The muon g-2 discrepancy: new physics or a relatively light Higgs?

0.16 MB·English

by  M. Passera

CPC(HEP&NP),2009,33(X):1—6 Chinese Physics C Vol.33,No.X, Xxx,2009
−
The muon g 2 discrepancy: new physics
or a relatively light Higgs?
M. Passera1 W. J. Marciano2 A. Sirlin3
1 Istituto Nazionale Fisica Nucleare, Sezione di Padova, I-35131, Padova, Italy
2 Brookhaven National Laboratory, Upton, NewYork 11973, USA
3 Department of Physics, New York University,10003 NewYork NY, USA
0
1 Abstract Afterabriefreviewofthemuong−2status,wediscusshypotheticalerrorsintheStandardModel
0
predictionthatmightexplainthepresentdiscrepancywiththeexperimentalvalue. Noneofthemseemslikely.
2
Inparticular, a hypotheticalincrease of thehadroproduction cross section in low-energy e+e− collisions could
n
bridgethemuong−2discrepancy,butitisshowntobeunlikelyinviewofcurrentexperimentalerrorestimates.
a
J If, nonetheless, this turnsout to be theexplanation of the discrepancy,then the 95% CL upperbound on the
6 Higgs boson mass is reduced to about 135 GeV which, in conjunction with the experimental 114.4 GeV 95%
2 CL lower bound, leaves a narrow window for themass of this fundamental particle.

Key words Muon anomalous magnetic moment, Standard Model Higgs boson
]
h
PACS 13.40.Em, 14.60.Ef, 12.15.Lk, 14.80.Bn
p
-
p
he 1 Introduction: status of aµ Thehigher-orderhadronictermisfurtherdivided
[ intotwoparts: aHHO=aHHO(vp)+aHHO(lbl). Thefirst
µ µ µ
one, 98(1) 10 11[8],istheO(α3)contributionofdi-
1 The anomalous magnetic moment of the muon, − × −
v agrams containing hadronic vacuum polarization in-
a , is one of the most interesting observables in par-
8 µ sertions [14]. The second term, also of O(α3), is the
2 ticle physics. Indeed, as each sector of the Standard
hadronic light-by-light contribution; as it cannot be
5
Model (SM) contributes in a significant way to its
4 determined from data, its evaluation relies on spe-
theoreticalprediction,theprecisea measurementby
. µ cific models. The latest determinations of this term,
1
theE821experimentatBrookhaven[1,2]allowsusto
0 116(39) 10−11[9, 15] and 105(26) 10−11[16], are in
0 testtheentireSMandscrutinizeviable“newphysics” verygoo×dagreement. Ifweaddthe×lattertoaHLO,for
1 appendages to this theory [3, 4]. µ
example the value of Ref. [7], and the rest of the SM
:
iv veniTenhtelySMspliptreidnitcotioQnEoDf, tehleectmrouwoenakg−(E2Wis) caonnd- contributions,weobtainaSµM=116591773(48)×10−11.
X The difference with the experimental value aEXP =
hadronic (leading- and higher-order) contributions: µ
r 116592089(63) 10 11[2](notethetinyshiftupwards,
−
a aSM = aQED+aEW+aHLO+aHHO. The QED predic- ×
µ µ µ µ µ with respect to the value reported in [1], due to the
tion, computed up to four (and estimated at five)
updated value of the muon-protonmagnetic moment
loops, currently stands at aQED=116584718.08(15)
10 11[5],whilethe EW effecµts provideaEW=154(2)× ratio[17])is∆aµ=aEµXP−aSµM=+316(79)×10−11,i.e.,
− µ × 4.0σ (all errors were added in quadrature). Slightly
10 11[6]. The latest calculations of the hadronic
− smaller discrepancies are found employing the aHLO
leading-ordercontribution,viathehadronice+e an- µ
− values reported in [12] (which also includes the re-
nihilation data, are in agreement: aHLO=6894(40)
µ × centπ+π−(γ)data of BaBar)and[9]: 3.2σ and 3.6σ,
10 11[7] (this preliminary result, presented at this
− respectively. We will use the aHLO value of Ref. [7]
workshop, updates the value 6894(46) 10 11 of µ
× − (which also provides the hadronic contribution to
Ref. [8]) and 6903(53) 10 11[9]. These determina-
× − the effective fine-structureconstantlaterrequiredfor
tions include the 2008 e+e π+π (γ) cross sec-
− → − our analysis), but we expect that a consistent inclu-
tion data from KLOE [10] (see also [11]). A some-
sion of the recent π+π (γ) BaBar data would not
−
what larger value, 6955(41) 10 11[12], was recently
obtained including also the×200−9 π+π (γ) data of change our basic conclusions. For reviews of aµ see
− Refs. [7, 9, 18].
BaBar [13].
ReceivedJanuary25th 2010
(cid:13)c2009ChinesePhysicalSocietyandtheInstituteofHighEnergyPhysicsoftheChineseAcademyofSciencesandtheInstitute
ofModernPhysicsoftheChineseAcademyofSciences andIOPPublishingLtd
No.X M.Passera,W.J.Marciano,A.Sirlin: Themuong−2discrepancy: newphysicsorarelativelylightHiggs? 2
The term aHLO can alternatively be computed in- off (for a detailed discussion of these radiative cor-
µ
corporating hadronic τ-decay data, related to those rections and the precision of the Monte Carlo gen-
ofhadroproductionine+e collisionsviaisospinsym- erators used to analyze the hadronic cross section
−
metry [19]. The long-standingdifference betweenthe measurementssee[27]), ands isthe squaredmomen-
e+e - and τ-based determinations of aHLO [20] has tum transfer. The well-known kernel function K(s)
− µ
beenrecentlysomewhatlessenedbyare-analysis[21] (see[28])ispositivedefinite,decreasesmonotonically
wheretheisospin-breakingcorrections[22]wererevis- for increasing s and, for large s, behaves as m2/(3s)
µ
itedtakingadvantageofmoreaccuratedataandnew to a good approximation. About 90% of the total
theoretical investigations (recent τ π π0ν data contributiontoaHLO isaccumulatedatcenter-of-mass
− → − τ µ
fromtheBelleexperiment[23]werealsoincluded). In energies√sbelow1.8GeVandroughlythree-fourths
spite of this, the τ-based value remains higher than of aHLO is covered by the two-pion final state which
µ
the e+e -based one, leading to a smaller (1.9σ) dif- is dominated by the ρ(770) resonance [12]. Exclusive
−
ference ∆a . On the other hand, recent analyses of low-energye+e crosssections weremeasuredatcol-
µ −
the pion form factor claim that the τ and e+e data liders in Frascati, Novosibirsk, Orsay, and Stanford,
−
are consistent after isospin violation effects and vec- while at higher energies the total cross section was
tor mesonmixings areconsidered,further confirming determined inclusively.
the e+e -based discrepancy [24]. Let’s now assume that the discrepancy ∆a =
− µ
The 3–4σ discrepancy between the theoretical aEXP aSM =[phone], is due to – and only
µ − µ × −
prediction and the experimental value of the muon to – hypothetical errors in σ(s), and let us increase
g 2canbeexplainedinseveralways. Itcouldbedue, this cross section in order to raise aHLO, thus reduc-
− µ
at least in part, to an error in the determination of ing∆a . This simple assumptionleadsto interesting
µ
the hadronic light-by-light contribution. However, if consequences. An upward shift of the hadronic cross
this werethe onlycause ofthe discrepancy,aHHO(lbl) section also induces an increase of the value of the
µ
would have to move up by many standarddeviations hadronic contribution to the effective fine-structure
(roughly ten) to fix it. Although the errors assigned constant at M [29],
Z
to aHHO(lbl) are only educated guesses, this solution
µ
seems unlikely, at least as the dominant one. ∆α(5) (M )= MZ2 P ∞ ds σ(s) (2)
Another possibility is to explain the discrepancy had Z 4απ2 Z4m2π MZ2−s
∆a viatheQED,EWandhadronichigher-ordervac-
µ
(P stands for Cauchy’s principal value). This inte-
uum polarization contributions; this looks very im-
gral is similar to the one we encountered in Eq. (1)
probable,asonecanimmediatelyconcludeinspecting
for aHLO. There, however, the weight function in the
their values and uncertainties reported above. If we µ
integrandgivesa strongerweightto low-energydata.
assume that the g 2 experiment E821 is correct, we
− Let us define
are left with two options: possible contributions of
physics beyond the SM, or an erroneous determina- su
tion of the leading-order hadronic contribution aHLO ai= dsfi(s)σ(s) (3)
µ Z
(or both). The first of these two explanations has 4m2π
beenextensivelydiscussedintheliterature;updating
(i = 1,2), where the upper limit of integration is
Ref. [25] we will study whether the second one is re- s < M2, and the kernels are f (s) = K(s)/(4π3)
u Z 1
alisticornot,andanalyzeitsimplicationsfortheEW and f (s) = [M2/(M2 s)]/(4απ2). The integrals
bounds on the mass of the Higgs boson. 2 Z Z−
a with i = 1,2 provide the contributions to aHLO
i µ
and∆α(5) (M ),respectively,from4m2 uptos (see
had Z π u
2 Connection with the Higgs mass Eqs. (1,2)). An increase of the cross section σ(s) of
the form
Thehadronicleading-ordercontributionaHLO can
µ ∆σ(s)=ǫσ(s) (4)
be computed via the dispersion integral [26]
in the energy range √s [√s δ/2,√s +δ/2],
1 ∞ ∈ 0 − 0
aHLO= dsK(s)σ(s), (1) whereǫ andδ arepositive constantsand2m +δ/2<
µ 4π3Z π
4m2π √s < √s δ/2, increases a by ∆a (√s ,δ,ǫ) =
0 u− 1 1 0
where σ(s) is the total cross section for e+e anni- ǫ √s0+δ/22tσ(t2)f (t2)dt. If we assume that the
hilation into any hadronic state, with vacuum−polar- mRu√osn0−gδ/22discrepa1ncyisentirelyduetothisincrease
−
ization and initial state QED corrections subtracted in σ(s), so that ∆a (√s ,δ,ǫ)=∆a , the parameter
1 0 µ
No.X M.Passera,W.J.Marciano,A.Sirlin: Themuong−2discrepancy: newphysicsorarelativelylightHiggs? 3
ǫ becomes ues reported in [7]). Our results show that an in-
crease ǫσ(s) of the hadronic cross section (in √s
∆a
ǫ= µ , (5) ∈
[√s δ/2,√s +δ/2]), adjusted to bridge the muon
√s0+δ/22tf (t2)σ(t2)dt 0− 0
R√s0−δ/2 1 g−2 discrepancy ∆aµ, decreases MHUB, further re-
and the corresponding increase in ∆α(5) (M ) is stricting the already narrow allowed region for MH.
had Z
We conclude that these hypothetical shifts conflict
√s0+δ/2f (t2)σ(t2)tdt with the lower limit MLB when √s &1.2 GeV, for
∆a2(√s0,δ)=∆aµR√√ss00−+δδ//22f2(t2)σ(t2)tdt. (6) values of δ up to severHal hundreds0of MeV. In [25]
R√s0−δ/2 1 we pointed out that there are more complex scenar-
The shifts ∆a2(√s0,δ) were studied in Ref. [25] for ioswhereitispossibletobridgethe∆aµ discrepancy
several bin widths δ and central values √s . withoutsignificantlyaffectingMUB,buttheyarecon-
0 H
The present global fit of the LEP Electroweak siderably more unlikely than those discussed above.
Working Group (EWWG) leads to the Higgs boson If τ data are used instead of e+e− ones in the
mass M =87+35 GeV and the 95% confidence level calculation of the dispersive integral in Eq. (1), aHLO
H 26 µ
(CL) upper bo−und MUB 157 GeV [30]. This result increases to 7053(45) 10−11[21] and the discrepancy
H ≃ ×
is based on the recent preliminary top quark mass drops to ∆aµ = +157(82)×10−11, i.e. 1.9σ. While
Mt=173.1(1.3)GeV [31] and the value ∆α(h5a)d(MZ)= using τ data reduces the ∆aµ discrepancy, it in-
0.02758(35)[32]. TheLEPdirect-search95%CLlower creases∆α(h5a)d(MZ) by approximately2×10−4,∗ lead-
bound is MHLB=114.4GeV [33]. Although the global ingtoasharplylowerMH prediction[38]. Indeed,in-
EW fit employs a large set of observables, MHUB is creasingthepreviouslyemployedvalue∆α(h5a)d(MZ)=
strongly driven by the comparison of the theoreti- 0.02760(15)[7]by2 10−4 andusingthesameabove-
×
cal predictions of the W boson mass and the effec- discussed previous inputs of the χ2-analysis, we find
tive EW mixing angle sin2θlept with their precisely an MUB value of only 138 GeV. If the remaining
eff H
measured values. Convenient formulae providing the 1.9σdiscrepancy∆aµ isbridgedbyafurtherincrease
M andsin2θlept SMpredictionsintermsofM ,M , ∆σ(s)=ǫσ(s) ofthe hadronic cross section, MUB de-
W eff H t H
∆α(5) (M ), and α (M ), the strong coupling con- creases to even lower values, leading to a scenario in
had Z s Z
stant at the scale M , are given in [34]. Combin- near conflict with MLB.
Z H
ing these two predictions via a numerical χ2-analysis Recent analyses of the pion form factor below
and using the present world-average values MW = 1GeVclaimthatτ dataareconsistentwiththee+e−
80.399(23) GeV [35], sin2θlept = 0.23153(16) [36], ones after isospin violation effects and vector meson
eff
M =173.1(1.3)GeV[31], α (M )=0.118(2)[37],and mixings are considered [24]. In this case one could
t s Z
the determination ∆α(h5a)d(MZ)=0.02758(35)[32], we usethee+e− databelow∼1GeV,confirmedbytheτ
get MH =92+3278 GeV and MHUB =158 GeV. We see ones, and assume that ∆aµ is accommodated by hy-
thatindeedth−eMH valuesobtainedfromtheMW and potheticalerrorsinthee+e− measurementsoccurring
sin2θlept predictions are quite close to the results of above 1 GeV, where disagreementpersists between
eff ∼
the global analysis. these two data sets. Our analysis shows that this as-
The M dependence of aSM is too weak to pro- sumptionwouldleadtoMUB valuesinconsistentwith
H µ H
vide M bounds from the comparison with the mea- MLB.
H H
sured value. On the other hand, ∆α(5) (M ) is one In the above analysis, the hadronic cross section
had Z
of the key inputs of the EW fits. For example, em- σ(s) was shifted up by amounts ∆σ(s) = ǫσ(s) ad-
ploying the latest (preliminary) value ∆α(h5a)d(MZ)= justedtobridge∆aµ. Apartfromtheimplicationsfor
0.02760(15)presented at this workshop[7] instead of MH, these shifts may actually be inadmissibly large
0.02758(35)[32],theM predictionderivedfromM when comparedwith the quoted experimentaluncer-
H W
and sin2θlept shifts to M =96+32 GeV and MUB = tainties. Consider the parameter ǫ = ∆σ(s)/σ(s).
eff H 25 H
153 GeV. To update the analysi−s of Ref. [25] we con- Clearly,its value depends onthe choiceofthe energy
sidered the new values of ∆α(5) (M ) obtained shift- range [√s δ/2,√s +δ/2] where σ(s) is increased
had Z 0− 0
ing 0.02760(15) [7] by ∆a (√s ,δ) (including their and, for fixed √s , it decreases when δ increases. Its
2 0 0
uncertainties, as discussed in [25]), and computed minimum value, 5%,occurs if σ(s) is multiplied by
∼
the corresponding new values of MHUB via the com- (1+ǫ) in the whole integration region, from 2mπ to
bined χ2-analysis based on the M and sin2θlept in- infinity. Such a shift would lead to MUB 75 GeV,
W eff H ∼
puts (for both ∆α(5) (M ) and aHLO we used the val- well below MLB. Higher values of ǫ are obtained for
had Z µ H
∗ Thisnumberrepresentsourroughupdate ofthevaluereportedinRef.[20].
No.X M.Passera,W.J.Marciano,A.Sirlin: Themuong−2discrepancy: newphysicsorarelativelylightHiggs? 4
narrower energy bins, particularly if they do not in- two-loop effects)
clude the ρ-ω resonance region. For example, a huge
100 GeV 2
ǫ 55%increaseisneededtoaccommodate∆a with asusy sgn(µ) 130 10 11 tanβ, (7)
∼ µ µ ≃ × × − (cid:18) m (cid:19)
ashiftofσ(s)inthe regionfrom2m upto500MeV susy
π
(reducingMUBto146GeV),whileanincreaseinabin where sgn(µ) = is the sign of the µ term in su-
H ±
of the same size but centered at the ρ peak requires persymmetry models and tanβ >3–4 is the ratio of
ǫ 9% (lowering MUB to 135 GeV). As the quoted the two scalar vacuum expectation values, tanβ =
∼ H
experimental uncertainty of σ(s) below 1 GeV is of φ / φ . The tanβ factor is an important source
h 2i h 1i
the order of a few per cent (or less, in some specific of enhancement. As experimental constraints on the
energy regions), the possibility to explain ∆a with Higgs mass have increased, so has the lower bound
µ
these shifts ∆σ(s) appears to be unlikely. Lower val- on tanβ. With larger tanβ now required, it ap-
ues of ǫ are obtained if the shifts occur in energy pears inevitable that supersymmetric loops have a
ranges centered around the ρ-ω resonances, but also fairlymajoreffectonthetheoreticalpredictionofthe
this possibility looks unlikely, since it requires varia- muon g 2 if m is not too large. In fact, equating
− susy
tionsofσ(s)ofatleast 6%. If,however,suchshifts (7) and the discrepancy ∆a , for example the value
∼ µ
∆σ(s) indeed turn out to be the solution of the ∆a ∆a =[phone] obtained using the aHLO de-
µ µ × − µ
discrepancy,then MUB is reducedto about 135GeV. termination of Ref. [7], one finds sgn(µ)=+ and
H
Itisinterestingtonotethatinthescenariowhere
m 64+10 tanβ GeV. (8)
∆aµ isduetohypotheticalerrorsinσ(s),ratherthan susy≃ −7 p
“new physics”, the reduced MUB.135 GeV induces For tanβ 4–50, these values are in keeping with
H
∼
some tension with the approximate 95% CL lower mainstream supersymmetric expectations. Several
bound MH & 120 GeV required to ensure vacuum alternative“newphysics”explanationshavealsobeen
stability under the assumption that the SM is valid suggested [3].
up to the Planck scale [39] (note, however, that this
lower bound somewhat decreases when the vacuum 3 Conclusions
is allowed to be metastable, provided its lifetime is
longer than the age of the universe [40]). Thus, one
We examined a number of hypothetical errors in
couldarguethatthistensionis,onitsown,suggestive
the SM prediction of the muon g 2 that could be re-
of physics beyond the SM. −
sponsible for the present 3–4σ discrepancy ∆a with
µ
We remind the reader that the present values of
theexperimentalvalue. Noneofthemlookslikely. In
sin2θlept derived from the leptonic and hadronic ob-
eff particular, updating Ref. [25] we showed how an in-
servables are respectively (sin2θlept) = 0.23113(21)
eff l crease∆σ(s)=ǫσ(s)ofthehadroproductioncrosssec-
and (sin2θlept) =0.23222(27) [36]. In Ref. [25] we
eff h tion in low-energy e+e collisions could bridge ∆a .
− µ
pointed out that the use of either of these values as
However, such increases lead to reduced M upper
H
an input parameter leads to inconsistencies in the
bounds – even lower than 114.4 GeV (the LEP lower
SM framework that already require the presence of
bound)iftheyoccurinenergyregionscenteredabove
“new physics”. For this reason,we followedthe stan-
1.2 GeV). Moreover, their amounts are generally
dardpracticeofemployingasinputtheworld-average ∼
very large when compared with the quoted experi-
value forsin2θlept determinedinthe SMglobalanaly-
eff mental uncertainties, even if the latter were signifi-
sis. SinceMUB alsodependssensitivelyonM ,in[25]
H t cantly underestimated. The possibility to bridge the
weprovidedsimpleformulaetoobtainthenewvalues
muon g 2 discrepancy with shifts of the hadronic
derived from different M inputs. −
t cross section therefore appears to be unlikely. If,
A 3–4σ discrepancy between the theoretical pre-
nonetheless,thisturnsouttobethesolution,thenthe
diction and the experimental value of the muon
95% CL upper bound MUB drops to about 135 GeV.
g 2 would have interesting implications if truly due H
− If τ-decay data are used instead of e+e− ones in
to “new physics” (i.e. beyond the SM expectations).
thecalculationofaSM,themuong 2discrepancyde-
Supersymmetry provides a natural interpretation of µ −
creasesto 2σ. While this reduces∆a , it raisesthe
this discrepancy (see Ref. [4] for a review). For illus- ∼ µ
value of ∆α(5) (M ) leading to MUB=138 GeV, thus
tration purposes, we assume a single mass m for had Z H
susy increasingthe tensionwiththe LEPlowerboundand
sleptons,sneutrinosandgauginosthatentertheasusy
µ suggesting a near conflict with it should one try to
calculation. Then one finds [41] (including leading
overcome the full discrepancy. One could also con-
sider a scenario, suggested by recent studies, where
No.X M.Passera,W.J.Marciano,A.Sirlin: Themuong−2discrepancy: newphysicsorarelativelylightHiggs? 5
the τ data confirm the e+e ones below 1 GeV, ues. This reduction, together with the LEP lower
−
∼
while a discrepancy between them persists at higher bound, leaves a narrow window for the mass of this
energies. If, in this case, ∆a is fixed by hypotheti- fundamental particle. Interestingly, it also raises the
µ
cal errors in the e+e measurements above 1 GeV, tension with the M lower bound derived in the SM
− H
∼
where the data sets disagree, one also finds values of from the vacuum stability requirement.
MUB inconsistent with the LEP lower bound.
H
If the ∆a discrepancy is real, it points to “new Acknowledgments: We thank D. Nomura and
µ
physics”, like low-energy supersymmetry where ∆a T. Teubner for discussions and communications, and
µ
isreconciledbytheadditionalcontributionsofsuper- the organizers of this workshop for the very pleas-
symmetric partners and one expects M .135 GeV ant and stimulating atmosphere. This work was
H
for the mass of the lightest scalar [42]. If, instead, supported in part by U.S. DOE grant DE-AC02-
the deviation is caused by an incorrect leading-order 76CH00016, E.C. contractMRTN-CT2006-035505, and
hadronic contribution, it leads to reduced MUB val- U.S.NSF grant PHY-0758032.
H
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