The "Goldstone Exception" II: Absence of a Higgs Fine-Tuning Problem in the Spontaneously Broken Limit of the Gell Mann Levy Linear Sigma Model: O(4) with PCAC and SU(2)_L with PCAC and Standard Model Quarks and Leptons

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by  Bryan W. Lynn

A Goldstone “Miracle”:
The Absence of a Higgs Fine Tuning Problem
in the Spontaneously Broken O(4) Linear Sigma Model
Bryan W. Lynn1, Glenn D. Starkman2, Katherine Freese3 and Dmitry I. Podolsky2
1 University College London, London WC1E 6BT, UK
2 ISO/CERCA/Department of Physics, Case Western Reserve University, Cleveland, OH 44106-7079 and
3 Department of Physics, University of Michigan, Ann Arbor, MI ∗
More than four decades ago, B.W. Lee and K. Symanzik proved that, in a generic set of O(4)
linear sigma models (LΣM) in the fπ −m2π/λ2 half-plane, Ward-Takahashi identities, along with
tadpole renormalization, i.e. a Higgs Vacuum Stability Condition, force all S-matrix ultra-violet
quadratic divergences (UV-QD) to be absorbed into the physical renormalized pseudo-scalar pion
2 mass squared, m2. We show that all such UV-QD, together with any finite remnants, therefore
π
1 vanish identically in the Goldstone-mode limit, m2 →m2 ≡0, where pions are Nambu-
π π;NGB;LΣM
20 GoldstoneBosons(NGB)andaxial-vectorcurrentconservationisrestored,∂µA~µ→0(i.e. whenLee
andSymanzik’sGoldstoneSymmetryRestorationConditionisenforced,asrequiredbyGoldstone’s

n theoremforthespontaneouslybrokentheory). ThescalarHiggsmassisthereforenotquadratically
a fine-tunedin the spontaneously broken theory. Hence Goldstone-mode O(4) LΣM symmetries are
J sufficient to ensure that any finite remnant, after UV-QD cancellation, does not suffer from the

1 Higgs Fine Tuning Problem or Naturalness Problem. This is contrary to the lore that quadratic
3 divergencesintheHiggsmass,observablealreadyat1-loop,leadtosuchproblemsintheO(4)LΣM,
andarethendirectlyinheritedbytheStandardModel;butitisconsistentwith1-loopresultsfound

] in canonical introductory quantum field theory textbooks where the cancellation of such 1-loop
h
quadraticdivergences is noted but its importance is not.
p
Itwasrecentlyshownthat,includingall-ordersperturbativeelectro-weakandQCDloops,andin-
-
p dependentof1-loopultravioletregularizationscheme,allfiniteremnantsofUV-QDintheStandard
e Model (SM) S-Matrix are absorbed into NGB mass-squareds m2 , and vanish identically
π;NGB;SM
h when m2 →0 , i.e. in the spontaneously broken SM. The SM Higgs mass is therefore not
[ π;NGB;SM
fine-tuned. This paper’s results on theO(4) LΣM lead us to a simple and intuitiveunderstanding
2 of that SM result, arising from theembedding of O(4) LΣM into theSM.
v
0 PACSnumbers: 11.10.Gh
5
1
2 I. INTRODUCTION (NGB)[1–3]. ThisisbecauseallUV-QDarecontainedin
. themasssquaredoftheNGB,soenforcingtheGoldstone
2
1 The idea that ultra-violet quadratic divergences (UV- Theorem (i.e. that the NGBs are indeed massless) [2, 3]
1 QD) in the Standard Model (SM) require extreme fine- eliminates all UV-QD including that in the Higgs mass
1 tunings in order to result in weak scale physics, has without fine-tuning. It also ensures the Higgs Vacuum
: had enormous impact in motivating the possible exis- Stability Condition (Higgs-VSC), that the physical vac-
v
i tence of Beyond the Standard Model physics, such as uum is stable against the spontaneous creationor disap-
X supersymmetry, technicolor, large extra dimensions, etc. pearanceofHiggsparticles,althoughthatisbynecessity
r These UV-QD are canonically demonstrated (at one- enforcedevenofftheGoldstonemodelimit. Notably,the
a loop) in the O(4) LΣM(eg. [52]), which is embedded Goldstonemodelimitispreciselytheappropriatelimitof
in the Electroweak sector of the SM. It is then ar- the O(4) LΣM for the SM, where those three massless
gued that these (Λ2) contributions to, say the Higgs NGBs of the O(4) LΣM are eaten by the three gauge
mass-squared,muOst be fine-tuned against similarly large bosonsof the three brokengeneratorsof the SU(2)xU(1)
counter-termstoobtainasmall (104GeV2)Higgsmass- symmetry. Forthespecific caseofweak-scaleGoldstone-
squared (within the LΣM) andOsimilar magnitude weak modeO(4)LΣM,theNaturalnessproblemandtheHiggs
gauge boson masses. In this paper, we will show that, Fine-Tuning problem would have been the same prob-
contrary to this lore the O(4) LΣM is perfectly well be- lem, which we here call the Higgs Fine-Tuning Problem
haved, i.e. there is no fine-tuning necessary to keep the (HFTP). We show that the O(4) LΣM theory does not
Higgs mass small, in the “Goldstone mode limit” where suffer from a HFTP in Goldstone mode.
the O(4) symmetry is spontaneously broken, leaving be-
Although the recent trend has been to argue that the
hind three precisely massless Nambu-Goldstone Bosons
SM is only a low-energy effective theory of some more
completetheory,ourgoalistounderstandtheproperties
of the actual SM. If the SM lacks fine-tuning problems
∗ [email protected],[email protected], relatedto quadratic divergences,then clearlythe onus is
[email protected],[email protected] on those proposing any extension of the SM to demon-
2
stratethattheymaintainthatproblem-freeexistence;for in 1PI1-loopandmulti-loopnested UV-QD. The result-
it would not be a problem of the SM, but of the exten- ing Goldstone-mode O(4) LΣM, which is embedded in
sion. Becauseweareinterestedonlyinthedispositionof theSM,providesasimpleandintuitiveunderstandingof
the the UV-QD of the SM, we turn our attention to the the results in Ref. [5]. Fortunately for most readers, all
much simpler O(4) LΣM, which has the same UV-QD relevant O(4) LΣM 1-loop Feynman diagrams were cal-
structure as the SM in the appropriate zero gauge cou- culated and agreed on long ago. We refer the interested
pling limit. The taming of logarithmic divergences and reader to that vast literature [6–45] for specifics. Moti-
the calculation of finite renormalization contributions is vatedbythatliterature,weuseEuclideanmetricandthe
not the subject of this paper, and it is important that Feynman-diagramnaming convention of Ref. [6].
those are sensitive to the difference between the SM and The structure of the remainder of this paper is:
the O(4) LΣM. Because we are interested in the dispo-
Section 2 clarifies the correct renormalization of
sition only of UV-QD arising from quantum loops in the
•
spontaneouslybrokenO(4)LΣM coupledto Stan-
O(4) LΣM, and not in the logarithmic divergences or
dard Model (SM) quarks and leptons. Most of the
finite parts,we are able to treat the O(4) LΣM in isola-
calculations(ifnottheeffectiveLagrangianpresen-
tion, as a stand-alone flat-space quantum field theory:
tation)inSection2Athrough2E(andprobably2F)
It is not embedded or integrated into any higher- are not new and have been common knowledge for
•
scale “Beyond the O(4) LΣM” physics. more than four decades. Specifically:
Loop integrals are cut off at a short-distance UV – Section2Aintroduces the (fermion-free)O(4)
•
scale, Λ. LΣM in the f vs. m2/λ2 half-plane.
π π
Although the cut-off can be taken to be near the – Section 2B calculates 1-loop UV-QD in 2-
•
Planck scale Λ M , quantum gravitational point self-energies and 1-point tadpoles and
Pl
≃
loops are not included. shows that Ward-Takahashi identities, to-
gether with tadpole renormalization, i.e. a
For pedagogical simplicity, we will ignore in all calcu- Higgs Vacuum Stability Condition (Higgs-
lations, discussions and formulae any logarithmic diver- VSC), force all these to be absorbed into
gences and finite contributions from loops, as these are the physical renormalized pseudo-scalar pion
unnecessaryto(anddistractingfrom)ourexplanationof (pole) mass-squaredm2.
π
the absence of any fine-tuning demanded by quadratic
– Section 2C defines the HFTP which emerges
divergences. Thus, while for dimension-2 coefficients of
in 1-loop-corrected m2 = 0 O(4) LΣM and,
rreelneovramntaloizpeedravtaolrusewseofwqilulacnatrietfiueslly(ed.gis.tinhgHuiis2h, bfπe2t)waenend imnopdaer”tilcimuliatr., the m2ππ6=6 0, fπ → 0 “Wigner-
the corresponding quadratically divergent bare counter-
term coefficient δ H 2, we will not distinguish between – Section 2D studies the opposite Goldstone-
things that are onhly liogarithmicallyor finitely renormal- mode m2 = 0 f = 0 limit, and shows that
π π 6
ized,suchasbetweenbarefields/dimensionlesscouplings the Goldstone Symmetry Restoration Condi-
and their renormalized values. We will also drop all tion (Goldstone-SRC) embedded there causes
vacuum energy and vacuum bubble contributions as be- all remnants of 1-loopUV-QD to canceliden-
yond the scope of this paper. Furthermore, this paper tically as m2 m2 0 (exactly
π → π;NGB;LΣM ≡
concerns stability and symmetry restoration protection zero!),andthetheorytohavenoHFTPthere.
againstonlyUV-QD.Itdoesnotaddressanyoftheother,
– Section 2E extends our bosonic Goldstone-
more usual, stability issues of the SM (cf. the discussion
mode O(4)LΣM resultsto all-ordersinloop-
in, for example, [4], and references therein) or the O(4)
perturbation theory.
LΣM (in part because the solutions to those issues are
generically different in the SM than in the O(4) LΣM). – Section 2F further extends Goldstone-mode
In a recent paper [5], it was shown that, including all- O(4) LΣM results to include virtual SM
orders perturbative electro-weak and QCD loops, and quarks and leptons.
independent of 1-loop UV regularization scheme, all fi-
Section3presentssomeconclusionsdrawnfromthe
nite remnants of ultra-violetquadratic divergences (UV- •
arguments and calculations of the other sections.
QD)intheStandardModel(SM)S-Matrixareabsorbed
into Nambu Goldstone Boson (NGB) mass-squareds
AppendixAshowsthatthevanishingof1-loopUV-
m2 and vanish identically when m2 •
π;NGB;SM π;NGB;SM → QD in Goldstone-mode O(4) LΣM does not de-
0 , i.e. in the spontaneously broken SM. Therefore, the
pend on the choice of n-dimensional regularization
SM Higgs mass is not fine-tuned. But, as a successful
or Pauli-Villars UV cut-off regularization.
theory of nature, the SM is necessarily quite baroque,
and sometimes even opaque. In this paper, we clarify Appendices B and E show detailed calculations of
•
that SM result by ignoring most SM parameters, gauge 1-loopUV-QDinO(4)LΣM withbosonsandwith
bosonsandghosts. Wetakethezero-gauge-couplinglimit virtual SM quarks and leptons, respectively.
3
Appendix C discusses a subtlety which arises in fπ
•
Goldstone-mode O(4) LΣM when calculations be-
gin using the Goldstone-mode Lagrangian. This µ<20
results in the 1-loop UV-QD “Goldstone-mode
RenormalizationPrescription”(GMRP),whichen- µ2=0
forces spontaneous symmetry breaking (SSB) and
the Goldstone Theorem. The GMRP, when ex- µ2>0
tended to the SM, clarifies its no-HFTP result [5].
Wigner mode
Goldstone
mode
Appendix D shows details for the extension of our 2/λ2
• mπ
results to all-loop-orders for Goldstone-mode O(4)
LΣM.
II. O(4) LINEAR SIGMA MODEL, WITH
EITHER MASSIVE PIONS m2 6=0, OR NGB
π
PIONS m2 →m2 ≡0
π π;NGB;LΣM
A. O(4) LΣM in the fπ vs. m2π/λ2 half-plane FIG. 1. fπ vs. m2π/λ2 half-plane; µ2/λ2 ≡ m2π/λ2 − fπ2.
(m2/λ2 ≥ 0 because λ2 > 0 and m2 ≥ 0 is the physical
π π
Wefollowcloselythepedagogy(andmuchofthenota- renormalizedpseudo-scalarpion(pole)mass-squared,asseen
tion)ofBenjaminW.Lee[40]andusethelinearrepresen- from Eq.7.) O(4)symmetryisexplicitly brokenofftheaxes.
Axial vector current conservation is restored on both the x-
tationofΦtomakemanifestthephysicalcontentofHiggs
axis(Wignermode)andthey-axis(Goldstonemode). Onthe
Vacuum Stability Condition (Higgs-VSC)tadpole renor-
x-axis the vacuum also respects the O(4) symmetry. On the
malization, as well as the exact not-fine-tuned cancel-
y-axisthevacuum breaks theO(4) symmetry spontaneously.
lation of UV-QD, when a Goldstone Symmetry Restora-
tionCondition(Goldstone-SRC)enforcesthewell-known
gGinoldwsittohntehTehpeuorreemsca[2la,r3]baasremL2πa→gramng2πi;aNnG:B =0. Webe- partially conserved axial-vector currents (PCAC) A~µ:
LBLΣarMe =−|∂µΦ|2−VLBΣaMre V~µ ≡~π×∂µ~π (4)
2
1
VLBΣaMre =λ2 Φ†Φ− 2 hHi2+δhHi2 −ǫH (1) A~µ ≡~π∂µH −H∂µ~π.
(cid:18) (cid:19)
(cid:0) 2 (cid:1)
1 Inclusion of explicit symmetry breaking yields
=λ2 Φ†Φ H 2
− 2h i
(cid:18) (cid:19) ∂ V~ =0 (5)
LCounterTerm LSymmetryBreaking; µ µ
− LΣM − LΣM ∂ A~ = ǫ~π
µ µ
−
where
The first of the connected Green’s function Ward-
1 Takahashiidentities,connectingthevacuumwiththeon-
LCounterTerm =λ2δ H 2 Φ†Φ H 2 ,
LΣM h i − 2h i shell one-pion state of momentum qµ reads [40–51]
(cid:18) (cid:19)
LSymmetryBreaking =ǫH. (2)
LΣM 0Ai(x)πj(q) iδijf q πˆieiqµxµ, (6)
h | µ | i≡− π µ
Here λ2 >0, while
wheref isdefinedastheexperimentalvalueoftherenor-
π
1 H +iπ 1 malized pion decay constant [40–51]. Its divergence is
Φ 3 , π (π iπ );
≡ √2(cid:20)−π2+iπ1 (cid:21) ± ≡ √2 1∓ 2 ∂ 0Ai(x)πi(q) δijf q2πˆieiqµxµ (7)
(3) µh | µ | i≡ π
= δijf m2πˆieiqµxµ.
H − π π
H h+ H , Φ = h i ,and h =0.
≡ h i h i 0 h i The final equality is due to the pion being on-shell, so
(cid:20) (cid:21)
m2 0 is the physical renormalized pseudo-scalar pion
Vacuum energy/bubble contributions are ignored. We π ≥
(pole) mass-squared.
assume λ2 = (1) and loosely refer to the real scalar h
To all-loop-orders of perturbation theory, equations 6
O
as the physical “Higgs.”
and 7 receive only logarithmically divergent and finite
Apart from LSLyΣmMmetryBreaking , the theory has O(4) corrections[40–45], sothat,including all-ordersUV-QD,
symmetry with conservedvector currents (CVC) V~ and butignoring(asun-interestingforthispaper)logarithmic
µ
4
divergences and finite contributions: n-dimensional regularization or Pauli-Villars regulariza-
tion is used. To fully exacerbate and reveal any HFTP,
∂µA~µ =−fπm2π~π we imagine Λ≃MPl near the Planck scale.
ǫ=f m2 (8) Using the bare Lagrangian LBare, we form a 1-loop-
π π LΣM
LSymmetryBreaking =f m2H. UV-QD-improved effective Lagrangian, which includes
LΣM π π all scalar 2-point self-energy and 1-point tadpole 1-loop
B.W.Lee and K. Symanzik then analyze the generic set UV-QD (but ignores, for us un-interesting, 1-loop log-
of quantum field theories in the f vs. m2/λ2 half-plane arithmically divergent, finite contributions and vacuum
π π
(cf. Figure 1). energy/bubbles):
LEffective;1−loop;Λ2 LBare;Λ2 +L1−loop;Λ2 (11)
LΣM ≡ LΣM LΣM
B. Inclusion of 1-loop ultra-violet quadratic = ∂ Φ2 VRenorm;1−loop;Λ2,
divergences −| µ | − LΣM
where
The UV-QD 1-loopLagrangian,evaluatedat zeromo-
mentum,includingall1-loop2-pointself-energy(Figures VRenorm;1−loop;Λ2=λ2 Φ†Φ hHi2 2 f m2h
3, 4 and 5) and 1-loop 1-point tadpole-function (Figure LΣM − 2 − π π
(cid:20) (cid:21)
6) UV-QDs, is (cf. Appendix B):
λ2δ H 2+CUnrenorm;1−loop;Λ2Λ2 Φ†Φ hHi2 (12)
LL1−ΣlMoop;Λ2=CLUΣnMrenorm;1−loop;Λ2Λ2 Φ†Φ− hH2i2 , (9) −(cid:0) h ih2 LπΣ2M (cid:1)(cid:20)2 − 2 (cid:21)
(cid:18) (cid:19) =λ2 + 3 +π π + H h +( H f )m2h
2 2 + − h i h i− π π
with CUnrenorm;1−loop;Λ2 a dimensionless constant. The (cid:20) (cid:21)
LΣM h2 π2
form of Eq. 9, explicitly calculated in Appendix B, fol- +m2 + 3 +π π ,
π 2 2 + −
lows from Lee/Symanziks proof [40–51] that the theory (cid:20) (cid:21)
is properly renormalized, throughout the f vs. m2/λ2
π π and
half-plane, with the same UV-QD graphs and counter-
terms as the symmetric Wigner-mode limit: H 2 0 m2 λ2δ H 2+CUnrenorm;1−loop;Λ2Λ2 . (13)
holding m2 = 0. Noticing that at tree levelhthei H→iggs π ≡− h i LΣM
[m6]a1s,s[3m82h, 3≡9π]26)λr2ehwHriit2e, wEeq.sBug8g(ecsft.ivAelpyp(eannddixrBec)o:gnizably Because of th(cid:0)e way it enters Eq. 12, m2π is th(cid:1)e physical
renormalized pseudo-scalar pion (pole) mass-squared of
CUnrenorm;1−loop;Λ2Λ2 = 6λ2 Λ2 (10) EqIsn.si7ghatndint8o. thetadpoleterm( H f )m2hinEq.12
LΣM − 16π2 h i− π π
(cid:18) (cid:19) followsbycarefullydefiningthepropertiesofthevacuum,
“=′′ (cid:0) (cid:1)3m2h Λ2 including all effects of UV-QD, and imposing the Higgs-
− 16π2 H 2 VSC on the vacuum and excited states:
(cid:18) h i (cid:19)
Higgs VSC: The physical Higgs particle must
(Here “ =′′ indicates that this is currently true only
notsimplydisappearintotheexact UV-corrected
at tree level, although we shall find below that m2 =
h vacuum.
2λ2 H 2 holds to all orders in loops.)
h i For the purposes of this paper, we take the UV-
The readeris remindedto takethe Wigner-modelimit
corrected vacuum to be the UV-QD-corrected vacuum.
carefully, taking H 2 0 but holding λ2 fixed. Ap-
h i → It is important to note several things:
pendixAshowsthatthisresultisindependentofwhether
1. The UV-QD-corrected vacuum includes all pertur-
bative UV-QD corrections, including
1 We agree with Veltmans [6] overall 1-loop SM UV-QD coeffi- 1-loop when referenced in subsections IIB,
cient, but his expression Eq. 6.1 for the relevant dimension-2 • IIC and IID,
operators with1-loopUV-QDcoefficients arisinginO(4) LΣM,
from virtual scalars, would read (i.e. if taken at face value 1PI multi-loop when referenced in subsection
and written in the language of this paper): L1−loop;Λ2 ∝ • IIE and Appendix D, and
SM,ΦSector
(cid:0)−21h2+ 21π2+π+π−−fπh(cid:1)Λ2. ComparisonwithourUV-QD 1-loopO(4)LΣM withfermionsinsubsection
counter-term Eq. 2 reveals that Veltman’s term is not O(4) in- •
IIF.
variantbeforeSSBbecauseoftherelativeminussign(withwhich
wedisagree)betweentheHiggsself-energy2-pointoperatorand
2. By definition, H = H +h,and Φ†Φ = 1 H 2 is
that of the Nambu-Goldstone Bosons. In contrast, our analo- h i h i 2h i
gous expression and counter-term aremanifestly O(4) invariant the exact vacuum expectation value (VEV).
before SSB, as they must be. If the relative signin his Eq. 6.1
isnotatypographicalerror,itisimpossibletocancelexactlyall 3. ThephysicalHiggsparticlehhasexactlyzeroVEV:
1-loopUV-QDinVeltman’spaper. h 0.
h i≡
5
4. Exact tadpole renormalization is to be imposed to VRenorm
all orders in perturbation theory.
It is important to realize that Higgs-VSC tadpole renor-
malization does not constitute fine-tuning; rather it is a
stability condition on the vacuum and excited states of
the theory.
Lee/Symanzik impose the Higgs-VSC essentially by
hand [40–45]. They ensure that
H
( H f )m2h=0 (14)
h i− π π
in the entire f m2/λ2 half-plane by enforcing2
π− π
H =f . (15)
π
h i
After tadpole renormalization, the effective 1-loop
Lagrangian (keeping 1-loop UV-QD, but ignoring un-
interesting logarithmic divergences, finite parts and vac-
uum energy/bubbles), can be re-written:
LLEΣffMective;1−loop;Λ2 =−|∂µΦ|2−VLRΣeMnorm;1−loop;Λ2, FesItGp.o2in.tCarnodssitssecloticoanl mofaVxLiRmΣeMnuomrm,;f1o−rloµop2,Λ<20t,hrmou2πg>h0it.sWdeheepn-
VRenorm;1−loop;Λ2 = (16) m2π =0,theminimaaredegenerate,andV acquiresthe“bot-
LΣM tom of thewine-bottle” symmetry.
λ2 Φ†Φ 1 f2 m2π 2 f m2H
− 2 π − λ2 − π π
(cid:18) (cid:19)
(cid:2) (cid:3) C. m2 6=0: A Definition of the Naturalness and
with π
Higgs Fine-Tuning Problems
H =h+f . (17)
π
For m2 = 0, the 1-loop-improved O(4) LΣM effec-
Here m2 is still the physical renormalized pseudo-scalar π 6
π tive Lagrangian (Eq. 11 or 16) has 1-loop UV-QD con-
pion (pole) mass-squaredof Eq. 13, and tributions from L1−loop;Λ2 (Eq. 9) to both the pion and
LΣM
m2 m2 +2λ2f2 m2. (18) Higgs masses. These are cancelled by the counter-term
h ≡ π π ≥ π LCounterTerm;Λ2 (Eq. 1 and 2), but may leave very large
A0 acnrodssm-s2ec>tio0niosfpVlLoRΣtetMneodrmin;1−Fliogoup;rΛe2,2.wiWthemo2πb/seλr2v−e tfhπ2a<t, finLiΣteMresidual contributions:
binecEaqu.se16oπ,fitthiesenxoptlipcoitsssiybmlemtoetcrhyabnrgeeakthinegretesurmltsfoπfms2πuHb- m2π =−λ2δhHi2+ 6λ2 1Λ6π22 (19)
sections IIC through IIF, by changing to the unitary Φ (cid:0) (cid:1)
and
representation[4]orby re-scaling H (cf. Appendix C).
The results (except the effectivhe Liagrangian presen- m2 =2λ2f2 λ2δ H 2+ 6λ2 Λ2 . (20)
tation and the inclusion of UV-QD into m2 ) in this h π − h i 16π2
π
subsection are not new [40–45]. We simply observe here As expected, the natural scale of (cid:0)the c(cid:1)oefficient of the
thatWard-Takahashi identities,togetherwithHiggs-VSC relevant scalar 2-point operator is δ H 2 Λ2. Now
tadpole renormalization, are sufficient to force all UV- imagine weak interaction experimentshreiquir∼e GExpf2 =
µ π
QD in the S-Matrix of the O(4) LΣM to be absorbed
(1), where the weak scale is set by the experimental
into the physical renormalized pseudo-scalar pion (pole) Omuon decay constant GExp = 1.16637 10−5GeV−2 =
mass-squared, m2π. We also remark that this is not sur- (292.807GeV)−2. µ ×
prising. The quadratic divergences are ultra-violet con-
Since f isafree parameter(definedbyEq.6but oth-
π
stants. They therefore retain no “memory” of the low-
erwise not determined by the internal self-consistency of
energy ( H Λ) breaking of the O(4) symmetry. H
h i ≪ the theory), there is no problem in choosing its numer-
andharethe sametoUV-QD,and,bythe O(4)symme-
ical value from experiment. But problems appear if ex-
try, indistinguishable from the πi. periments also require weak-scale GExpm2 = (1) and
µ π O
GExpm2 = (1). For Planck scale ultra-violet cut-off
µ h O
Λ M , absorption of the UV-QD into m2 and m2
2 ThisisdistinctfromtheobservationthatintheGoldstonemode req≃uiresPfilne-tuning δ H 2 to within GExpMh2 −1 π
this Higgs-VSC condition (Eq. 14) is also guaranteed by the h i µ Pl ≃
Goldstone-SRC, i.e. by m2 = 0. Whether or not Eq. 15 can 10−32ofitsnaturalvalue[6,52–54]. ThisisaHiggsFine-
π (cid:0) (cid:1)
therefore be relaxed in Goldstone mode, does not change the TuningProblemandaviolationofNaturalnesswhichhas
conclusionsofthispaper. been variously defined to be the demand that:
6
observable properties of the theory be stable which ǫ f m2 0 [40–45]. Namely f = 0 while
• against minute variations of fundamental param- m2 0≡(i.e.π thπe→y-axis in Figure 1). Inclπud6ing 1-loop
π →
eters [52, 53]; and UV-QD (but ignoring un-interesting logarithmic diver-
gences,finite contributionsandvacuumenergy/bubbles)
electroweakradiativecorrectionsbethesameorder
the 1-loop-correctedeffective Lagrangianis:
•
(ormuchsmaller)thantheactuallyobservedvalues
[6]. LEffective;1−loop;Λ2 LEffective;1−loop;Λ2 (24)
The problem is widely regarded as exacerbated in O(4) LΣM −f−π−6=−0−,m−2π−=−→0 Goldstone;LΣM
LΣM (and the SM) because even if one fine tunes the where
parametersat1-loop,onemustretunethemfromscratch
at 2-loop, and again at each order in loops. This is LEffective;1−loop;Λ2 = ∂ Φ2 λ2 Φ†Φ fπ2 2 . (25)
sometimescalledthe TechnicalNaturalnessProblemFor Goldstone;LΣM −| µ | − − 2
weak-scaleO(4)LΣM withm2 =0, these problemsare (cid:20) (cid:21)
all the same problem, the HFTπP6.3 with
O(4) LΣM in Wigner mode [40–45], with unbroken
O(4) symmetry and four degenerate massive scalars (i.e. H = h+fπ,
1 scalar + 3 pseudo-scalars), is defined as the limit ǫ = m2 = λ2δ H 2+CUn−renorm;1−loop;Λ2Λ2
f m2 0,holding m2 =0 while f 0(i.e. the x-axis π − h i LΣM
inπFiπgu→re 1). The axiaπl6current is cπo→nserved, ∂µA~µ = 0, →m2π(cid:16);NGB;LΣM ≡0, (cid:17) (26)
in Wigner mode. m2 = m2 +2λ2f2 2λ2f2
Including 1-loopUV-QD (but ignoring logarithmicdi- h π;NGB;LΣM π ≡ π
vergences, un-interesting finite contributions and vac- Insight into these equations requires carefully defin-
uum energy/bubbles) the 1-loop-corrected effective La- ing the properties of the vacuum, including all effects
grangian is: of 1-loop-induced UV-QD, after spontaneous symmetry
breaking (SSB). The famous theorem due to Goldstone
LEffective;1−loop;Λ2 LEffective;1−loop;Λ2 (21) [2] and proved by Goldstone, Salam and Weinberg [3]
LΣM −f−π−=−0−,m−−2π6=−→0 WignerLΣM states that: “if there is continuous symmetry transfor-
mation under which the Lagrangian is invariant, then
where
eitherthevacuumstateisalsoinvariantunderthetrans-
LEffective;1−loop;Λ2 = ∂ Φ2 λ2 Φ†Φ+ m2π 2 (22) formation, or there must exist spinless particles of zero
WignerLΣM −| µ | − 2λ2 mass.” The O(4) invariance of the LΣM Lagrangian
(cid:20) (cid:21) Eq. 1 is explicitly broken by the ǫH term everywhere in
with the f m2/λ2 plane, except of course for ǫ = 0 where
it is rπes−toreπd. In Wigner mode (ǫ = 0 because m2 = 0,
H h; H =f 0 π
π but f =0), the vacuum retains the O(4) invariance. In
→ h i → π
6
(23) Goldstone mode, the presence of a VEV for Φ spoils the
Λ2 invariance of the vacuum state under the O(4) symme-
m2h =m2π =−λ2δhHi2+ 6λ2 16π2 . try, hence, by the Goldstone Theorem, there must exist
threespinlessparticlesofzeromass,theNGBs[1–3]with
(cid:0) (cid:1)
Wigner mode suffers HFTP while fine-tuning the Higgs m2 =0.
π;NGB;LΣM
and pion masses to an un-natural experimental value,
As shown by Lee [40], and Symanzik [41, 42], the
m2 Λ2.
π ≪ Goldstone Theorem is not manifest order-by-order in
quantum loops as the O(4) LΣM is renormalized, and
so in order to obey the Goldstone Theorem one must
D. Goldstone-mode weak-scale spontaneously
impose the Goldstone Symmetry Restoration Condition
broken O(4) LΣM is HFTP-free at 1-loop
(Goldstone-SRC): After SSB, the Goldstone Theorem
must be an exact property (to all loop orders) of the
Goldstone-mode O(4) LΣM, with spontaneously bro-
Goldstone-mode O(4) LΣM vacuum and its excited
kenO(4)symmetry,1massivescalarand3exactlymass-
states. It is noteworthy that:
less pseudo-scalar Nambu-Goldstone Bosons, is defined
astheother(notWigner-mode)limitoftheO(4)LΣM in
1. The Goldstone-SRC includes all perturbative
quadratically divergent corrections,including
3 Weak-scaleO(4)LΣM alsoseparatelysuffersaWeakHierarchy 1-loopwhenreferencedinsubsectionsIIDand
•
problembecauseitisunabletopredictorexplaintheenormous IIF;
splitting between the weak scale and the next larger scale (e.g.
classical gravitational physics Planck scale MPl). We make no • 1PI Multi-loop when referenced in subsection
claimtoaddressthataestheticproblemhere. IIE and in Appendix D.
7
2. In generic theories spanning the f vs. m2/λ2 The crucial observation here is that proper 1-
π π
half-plane (Figure 1), it would be necessary to loop enforcement of the Goldstone Theorem
impose this Goldstone-SRC (essentially by hand) m2 0, requires imposition (essentially
π;NGB;LΣM ≡
in order to force the theory to the Goldstone- by hand) of Lee/Symanziks Goldstone-SRC [40–45]
mode O(4) LΣM limit, i.e. onto the y-axis,
m2 m2 0. (Where m2 is the loop- Λ2
indπu→cedNπa;NmGbBu;-LGΣoMld≡stoneBoson(NGπB)mass(see 0≡m2π;NGB;LΣM = −λ2δhHi2+6λ216π2 . (30)
(cid:18) (cid:19)
Appendix C)).
exactly and identically. Onlyatthatpointcanwewrite
3. In more modern language, for spontaneously bro-
ken O(4) LΣM , it is necessary to explicitly en- VRenorm;1−loop;Λ2
Goldstone;LΣM
force the Goldstone Theorem order-by-quantum- h2 π2 2
loop-order so that the pions remain exactly mass- =λ2 + 3 +π π +f h (31)
+ − π
2 2
less to allordersof loopperturbationtheory. That (cid:20) (cid:21)
is the purpose of Lee/Symanziks Goldstone-SRC. f2 2
=λ2 Φ†Φ2 π .
One thus has no choice but to operate in the
− 2
f m2/λ2 half plane, because at each order in (cid:20) (cid:21)
loπop−s, mπ2 reappears as an UV-QD plus countert- A central observation of this paper is that, in conse-
π
erm,andateachorder(oratleastatthefinalorder quence of SSB, careful definition of the 1-loop-UV-QD-
to which one calculates) one must explicitly take improved vacuum, Higgs-VSC tadpole renormalization
m2 m2 0 to realize the Goldstone andGoldstone-SRCenforcementofthe Goldstone Theo-
Thπeo→rem.π;NGB;LΣM ≡ rem, all finite remnants of 1-loop UV-QD contributions
to the Higgs mass term in Eq. 28,
4. Since UV-QD (in the generic LΣM theory’s S-
Matrix) are all packed into the physical renormal- h2
m2 , (32)
izedpseudo-scalarpion(pole)mass-squared(asde- π;NGB;LΣM 2
(cid:20) (cid:21)
manded by O(4) symmetry, and as shown by ex-
plicit calculation[40]), the Goldstone Theorem also cancelidentically,withoutfine-tuning,thusavoidingany
forces any finite remnant of UV-QD to be exactly Higgs Fine Tuning Problem.
zero in the Goldstone-mode (i.e. spontaneously LEffective;1−loop;Λ2 in Eq. 28 and 31 gives the sensi-
Goldstone;LΣM
broken) limit of the theory. ble,atworstlogarithmicallydivergentandnot-fine-tuned
Higgs mass:
5. Wehaveignoredinfra-red(IR)subtletiesasbeyond
the scope of this paper. m2 =2λ2f2. (33)
h π
It is easy to see that VLRΣeMnorm;1−loop;Λ2 in Eq. 16 Before going forward, let us now summarize the gen-
(shown in Figure 2 with m2π 6= 0) only has NGB when eral 1-loop and specific 1-loop Goldstone-mode observa-
“bottom-of-the-wine-bottle Goldstone symmetry is re- tions so far, and which are extended to all-loop-orders
stored: i.e. in Goldstone mode with in subsection IIE and Appendix D. We emphasize that
mostofthecalculationsinSectionIIarenotnew;rather,
m2 m2 0 (27)
π → π;NGB;LΣM ≡ they have been common knowledge for more than four
is restored exactly and identically. Re-writing Eq. 11 decades. What is new are certain observations that
apply specifically to spontaneously broken O(4)
after tadpole renormalization
LΣM (i.e. in Goldstone mode):
LEffective;1−loop;Λ2 = ∂ Φ2 VRenorm;1−loop;Λ2,
Goldstone;LΣM −| µ | − Goldstone;LΣM 1. Spontaneously broken O(4) LΣM must be viewed
as a limiting case of Lee/Symanziks generic set of
where
m2/λ2 0 theories (i.e. the m2/λ2 = 0 line (y-
VRenorm;1−loop;Λ2 axπis) in≥the f vs. m2/λ2 half-plπane shown in Fig-
Goldstone;LΣM π π
h2 π2 2 ure1),wheretwoconditionsmustbe explicitlyim-
=λ2 + 3 +π π +f h (28) posedontheO(4)LΣMvacuumandonGoldstone-
+ − π
2 2
(cid:20) (cid:21) mode excited states:
h2 π2
+m2π;NGB;LΣM 2 + 23 +π+π− . • Higgs VSC: everywhere in the fπ −m2π/λ2
(cid:20) (cid:21) half-plane the Higgs must not simply disap-
Here pear into the vacuum, so H f ;
π
h i≡
m2 = λ2δ H 2 CUnrenorm;1−loop;Λ2Λ2 Goldstone-SRC: the masses of Nambu-
π;NGB;LΣM − h i − LΣM • Goldstone Bosons must be fixed to exactly
(cid:16) Λ2 (cid:17) zerointheGoldstone-mode(y-axis)toenforce
= λ2δ H 2+6λ2 . (29)
− h i 16π2 the Goldstone Theorem.
(cid:18) (cid:19)
8
2. Ward-Takahashi identities and tadpole renormal- 10. The fact that all 1-loop UV-QD in the SM Higgs
ization force all UV-QD in the S-Matrix to be ab- self-energy and mass cancel exactly after tadpole
sorbedintothephysicalrenormalizedpseudo-scalar renormalization has been known [30, 39] for more
pion (pole) mass-squared m2 everywhere in the than 3 decades. We have traced that SM result
π
half-plane. to the Goldstone mode O(4) linear sigma model
embedded in the spontaneously broken SM, giving
3. All 1-loop UV-QD divergences can be obtained asimpleandintuitiveunderstandingoftheresultof
by first calculating them in the unbroken theory [5], i.e. that the SM Higgs masss is not fine-tuned.
(Wigner mode) and then breaking the symmetry
An importantsubtlety, which givesa clear 1-loopUV-
throughout the half-plane, and specifically on the
QD “Goldstone-Mode Renormalization Prescription”
y-axis(Goldstonemode),whereO(4)isbrokenonly
(GMRP) for both Goldstone-mode O(4) LΣM and the
spontaneously. These 1-loop results do not de-
Standard Model, is discussed in Appendix C. There we
pend on choice of UV regularization scheme (e.g.
re-do the calculations and analysis in this section, but
n-dimensional or Pauli-Villars cut-off, as shown in
beginning with the Goldstone-mode O(4)LΣM bare La-
Appendix A).
grangian,Eq. 1 with ǫ=0:
4. Allrelevantdimension-2operators(i.e. with1-loop LBare;Λ2 = (34)
UV-QDcoefficients)formarenormalizedHiggspo- Goldstone;LΣM
tential (Figure 2) that is well-defined everywhere 1 2
in the half-plane and can be minimized at the 1- ∂µΦ2 λ2Φ†Φ H 2+δ H 2
−| | − − 2 h i h i
loop-corrected VEV. Although the coefficients of (cid:20) (cid:21)
(cid:0) (cid:1)
the relevant dimension-2 scalar self-energy-2-point The subtlety arises after UV-QD cancellation, when en-
and tadpole-1-point relevant operators take their forcingm2π →m2π;NGB;LΣM ≡0 and SSB andavoidinga
natural scale δ H 2 Λ2, there is still no need “fine-tuning discontinuity.”
to fine-tune awhayithe∼UV-QD to get a weak-scale The same thing happens when SM calculations begin
Higgs mass-squaredm2GExp O(1) if experiment withthespontaneouslybrokenSMbareLagrangian: one
sodemands,oncem2π →h mµ2π;N≃GB;LΣM ≡0,isfixed. wmhuisctheinsfocrocreremct2πf;oSrMth→e SmM2π;N[5G].B;SM ≡0 in that GMRP,
5. The Goldstone Theorem insists that the NGBs are
exactly massless, m2 0. All UV-QD in m2, to-
π ≡ π
gether with all logarithmically divergent and all E. Goldstone-mode weak-scale spontaneously
finite remnants, vanish identically and exactly as broken O(4) LΣM has no HFTP at any loop-order
m2 = 0. This forces all UV-QD in the S-Matrix
π
to cancel exactly, with finite remnant exactly zero. The reader should worry that cancellation of 1-loop
They are not absorbed into renormalized parame- UV-QD is insufficient to demonstrate that the theory
ters, including, but especially m2. does not require fine-tuning. UV-QD certainly appear
h
at multi-loop orders and fine-tuning δ H 2 might yet
6. The root cause of this fine-tuning-free disappear- be required. If each loop order contrhibuited a factor
ance of UV-QD is imposition of a Higgs-VSC and ~/16π2 10−2, then cancellation of UV-QD to more
aGoldstone-SRConthespontaneouslybrokenvac- than161≃PIloopswouldbe requiredtodefeatafactorof
uumanditsexcitedstatesasdemandedbyLeeand GExpΛ2 10−32 for Λ M .
Symanzik. µAs we≤remarked abo≥ve, BP.lW. Lee, and K. Symanzik
renormalized generic O(4) LΣM (i.e. in the full f
7. Our no-fine-tuning-theorem for a weak-scale Higgs m2/λ2 half-plane of Figure 1) to all-loop-orders mπor−e
mass (in the spontaneously broken O(4) LΣM) π
than forty years ago. We simply remind the reader of
is then simply another (albeit un-familiar) conse-
their results [40–45], of important follow-up results [46–
quence of the Goldstone Theorem, an exact prop-
48]andofsomeaccessibletextbookpresentations[49–51]
ertyofthespontaneouslybrokenvacuumandspec-
relevant to this paper. In particular:
trum.
The UV divergence and counter-term structure is
8. There is no possibility (or need) to cancel 1-loop • the same throughout the f vs. m2/λ2 half-plane
π π
UV-QDbetweenvirtualbosonsandfermionsinthe (Figure1),givingproperinterpolationbetweenthe
O(4) LΣM. (After all, we have so far not allowed symmetric Wigner mode (f = 0, m2 = 0) and
for any fermions.) thespontaneouslybrokenGoπldstonemoπde6 (f =0,
π
m2 =0). 6
9. It is un-necessary to impose any new Beyond the π
O(4)LΣMsymmetriesorotherphysics: weak-scale The Ward-Takahashi Identity, CVC and PCAC
•
spontaneously broken O(4) LΣM already has suf- (Eqs. 5,6 and 7) receive only logarithmicallydiver-
ficient symmetry to force all S-Matrix UV-QD to gent and finite corrections. Ignoring those, but in-
vanishandtoensurethatitdoesnotsufferaHFTP. cludingall-ordersUV-QD,Eq.7isstilltrueandm2
π
9
is still the all-loop-corrected physical renormalized with
pion (pole) mass-squared. CUnrenorm;1−loop;Λ2 = (37)
LΣM+SMq&l
6λ2f2+4 flavor,colorm2 +4 flavor m2
Appendix D demonstrates just such exact cancella- − π quarks quark leptons lepton
tion and shows that 1PI multi-loop fine-tuning is un- " P 16π2hHi2 P #
necessary. AppendixDthenshowsthatSSB,carefuldefi-
recognizable as the zero-gauge-coupling limit of 1-loop
nitionofthevacuum–Goldstone-SRCenforcementofthe
UV-QD in the SM [6, 38, 39], where to this order
GoldstoneTheoremandHiggs-VSCtadpolerenormaliza-
6λ2f2 = 3m2.
tion–aresufficienttoforceall-loop-ordersS-matrixUV- − π − h
After Higgs-VSC tadpole renormalization (cf. subsec-
QDinto m2 (inLEff,All−loop;Λ2)andtothereforevanish
π LΣM tionIIB),the effective scalar-sectorLagrangian,keeping
identically and exactly in the m2 m2 0
π → π;NGB;LΣM ≡ 1-loop UV-QD (but ignoring un-interesting logarithmic
limit. They are not absorbed into renormalized param-
divergences, finite parts and vacuum energy/bubbles)
eters, such as the Higgs mass squared. There is no sur-
can again be written:
viving remnant of UV-QD at any order of perturbation
theory, nor is there a HFTP in UV-QD-corrected spon- LLEΣffMec+tSivMe;1q−&ll;oΦop;Λ2 =−|∂µΦ|2−VLRΣeMno+rmSM;1−q&lolop;Λ2,
taneously broken O(4) LΣM. VRenorm;1−loop;Λ2 = (38)
It is easy to see from Appendix D that the observa- LΣM+SMq&l
tions and lessons at the end of subsection IID can be λ2 Φ†Φ 1 f2 m2π 2 f m2H
extended to all orders of bosonic O(4) LΣM, at least − 2 π − λ2 − π π
in loop-perturbation theory. We now add fermions to h (cid:18) (cid:19)i
with
theO(4)LΣM,andshowthatthespontaneouslybroken,
H =h+f . (39)
O(4) LΣM with Standard Model fermions remains free π
of any HFTP. Here
m2 λ2δ H 2+CUnrenorm;1−loop;Λ2Λ2 (40)
π≡− h i LΣM+SMq&l hHi=fπ
(cid:16) (cid:17)(cid:12)
F. O(4) LΣM in the fπ vs. m2π/λ2 half-plane with is still the physical renormalized pseudo(cid:12)(cid:12)-scalar pion
(pole) mass-squared,and
SM quarks and leptons
m2 m2 +2λ2f2 m2. (41)
h ≡ π π ≥ π
All-ordersrenormalizationof the generic bosonic O(4) Onceagain,thephysicalrenormalizedpseudo-scalarpion
LΣM was long ago extended to include parity-doublet pole mass-squared m2 has absorbed all 1-loop UV-QD,
π
fermions by J.L Gervais and B.W.Lee [44]. The full rea- which now includes those from virtual SM quarks and
soning necessary to secure renormalized inclusion of SM leptons. Therefore, the results in each of the sections
quarks and leptons in the fπ vs. m2π/λ2 half-plane is above are generalized to include all UV-QD traceable to
beyond the scope of this paper (i.e. involves infra-red virtual SM quarks and leptons:
and regularization subtleties [40–51]). We merely re-
Section IIC (1-loop): At 1-loop the generic O(4)
mindthereaderofthekeycomplication,thatSMfermion • LΣM in the f vs. m2/λ2 half-plane of Figure 1
masses vanish in the symmetric Wigner mode (f = 0, π π
π (i.e. away from the y-axis, m2 =0), and including
m2/λ2 =0, as π
π 6 Wigner mode (x-axis), has (additional) surviving
finiteremnantsofUV-QDduetovirtualSMquarks
m ,m H =f ,. (35)
quark lepton π and leptons, and continues to have a HFTP.
∝h i
Section IID (1-loop): It is obvious that
Iufsuwaelcwoauyp,lethSeMnnqeuwarUksVa-QndDleaprteopnrsotpoorOti(o4n)aLlΣtoMYuinkatwhae • VRenorm;1−loop;Λ2 in Eq. 38 and Figure 2 has
LΣM+SMq&l
couplings squared, and appear only in coefficients of rel- Nambu-Goldstone Bosons only when “bottom-of-
evant dimension-2 scalar-sector operators. They are as- the-wine-bottle Goldstone symmetry is restored,
sembled and included in the scalar-sector effective La- i.e. in Goldstone mode on the m2 = 0 line. After
π
grangian as in Section 2B. Goldstone-SRC, the Goldstone Theorem forces all
1-loop UV-QD and their finite remnants to vanish
Wefocusonexplicitcalculationof1-loopUV-QDcon-
exactly and identically:
tributions, due to SM quarks and leptons, in the generic
fπ vs. m2π/λ2 half-plane. The total UV-QD 1-loop La- m2 = λ2δ H 2 (42)
grangian, calculated in Appendix E (Eq. E7) is π − h i −
6λ2f2+4 m2 +4 m2
− π quark lepton Λ2
L1−loop;Λ2 = (36) " P16π2hHi2 P #
LΣM+SMq&l
m2 0
CUnrenorm;1−loop;Λ2Λ2 Φ†Φ 1 H 2 → π;NGB;LΣM+SMq&l ≡
LΣM+SMq&l − 2h i m2 = 2λ2f2,
(cid:18) (cid:19) h π
10
where the sum is over all flavors, and for quarks aswehaveshowed,theGoldstoneTheoremcanprotecta
over all colors. limitednumberofscalars(atleastthoseinthesamemul-
tiplet as the Nambu-Goldstone Bosons) from such fine-
Eq. 42 fixes the numerical value of δ H 2 so that
tuning problems. The O(4) Linear Sigma Model (LΣM)
• h i
there is exactly zero finite remnant after UV-QD
with one Higgs doublet, the same scalarstructure as the
cancellation. m2 is still the physical renormalized
h standard model (SM), is one example of such a theory
scalar Higgs pole mass-squared: it is still not fine-
that is protected in Goldstone mode, i.e. when spon-
tuned. Therefore, including virtual SM quarks,
taneously broken. This is an important loophole in the
leptons and scalars, spontaneously broken O(4)
canonical objection to fundamental scalars. More par-
LΣM hasnosurvivingfinite remnantsofcancelled
ticularly, it is a loophole that the stand-alone Standard
1-loop UV-QD and therefore has no HFTP.
Model, unadorned by BSM physics, makes use of [5] so
LEffective;1−loop;Λ2 (1-loop): The 1-loop that the SM Higgs mass is not-fine-tuned.
• Goldstone;LΣM+SMq&l;Φ Low energy supersymmetry (SUSY) aims to solve the
Goldstone-modescalar-sectoreffectiveLagrangian,
Higgsfine-tuningproblembythemiraculouscancellation
including all 1-loop UV-QD effects due to virtual
of the quadraticallydivergentportion of the loopcontri-
scalars and SM quarks and leptons, is as given in
butions of particles against those of their superpartners.
Eq. 38 with m2 = 0. This is the same as Eq. 31.
π The key observation is that if the particle is a boson it’s
Therefore
superpartner is a fermion, and vice versa – the 1 that
−
LEffective;1−loop;Λ2 =LEffective;1−loop;Λ2. (43) fermions pick up in loops relativeto bosonsis key to un-
Goldstone;LΣM+SMq&l;Φ Goldstone;LΣM;Φ derstanding this cancellation. (But not sufficient – the
numerical values of the coefficients must be equal, and
Section IIE (1PI multi-loops): Calculation of 1PI
are as a consequence of the supersymmetry.) Low en-
•
naivedegreesofdivergencereveals[40–51]thatUV-
ergy SUSY might be used to solve the scalar fine-tuning
QD can only appear in the scalar sector efective
problem of theories that have one. However, since the
Lagrangian, where they are exactly cancelled by
spontaneously broken O(4) LΣM(and the SM [5]) does
scalar sector counterterms. Therefore, when in-
not have this problem, low energy SUSY can only solve
cludingUV-QDtraceabletovirtualSMquarksand
the scalar fine-tuning problems of BSM physics. For ex-
leptons, the 1-loop observations and lesson in Sec-
ample, Grand Unified Theories will generically have a
tionIIDaregeneralizedintoall-perturbative-loop-
Higgs fine-tuning problem, as finite contributions to the
orders.
Higgs mass-squared will be of order the GUT scale, and
will need to be fine tuned down to the weak scale. Thus
Thatall1-loopUV-QDduetovirtualSMfermions
• SUSYGUTs can make use of low energy SUSY to avoid
cancel exactly in the SM Higgs self-energy and
a Higgs Fine-Tuning Problem.
masscancelaftertadpolerenormalizationhasbeen
The role of the Goldstone Theorem in taming the
known [30, 39] for more than 3 decades.
Higgs fine-tuning problem in spontaneously brokenO(4)
LΣM (and the Standard Model [5]) appears to have
III. CONCLUSION gone unnoticed. As we have shown, because quadratic
divergences are sensitive only to ultraviolet physics, the
O(4)symmetryforcesthequadraticallydivergentcontri-
Belief in huge non-vanishing remnants of cancelled
butionstotheHiggsmass-squaredtobeidenticaltothose
ultra-violet quadratic divergences, and consequent fine-
of the other three bosons in its multiplet. This is true
tuning problems in field theories with fundamental
whethertheO(4)symmetryisspontaneouslybroken,ex-
scalars,fruitfullyformedmuchoftheoriginalmotivation
plicitly broken or unbroken. In the unbroken (Wigner-
for certainproposedBeyondthe StandardModel (BSM)
mode) and explicitly broken phases of the theory, this
physics. A partial list would include: Technicolor [52],
stillleavesafine-tuningifthebosonmassesaremeantto
low-energy supersymmetry [55], little Higgs theories [56]
be small compared to whatever scale the cutoff is taken
and Lee-Wick theories [57].
to be. However, in the spontaneously broken phase the
Susskind [52] was probably the first to motivate BSM
Goldstone Theorem guarantees that the companions of
physicsoutofconcernforfine-tuningsinthescalarsector.
theHiggsareexactlymasslessNambu-GoldstoneBosons,
Hearguedthatfundamentalscalarsarebadbecausethey
and thus automatically eliminates any common UV-QD
lead to quadratic divergences that must be fine tuned
contributions to the masses of the NGBs and the Higgs.
away. The motivation for technicolor was to replace the
ItappearsthattherequirementthattheHiggsvacuum
fundamental scalar Higgs doublet with a composite con-
is stable (against spontaneous appearance or disappear-
densateofnewfermions,calledtechniquarks,withQCD-
ance of physical Higgs) fixes the natural scale of the re-
likeconfininginteractions. Fundamentalscalarsdoresult
maining contributions to m2 to be f2 by requiring that
in quadratic divergences that must be fine tuned away h π
theHiggsvacuumexpectationvalue H beexactlyequal
under many circumstances (e.g. in Section IIC) , and
h i
so most renormalizable field theories with fundamental to f = GExp/√2 −1/2. This leavesm2 to pick up only
π µ h
scalars do have a scalar fine-tuning problem. However, finite and logarithmically divergent contributions.
(cid:0) (cid:1)