 12, 1933. -- e) W. Heisenberg, ZS. f. Phys. 77, 1, 1932. ! schiedene Werte yon ~o sind in der Fig. 9. zusammengestellt, wobei ffir die Bequemlichkeit der Zeichnung die Ordinateneinheit in den versehiedenen11" 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… 1 m2 m2 +ln4⇡ +1 ln W + (1) W 4 n E µ2 ··· g = 4 x 10✓ -50 cm3 erg = 3.25 x 10-6 GeV-2 ◆
-1/2 This translates into scale MF = g = 555 GeV. 2 2 = H (2) 6 L | | | | MF/me = 10 Now that’s star ng to be concerning!
2 2 Today, we quote Fermi’s constant as mH m ln m (3) / -5 -2 GF = 1.166 x 10 GeV where the normaliza on is set by
GF µ (1 5) (1 5) u¯ d e¯ µ ⌫ (4) p2 2 2 22
GF µ (¯uL dL e¯L µ⌫ + h.c.) (5) p2
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.
23 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:
24 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.
à
This is the introduc on of chirality into the theory.
25 3
3 where 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 is the' spin-1/2 electron field, Aµ is the spin- lagrangian1 in the constituentµ⌫ µ two-component formalism: and m =0.511 MeV is the electron mass. The coupling = F F + i † (@ ieA ) (3) 1e photon field, e 0.31 is the gauge coupling constant L 4 µ⌫ L µ µ L constant appears to satisfy' the demands of Absolute Nat- 1 µ⌫ µ and me =0.511 MeV is the electron mass. The coupling = Fµ⌫ F µ + i L† (@µ ieAµ) L (3) L 4 † † † uralness,constant but the appears electron to satisfy mass the does demands not. of Absolute Nat- + R (@µ ieAµ) R + me( L R + R L) µ + † (@µ ieAµ) R + me( † R + † L) Oneuralness, violation but of the the electron electron mass mass does with not. respect to Ab- We can accomplishR our first task byL somehowR treating One violation of the electron mass with23 respect to Ab- solute Naturalness is that me/MPl 10 . However, WeL di can↵erently accomplish than our R first. The task electric by somehow charge treating for both we solute Naturalness is that m /M ' 10 23. However, one might object that this involvese thePl mysteries of gravi- must L di↵ keeperently at than1, butR. The we electricmust assign charge di for↵erent both we charges one might object that this involves the' mysteries of gravi- ties which are too di cult to sort out and that perhaps all formust each keep under at 1, the but new we symmetry must assignG di.↵ Aerent simple charges concrete ties which are too di cult to sort out and that perhaps all for each under the new symmetry G. A simple concrete of the Naturalness considerations of the Standard Model start to this would be to let G be some new abelian group of the Naturalness considerations of the Standard Model 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)0 and assign R double the charge of L. In other can be satisfied at mass scales far removed from MPl.I U(1)0 and assign R double the charge of L. In other do notdo agree not agree with with this this statement, statement, and and I believe I believe a a theory theory words,words, our our spectrum spectrum is is that trulythat truly possesses possesses Absolute Absolute Naturalness Naturalness will will have have no no UnderUnderU(1)U(1)EMU(1)U0 (1): 0 : unexplainedunexplained large large hierarchies, hierarchies, including including with with respect respect to to EM ⇥ ⇥ MPl.M However,Pl. However, the problem the problem with with the the electron electron mass mass is is L =(=(1, 1,1),1) , R =(=(1, 2)1, 2) (4) (4) L R alreadyalready transparent transparent without without invoking invokingMMPl. . Pl With these charge assignments the ¯ term¯ is no longer In 1933 Fermi proposed his theory of nuclear beta de- With these charge assignments the term is no longer In 1933 Fermi proposed his theory of nuclear beta de- allowed. cay, which in modern language is governed by the four allowed. cay, which in modern languageµ is governed by the four The next step is to somehow regain the electron mass fermion operator GF uµ¯L dLe¯ µ⌫,whereGF =1.17 The next step is to somehow regain the electron mass 26 fermion operator5 2 GF u¯L dLe¯ µ⌫,whereGF =1.17 ⇥ through some other means. Everything would be tried, 10 GeV is the Fermi constant. This constant defines 5 2 ⇥ throughand in time some it would other be means. recognized Everything that one would could form be tried, 10 GeVa new massis the scale FermiM constant.=(G ) 1/ This2 = 293 constant GeV. The defines vio- F F1/ 2 anda renormalizable in time it would operator be with recognized and that if one a scalar could is form a newlations mass scale of AbsoluteMF =( NaturalnessGF ) can= 293 now GeV. be phrased The vio- as a L R aadded. renormalizable The operator operator is y † with where L andy isR aif dimen- a scalar is lationsratio of Absolute that is unnaturally Naturalness too can small: now be phrased as a e L R e added.sionless coupling The operator constant is andy † the quantumwhere numbersy is a of dimen- ratio that is unnaturally too small: e L R e sionless 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) downare because (0, 1). I there believe would it would be an attempt be inevitable to get to L writeand this M ⇥ me F 6 together in an operator somehow since a mass term =1.7 10 (Absolute Naturalness problem) (2) downR because there would be an attempt to get L and MF ⇥ has to connect the two together by some means. The R together in an operator somehow since a mass 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- to add this scalar. able to apply the principle to this problem and see that easiest way to do that, which surely would be found, is structing more fundamental theories, then we should be The next inevitable step is to enhance the Lagrangian it could have guided one to deeper insights if adhered to toto include add this kinetic scalar. 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 2 2 2 it could have guided one to deeper insights if adhered to = @ µ † ( † ) (5) We should reiterate that small me is Technical Natu- to include kineticL | µ terms| and potential for the scalar field uncompromisingly.ral, as a chiral symmetry enhancement develops in the I believe that once this step2 occurs2 the Higgs mechanism2 Welimit should of m reiteratee 0. The that small small massm valueis Technical is of course Natu- sta- = @µ µ † ( † ) (5) ! e follows almostL immediately.| | A systematic study of the ral, asble a tochiral quantum symmetry corrections enhancement of the theory. develops But Absolute in the ¯ ¯ Iparameters believe that of this once potential this step would occurs easily the recognize Higgs mechanism that Naturalness says that me really should be MF . 2 limit of me 0. The small mass value is of course sta- when µ < 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 Absolute is the Higgs boson and stability of the potential requires > 0 problem. This operator¯ is dimension three and¯ so to parameters of this potential would easily recognize that Naturalness says that me really should be MF . at the renormalizable2 level. This leads to a minimum of round out the dimensions to four one needs a massive whenthe potentialµ < 0 where there the is afield vacuum has a expectation vacuum expectation value of the The QED gauge invariance of the operator ¯ is the coupling, which Absolute Naturalness demands should be Higgsvalue of boson = andµ2 stability/ . of the potential requires > 0 h i problem.similar This to M operatorF . A reasonable is dimension starting path three is andto somehow so to 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 four onenot anneeds invariant a massive in the theassumption potential that where the Higgs the mechanismfield has a would vacuum be expectation an in- coupling,spirit which that Absoluteme is “closer Naturalness to zero than demands it is to shouldMF .” One be valueevitable of breakthrough, = µ2/ . given that it is so highly cele- can then look to set the mass to zero and then find a h i similar to MF . A reasonable starting path is to somehow bratedThe readertoday, but might recall initially that many recoil people at such simultane- a nonchalant new means by which to recover¯ a correction to that and augment the theory to make not an invariant in the assumptionously understood that it the in di Higgs↵erent mechanismcontexts historically. would be The an in- a give a finite value. biggest leap is not the Higgs mechanism but coming to spirit that me is “closer to zero than it is to MF .” One evitable breakthrough, given that it is so highly cele- If adherence to Absolute Naturalness is religious, no the realization that it is important to study a scalar po- can then look to set the mass to zero and then find a brated today, but recall that many people simultane- new meansugliness by or which complexity to recover should a stop correction us from tofinding that a and way tential. Thus, in our problem at hand, the biggest leap to banish this o↵ending operator ¯ . In time it would ouslyin all understood of our discussion it in di in↵ myerent view contexts is first historically. agreeing to The a give a finite value. 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 the ¯ 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 usleft from handed finding components a way tential.the spinor Thus, di↵erently. in our problem at hand, the biggest leap to banishof the this spinor. o↵ending The kinetic operator term ¯ 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 ¯ can be banished by treating the two components of others: it mixes the right and left handed components the spinor di↵erently. 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. 27 Building on Landau's theory of phase transi ons (1937), Ginzburg- Landau theory of superconduc vity (1950) looks like the Higgs boson.
28 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. 29 4
fermion mass is obtained by m = y .Thus,the a Higgs boson, but there will also be a massless pseu- e eh i electron mass is generated and for suitable choice of ye doscalar particle in the spectrum from the spontaneous and the correct mass is obtained. breaking of the global symmetry. I believe it is also in- Theh i 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)0 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)0 giving it mass of the objections are also based on debatable philosophical M 2 = 1 g 2 2. A0 2 0 notions so we should carry on. More damaging is that h i we have replaced one problem with Absolute Naturalness (the electron mass) for another problem. If the 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)0-graviton-graviton and U(1)0 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)0. 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 : 6Q10 /3 +3Q40 /3 +3Q0 2/3 +1Q0 1 fermions to cancel anomalies. These inferences are ne- where nQq0 means n copies of fermions with charge q. cessitated by the approach we took. What’s also possible These add to our original fermions 1Q0 1 +1Q0 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]). The30 steps along the way required would have required There was infinite freedom in how I chose U(1)0 charge creativity and dedication to discovery, and also would assignments for L and R, 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 R charge being twice that of L. 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 o↵ 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, L could transform as a fundamental under SU(N), pressure-strained research to recast particle physics in a R as a singlet, and 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)0 as a global [2] G. F. Giudice, “Naturally Speaking: The Naturalness Cri- symmetry, and once 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 m' = Pierce, Aaron (ed.): Perspectives on LHC physics* 155- 2 , which is the analog of what we know today to be 178 [arXiv:0801.2562 [hep-ph]]. h i 4 fermion mass is obtained by m = y .Thus,the a Higgs boson, but there will also be a massless pseu- e eh i electron mass is generated and for suitable choice of ye doscalar particle in the spectrum from the spontaneous and the correct mass is obtained. breaking of the global symmetry. I believe it is also in- Theh i 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)0 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)0 giving it mass of the objections are also based on debatable philosophical M 2 = 1 g 2 2. A0 2 0 notions so we should carry on. More damaging is that h i we have replaced one problem with Absolute Naturalness (the electron mass) for another problem. If the 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)0-graviton-graviton and U(1)0 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)0. 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 : 6Q10 /3 +3Q40 /3 +3Q0 2/3 +1Q0 1 fermions to cancel anomalies. These inferences are ne- where nQq0 means n copies of fermions with charge q. cessitated by the approach we took. What’s also possible These add to our original fermions 1Q0 1 +1Q0 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 03 Y creativity and dedication to discovery, and also would Field SU(3) SU(2)L T 2 Q = T + 2 assignmentsa for L and R, 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 ± ± charge0 being twice that of . We did not have to R Bµ 11L 00 0 faith in Absolute Naturalness would have been easy, but choose thoseu charges, of course. Furthermore,1 we2 could L 32 2 1 3 unwavering devotion to the concept would have paid o↵ QL = 1 6 1 have even madedL G a non-abelian group, as that would do ! " ! − 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, L couldR transform as a fundamental3 under SU3 (N), 1 − − pressure-strained research to recast particle physics in a νL 2 1 0 R as aE singlet,L = and 12as 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 ! " ! − 2 " ! " cancelation conditionsφ0 can be repeated1 many times0 and 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.31 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)0 as a global [2] G. F. Giudice, “Naturally Speaking: The Naturalness Cri- interaction:symmetry, and once 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 m' = Pierce, Aaron (ed.): Perspectives on LHC physics* 155- c yt v + h 2 , 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]]. h i 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 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.
32 Conclusion 2/2
Same situa on today? Naturalness for Higgs leads to many exo c implica ons.
Analogies: Supersymmetry & Extra Dimensions & 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. 33