A. Goshaw Physics 846 - Duke Universitywebhome.phy.duke.edu/~goshaw/Lecture20_2017.pdf · 2017-11-07 · A. Goshaw Physics 846 PandaX-II experiment OFHC copper vessel. Surrounding

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A. Goshaw Physics 846

Lecture 20 November 7, 2017

Developing the SM’s electroweak theory

●  Research News

●  Mass generation using the Higgs mechanism Ø  Real scalar fields generating fermion masses TODAY Ø  Complex scalar fields and Goldstone bosons TODAY Ø  Complex scalar fields generating boson masses

●  Complex scalar fields in isospin doublets: the Standard Model’s electroweak theory (the Weinberg- Salam Model)

●  Detection of axions Long Li

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A. Goshaw Physics 846Class Plans

Date Lecture Essay presentation (15 minutes) Nov. 7 20 Long Detection of axions

Nov. 9 21 Baran Neutrino mass from tritium decay Nov. 14 22 Sourav The strong CP problem

Nov. 16 23 Michael Composite Higgs models Nov. 21 24 Ping Family/flavor symmetry in the SM

Nov. 23 Thanksgiving break Nov. 28 25 Nov. 30 26 (last class)

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 News  Recent results from

 dark matter searches

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●  We will hear today from Long about searches for axions – a dark matter candidate that could be part of the SM.

●  The other broad domain of dark matter searches are for Weakly Interacting Massive Particles (WIMPS). These look for the recoil of nuclei from a hit by massive DM particles. ●  In the most recent issue of PRL two experiments report on the most sensitive searches for WIMPS ( see PRL 119, 181301 and 181301 (2017) )

Ø  First Dark Matter Search Results from the XENON1T Experiment Ø  Dark Matter Results from 54-Ton-Day Exposure of PandaX-II Experiment

Research news: dark matter

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S2 transverse position is given by maximizing a likelihoodbased on an optical simulation of the photons producedin the S2 amplification region. The simulation-derivedtransverse resolution is ∼2 cm at our S2 analysis thresholdof 200 PE (uncorrected). The interaction position iscorrected for drift field nonuniformities derived from afinite element simulation, which is validated using 83mKrcalibration data. We correct S2s for electron losses duringdrift, and both S1s and S2s for spatial variations of up to30% and 15%, respectively, inferred from 83mKr calibrationdata. These spatial variations are mostly due to geometriclight-collection effects. The resulting corrected quantitiesare called cS1 and cS2. As the bottom PMT array has amore homogeneous response to S2 light than the top, thisanalysis uses cS2b, a quantity similar to cS2 based on theS2 signal seen only by the bottom PMTs.To calibrate XENON1T, we acquired 3.0 days of data

with 220Rn injected into the LXe (for low-energy ERs),3.3 days with 83mKr injected into the LXe (for the spatialresponse) and 16.3 days with an external 241AmBe source(for low-energy NRs). The data from the 220Rn [19] and241AmBe calibrations are shown in Figs. 2(a) and 2(b),respectively. Following the method described in Ref. [20]with a W value of 13.7 eV, we extracted the photon gaing1 ¼ ð 0.144# 0.007Þ PE per photon and the electron gaing2 ¼ ð11.5# 0.8Þ PE (in the bottom array, 2.86 timeslower than if both arrays are used) per electron in thefiducial mass by fitting the anticorrelation of cS2b and cS1for signals with known energy from 83mKr (41.5 keV), 60Cofrom detector materials (1.173 and 1.332 MeV), and fromdecays of metastable 131mXe (164 keV) and 129mXe(236 keV) produced during the 241AmBe calibration. The

cS1 and cS2b yields are stable in time within 0.77% and1.2%, respectively, as determined by the 83mKr calibrations.WIMPs are expected to induce low-energy single-scatter

NRs. Events that are not single scatters in the LXe areremoved by several event-selection cuts: (1) a single S2above 200 PE must be present and any other S2s must becompatible with single electrons from photoionization ofimpurities or delayed extraction; (2) an event must notclosely follow a high-energy event (e.g., within 8 ms after a3 × 105 PE S2), which can cause long tails of singleelectrons; (3) the S2 signal’s duration must be consistentwith the depth of the interaction as inferred from the drifttime; (4) the S1 and S2 hit patterns must be consistent with

FIG. 1. NR detection efficiency in the fiducial mass at succes-sive analysis stages as a function of recoil energy. At low energy,the detection efficiency (blue line) dominates. At 20 keV, theefficiency is 82%, primarily due to event selection losses (greenline). At high energies, the effect of restricting our data to thesearch region described in the text (black line) is dominant. Theblack line is our final NR efficiency, with uncertainties shown ingray. The NR energy spectrum shape of a 50-GeV=c2 WIMP (ina.u.) is shown in red for reference.

FIG. 2. Observed data in cS2b vs cS1 for (a) 220Rn ERcalibration, (b) 241AmBe NR calibration, and (c) the 34.2-daydark matter search. Solid and dotted lines indicate the median and#2σ quantiles, respectively, of simulated event distributions(with the simulation fitted to calibration data). Red lines showNR (fitted to 241AmBe) and blue ER (fitted to 220Rn). In (c), thepurple distribution indicates the signal model of a 50-GeV=c2

WIMP. Thin gray lines and labels indicate contours of theconstant combined energy scale in keV for (a) ER and (b),(c)NR. Data below cS1 ¼ 3 PE (the gray region) are not in ouranalysis region of interest and are shown only for completeness.

PRL 119, 181301 (2017) P HY S I CA L R EV I EW LE T T ER S week ending3 NOVEMBER 2017

181301-3

VIEWPOINT

The Relentless Hunt for Dark MatterThe latest results from two dark matter searches have further ruled out many theoreticallyattractive dark matter particle candidates.

by Dan Hooper⇤

Dark matter does not emit, radiate, or absorb light,but most models predict that dark matter par-ticles should—on rare occasions—interact withordinary matter. Since the late 1980s, physicists

have been deploying experiments deep underground in aneffort to detect the gentle impacts of individual particles ofdark matter. Over the past fifteen years or so, the collec-tive sensitivity of these experiments has been increasing atan exponential rate, doubling each year or so on average.This staggering rate makes Moore’s Law seem stagnant incomparison.

Two independent experimental collaborations—XENON[1] and PandaX-II [2]—have recently taken the next steps inthis relentless march of progress. The former group has builtthe world’s largest dark matter detector, called XENON1T.It utilizes a 2000-kg target of liquid xenon, housed in a10-m-tall water tank located 1.4 km underground in thelow-background environment of central Italy’s Gran SassoNational Laboratory (see Fig. 1). PandaX-II, by contrast,is located 2.4 km below ground in the China Jinping Un-derground Laboratory in Sichuan, China, and consists of584 kg of liquid xenon. The reason both experiments usexenon is twofold. First, it is highly unreactive, helping tomaintain the required low rate of background events. Sec-ond, its nucleus is relatively high in mass (containing 131nucleons on average), providing a big target for incomingdark matter particles. If one of these particles were to passthrough the Earth and then collide with a xenon nucleus inXENON1T or PandaX-II, that interaction could produce afaint but detectable signal of light (scintillation) and electriccharge (ionization). Observing even a handful of such eventswould put us well on the way to identifying the nature ofthe mysterious substance that makes up our Universe’s darkmatter. But even nondetections constitute progress, in thatthey reveal to us what the dark matter is not.

The recent publications from the XENON and PandaX-IIcollaborations do not claim to have detected any particlesof dark matter, but the lack of such events can be used toplace upper limits on the likelihood—or cross section—withwhich dark matter particles interact with ordinary matter.

⇤Fermi National Accelerator Laboratory, Batavia, IL 60510, USA

Figure 1: A cut-out view of the XENON1T experiment at the GranSasso National Laboratory. The 2-ton liquid xenon target hangs inthe middle of the large water tank on the left. Auxiliary systems arelocated in the three-story building on the right. (Roberto Corrieriand Patrick De Perio)

Because of the unprecedented size of these experiments, thereported limits are the most stringent to date on the darkmatter’s proclivity to interact with nuclei. For a dark mat-ter particle with a mass of 100 GeV, for example, each ofthese collaborations rule out cross sections for such interac-tions that are larger than about 10�46 cm2 per nucleon. Forcomparison, this upper limit is millions of times smaller thanthe cross section predicted for a hypothetical 100-GeV neu-trino interacting with a nucleus through the weak force. Athigher masses, the limits presented by PandaX-II are slightlymore stringent than those from XENON1T, while at lowermasses the opposite is the case. And not too far behindthese two is yet another limit, which was placed last yearby the LUX experiment, located at the Sanford UndergroundResearch Facility in South Dakota [3]. These three collabora-tions—representing the cutting edge in the search for darkmatter— are in a race to build bigger and bigger detectors,with as low of a background as possible.

The lack of a definitive detection of dark matter particles,in both underground experiments and at the Large HadronCollider [4–7], has had a palpable effect on the community

physics.aps.org c� 2017 American Physical Society 30 October 2017 Physics 10, 119

XENON1T experiment

Target is 3200 kg of liquid xenon. Located in Gran Sasso Laboatory 1.4 km under ground.

Detect scintillation light and ionization electrons

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PandaX-II experimentOFHC copper vessel. Surrounding the shield structure is a steel platform that supportsthe cryogenic system and the electronics. The total weight of the passive shield is 93 tons,composed of 12 ton copper, 58 ton lead, 20 ton polyethylene, and 3 ton steel.

The shield was designed with two important requirements: (i) allow less than one neutronor gamma induced background event per year in the energy region useful for dark matterdetection, and (ii) satisfy the space constraint of CJPL, since the space allocation for PandaXwas limited to a length of about 10m, and a width of 3.5m.

FIG. 17: Left panel: Passive shield cross sectional view. Right panel: Top view of the shield.

To satisfy these requirements, the shield was made of a 5 layered structure of, startingfrom the outside, 40 cm polyethylene, 20 cm lead, 20 cm polyethylene, and 5 cm copper. Theinnermost layer is a cylindrical 5-cm thick copper vessel, used both as the last layer of theshielding and as the wall of the cryostat. The cross-sectional and top views of the shield areshown in Fig. 17. The horizontal leveling of the copper vessel is adjustable with two rotarydegrees of freedom with the range 0.25�, used to adjust the leveling between liquid xenonsurface and TPC grids. In the vertical direction, the maximum height of the crane’s liftinghook is 510 cm, which must accommodate the length of the lifting rope (50 cm), the heightof the inner vessel plus the outer vessel, and the height of the shield base ( 90 cm). Thusthe maximum height of the outer vessel is about 185 cm. In the horizontal direction, aftersubtracting the shield thickness and construction space, the diameter of the outer vessel islimited to 135 cm. There are four 6-inch side openings around the outer vessel as shownin Fig. 17, designed for the cooling bus tubes, the signal cables, and the high voltage andcalibration feedthroughs.

Because polyethylene and lead have poor mechanical strengths, a steel structure had tobe installed to support the weight of the shield. To contain radioactivity, most of the steelis used outside the lead shield. Each side of the polyethylene plate is made with 45�-slopededges, and the small gaps between the adjacent plates are tangential to the inner space toprevent neutron leakage.

The shield covers are easily removal for detector installation and maintenance. Theyconsist of one copper cover, one assembly cover, three lead covers and two outer polyethylenecovers. In the cover design, the payload of the crane is an important factor. The assemblycover weighes about 2,800 kg, consisting of an outer top copper plate, an inner polyethylenelayer, and some lead bricks. The lead cover is comprised of a steel box and 14 layers of lead

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Target is 580 kg of liquid xenon. China Jinping Laboratory 2.4 km under ground.

Photomultipliers collect prompt and delayed scintillation photns.

The final candidates in Runs 9 and 10 were combined tosearch for WIMPs. An unbinned likelihood function wasconstructed as

Lpandax ¼!Ynset

n¼1

Ln

"×!GðδDM; σDMÞ

Y

b

Gðδb; σbÞ"; ð2Þ

where

Ln ¼ PoissðNnmeasjNn

fitÞ

×!YNn

meas

i¼1

#Nn

DMð1þ δDMÞPnDMðS1i; S2iÞ

Nnfit

þX

b

Nnbð1þ δbÞPn

bðS1i; S2iÞNn

fit

$": ð3Þ

As in Ref. [3], the data were divided into 14 sets in Run 9,and four sets in Run 10 (nset ¼ 18) to reflect differentoperation conditions in the TPC fields and electron lifetime.For each data set n, Nn

meas and Nnfit represent the measured

and fitted total numbers of detected events;NnDM andNn

b arethe numbers of WIMP and background events, with theircorresponding PDFs Pn

DMðS1; S2Þ and PnbðS1; S2Þ. The

detection efficiencies needed for determining the detectednumbers of events are either contained in the PDF, or

S1 [PE]3 4 5 6 7 8 9 10 20 30 40

(S2/

S1)

10lo

g

1

1.5

2

2.5

3

3 4 5 7 10 15 20 25

30 40 60 90nr

keV

FIG. 4. The distribution of log10ðS2=S1Þ vs S1 for the DMsearch data in Run 10, overlaid with the corresponding median,10% quantile, and 90% quantile of the ER background PDFs. Thered curve is the NR median from the AmBe calibration.

)2WIMP mass (GeV/c10 210 310 410

)2W

IMP-

nucl

eon

cros

s se

ctio

n (c

m

46−10

45−10

44−10

This work

LUX 2017

XENON1T 2017

PandaX-II 2016

(a) log scale in mx

mx

)2WIMP mass (GeV/c1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

)2W

IMP-

nucl

eon

cros

s se

ctio

n (c

m

46−10

45−10

44−10

43−10

This work

LUX 2017

XENON1T 2017

PandaX-II 2016

(b) linear scale in

FIG. 5. The 90% C.L. upper limits versus mχ [(a) log scale,(b) linear scale between 40 GeV=c2 and 10 TeV=c2] for the spinindependent WIMP-nucleon elastic cross sections from thecombined PandaX-II Run 9 and Run 10 data (red), overlaidwith that from PandaX-II 2016 [3] (blue), LUX 2017 [2](magenta), and XENON1T 2017 [4] (black). The green bandrepresents the %1σ sensitivity band.

TABLE III. The best fit total and below-NR-median back-ground events in Run 9 and Run 10 in the FV. The fractionaluncertainties of expected events in the table are 13% (Run 9 ER),20% (Run 10 ER), 45% (accidental), and 50% (neutron),respectively, and are propagated into that for the total fittedevents. The below-NR-median ER background for Run 9 wasupdated using the new ER calibration. The corresponding best fitbackground nuisance parameters [δb’s in Eq. (2)] are 0.123(127Xe), 0.135 (tritium), −0.105 (flat ER), 0.111 (accidental), and−0.098 (neutron). The number of events from the data are shownin the last column.

ER Accidental NeutronTotalFitted

TotalObserved

Run 9 376.1 13.5 0.85 390% 50 389Below NRmedian

2.0 0.9 0.35 3.2% 0.9 1


0.9 0.6 0.33 1.8% 0.5 0


181302-5

7


The final candidates in Runs 9 and 10 were combined tosearch for WIMPs. An unbinned likelihood function wasconstructed as

Lpandax ¼!Ynset

n¼1

Ln

"×!GðδDM; σDMÞ

Y

b

Gðδb; σbÞ"; ð2Þ

where

Ln ¼ PoissðNnmeasjNn

fitÞ

×!YNn

meas

i¼1

#Nn

DMð1þ δDMÞPnDMðS1i; S2iÞ

Nnfit

þX

b

Nnbð1þ δbÞPn

bðS1i; S2iÞNn

fit

$": ð3Þ

As in Ref. [3], the data were divided into 14 sets in Run 9,and four sets in Run 10 (nset ¼ 18) to reflect differentoperation conditions in the TPC fields and electron lifetime.For each data set n, Nn

meas and Nnfit represent the measured

and fitted total numbers of detected events;NnDM andNn

b arethe numbers of WIMP and background events, with theircorresponding PDFs Pn

DMðS1; S2Þ and PnbðS1; S2Þ. The

detection efficiencies needed for determining the detectednumbers of events are either contained in the PDF, or

S1 [PE]3 4 5 6 7 8 9 10 20 30 40

(S2/

S1)

10lo

g

1

1.5

2

2.5

3

3 4 5 7 10 15 20 25

30 40 60 90nr

keV

FIG. 4. The distribution of log10ðS2=S1Þ vs S1 for the DMsearch data in Run 10, overlaid with the corresponding median,10% quantile, and 90% quantile of the ER background PDFs. Thered curve is the NR median from the AmBe calibration.

)2WIMP mass (GeV/c10 210 310 410

)2W

IMP-

nucl

eon

cros

s se

ctio

n (c

m

46−10

45−10

44−10

This work

LUX 2017

XENON1T 2017

PandaX-II 2016

(a) log scale in mx

mx

)2WIMP mass (GeV/c1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

)2W

IMP-

nucl

eon

cros

s se

ctio

n (c

m

46−10

45−10

44−10

43−10

This work

LUX 2017

XENON1T 2017

PandaX-II 2016

(b) linear scale in

FIG. 5. The 90% C.L. upper limits versus mχ [(a) log scale,(b) linear scale between 40 GeV=c2 and 10 TeV=c2] for the spinindependent WIMP-nucleon elastic cross sections from thecombined PandaX-II Run 9 and Run 10 data (red), overlaidwith that from PandaX-II 2016 [3] (blue), LUX 2017 [2](magenta), and XENON1T 2017 [4] (black). The green bandrepresents the %1σ sensitivity band.

TABLE III. The best fit total and below-NR-median back-ground events in Run 9 and Run 10 in the FV. The fractionaluncertainties of expected events in the table are 13% (Run 9 ER),20% (Run 10 ER), 45% (accidental), and 50% (neutron),respectively, and are propagated into that for the total fittedevents. The below-NR-median ER background for Run 9 wasupdated using the new ER calibration. The corresponding best fitbackground nuisance parameters [δb’s in Eq. (2)] are 0.123(127Xe), 0.135 (tritium), −0.105 (flat ER), 0.111 (accidental), and−0.098 (neutron). The number of events from the data are shownin the last column.

ER Accidental NeutronTotalFitted

TotalObserved


2.0 0.9 0.35 3.2% 0.9 1


0.9 0.6 0.33 1.8% 0.5 0


181302-5

similar to Ref. [23]. Isolated S1s may arise from inter-actions in regions of the detector with poor chargecollection, such as below the cathode, suppressing anassociated cS2 signal. Isolated S2s might arise fromphotoionization at the electrodes, from regions with poorlight collection, or from delayed extraction [24]. Mostaccidental events are expected at low cS1 and at lower cS2bthan at typical NRs.Fifth, inward-reconstructed events from near the TPC’s

polytetrafluoroethylene wall are expected to contributeð0.5" 0.3Þ events, with the rate and (cS1, cS2b) spectrumextrapolated from events outside the fiducial mass. Most ofthese events would appear at unusually low cS2b due tocharge losses near the wall. The inward reconstructionis due to limited position reconstruction resolution, limitedespecially for small S2s, near the 5 (out of 36) top PMTsin the outermost ring that are unavailable in thisanalysis.Sixth and last, we add a small uniform background in the

(cS1, log cS2b) space for ER events with an anomalouscS2b. Such anomalous leakage beyond accidental coinci-dences was observed in XENON100 [23], and one suchevent is seen in the 220Rn calibration data [Fig. 2(a)]. Ifthese were not 220Rn-induced events, their rate would scalewith exposure and we would see numerous such events inthe WIMP search data. We do not observe this andtherefore assume their rate is proportional to the ER rate,at 0.10þ0.10

−0.07 events based on the outliers observed in the220Rn calibration data. The physical origin of these events isunder investigation.The WIMP search data in a predefined signal box were

blinded (99% of ERs were accessible) until the eventselection and the fiducial mass boundaries were finalized.We performed a staged unblinding, starting with anexposure of four live days distributed evenly throughoutthe search period. No changes to either the event-selectionor background types were made at any stage.A total of 63 events in the 34.2-day dark matter search

data pass the selection criteria and are within the cS1 ∈½3; 70& PE, cS2b ∈ ½50; 8000& PE search region used in thelikelihood analysis [Fig. 2(c)]. None are within 10 ms of amuon-veto trigger. The data are compatible with the ERenergy spectrum in Ref. [9] and implies an ER rate ofð1.93" 0.25Þ × 10−4 events=ðkg × day × keVeeÞ, compat-ible with our prediction of ð2.3" 0.2Þ × 10−4 events=ðkg × day × keVeeÞ [9] updated with the lower Kr concen-trationmeasured in the current science run. This is the lowestER background ever achieved in such a dark matter experi-ment. A single event far from the bulk distribution wasobserved at cS1 ¼ 68.0 PE in the initial 4-day unblindingstage. This appears to be a bona fide event, though itslocation in ðcS1; cS2bÞ [see Fig. 2(c)] is extreme for allWIMP signal models and background models other thananomalous leakage and accidental coincidence. One event atcS1 ¼ 26.7 PE is at the −2.4σ ER quantile.

For the statistical interpretation of the results, we use anextended unbinned profile likelihood test statistic in (cS1,cS2b). We propagate the uncertainties on the most signifi-cant shape parameters (two for NR, two for ER) inferredfrom the posteriors of the calibration fits to the likelihood.The uncertainties on the rate of each background compo-nent mentioned above are also included. The likelihoodratio distribution is approximated by its asymptotic dis-tribution [25]; preliminary toy Monte Carlo checks showthat the effect on the exclusion significance of this conven-tional approximation is well within the result’s statisticaland systematic uncertainties. To account for mismodelingof the ER background, we also calculated the limit usingthe procedure in Ref. [26], which yields a similar result.The data are consistent with the background-only hypoth-

esis. Figure 4 shows the 90% confidence level upper limit onthe spin-independent WIMP-nucleon cross section, powerconstrained at the−1σ level of the sensitivity band [29]. Thefinal limit is within 10% of the unconstrained limit for allWIMP masses. For the WIMP energy spectrum, we assumea standard isothermal WIMP halo with v0 ¼ 220 km=s,ρDM ¼ 0.3 GeV=cm3, vesc ¼ 544 km=s, and the Helmform factor for the nuclear cross section [30]. No light orcharge emission is assumed for WIMPs below 1-keVrecoil energy. For all WIMP masses, the background-onlyhypothesis provides the best fit, with none of the nuisanceparameters representing the uncertainties discussed abovedeviating appreciably from their nominal values. Ourresults improve upon the previously strongest spin-independent WIMP limit for masses above 10 GeV=c2.Our strongest exclusion limit is for 35-GeV=c2 WIMPs, at7.7 × 10−47 cm2.These first results demonstrate that XENON1T has the

lowest low-energy background level ever achieved by a

FIG. 4. The spin-independent WIMP-nucleon cross sectionlimits as a function of the WIMP mass at 90% confidence level(black line) for this run of XENON1T. In green and yellow are the1σ and 2σ sensitivity bands. Results from LUX [27] (the red line),PandaX-II [28] (the brown line), and XENON100 [23] (the grayline) are shown for reference.


181301-5

WIMP limits vs mass

Limits are placed on WIMP masses between about 10 GeV to 10 TeV. Below 10 GeV searches at the LHC place (model dependent) limits.

Assume moving through a DM field of characteristics deduced from astronomy gravitational measurements: density 0.3 GeV/cm3 and velocity 220 km/s.

8


 The Higgs mechanism:  Generating fermion masses

 using a real scalar field

9


The Higgs mechanism: fermions

●  The purpose of this discussion is to show how the introduction of a scalar field with particular properties can “induce” a mass term into the fermion field equation.

●  Start with a free real scalar (spin 0) field φ of mass m described by the standard Lagrangian density (units of energy/length3 ).

●  Introduce a quartic self-interaction of the field with dimensionless coupling strength λ. This is not as arbitrary as it might seem. The requirement of renormalizability restricts the choices.

  φ

 λ

L� = 12@µ�@

µ� - 12 [

mc~ ]2�2

The field � has unitsq

energylength

10



●  The Lagrangian then becomes:

●  Using the Euler-Lagrange equation the field equation is

(showing why the choice of factors of ½ and ¼ in the Lagrangian)

 kinetic energy  mass  field self-coupling

L� = 12@µ�@

µ� - 12 [

mc~ ]2�2 - �

4(~c)�4

●  Since the Lagrangian = K – V, the potential V associated with this field is: where again the units are energy/length3 .

V = 12 [

mc~ ]2�2 + �

4(~c)�4

@µ@µ� + [mc~ ]2� + �

~c�3 = 0

11


●  Some observations:

2 Parameters in this model are: m = mass of the scalar particle φ particle and λ = the φ self coupling 3. The Lagrangian for φ is invariant under φ -> -φ

1. The potential for φ has the form

4. The value of φ that minimizes V is quite naturally called the vacuum expectation value (vev) of φ . Let’s denote this by φvev . Define a quantity “v” with units of energy that specifies the vev of the potential:

Quite trivially in this case v = 0.


V = 12 [

mc~ ]2�2 + �

4(~c)�4

v =p~c �vev

12



●  Assume this scalar field couples to fermions. The coupling term will look like those we have previously introduced, but will be simpler since this is a scalar field.

●  This introduces more parameters “gf”, the coupling strength of φ to each fermion.

●  If a scalar field of this type existed in Nature, then this model could be used to make predictions that depend on the 3 parameters m (the scalar boson mass), λ, (the scalar boson self coupling), and gf (the scalar boson coupling to fermions). An interesting exercise, but this model does not “induce” mass terms for the fermion field.

where the fermion mass has been set = 0.

7


Lf = i(~c) �µ@µ - gf

p~c �

13



●  Next explore the consequences of modifying the potential in this example in a seemingly crazy way. Let mc2 = i µ . Clearly the “m” term would no longer represents a mass. The Lagrangian has simply been modified in an ad hoc manner.

●  The potential is described by two parameters: µ2 > 0 (with units energy2) and λ > 0 (dimensionless).

●  This mathematical manipulation is at the heart of the Higgs Mechanism.

●  This describes a massless scalar field with potential

●  The modified Lagrangian becomes:

V = - µ2

2(~c)2�2 + �

4(~c)�4

L0� = 1

2@µ�@µ� + µ2

2(~c)2�2 - �

4(~c)�4

14



●  This model for the field φ is symmetric under the change φ è -φ

●  However now break the symmetry by arbitrarily choosing one of the two potential minima as the vacuum expectation value of the scalar field. That is, assume the field φ gets “stuck” in one of the potential minima.

For example choose

Examine a model where there are field variations about this potential minimum.

●  The modified potential is of the form The vacuum expectation value is:

v =p~c �vev = ±

pµ2/�

21


V (�)

�

µ2 > 0µ2= 0

qµ2

(~c)�-q

µ2

(~c)�

�vev = +q

µ2

(~c)�

=) µ2 = �v2

15


●  Re-express the field φ(x) relative to the vacuum minimum by introducing a scalar field H(x) where:

●  Now re-write L f and L φ for the mass-less fermion and scalar boson


Lf = i(~c) �µ@µ - gfp~c �

L0� = 1

2@µ�@µ� + µ2

2(~c)2�2 - �

4(~c)�4

�(x) = �vev +H(x) = vp~c +H(x)

in terms of H(x).

16



●  So far this is just a mathematical game. Obviously it was done originally with an eye on the result you wanted to get, namely the generation of masses for fermions and the scalar boson field H(x).

●  So perhaps not surprisingly this is exactly what happens when you make the substitution ●  You should verify the result on the next page (also see HW4).

�(x) = vp~c +H(x)

17



●  These equations describe a massive fermion and a massive scalar particle with the usual form of the Lagrangians expected for these particles. ●  Use the general fermion and scalar Lagrangians to identify.

Lf = i(~c) �µ@µ - (gfv) - gfp~c H

LH = 12@µH@µH - �v2

(~c)2H2 - �v

(~c)3/2H3 - �

4(~c)H4 + �v4

4(~c)3

mfc2 = gfv

mHc2 =p2�v2

Substitute �(x) = vp~c + H(x) and µ

2 = �v2.

18


●  The two Lagrangians also specify the coupling of the Higgs field to the fermions and the self-coupling of the Higgs field.

●  The 3 parameters of the model ( µ2, gf, and λ ) can be replaced by the fermion masses mf and the vacuum expectation value v of the Higgs field. Then the model is completely predictive given

mf mH v


7


 ∼ λ v  ∼ λ  gf

19


The Higgs mechanism: fermions Taking Stock

●  At the time in history when these ideas were circulating, there was no evidence for this scalar field. Now of course we know a Higgs boson exists with the requisite properties.

●  The one parameter remaining at this stage of the development of the theory is the vacuum expectation value of the field:

v =p~c �vev

●  Let’s anticipate that in the full theory this will be determined. It has the shockingly large value of about 250 GeV.

20


The Higgs mechanism: fermions Taking Stock

●  With the known value of the Higgs boson mass, and the yet-to-be-determined value of v, the properties of the first generation of SM fermions are described as follows.

particle mass gf couplings u 2.2 MeV 8.8x10-6 gs , gw , e d 4.7 MeV 1.9x10-5 gs , gw , e νe < 2 eV < 8x10-12 gw

e- 0.51 MeV 2.0x10-6 gw , e H 125 GeV ---- gf

With Higgs field parameters: v = 250 GeV µ = 88 GeV λ = 0.125

21


 Complex scalar fields  and Goldstone bosons

22


Complex scalar fields and Goldstone bosons

●  By introducing different coupling strengths of the Higgs field to each fermion, the procedure just described can be used to generate all the fermion masses.

●  But the model must be modified to allow for the flexibility needed for the electroweak bosons W, Z and γ .

●  The next step is to generalize to a complex scalar field. This will allow for the generation of boson masses for Abelian vector fields Still not what we want but a useful pedagogical step to the final solution needed for the SM EWK theory.

●  The results already obtained for fermion mass generation will not change from those already derived.

23



●  where φ1 and φ2 are real scalar fields.

●  Then if we maintain the same form for the potential as used in for a single real scalar field:

(this reduces to the L φ on page 10 if φ2 = 0)

●  This has more symmetry than the reflection symmetry for the real field. It is invariant under a global phase transformation:

●  Now repeat the steps used for the Higgs mechanism with a real scalar field, but use this complex scalar field

� ! �0= exp(i↵0)� =) a global U(1) symmetry.

Let � = (�1 + i�2)/p2

L� = @µ�@µ�⇤ - [mc~ ]2��⇤ - �

~c (��⇤)2

24



●  Again, change the form of the potential by letting mc2 = i µ . Also express the result in terms of the field components φ1 and φ2 .

●  Now the minima of the potential term occurs at

L0� = 1

2@µ�1@µ�1 + 12@µ�2@µ�2 + µ2

2(~c)2 (�21 + �2

2) -�

4(~c) (�21 + �2

2)2

(~c)(�21 + �2

2)vev = µ2/� = v2

Therefore in a plane of

p~c �1 vs

p~c i�2

the potential minima are on a circle

of radius

qµ2

� = v

p~c �1

p~c i �2

 Higgs potential V

25



Break the U(1) symmetry by choosing a point P on the potential minimum circle to define the field vacuum expectation value.

●  Expand the field φ(x) about the minimum point φvev 33


p~c �1

p~c i�2

v

minimum

of V (�)

✓P

P

p~c �vev = (v cos✓P + i v sin✓P )/

p2

�(x) = �vev +H(x)where we chose to write H(x) = [⌘(x) + i⇢(x)]/

p2

Therefore

� = [(

vp~c cos✓P + ⌘) + i (

vp~c sin✓P +⇢)]/

p2

26



●  Following the procedure used for a single real scalar field. Substitute the expression for φ(x) expanded about point P back into the Lagrangian:

L0� = 1

2@µ�1@µ�1 + 12@µ�2@µ�2 + µ2

2(~c)2 (�21 + �2

2) -�

4(~c) (�21 + �2

2)2

●  The result can not depend on which point P is chosen, so take P on the real field axis at (v + i 0). That is make the substitution:

�1 = [ vp~c + ⌘(x)] and �2 = ⇢(x)

27



●  After some manipulation the result is:

LH = 12@µ⌘@

µ⌘ + 12@µ⇢@

µ⇢

- �v2

(~c)2 ⌘2 - (0)⇢2

- �4(~c)⌘

4 - �v(~c)3/2 ⌘

3

kinetic energy terms

mass terms

triple/quartic η field couplings

+ terms involving the ρ field couplings.

( note that if the ρ(x) = 0 this is identical to the result on page 17)

●  This Lagrangian describes two scalar fields

1. A field η(x) with mass as found previously. and called a Higgs boson. 2 . An additional field ρ(x) of mass 0, called a Goldstone boson.

m⌘c2 =p2�v2

28



●  Why is the ρ field of mass zero and the η field not? This arises from the symmetry breaking introduced by the choice of the vacuum reference v +i 0 . The potential well is ~ parabolic when expanded in η about v, but flat when expanded in ρ about v. Other choices of the point to break the global U(1) symmetry will lead to the same result: one massive scalar field and one mass-less field.

●  This result is a general phenomena when a continuous global symmetry is broken. This is stated in a theorem by Goldstone: “Breaking of a continuous global symmetry is always accompanied by the appearance of one or more massless scalar (spin 0) bosons.”

29



●  In the process of trying to extend the flexibility of the Higgs mechanism we have conjured up a disaster. No massless scalar fields have been observed experimentally.

●  The good news is that this “problem” is the solution to generation of masses for the vector bosons W+ , W- and Zo .

●  When the above analysis of a complex scalar field is extended by demanding a local gauge symmetry (such as Ug(1) = exp[i g α(x)] ), and combined with coupling to a mass-less vector field, a mass term is generated for the vector fields.

●  The goldstone boson is absorbed and appears as a transverse spin (longitudinal polarization) of vector bosons.

30


End Lecture 20 Next: The generation of boson masses

31


32


V (�)

�

µ2 > 0µ2= 0

qµ2

(~c)�-q

µ2

(~c)�

33


p~c �1

p~c i�2

v

minimum

of V (�)

✓P

P

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A. Goshaw Physics 846 - Duke Universitywebhome.phy.duke.edu/~goshaw/Lecture20_2017.pdf · 2017-11-07 · A. Goshaw Physics 846 PandaX-II experiment OFHC copper vessel. Surrounding