[PDF] Bread & Butter Physics at Muon Colliders

39509_72021_IPPP.pdf

Tao Han1Bread & Butter Physics at Muon CollidersIPPP Topical Meeting on Physics with High-Brightness Stored Muon Beams•A Higgs factory•EW physics at high energies•Precision Higgs physics

2

Target Energy and LuminosityarXiv:1901.06150 Energy: For a striking Direct Exploration program, after HL-LHC*, energy should be close or above 10 TeV!At few TeV energy one can still exploit high partonic energy for a striking Indirect Exploration program, by High-Energy Precision!We can borrow CLIC physics case (see below)*see arXiv:1910.11775 for HL-LHC and F.C. projections summaryLuminosity: Set by asking for 100K SM "hard" SM pair-production events.!Compatible with other projects (e.g. CLIC = )!If much less, we could only bet on Direct Discoveries !!Could be reduced by running longer than 5yrs and > 1 I.P.(3TeV/10TeV)

2 6!10 35
L"

5years

time s ! 10TeV 2 2!10 35
cm #2 s #1

81 ab-1/yr

hh & ! s=3,6,10,14,30and 100TeV,L=1,4,10,20,90,and1000ab !1 . s "

Lumi-scaling scheme: !L~ const.The aggressive choices:European Strategy, arXiv:1910.11775; arXiv:1901.06150; arXiv:2007.15684.Collider benchmark points: !Multi-TeVcolliders:!The Higgs factory:7

Table1:Mainparametersof theproton driver muonfacilities

ParameterUnitsHiggsMulti-TeV

CoMEnergy TeV0.1261.53.06.0

Avg.Luminosity10

34
cm !2 s !1

0.0081.254.412

BeamEnergy Spread%0.0040.10.10.1

HiggsProduction/10

7 sec13'50037'500200'000820'000

Circumferencekm0.32.54.56

No.ofIP' s1222

RepetitionRateHz1515 126

! " x,y cm1.710.50.25

No.muons/bunch 10

12 4222

Norm.Trans. Emittance,"

TN

µm-rad2002525 25

Norm.Long.Emittance, "

LN

µm-rad1.57070 70

BunchLength,#

S cm6.310.5 0.2

ProtonDriv erPowerMW44 41 .6

WallPlugPowerMW200216230270

Aschematiclayout ofaproton driv enmuoncollider facilityis sketchedin Figure2.Them ain parametersofthe enabledfacilities aresummarized inTable 1. Thefunctionalelements ofthe muonbeamgeneration andaccelerationsystems are: -aprotondri ver producingahigh-powermulti-GeV,multi-MWbunchedH ! beam, -abuncher madeofanaccumulatorand acompressorthat formsintenseand shortprotonb unches, -apionproduction target inahea vilyshieldedenclosureabletowithstand thehighproton beam power,whichisinsertedina highfieldsolenoid tocapturethe pionsandguide themintoa decay channel, -afront-endmade ofa solenoiddecaychannel equippedwithRF cavitiesthat capturesthemuons longitudinallyinto abunch train,andthen appliesatime-dependentaccelerationthatincreases the energyoftheslower (low-ener gy)bunches anddecreasestheenergyof thefaster(high-energy) bunches, -an"initial"cooling channelthat usesam oderateamountof ionizationcoolingto reducethe6D phasespaceoccupied bythe beambya factorof 50(5ineachtransv erseplane and2inthe longitudinalplane),so thatitfits withintheacceptance ofthefirst accelerationstage. Forhigh luminositycolliderapplications, furtherionizationcooling stagesarenecessary toreduce the6D phasespaceoccupied bythe beambyup tofiv eordersof magnitude, -thebeamis thenacceleratedby aseries offast accelerationstagessuch asRecirculatingLinacs Accelerators(RLA)or Fixed FieldAlternatingGradient (FFAG)andRapidCycling Synchrotron (RCS)totak ethe muonbeamstotherelev antenergy beforeinjectionin themuoncollider Ring.

3.2.2R&D

TheMAPR&D program(2011-2018)addressed many issuestow ardtechnicaland designfeasibilityof amuonbased neutrinofactory orcollider [19].Significant R&Dprogress,also summarizedin [1],was achieved. OperationofRFCavitiesin HighMa gneticFields Acceleratinggradientsin excess of50MV /mina

3Tmagneticfield have beendemonstrated intheFNALMuCoolTestArea(MT A).

Initialand6D IonizationCoolingDesigns andpioneeringdemonstr ationConceptswerede veloped for InitialCooling,and 6DCooling withRFca vitiesoperatingin vacuum(VCC) ,includinga varianton this designwherethe cavities werefilledwith gasusedasdiscrete absorber(hybrid scheme),anda Helical 6

Ecm=mHL~ 1 fb-1/yr"Ecm~ 5 MeV

31. A Higgs FactoryResonant Production:About O(40k)events produced per fb-1

SM Higgs is (very) narrow:At mh=125 GeV, !h= 4.2 MeV

10 V. Barger et al. I Physics Reports 286 (I 997) I-51

convoluting crh(S^) with the Gaussian distribution in & centered at & = 4: (1.8) Fig. 7 illustrates the effective cross section, h(d), as a function of fi for mh = 110 GeV and beam energy resolutions of R = 0.0 l%, R = 0.06%, and R = 0.1%. Results are given for the cases: IZsM, ho with tan /I = 10, and ho with tan p = 20. All channels X are summed over. In the case where the Higgs width is much smaller than the Gaussian width ad, the effective signal cross section result for fi = mh, denoted by ah, is (Th = (1.9)

Henceforth, we adopt the shorthand notation

G(X) = T(h --f j+)BF(h + X) (1.10)

for the numerator of Eq. (1.9). The increase of ah<+ = mh) with decreasing a~ when Gt:"' 4 06 is apparent from the h sM curves of Fig. 7. In the other extreme where the Higgs width is much broader than 04, then at ,,& = mh we obtain (T,, =

4rcBF(h -+ p,u)BF'(h ---f X)

- mi (ly W&) f (1.11) Note that this equation implies that if there is a large contribution to the Higgs width from some

channel other than yp, we will get a correspondingly smaller total event rate due to the small size of

BF(h + ,up). That ??h( fi = mh) is independent of the value of 04 when Gtot B ad is illustrated by

the tan j3 = 20 curves for the ho in Fig. 7. Raw signal rates (i.e. before applying cuts and including

other efficiency factors) are computed by multiplying ah by the total integrated luminosity L. The basic results of Eqs. (1.9) and (1.11) are modified by the effects of photon bremsstrahlung from the colliding muon beams. In the case of a narrow Higgs boson, the primary modification for

fi = mh is due to the fact that not all of the integrated luminosity remains in the central Gaussian

peak. These modifications are discussed in Appendix A; to a good approximation, the resulting signal

rate is obtained by multiplying ??h of Eq. (1.9) by the total luminosity L times the fraction f of

the peak luminosity in the Gaussian after including bremsstrahlung relative to that before (typically

f M 0.6). For a broad Higgs resonance, the lower energy tail in the luminosity distribution due to

bremsstrahlung makes some contribution as well. In the results to follow, we avoid any approximation

and numerically convolute the full effective luminosity distribution (including bremsstrahlung) with

the Higgs cross section of Eq. (1.7). In performing this convolution, we require that the effective $,L- c.m. energy be within 10 GeV of the nominal value. Such a requirement can be implemented

by reconstructing the mass of the final state as seen in the detector; planned detectors would have the

necessary resolution to impose the above fairly loose limit. This invariant mass selection is imposed

in order to reduce continuum (non-resonant) backgrounds that would otherwise accumulate from the entire low-energy bremsstrahlung tail of the luminosity distribution. As is apparent from Fig. 7, discovery and study of a Higgs boson with a very narrow width at the p+p- collider will require that the machine energy ,,& be within crd of mh. The amount of I/ Barger et al. /Physics Reports 286 (1997) l-51 h ---_ b (t> -m!J ""b (mt)

Effective Cross Sections: mh= 110 GeV

10-L -

no Bquark mixing

10-Z I 8 I ' L ' ' ' ' ' 1 ' ' ' ' ' D

100 109.5 110 1105 111

6 (CeV)

Fig. 6. s-channel diagram for production of a Higgs bosom

Fig. 7. The effective crass section, h, obtained after convoluting CJh with the Gaussian distributions for R = O-01%,

R = 0.06%, and R = O.l%, is plotted as a function of fi taking Mh = 110 GeV. Results are displayed in the cases: hsM,

ho with tan p = 10, and ho with tan /I = 20. In the MSSM ho cases, two-loop/RGE-improved radiative corrections have

been included for Higgs masses, mixing angles, and self-couplings assuming rni = 1 TeV and neglecting squark mixing.

The effects of bremsstrahlung are not included in this figure. The rms spread in fi (denoted by od) prior to including bremsstrahlung is given by where R is the resolution in the energy of each beam. A convenient formula for ah is ad = (7 MeV)(R/0.01%)(~/100 GeV) . (1.6) The critical issue is how this resolution compares to the calculated total widths of Higgs bosons when fi = mh. For R 5 O.Ol%, the energy resolution in Eq. ( 1.6) is smaller than the Higgs widths

in Fig. 3 for all but a light SM-like Higgs. We shall demonstrate that the smallest possible R allows

the best measurement of a narrow Higgs width, and that the total luminosity required for discovery by energy scanning when r 5 ad is minimized by employing the smallest possible R. For a Higgs boson with width larger than ah, results from a fine scan with small R can be combined without any increase in the luminosity required for discovery and width measurement. The Feynman diagram for s-channel Higgs production is illustrated in Fig. 6. The s-channel Higgs resonance cross section is (1.7) where i = (pp.+ + pp- )2 is the c.m. energy squared of a given p"'pu- annihilation, X denotes a final state and Gtot is the total width. 1 The sharpness of the resonance peak is determined by Pot. Neglecting bremsstrahlung for the moment, the effective signal cross section is obtained by h

' Effects arising from implementing an energy-dependent generalization of the rnhGtot denominator component of this

simple resonance form are of negligible importance for our studies, especially for a Higgs boson with GtO* 4mh.

"Muon Collider Quartet":Barger-Berger-Gunion-HanPRL & Phys. Report (1995)TH, Liu: 1210.7803;Greco, TH, Liu: 1607.03210

ARTICLEINPRESS

UNCORRECTEDPROOF

Pleasecitethisart icleinpress as:M.Grecoet al.,ISReffectsforr esonantHi ggsproductionatfuturele ptoncolliders,P hys.Lett.B(2016) ,

http://dx.doi.org/10.1016/j.physletb.2016.10.078 JID:PLBAID:32399/SCODoc topic:Phenomenology[m5Gv1.3;v1.190;Prn:4/11/2016;17:09]P .4(1-7)

4M.Grec oetal./PhysicsL ettersB •••(••••)•••-•••

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60125
61126
62127
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65130
Fig.2.Theline shapesofthe resonancesproduct ionoft heSMHiggs bosonasa functi ono fthebeamenergy ! sataµ + µ " collider(leftpanel) andane + e " collider(right

panel).Thebluecurveist heBre it-Wignerres onancelineshape.Theoran gelin eshapeincludestheI SReffectalo neforJadach-Ward-Was(b).Thegreencurvesinclu dethe

BESonlywi thtwodiffer entenergyspr eads.Ther edlineshapestakeintoaccountalltheBreit-Wignerresonance,ISReffectand BESinsoli dand dashedli nes,respective ly.

(Forinterpretat ionofthereferencestocolorinthisfigu re,thereaderisreferredtothewebversionofthisarticle.)

effectincreasesth eproductionratevia"radiativ ereturn"mecha- nism.Still,theo veralleffectisthe reduction ofon-shellrateas clearlyindicatedin theplot.Inredlinesweshowtheli neshapes oftheH iggsbosonwi thbothth eBESandtheISRef fect.Wecan seether esultingli neshapeisnotmerelyaproduc toftw oeffect butrather complexcon volution,justifyingnecessityo fournumer- icalevalu ation. Havingunderstood theISRandBESeffectsonthesignal pro- ductionrat esandlineshapes ,wenowprocee dtoundersta ndthe effectonthebackgrou nd.Fo rthemuonc olliderstudy,themain searchchannelsfort heHiggsbosonwillbetheexcl usivemod eof b ¯ bandWW # .Fortheb ¯ bfinalstateth emainbackgroundi sfrom theoff-sh ellZ/!s-channelproduction.Th eISRandBESeffects barelychangetherat efromsuchoff-shell process.H owever,the

ISReffect doesincreasetheon -shellZ$b

¯ bbackgroundthrough the" radiativereturn"mechanism.Ournumer icalstudyshowsthat the"r adiativereturn"oftheZbosontob ¯ bincreasetheinclusive b ¯ bbackgroundbyafactorofseven. Sincew eu nderstandthatthe increaseofthebackgroundi sdomin antlyfro mtheon-shellZbo- son,thenewba ckgroundra tesafter imposingab ¯ binvariantmass cutof95,1 00,110 GeV,chang eto17,20,25p b,respect ively.Given thefinit eresolutionofthe b-jetenergy reconstruction, wepropose ani nvariantmasscutoftheb ¯ bsystemof100GeV,whi chlead s toarou nd20%increaseinsu chbackgrou ndcomparingtothetree- levelestimate.S ofarwehavesuggestedtheinvar iantma sscutfor theb ¯ bpair,asanexamp leofdi scrimi nationfromthebackgrou nd. Onecould alsoforeseeacut ontheangleb etweenthetwob-jets, whichcouldb emeasuredmorepre ciselythan theinvariantmass. 2

Beyondtheb

¯ bfinals tate,anothermajorchannelfo rmuoncol- liderHiggsphysic sistheWW # channel.Thischannelenj oyslittle (irreducible)backgroundformtheSMproces s.TheISReffectin- troducesno"radiativ ere turn"forsuchprocess.Consequently,the backgroundratedoesnotchange fromthetree-l evelestimat e.We summarizeinTable2theon-shel lHiggspr oductionratean dba ck- groundrateinthe setwoleading chann elswiththeinclu sionofthe ISRandBES eff ects.We canseefromthetablethatatthemuon colliderHiggsfactory,thes ignalbackgr oundratioisprettylarge andthe obser vabilityissimplydominatedbythest atistics .The "radiativereturn"fromtheISRef fect,however,does impactsev- eralotherHi ggsdecaychannelse archmore .Forexample,sea rches ofHiggsr aredecayo fh$Z!,Hi ggsdecayof h$ZZ # with 2 Wet hanktheEditorGi giRolandi forsuggestingthis discriminationproce dure.

Table2

Signalandbackgr oundeffectiv ecrosssectionsattheresonance ! s=m h =125G eV ataµ + µ " collider(upperpanel,in pb)andane + e " collider(lowerpanel ,in ab)for twochoiceso fbeamenergy resolutionsRandtw oleadingdec aychannels withISRef fectstake nintoaccount,withtheSMbr anchingfractionsBr b ¯ b =58%a nd Br WW #=21%.For theb ¯ bbackground,aconservativecuton the b ¯ binvariantmass tob egreate rthan100GeVisappli ed.

R(%)µ

+ µ " $h " eff (pb) h$b ¯ bh$WW # " Sig " Bkg " Sig " Bkg

0.0110 5.6202.10.0 51

0.00322 124.6

R(%)e + e " $h " eff (ab) h$b ¯ bh$WW # " Sig S/B" Sig S/B

0.0448 27O(10

"6 )10O(10 "3 )

0.0115 08131

Table3

Fittingaccuraciesf oronestandarddeviationof#

h ,Bandm h oft heSMHiggs with thesca nningschemefortworepresentativ eluminositiesperstepandtwobench- markbeame nergyspreadpara meters. # h =4.07MeVL step (fb "1 )$# h (MeV)$B$m h (MeV)

R=0.01%0.050 .793.0%0.36

0.20.391.1%0.18

R=0.003%0.050.30 2.5%0.14

0.20.140.8%0.07

Z # $%¯%,etcarefacingmorechallengesandnewselectioncuts needtobedes ignedan dappli ed. Finally,weperformastud yonthe potentialprecisionon the Higgsproper tiesatafuturemuoncolliderthro ughalin es hape scan.Wefo llowthebe nchmarks,statisti caltreat mentandpro- ceduredefinedinRef .[5],wherea 21steps scaninthemas s windowof±30MeVa roundthe Higgsmasswithequalin tegrated luminosities. 3

Afittotheresultofsuchlineshapescancansi-

multaneouslydeterminetheHiggstotalwi dth# h ,th eHiggsmass m h andinter actionstrengthBwithgreatpre cision.Thein terac- tionstrengthBcanbedir ectlyt ranslatedintotheHiggsmuon Yukawaafterfixing thedecaybranch ingfractionsorp erforming aglobalfit.Wetabulatetheprojectedprecisionsonthesequanti- tiesinTable3 forthetwo benchmarkBES value sofR=0.01%and 3 TheHigg smassmaynotkno wntothe±30Me Vlevelby thetimeofthemuon collider,andapre-scansta getodet erm inetheHiggsmassw illbe require d[30]. Ideal, conceivable case: (Δ= 5 MeV, Γh≈ 4.2 MeV) An optimal fitting would reveal Γh 6 ! eff !s"# Z d !!! ^s p dL! !!! s p " d !!! ^s p!!" $ " % !h!X" / 8 > < > : ! 2 h

B=&!s%m

2 h " 2 $! 2 h m 2 h '!"(! h "; Bexp h %!m h % !! s p " 2 2" 2 i" ! h " # =m 2 h !")! h ": (2.2)

For"(!

h ,theline shapeof aBreit-Wigner resonance canbemapped outby scanningov ertheener gyasgi ven inthefirst equation.For ")! h ontheother hand, thephysicalline shapeis smearedoutby theGaussian distributionofthebeam energyspread, andthesignal rate willbedetermined bytheo verlapof theBreit-Wigner and theluminositydistrib utions,as seeninthesecondequation above.

Unlessstatedotherwise, wefocuson theSMHiggs

bosonwiththe massand totalwidthas m h #126GeV;! h #4:21MeV:(2.3)

Fordefinitivenessin thisstudy,weassumetwosetsof

representativevaluesforthemachineparameters [8]

CaseA:R#0:01%!"#8:9MeV";L#0:5fb

%1 ;(2.4)

CaseB:R#0:003%!"#2:7MeV";L#1fb

%1 :(2.5)

Weseethattheircorresponding beamenergy spread"is

comparabletothe Higgstotal width.InFig. 1,wesho w theeffecti vecrosssectionversusthe" $ " % colliderc.m. energyfortheSM Higgsbosonproduction. ApureBreit- Wignerresonanceissho wnbythe dottedcurve. Thesolid anddashedcurv esincludethe convolutionoftheluminos- itydistribution forthetwobeamenergy resolutionsandare integratedover !!! ^s p .For simplicity,wehavetaken the branchingfractionsh!" $ " % tobethe SMvalue and thefinalstate h!Xtobe100%. Thebeam energy resolutionmanifestsits greatimportancein comparison betweenthesolid anddashed curvesin thisfigure.

III.WIDTHDETERMIN ATION FORTHESM

HIGGSBOSON

Anexcellent beamenergyresolutionfor amuoncollider wouldmakea directdetermination oftheHiggs boson widthpossiblein contrastto thesituationsin theLHC andILC.Because ofthe expectednarro wwidthfor aSM Higgsboson,one stillneedsto convolute theidealistic

Breit-Wignerresonancewiththerealistic beamenergy

spectrumasillustrated inEq. (2.2).We firstcalculatethe effectivecrosssectionsatthepeakforthe twocasesof energyresolutionsAandB. Wefurther evaluate thesignal andSMbackground forthe leadingchannels h!b # b;WW * :(3.1)

Weimposeapolarangle acceptanceforthe final-state

particles, 10 + <#<170 + :(3.2)

Tighteningupthepolar angleto20

+ -160 + willfurther reducethesignal by4.6% andthebackground by6.7% (15%)forthe b # b(WW * )finalstates. Weassume a60% singleb-taggingefficiency andrequireatleastonetagged bjetforthe b # bfinalstate.The backgroundsare assumedto beflatwith crosssectionse valuatedright at126GeV using Madgraph5[10].Thisappears tobe anexcellent approxi- mationover theenergyrangeof thecurrentinterest about

100MeV. Wetabulatethe resultsinTableI.Theback-

groundrateof " $ " % !Z * =$ * !b # bis15pb, andthe rateof" $ " % !WW * !4fermionsisonly 51fb, as showninTable I.Here,we considerallthe decaymodes ofWW * becauseofits clearsignatureat amuoncollider .

Thefour-fermion backgroundsfromZ$

* and$ * $ * are smallertobeg inwithand canbegreatlyreducedby kinematicalconsiderationssuch asbyrequiring the invariantmassofonepairof jetstobe nearm W andsetting alower cutfortheinvariant massofthe otherpair.

Whiletheb

# bfinalstatehas alarger signalratethan that forWW * byabouta factorofthree, thelatterhas amuch improvedsignal(S)tobackground (B)ratio,about 100:1 nearthepeak.

125.97125.98125.99 126126.01126.02 126.03

0 10 20 30
40
50
60
70
sGeV eff spb

BreitW igner

h

4.21MeV

R0.003

R0.01 h

FIG.1(color online).Ef fectivecross sectionfor"

$ " % !h versusthecollider energy !!! s p fortheSM Higgsbosonproduction withm h #126GeV.ABreit-W ignerlineshape with! h #

4:21MeVisshown (dottedcurve).Thesolidand dashedcurves

comparethetwo beamenergy resolutionsofcases AandB. TABLEI.Effecti vecross sections(inpb)attheresonance !!! s p #m h fortwochoices ofbeam energyresolutions Rand twoleadingdecay channels,with theSMbranching fractions Br b # b #56%andBr WW *#23%[9]. " $ " % !hh!b # bh!WW * R(%)! eff (pb)! Sig ! Bkg ! Sig ! Bkg

0.01167.63.7

0.0033818155.5 0.051

TAOHANANDZHENLIUPHYSICAL REVIEWD87,033007(2013)

033007-2

a cone angle cut: 10o< !< 170oTH, Liu: 1210.7803; Greco, TH, Liu: 1607.03210

ARTICLEINPRESS

UNCORRECTEDPROOF

Pleasecitethisar ticleinpres sas:M.Grecoe tal.,ISReffectsforr esonantHi ggsproductionatfuturele ptoncolliders,P hys.Lett.B(2016) ,

http://dx.doi.org/10.1016/j.physletb.2016.10.078 JID:PLBAID:32399/SCODoc topic:Phenomenology[m5Gv1.3;v1.190;Prn:4/11/2016;17:09]P .4(1-7)

4M.Grec oetal./PhysicsL ettersB •••(••••)•••-•••

166
267
368
469
570
671
772
873
974
1075
1176
1277
1378
1479
1580
1681
1782
1883
1984
2085
2186
2287
2388
2489
2590
2691
2792
2893
2994
3095
3196
3297
3398
3499
35100
36101
37102
38103
39104
40105
41106
42107
43108
44109
45110
46111
47112
48113
49114
50115
51116
52117
53118
54119
55120
56121
57122
58123
59124
60125
61126
62127
63128
64129
65130
Fig.2.Theline shapesofthe resonancesproduct ionoft heSMHiggs bosonasa functi ono fthebeamenergy ! sataµ + µ " collider(leftpanel) andane + e " collider(right

panel).Thebluecurveist heBre it-Wignerres onancelineshape.Theora ngelin eshapeincludesthe ISReffectalo neforJadach-Ward-Was(b).Thegreencurvesinclud ethe

BESonlyw ithtwodiffer entenergysp reads.Ther edlineshapestakeintoaccountalltheBreit-Wignerresonance,ISReffectand BESinsol ida nddashedl ines,respective ly.

(Forinterpretatio nofthereferencestocolorinthisfigur e,thereaderisreferredtothewebversionofthisarticle.)

effectincreasesth eproductionratevia"radiativ ereturn"mecha- nism.Still,theo veralleffectisthe reduction ofon-shellrateas clearlyindicatedin theplot.Inredlinesweshowtheli neshapes oftheH iggsbosonwi thbothth eBESandtheISRef fect.Wecan seethere sultingline shapeisnotmerelyaproduc toftwo effect butrather complexcon volution,justifyingnecessity ofournumer- icalevalu ation. Havingunderstood theISRandBESeffectsonthesignal pro- ductionrat esandlineshapes ,wenowprocee dtoundersta ndthe effectonthebackgrou nd.Fo rthemuonc olliderstudy,themain searchchannelsfor theHiggsbosonwillbetheexc lusivemod eof b ¯ bandWW # .Fortheb ¯ bfinalstateth emainbackgroundi sfrom theoff-sh ellZ/!s-channelproduction.Th eISRandBESeffects barelychangetherat efromsuchoff-shell process.H owever,the

ISReffec tdoesincreasetheon -shellZ$b

¯ bbackgroundthrough the" radiativereturn"mechanism.Ournumer icalstudyshowsthat the"r adiativereturn"oftheZbosontob ¯ bincreasetheinclusive b ¯ bbackgroundbyafactorofseven. Sincew eu nderstandthatthe increaseofthebackgroundi sdomin antlyfro mtheon-shellZbo- son,thenewba ckgroundra tesafter imposingab ¯ binvariantmass cutof95,1 00,110 GeV,chang eto17,20,25p b,respect ively.Given thefinit eresolutionof theb-jetenergyr econstruction, wepropose anin variantmasscutoftheb ¯ bsystemof100GeV,whi chlead s toarou nd20%increaseinsu chbackgrou ndcomparingtothetree- levelestimate.S ofarwehavesuggestedtheinvar iantm asscutfor theb ¯ bpair,asanexamp leofdi scrimi nationfromthebackgrou nd. Onecould alsoforeseeacut ontheangleb etweenthetwob-jets, whichcouldb emeasuredmorepre ciselythan theinvariantmass. 2

Beyondtheb

¯ bfinals tate,anothermajorchannelfo rmuoncol- liderHiggsphysic sistheWW # channel.Thischannelenj oyslittle (irreducible)backgroundformtheSMproces s.TheISReffectin- troducesno"radiativ ere turn"forsuchprocess.Consequently,the backgroundratedoesnotchange fromthetree-l evelestimat e.We summarizeinTable2theon-shel lHiggspr oductionratean dba ck- groundrateinthe setwoleading chann elswiththeinclu sionofthe ISRandBES eff ects.We canseefromthetablethatatthemuon colliderHiggsfactory,thes ignalbackgr oundratioisprettylarge andthe obser vabilityissimplydominatedbythest atistics .The "radiativereturn"fromtheISRef fect,however,does impactsev- eralotherHi ggsdecaychannelse archmore .Forexample,sea rches ofHiggsr aredecayo fh$Z!,Hi ggsdecayof h$ZZ # with 2 Wet hanktheEditorGi giRolandi forsuggestingthis discriminationproce dure.

Table2

Signalandbackgr oundeffectiv ecrosssectionsattheresonance ! s=m h =125G eV ataµ + µ " collider(upperpanel,in pb)andane + e " collider(lowerpanel ,in ab)for twochoiceso fbeamenergy resolutionsRandtw oleadingdec aychannels withISRef fectstake nintoaccount,withtheSMbr anchingfractionsBr b ¯ b =58%a nd Br WW #=21%.For theb ¯ bbackground,aconservativecuton the b ¯ binvariantmass tob egreate rthan100GeVisappli ed.

R(%)µ

+ µ " $h " eff (pb) h$b ¯ bh$WW # " Sig " Bkg " Sig " Bkg

0.0110 5.6202.10.0 51

0.00322 124.6

R(%)e + e " $h " eff (ab) h$b ¯ bh$WW # " Sig S/B" Sig S/B

0.0448 27O(10

"6 )10O(10 "3 )

0.011 508131

Table3

Fittingaccuraciesf oronestandarddeviationof#

h ,Bandm h oft heSMHiggs with thesca nningschemefortworepresentativ eluminositiesperstepandtwobench- markbeame nergyspreadpara meters. # h =4.07MeVL step (fb "1 )$# h (MeV)$B$m h (MeV)

R=0.01%0.050 .793.0%0.36

0.20.391.1%0.18

R=0.003%0.050.30 2.5%0.14

0.20.140.8%0.07

Z # $%¯%,etcarefacingmorechallengesandnewselectioncuts needtobedes ignedan dappli ed. Finally,weperformastud yonthe potentialprecisionon the Higgsproper tiesatafuturemuoncolliderthro ughalin es hape scan.Wefo llowthebe nchmarks,statisti caltreat mentandpro- ceduredefinedinRef .[5],wherea 21steps scaninthemas s windowof±30MeVa roundthe Higgsmasswithequalin tegrated luminosities. 3

Afittotheresultofsuchlineshapescancansi-

multaneouslydeterminetheHiggstotalwi dth# h ,th eHiggsmass m h andinter actionstrengthBwithgreatpre cision.Thein terac- tionstrengthBcanbedir ectlyt ranslatedintotheHiggsmuon Yukawaafterfixing thedecaybranch ingfractionsorp erforming aglobalfit.Wetabulatetheprojectedprecisionsonthesequanti- tiesinTable3 forthetwo benchmarkBES value sofR=0.01%and 3 TheHig gsmassmaynotkno wntothe±30Me Vlevelby thetimeofthemuon collider,andapre-scansta getodet erm inetheHiggsmassw illbe require d[30]. Achievable accuracy at the Higgs factory:Good S/B, S/"B !% accuracies~ 3.5% 7 v/E,m t /E,M W /E!0!

!A massless theory: !splitting phenomena dominate!!EW symmetry restored: !SU(2)Lx U(1)Yunbroken gauge theory!v/Eas power corrections !Higher twist effects.J. Chen, TH, B. Tweedie, arXiv:1611.00788;G. Cuomo, A. Wulzer, arXiv:1703.08562; 1911.12366. !EW physics at ultra-high energies:2. A Multi-TeVCollider

v E : v(250GeV)

10Te V

! ! QCD (300MeV)

10Ge V

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Ciafaloniet al., hep-ph/0004071; 0007096; A. Manohar et al., 1803.06347. C. Bauer, Ferland, B. Webber et al., arXiv:1703.08562; 1808.08831.

8 X q p k Figure1.Sc hematicprocessinvolvingac ollinearsplittingA!B+C. thecrosss ectioncanbeexp ressedinafactorized form d! X,BC "d! X,A #dP A!B+C ,(2.1) wherePisthesplittingfunctionforA!B+C.Agi ven splittingcan alsoactasthe"hard" processforlatersplitti ngs,bui ldingupjets .Thefactorizationofcollinearsplittin gsapplies similarlyforinitial-statepar ticles ,leadingtothepictureofpartondistrib utionfunctions (PDFs)foraninitialst ateparton B(orC) d! AB ! !CX "dP A!B+C #d! BB ! !X ,(2.2)

Wewill discussthiss ituationinthenextsecti on.

Integratingouttheazimuthalorient ationofth eB+Csystem,thesplittingk inemat- icsareusual lyparame trizedwithtwovariab les:adimensionfulscaleandadimens ionless energy-sharingvariablez.Th epartonsho werorDGLAPequati onsareconstructedby usingthedimens ionfulscale asanevolutionvariable,thoughthechoiceisnot unique.

Commonchoicesi ncludethetransverse momentumk

T ofBorCrelativetoA'sthree- momentumvector,thevirt ualityoftheo!-shellleg( Aforfinal- stateshowering,BorCfor initial-stateshowering),theenergy-we ightedopeningangleofthesplit,orther enormal- izationscalewithi ndimensionalre gularization.Wewillmai nlyusek T -orderinginwhat follows,thoughwewillalso discusssomer esults withvirtualit y-orderin g.Theenergy- sharingvariablez(¯z$1%z)is commonl ytakentobetheenergyf ractionofAtaken upby B(C).Alter nately,zissomet imesdefinedasthelightconemom entumfraction, z$(E B +"p B

·ˆp

A )/(E A +|"p A |).Here, inpracticewe willus ethethree-momentumfrac tion z$ |"p B | |"p B |+|"p C | ,(2.3) whichgenerallyspan sfromzerotoone,evenina massiveshowe r.Inthere lativisticre gime, wherethecolline arfactorizati onisstrictlyvalid,alloft hesedefinitionsareequivalent. 1 1 Thereisunavo idably someframe-dependencetothissetup,as thereisinall partonshowersthatare definedstrictlyusing collinearapproximatio ns.Amorecomplet etreatmentwouldexhibitmanife stLorentz- -4-

Thesplit tingkinematicsthenbecome

E B !zE A ,E C !¯zE A ,k T !z¯zE A ! BC ,(2.4) where! BC isthe( small)angleb etweenBandC. Inthes implestc ases,generalizingthesplitt ingfunctioncalcul ationstoaccountfor massesisstraightf orward: dP A!B+C (z,k 2 T )" 1 16" 2 z¯z|M (split) | 2 (k 2 T +z¯m 2 +¯zm 2 #z¯zm 2 A ) 2 F(z,k 2 T ;E A )dzdk 2 T .(2.5)

Here,M

(split) istheA$B+Csplittingmatrix-element,whi chcanbecomputedfrom thecorresp ondingamputated1$2Fe ynmandiagramswithon-sh ellpolarizationvec tors (modulogaugeambiguities,w hichwe discusslater).Thismayormaynotbes pin-averaged, dependingonhowmuchinformationi sto bekeptin theshower.We havealsoemployedthe shorthandm%m B forthem assofthefir stdaughterp arti cle(with energy/momentum frac- tionz),and¯m%m C forthem assofthese conddaughter part icle(wit henergy/momentum fraction¯z).Thead ditionalfu nctionFcollectsphasespacefactors thatbecomerel evant inthen onrelativis ticlimit:

COMPUTEME!(2.6)

Insomec aseswherei nterference canbeimportant,dis cussedbelow,thefinalidentityof adau ghtermightnotbeimmed iatelyknown. Inthose cases,w edefaulttochoosingthe smallestpossiblemassvalu e,namelyzerointhecaseof amixed#/Zstate,orm Z inthe caseofamixe dh/Z long state.Thisallows thebroadestposs iblesplittingp hasespace.

Ondime nsionalgrounds,|M

(split) | 2 goeslikee itherk 2 T orsom ecombinati onofthe variousm 2 's.Thesp littingfu nctionsthustypicallyscalelike dk 2 T /k 2 T .Th erearealso mass-dependenttermslikem 2 dk 2 T /k 4 T ,th atleadstot heso-calledultracolline arbehavior. However,theintegratedsp littingrat eatagivenzbecomesasymptotically finiteathigh energies,proportionaltodimen sionlesscombinationsofcouplin gsandmass es,withthe vastmajorit yoftherateconcentratedne arth ek T cuto!.Thise!ectivelyactsasakindof thresholdcorrectionattheend oftheshower.Ineithercase, the remainingzdependence afterintegratin goverk T canbeei therdz/zordz&(regular).Theformeryieldsad ditional softlogarith ms(again,formallyregulatedby theparticlem asses),andappearsonlyi n splittingswhereBorCisagaugeb oson .

2.2Evolu tionequations

Thesplitt ingfunctionsdefinedintheprev ioussectionarerelatedtothep erturbative predictionfortheinitialstaterad iation( ISR)andthu sthepartondistribut ionfunctions invarianceandcontrolofthelow -momentum region,attheexpens eofmore complicat edbook-keepingof theglobal eventstructure,byus ingsuperposition sofdi!erent2!3dipolesplittings.Extendingour treatmentinthismannerisi nprinci plestraightfo rward,butbeyondthesc opeoft heprese ntwork. -5- X AB C X A B C B ! Figure1:SchematicprocessesinvolvingacollinearsplittingA!B+Cine itherthe finalsta te(left)orinitia lstate(right). brokenphase,wherewe introducetheGo ldstoneEquivale nceGauge.Section5explores someofthe consequ encesofel ectroweakshoweringinfinal-stateand initial-state splitting processes,includinginterleavingi ntoQCDshowers.We summarizeandconclu deinSec- tion6.AppendicesgivesupplementarydetailsofGoldstoneEquivalenceGaugeandthe correspondingFeynmanrulesinpracticalcal culations.

2ShoweringPreliminariesandNovelFeatureswithEWSB

Wefir stsummarize thegeneralformalismforthes plittingf unctionsandevolutio nequati ons withmassi veparticlesthatformst hebasisfortherestofthepresentation.Wethenlay outsomeot hernovel featuresdueto EWSB.

2.1Splittingf ormalism

Letusc onsider ageneric"hard"processn omi nallycontainingapa rti cleAinth efinal state,slightlyo!-shellandsubseq uentlyspli ttingtoBandC,asdepictedinFig.1.Inthe limitwhe rethedaughtersBandCarebothap proximate lycollineartotheparentparticle A,thecrosssectioncanbeexpressedinafactorizedform[2] d! X,BC "d! X,A #dP A!B+C ,(2.1) wherePisth esplittingfunctionforA!B+C.Agivensplittingcanalsoactasthe"hard" processforlaterspli ttings,buildi ngupjets.Thefacto rizationofcollinears plitti ngsapplies similarlyforinitial-statep articles, leadingtothepictureofpa rtondis tributionfunctions (PDFs)foranin itialstatepa rtonB(orC) d! AB ! !CX "dP A!B+C #d! BB ! !X .(2.2)

Wewi lldiscussth issituationinthenext subsection.

Integratingouttheazimuthalorie ntationo ftheB+Csystem,thesplitti ngkinemat ics areparame trizedwithtwovariables:adimensi onfulscale(usuallychosentobe approxi- matelycollinear boost-invariant)andadimensi onlessenergy-sharingvariablez.Common choicesforthedimen sionfulva riablear ethedaughtertransversemomentu mk T relativeto -5- thespli ttingaxis,thevirtualityQofth eo!-shellparticlein theprocess,andvariatio nspro- portionaltothedaughters' energy -weighte dopeningangle!E A .Ourdescriptionsherewill mainlyusek T ,asthismakesmoreobviousthecollinearphasespacee!ectsinthepresence ofmas ses.Forournumericalre sults inSect ion5,weswitchtovirtuality,whichallowsfor asimplermatchingontoresonances.Mappingbetweenbetweenanyofthesedi!erentscale choicesishoweverst raight forward.Theenergy-sh aringvariablez(¯z!1"z)iscommonly takentobetheene rgy fractio nofAtakenupbyB(C).T hesplitting kinematicstakes thefor m E B #zE A ,E C #¯zE A ,k T #z¯zE A !.(2.3) Whencons ideringsplittingsinvolvingm assiveorhighlyo!-shellparticl es,variouspossible definitionsofzexistwhichexh ibitdi!erentnon-r elativisticlimits.Besidesstr ictenergy fraction,acommonchoiceisth eli ght-conemoment umfraction,z!(E B + " k B · ˆ k A )/(E A + | " k A |).O urspecific implementationinSection5usesthethre e-mome ntumfractionz! | " k B |/(| " k B |+| " k C |),(Tao)("pto " kallchange d,tobeconsistentt hroughoutthe paper,includingAppendixD, belowEq .D2...)whichmakespha sespacesuppres sion inth enon-relat ivisticlimitsomewhatmoreobvious.However,inth erelat ivisticreg ime, wherethecolli nearfacto rizationisstrictlyvalid,al lofthesedefinitio nsareequivalent,and wedo notpr esentlyma keafurtherdistinction. 1 Inth esimplestca ses,generalizingthe collinearspli ttingfunctioncalculationstoac- countformass esisstraig htforward.Uptot henon -universalandconvention-dependent factorsthatcomeinto playinthenon -relativ isticlimit,thespl ittingfunctionscanbe expressedas dP A!B+C dzdk 2 T $ 1 16# 2 z¯z|M (split) | 2 (k 2 T +¯zm 2 B +zm 2 C "z¯zm 2 A ) 2 .(2.4)

Here,M

(split) ist heA%B+Csplittingmatrix-element, whichcanbecomputedfrom thecorr espondingamputated1%2Feynmandiagramswithon-shellpolarizationvectors (modulogaugeambigu ities,whichwedis cusslater).Thismayormaynotbespin-averaged, dependingonhowmuchinf ormati onisto bekeptint heshower.Dependinguponthe kinematics,themass-dependentfa ctorsinth edenominatoracttoe ithere !ectivelycut o!collineardivergencesatsmal lk T or,infi nal-stat eshowers,topossiblytransi tionthe systemintoareso nanceregion. Inc aseswhereinterferencebetweendi!erentmassei gen- statescanbeimpor tant,this basicf rameworkmustbefurthergeneralized.Resonanceand interferencee!ectsare introducedi nSection2.3.

Ondi mensionalgrounds,|M

(split) | 2 goeslikee itherk 2 T ors omecombinati onofthe variousm 2 's.Con ventionalsplittingfunctionstypicallyscale likedk 2 T /k 2 T ,whichisexhib- itedbyallo fthega ugeand Yukawa splittingsof themasslessunbrokenelectroweak theory, ast obesho wninSe ction3.Therecanalsobemass-dependentsplittingmatrixelements 1 Thereisunavo idably someframe-dependencetothissetup,a sthereisinallpartonshowersthatare definedstrict lyusingcollinearapproximatio ns.Amoreco mpletetreatme ntwouldexhibitmanifestLorentz- invarianceandcontrolofthelow- momentum region,attheexpenseofmorec omplic atedbook-keep ingof theglo baleventstructure,by usingsuperposi tionsofdi!erent2!3dipolesplittings.Extendingour treatmentinthismannerisi nprin ciplestrai ghtforward,butb eyondthescopeofthe present work. -6- !On the dimensional ground: |M split | 2 !k 2 T orm 2

!When SU(2) quantum numbers not summed/averaged,factorized formalism may NOT be valid:!Bloch-NordsiecktheoremviolationCiafaloniet al., hep-ph/0004071; 0007096 C. Bauer, Ferland, B. Webber et al., arXiv:1703.08562; 1808.08831.A. Manohar et al., 1803.06347, J. Chen, TH, B. Tweedie, arXiv:1611.00788. EW splitting physics: EW PDFs & showering

9e.g.: fermion splitting:Start from the unbroken phase -all massless.Infrared & collinear singularities (Pgq)Collinear singularity,Chirality-flip, Yukawa

! ! ! " 1 8! 2 1 k 2 T !

1+¯z

2 z " 1 8! 2 1 k 2 T # z 2 $ !V T f (!) s [BW] 0 T f s H 0(") f -s or" ± f ! -s f s=L,R g 2 V (Q V fs ) 2 g 1 g 2 Y fs T 3 fs y 2 f (!) R

Table1:ChiralfermionsplittingfunctionsdP/dzdk

2 T inth emassless limit,withz(¯z"

1#z)labelingtheenergyfractionofthefirst(second)producedparticle.Thefermion

helicityislabelledbys.Double-arrowsinFeynmandiagramsindicateexamplefermion helicityflows.Primeindica tesisospinpart ner(u ! s =d s ,etc,independentofs).Y ukawa couplingsarelabelledbythep articipati ngRH-helicityfermion.ThestateH 0" ist he"anti- H 0 ",p roducedwhentheRHfermionis down-typeand intheinit ial-state,orup-typein thefina l-state.ProcesseswithB 0 andW 0 implicitlyrepresent therespectivediagonal termsintheneutr algaug eboso n'sdensitymatrix,wh ereas[BW] 0 indicateseitherofthe o!-diagonalterms(seetext).Anti-fe rmionsplittin gsareobtainedbyCPconjugat ion.The conventionsforthecouplingsare giveninC.1. ! " 1 8! 2 1 k 2 T ! (1#z¯z) 2 z¯z " 1 8! 2 1 k 2 T ! z 2 +¯z 2 2 " 1 8! 2 1 k 2 T (z¯z) !W T W T f s ¯ f (!) -s " + " # or H 0 H 0" " + H 0" or " # H 0 V T 2g 2 2 (V=W

0,±

)N f g 2 V (Q V fs ) 21
4 g 2 V 1 2 g 2 2 [BW] 0 T 0N f g 1 g 2 Y fs T 3 fs 1 2 g 1 g 2 T 3 ! + ,H 0 0 Table2:TransversevectorbosonsplittingfunctionsdP/dzdk 2 T int hemassless limit, whereallowed byelectricchargeflow.N f isac olo rmultiplicityfact or(N f =1forleptons, N f =3forquarks).OtherconventionsasinTable1. ! ! 1 8! 2 1 k 2 T !

2¯z

z " 1 8! 2 1 k 2 T ! 1 2 " !V 0 T H[BW] 0 T HW ± T H ! u R ¯u (!) R ¯ d L d (!) L or ¯e L e (!) L H=" + ,H 01 4 g 2 V 1 2 g 1 g 2 T 3 ! + ,H 0 1 2 g 2 2 3y 2 u N d,e y 2 d,e

Table3:ScalarsplittingfunctionsdP/dzdk

2 T int hemassless limitviagaugecouplings andYuk awacouplings.Thesymb olHinth ecolumnhe adingsrepresents theappropriate state" + ,H 0 fortheg ivenspli tting,andH ! representstheSU(2) L isospinpartner(e .g., H 0! =" + ).A nti-particlesplittingsareobtainedbyC Pconjugation.Otherconventionsas inTa bles1and2. -15-

Chiral fermions: fs, gauge bosons: B,W0,W

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