THE HIGGS, VACUUM STABILITY, and OTHER CONSTRAINTS Matthew Reece Harvard University at the PCTS, April 26, 2013 THIS TALK

THE HIGGS, VACUUM STABILITY, AND OTHER CONSTRAINTS Matthew Reece Harvard University at the PCTS, April 26, 2013 THIS TALK

First, I’m going to explain some constraints on new physics scenarios that try to signiﬁcantly alter Higgs properties. The motivation for dwelling on this is diminished given recent CMS updates on diphoton.

Then I’ll talk a bit about Higgs vacuum stability in the Standard Model. LHC: WHERE WE STAND I looked over my slides from my PCTS talk almost exactly one year ago and still agree with what I said then:

•No hint of strong dynamics. Technicolor, composite Higgs, Randall-Sundrum all look even less plausible than they did pre-Higgs. •Higgs mass puts SUSY in an awkward spot •“Natural SUSY”: the MSSM must be very tuned to ﬁt a 125 GeV Higgs (despite many contrary claims in the literature in the last year). Models beyond the MSSM are typically awkward and/or complicated. •Semi-split SUSY: solve most of the hierarchy problem, put scalars at ~10 to 1000 TeV. Increasingly plausible. •Important to keep looking for small deviations from SM Higgs •Keep looking for naturalness signatures (stops, etc.), but bounds are already becoming very strong... HIGGS PROPERTIES

Looking increasingly Standard Model-like as more data comes in. Still room for 30 to 50% (large!) deviations! Let’s say, for instance, thatm ˜ D dominates, and is 1 TeV, with ⌅ =0.1, Mmess = 100 TeV (which is consistent with F 100 TeV as well), m = 2 TeV, and m = 150 GeV. Then ⇥ Y S we have: Let’s say, for instance, thatm ˜ D dominates, and is 1 TeV, with ⌅ =0.1, Mmess = 100 TeV Let’s say, for instance, thatm ˜ dominates, and is 1 TeV, with ⌅ =0.1, M = 100 TeV (which is consistentD with F 100 TeV as well), m = 2 TeV,mess and m = 150 GeV. Then 2 2 Y 2 S (which is consistent with F 100m TeV˜ = as⇥ well),2970m GeVY = 2 TeV, andm mS == 150150 GeV. Then2970 GeV 140 GeV. (2.4) we have: ⇥ S scalar we have: ⇧ ⇤ Let’s say, for instance, thatm ˜ D dominates, and is 1 TeV, with ⌅ =0.1, Mmess = 100 TeV ⇥ So, this2 model is easily2 compatible with2 the sort of splittings we’re interested in. (which is consistent with F 100 TeV as well), mY = 2 TeV,2 and mS =m˜ 150S = GeV.2 2970 Then GeV m2scalar = 150 2970 GeV 140 GeV. (2.4) Let’s say, for instance, thatm ˜ D dominates,⇥ and is 1 TeV, with ⌅ =0m˜.1,S M=mess2970= 100 GeV TeV mscalar =⇧ 150 2970 GeV 140 GeV. ⇤ (2.4) we have: ⇧ ⇤ (which is consistent with F 100 TeV as well), mY = 2 TeV, and mS = 150 GeV. Then ⇥ ⇥ Let’s say, for instance,So, that thism ˜ D modeldominates,2.2 is Couplings easily and is compatible 1 TeV, to with Gauge⇥⌅ with=0.1, the BosonsMmess sort= of 100 and splittings TeV Gauginos we’re interested in. we have: 2 2 So, this2 model is easily compatible with the sort of splittings we’re interested in. m˜ = 2970 GeV mscalar(which= is150 consistent2970 with GeVF 140100 GeV TeV. as well), m(2.4)Y = 2 TeV, and mS = 150 GeV. Then S ⇧ ⇤⇥ ¯ 2 2 we have:2 Now we will integrate out Y and Y and study the induced couplings of S to gauge bosons. m˜ S = 2970 GeV mscalar = 150 ⇥2.22970 Couplings GeV 2.2140 Couplings toGeV Gauge. Bosons(2.4) to Gauge and Bosons Gauginos and Gauginos So, this model is easily compatible⇧ with the sort of splittings⇤ we’re interested in. 2 2 To keep things2 simple, we will assume that the mass of the Y s is dominantly supersymmetric. ⇥ m˜ S = 2970Now GeV we will integratemscalar = ¯ out150 Y 2970and GeVY¯ and140 study GeV. the induced(2.4) couplings of S to gauge bosons. So, this model is easily compatible with the sort of splittingsNow we’re we will interested integrate in.⇧In out thatY and case,Y and we expect study the to⇤ induced find an couplings e⇤ective of operatorS to gauge of bosons. the form: 2.2 Couplings to Gauge Bosons and Gauginos ⇥ So, thisTo model keep is easily thingsTo compatible keep simple, things we with will simple, the assume sort of we splittings that will the assume we’re mass interested that of the theY in.s mass is dominantly of the Y s supersymmetric. is dominantly supersymmetric. 2.2 Couplings to Gauge Bosons and Gauginos Now we will integrate out Y and Y¯ and study theIn induced that case, couplingsIn we that expect case, of S to weto find gauge expect an bosons. e⇤ toective find operator an e⇤ective of the operator form:c of the2 form: Now we will integrate out Y and Y¯ and study the2.2 induced Couplings couplings to Gauge of S Bosonsto gauge and bosons. Gauginos d ⇤SWW + h.c. (2.5) To keep things simple, we will assume that the mass of the Y s is dominantly supersymmetric.c mY To keep things simple, we will assume that the massNow of we the willY s integrate is dominantly out Y supersymmetric.and Y¯ and study the induced2 couplingsc of S2 to gauge bosons. In that case, we expect to find an e⇤ective operator of the form: d ⇤SWW +dh.c.⇤SWW + h.c. (2.5) (2.5) In that case, we expect to find an e⇤ective operatorTo keep of the things form: simple, we will assume that the massmY of the Y s ism dominantly supersymmetric. Let’s work out the coe⌃Ycient c and then compute the resulting decays. c 2In that case, we expect to findHIGGS an e⇤ective LOOP-LEVEL operator of the form: c d ⇤SWWLet’s+ workh.c. out the coe⌃cientToc computeand then(2.5) compute the coe⌃ thecient, resulting let me decays. try to flesh out JiJi’s statement that it’s related to a md2⇤SW W + h.c. Let’s work out the(2.5) coe⌃cient c and then compute the resulting decays. Y c 2 mY To compute the coe⌃cient,d ⇤SW letW me+ tryh.c. to flesh out JiJi’s statement(2.5) that it’s related to a To computembetaCOUPLINGS function the coe ⌃ coecient,⌃cient. let me Suppose try to flesh we out have JiJi’s a Lagrangian statement that it’s related to a Let’s work out the coe⌃cient c and then computebeta the resulting function decays. coe⌃cient.Y Suppose we have a Lagrangian Let’s work out the coe⌃cient c and then compute the resulting decays.beta function coe⌃cient. Suppose we have a Lagrangian To compute the coe⌃cient, let me tryLet’s to flesh work out out JiJi’s the coe statement⌃cient c and that then it’s compute related the to resulting a decays. 1 To compute the coe⌃cient, let me try to flesh out JiJi’s statementA lot of interest that it’s in related this when to a both ATLAS1 and CMS a aµ⇧ To compute the coe⌃cient, let me try to flesh out JiJi’s statementa aµ that⇧ it’s related= to a 2 Gµ⇧G . (2.6) beta function coe⌃cient. Suppose we have a Lagrangian = Gµ⇧G . 1 a L aµ⇧4g (2.6) beta function coe⌃cient. Suppose we have a Lagrangianbeta function coeshowed⌃cient. a Suppose high diphoton we have a rate. Lagrangian Recall:L Higgs4g2 couplings= to G G . (2.6) photons, gluons related to low-energy theorem.L 4g2 µ⇧ 1 1 a aµ⇧ = a aµG⇧ G . Then in1 facta (2.6)gaµ⇧is a running coupling. In particular, suppose that we have fields at a scale M, = Gµ⇧G2 µ.Then⇧ in fact g is a running= coupling.(2.6)G G In. particular, suppose that we(2.6) have fields at a scale M, L L4g2 4g 2 µ⇧ with µ

2 In our case, we have fermions with mass mY +⌅S(x) and scalars with mass-squared2 mY + ⌅S(x) , In our case, we have fermions with mass mY +⌅S(x)In and particular, scalars with if M mass-squared( x ) depends onm Ythe+ Higgs,⌅S⇥(Mx()x )M , =⇤ S ( Mx ) ( h ( |x )) , | implying that to leading order for⇥M all(x) the fields⇤S(x we) integrate| out, | = . The contribu- implying that to leading order for all the fields we integrate out, = . The contribu-M mY tion of the chiralthen multiplets we extractY,Y¯ toM thean betaeffectivemY function coupling: of SU(3) is that of one supersymmetric ¯ tion of the chiral multiplets Y,Y to theflavor beta for function SQCD, i.e. ofb SU(3)= +1, so is wethat haveb of a one coupling: supersymmetric⇥ log M(v) hGa Gaµ flavor for SQCD, i.e. b = +1, so we have a coupling: 2 µ 32 ⌅ a aµ⇧ ⇥v 2 SGµ⇧G . (2.9) 32⌃ mY ⌅ a aµ⇧ 2 SGµ⇧G . (2.9) 32If⌃ Im thenY canonically normalize my gluon fields and change their name to F , this becomes:

⌅s a aµ⇧ If I then canonically normalize my gluon ﬁelds and change their name to FSF,µ this⇧F becomes:. (2.10) 8⌃mY

⌅s a aµ⇧ SFµ⇧F . (2.10) 8⌃mY –2–

–2– STOPS Things to note: 2 2 2 2 2 1 ⇥ log det Mt˜ m˜ Q +˜mu Xt sin ytmt 2 ⇥v ⇠ m˜ 2 m˜ 2 X2m2 sin2 Q u t t Small numerator factor Minus sign: large mixing (for heavy stops): leads to opposite-sign decoupling couplings Intuition: in the highly mixed case, larger VEV means more mixing, splitting light and heavy stops more. The light one contributes more, and is pushed lighter, so the overall sign reverses. MODIFIED GLUON AND PHOTON COUPLINGS

7 3, Q=2 3 Many papers last year: new 3, Q=1 3 6 3, Q=5 3 8, Q=0+1 particles with color and 6, Q=4ê3 6, Q=2ê3 5 ê¯: Best Fit charge running in loops can L SM H ê alter Higgs couplings to gg ê

Æ 4 h G

ê gluons and photons. gg Æ

h 3 G

2 ¯ Opposite-sign gluon coupling helps? 1

0.0 0.5 1.0 1.5 GhÆgg GhÆgg SM MR, 1208.1765: No.

H L Figure 2: Fit of the WW, ZZ, and channels in the ATLASê and CMS 7+8 TeV data, with 1 and 2 contours as in the left-hand plot of Figure 1, but now showing the values achieved by adding particles in the loop in a variety of representations of SU(3)c and U(1)EM.

In particular, because there are vectorlike masses that are split by the mixing terms proportional to the br Yukawas, we get a negative contribution to the amplitude. The factor is the ratio of the SU(3)c beta- b3 function coefficient of the representation that and transform under, relative to the beta-function coefficient of a triplet. Loops of fermions contribute a correction to the Higgs quartic, which in the special case m = m = m is: 2 Nc;F 4 4 m 1 2 2 2 F = y + y log + 5y 2y1 y2 + 5y y1 + y2 . (9) 16⇡2 1 2 µ2 6 1 2 ® ´ Ä ä Ä ä The result in the more general case m = m is listed in Appendix A. Notice that the logarithmic term here can be interpreted as encoding a beta6 function coefficient. Because the full renormalized potential must be independent of µ, the tree-level quartic must run in such a way as to cancel the µ-dependence of the Coleman-Weinberg potential.

2.3 New scalar states Assume a mass matrix 2 2 2 m1 + 1 v Av S = 2 2 . (10) Av m + 2 v M Ç 2 å The correction to the h gg amplitude relative to the Standard Model amplitude is ! 2 2 2 2 2 A(hGG) br v 1m2 + 2m1 + 212 v A , (11) A hGG = b 2 2 2 2 2 2 SM( ) 3 4 Äm1 + 1 v m2 + 2 v A vä ÄÄ äÄ ä ä 5 3.1 Inverting hGG with Stops Given that we are looking for large changes to the Higgs potential that require light new colored and charged particles, it is reasonable to ﬁrst consider whether stops can be responsible, since naturalness of electroweak symmetry breaking in supersymmetric theories favors light stops [29, 30]. In the case of stops, the general results discussed in the previous section imply a correction to the hGG amplitude (specializing the general result Eq. 11):

2 2 2 2 A(hGG) 1 mt mt mt X t = 1 + 2 + 2 2 2 , (14) ASM(hGG) 4 m m m m 0 ˜t1 ˜t2 ˜t1 ˜t2 1 up to small D-term corrections (taken into account@ in the plots below).A Here m and m are mass ˜t1 ˜t2 eigenvalues, not Lagrangian parameters. The effect of stops on Higgs branching ratios has been discussed insplitting several papers between in the the two recent stop literature mass eigenstates.[6,17,18,24,39 In this region], which of parameter reach a variety space, of the conclusions. lightest stop eigenvalue tends to be fairly light. For example, pushing the light eigenstate up to 450 GeV implies 20 As emphasized by Ref. [39], light unmixed stops tend to increase the hGG coupling and decrease the h coupling,TeV A-terms, whereas which highly is an mixed enormously stops finely-tuned contribute large scenario, corrections both from to them point2 (thus of view requiring of electroweak more symmetry breaking and of b s. In fact, from the Coleman-Weinberg discussionHu in Section 2.3, one tuning for EWSB) and lead to large corrections to b s that must also be tuned away. The same can readily see that such large! A-terms lead to very large negative threshold corrections to the Higgs considerations led Ref. 24 to focus on the “funnel” region! in which the stop corrections to hGG are Whatmass. This it implies takes[ the] need to for get very large hGG beyond-MSSM coupling couplings equal of the Higgs and boson thatopposite are capable to small. Onof the lifting other its hand, mass up Refs. to 125[18, GeV. 22] Whenargued such for couplings light and become highly mixed large enough, stops in it the is difficult region to with imagine the invertedits signthat SM of otherhGG value: Higgs, which properties could remainimprove unmodified, the fit to data. so that considering only stop-loop modifications to the partial widths is dubious. On the other hand, one may wonder if the lower-left corner of the plot, with 4 4 a light stopLightest eigenstate,Stop canMass fitand theXt data, with large but no longer unreasonablyTree-Level largePotentialA2q-terms.é TeV It is still Tree-Level Potential TeV Heaviest Stop Mass and sin t2750 2000rather tuned. Recent experimental searches450 for direct production of light stops [2040–4510 ] constrain much 0.25 2000 0.3 0.4 of the stop parameter space with m < 500 GeV, but only for sufficiently0.2 light neutralinos.0.5 The more0 ˜t1 20 000 ⇠ 3.00.1 -0.01 squeezed regime will be probed by a combination of traditional missing-ET signatures2250 [46–552500 @ ] andD spin @ D 17 500 0.20 correlations [56], and even350 the case of R-parity violation may be constrained soon [57]. Nonetheless, 2.5 0.6 0.001 1500for the moment, these considerations still allow as1500 a logical possibility that light,0 highly mixed stops -0.005 15 000 D D 10 000 significantly alter the Higgs properties. 2.0 0.7 0.15 TeV 15 1750 TeV @ @

U However, vacuum250 instability poses an even more serious problem for this scenario than ﬁne-tuning. 7500 12 500 U 2000 -5 R R

m -0.003 m t t ˜ ˜ é é The large A-term mixing is a trilinear scalar coupling t L tR⇤h, so1.5 the1250 potential1500 can acquire large negative

1000 = = 1000 5 0.8 0.10 values when all three5000 of these fields have VEVs. Because theL Higgs and one stop eigenstate are relatively L t t é é light, the barrier separating our EWSB vacuum from a color-10001.0 and charge-breaking minimum can be 150 0 relatively low. At large enough field values, quartic couplings arising from the Yukawa coupling0.9 will -0.000115 2500 750 0.05 -0.00025 500prevent the potential from being unbounded from below,500 even0.5 in the D-flat direction where the stop and Higgs VEVs are equal. Nonetheless, a deep charge- and color-breaking vacuum will exist when the A-term is large. This is illustrated with contour plots of the0.0 potential in Figure 4. It remains to check 0.00 500 1000 1500 2000 500 1000 1500 2000 whether the vacuum decay to this deep minimum happens fast- enough1 to0 rule out1 this scenario.2 3 0.15 0.20 0.25 0.30 0.35 0.40 m For this calculationQ we use the tree-level potential for the up-type HiggsmQH andh TeV the third generation h TeV squark superfields:

2 2 2 2 2 V H,Q˜ , u˜c m2 H 2 m2 Q˜ m2 u˜c y2 Q˜ u˜c HQ˜ Hu˜c Figure 3: Stop( parameter3 3)= space thatH achieves+ Q a3hGG+ couplingU 3 + thatt is 313 times+ its Standard3 + @ 3 ModelD value. This @ D | | Figure 4: Tree-level potential V (h, ˜t L, ˜tR) along the subspace ˜t L = ˜tR. We have fixed mQ = mU = 800 GeV and condition reduces the three-dimensional parameter space m , m2 , X to two dimensions,2 which we parametrize2 1 2 2 1 2 4 ( cadjustedQ 2 U ⇣1t )X t2 to produce 2 A(2hGG )=4 ⌘ASM2(hGGc).2 The right-hand plot zooms in near the good EWSB vacuum where g H Q˜ u˜ g H Q˜ Q˜ u˜ with mQ and mU . At left:+ contours8 0 of the+ lightest3 3 stop3 mass3 h (orange,+2468 GeV dashed) and the and3 stops the+ value have3 of no3X VEV.t needed A3 much to deeper minimum is located near the D-flat direction where achieve the desired coupling (purple, solid).| | At right: contours of the heavy| stop| mass (orange, dashed) and the ✓4 c theh i⇡◆ Higgsc and stop VEVs are all equal. The barrier separating the two minima is shallow. 2 ˜ ˜ ⇣ ⌘ ⇣ ⌘ corresponding stop mixing sin+ ✓ parametrizingH yt X t HQ the3u˜3 right-handednessyt Xt HQ3u˜3 ⇤ of. the stop (purple, solid). (15) ˜t | | 2 1 2 We take m = m with mh = 125 GeV the measuredÄ Higgsä mass. Here represents the corrections H 2 h > We illustraterequired the to achieve parameter the appropriate space that measured can achieve HiggsaA( bounce VEV;hGG we)= action remainASM(S agnostichGG0 )400,in about Figure necessary what 3. model As for is clear gener- a sufficiently long-lived metastable vacuum to describe our ⇠ from equationates these 14, corrections this occurs (in at particular, very large we values do not of tie theuniverse, them mixing to the occurs parameterstop masses onlyX and fort . athe This light MSSM leads stop radiative to mass a large eigenstate cor- below 70 GeV. Such a light stop is excluded by 1 0 1 rections). In the plot in Figure 4, we have taken theLEP,even fields to in be the real case valued, of small with H˜t ˜h,Qmass˜ 3 splitting˜t L, 59, 60 . 1= p2 1 = p2 [ ] c 1 and u˜ ˜tR. We ignore the down-type Higgs; at large tan , it should not be important, and more 3 = p2 7 generally we don’t expect that it will qualitatively3.2 alter the Inverting results. hGG with charged scalar color octets Because the results of Ref. [38] are expressed as a scatter plot of points that are viable or not, it is not possible to do a systematic check from their resultsHere we of whether will consider the parameter a different space possibility for which thethat does not involve large mixing effects. If we drop the hGG amplitude is inverted (as displayed in Figureassumption 3) is ruled out. of supersymmetry, Thus, we perform we a new can numerical consider charged scalar octets that have a mass that decreases calculation of the zero-temperature tunneling rate,with using increasing a slightly modified Higgs mass, version of the CosmoTran- 1 sitions software [58]. The result is depicted in Figure 5. In the right-hand panel, one can see that 2 2 1The main change was to replace a call to scipy.optimize.fmin with one to scipy.optimize.fminbound2 † † to pre- 2 † † † V = µ H H + H H H + mO HOH H O O + O O O , (16) with8 HO > 0. This is a simplifiedÄ subsetä of theÄ interactions thatä arise, forÄ example,ä for the Manohar- Wise scalar in the (8, 2)1/2 representation of the Standard Model gauge group [61]. Other interactions contract the SU(2) indices of H with those of O. There is no principled reason to ignore them, but we restrict to a low-dimensional parameter space for ease of plotting the results and because we expect it will capture the qualitative story of the interplay between vacuum stability and Higgs corrections. Quantitatively, it could be worthwhile to explore the full set of operators, but this is beyond the scope of this paper. The Manohar-Wise representation contains both a neutral scalar O0 and a charged scalar O+; assuming they have the same mass, as they do with this simplified set of interactions with the Higgs, one finds that they affect the Higgs decay widths as shown by the dashed purple curve in Figure 2,

vent a minimum-ﬁnding step from skipping over a shallow minimum and falling into a deep one.

9 3.1 Inverting hGG with Stops Given that we are looking for large changes to the Higgs potential that require light new colored and charged particles, it is reasonable to ﬁrst consider whether stops can be responsible, since naturalness of electroweak symmetry breaking in supersymmetric theories favors light stops [29, 30]. In the case of stops, the general results discussed in the previous section imply a correction to the hGG amplitude (specializing the general result Eq. 11):

2 2 2 2 A(hGG) 1 mt mt mt X t = 1 + 2 + 2 2 2 , (14) ASM(hGG) 4 m m m m 0 ˜t1 ˜t2 ˜t1 ˜t2 1 up to small D-term corrections (taken into account@ in the plots below).A Here m and m are mass ˜t1 ˜t2 eigenvalues, not Lagrangian parameters. The effect of stops on Higgs branching ratios has been discussed insplitting several papers between in the the two recent stop literature mass eigenstates.[6,17,18,24,39 In this region], which of parameter reach a variety space, of the conclusions. lightest stop eigenvalue tends to be fairly light. For example, pushing the light eigenstate up to 450 GeV implies 20 As emphasized by Ref. [39], light unmixed stops tend to increase the hGG coupling and decrease the h coupling,TeV A-terms, whereas which highly is an mixed enormously stops finely-tuned contribute large scenario, corrections both from to them point2 (thus of view requiring of electroweak more symmetry breaking and of b s. In fact, from the Coleman-Weinberg discussionHu in Section 2.3, one tuning for EWSB) and lead to large corrections to b s that must also be tuned away. The same can readily see that such large! A-terms lead to very large negative threshold corrections to the Higgs considerations led Ref. 24 to focus on the “funnel” region! in which the stop corrections to hGG are Whatmass. This it implies takes[ the] need to for get very large hGG beyond-MSSM coupling couplings equal of the Higgs and boson thatopposite are capable to small. Onof the lifting other its hand, mass up Refs. to 125[18, GeV. 22] Whenargued such for couplings light and become highly mixed large enough, stops in it the is difficult region to with imagine the invertedits signthat SM of otherhGG value: Higgs, which properties could remainimprove unmodified, the fit to data. so that considering only stop-loop modifications to the partial widths is dubious. On the other hand, one may wonder if the lower-left corner of the plot, with 4 4 a light stopLightest eigenstate,Stop canMass fitand theXt data, with large but no longer unreasonablyTree-Level largePotentialA2q-terms.é TeV It is still Tree-Level Potential TeV Heaviest Stop Mass and sin t2750 2000rather tuned. Recent experimental searches450 for direct production of light stops [2040–4510 ] constrain much 0.25 2000 0.3 0.4 of the stop parameter space with m < 500 GeV, but only for sufficiently0.2 light neutralinos.0.5 The more0 ˜t1 20 000 ⇠ 3.00.1 -0.01 squeezed regime will be probed by a combination of traditional missing-ET signatures2250 [46–552500 @ ] andD spin @ D 17 500 0.20 correlations [56], and even350 the case of R-parity violation may be constrainedDeep soon [57]. Nonetheless,minimum 2.5 0.6 0.001 1500for the moment, these considerations still allow as1500 a logical possibility that light,0 highly mixed stops -0.005 15 000 D D 10 000 from A-terms significantly alter the Higgs properties. 2.0 0.7 0.15 TeV 15 1750 TeV @ @

U However, vacuum250 instability poses an even more serious problem for this scenario than ﬁne-tuning. 7500 12 500 U 2000 -5 R R

m -0.003 m t t ˜ ˜ é é The large A-term mixing is a trilinear scalar coupling t L tR⇤h, so1.5 the1250 potential1500 can acquire large negative

1000 = = 1000 5 0.8 0.10 values when all three5000 of these fields have VEVs. Because theL Higgs and one stop eigenstate are relatively L t t é é light, the barrier separating our EWSB vacuum from a color-10001.0 and charge-breaking minimum can be 150 0 relatively low. At large enough field values, quartic couplings arising from the Yukawa coupling0.9 will -0.000115 2500 750 0.05 -0.00025 500prevent the potential from being unbounded from below,500 even0.5 in the D-flat direction where the stop and Higgs VEVs are equal. Nonetheless, a deep charge- and color-breaking vacuum will exist when the A-term is large. This is illustrated with contour plots of the0.0 potential in Figure 4. It remains to check 0.00 500 1000 1500 2000 500 1000 1500 2000 whether the vacuum decay to this deep minimum happens fast- enough1 to0 rule out1 this scenario.2 3 0.15 0.20 0.25 0.30 0.35 0.40 m For this calculationQ we use the tree-level potentialShallow for the up-type HiggshillmQH andh TeV the third generation h TeV squark superfields:

vent a minimum-ﬁnding step from skipping over a shallow minimum and falling into a deep one.

9 RAPID VACUUM DECAY The large trilinear terms in this region lead to rapid vacuum decay (known already in 90s: Kusenko, Langacker, Segre, hep-ph/9602414).

Bounce Action Bounce Action Bounce1000 action computed 100 GeV with CosmoTransitions900 5 450 800 90 GeV D code D by Max Wainwright 400 700 10 GeV GeV @ @

(UCSC student) U 100 U 80 GeV m m 600 350 20 30 500 200 70 GeV Excludes50 these 300 60 GeV 400 400 scenarios400 500 600 700 800 900 1000 300 350 400 450

mQ GeV mQ GeV

@ D @ D Figure 5: Contours of the bounce action S0 as calculated by CosmoTransitions [58]. The requirement for a > sufﬁciently long-lived vacuum is S0 400. The left-hand plot shows that the bulk of the parameter space fails this requirement by a wide margin.⇠ The right-hand plot zooms in on the low-mass region, overlaying contours

of the mass of the light stop eigenstate ˜t1 (orange, dashed). The bounce action exceeds 400 only when the light stop eigenstate is below 70 GeV,and thus cleanly excluded by LEP constraints.

which comes rather close to the best-fit point of our simplified 2 fit. Effects of such an octet scalar on the hGG amplitude were considered recently in Refs. [62–64] in the regime with relatively small corrections that would lead to a reduced gg H cross section. The possibility that HO < 0 could lead to a reasonable fit of the data with enhanced! diphoton rate was observed in Ref. [65]. Furthermore, as emphasized in Ref. [66], this regime of parameter space makes a striking prediction of a di-Higgs production rate hundreds or thousands of times larger than the rate in the Standard Model. 2 2 In this case, the condition ANP(hGG)= 2ASM(hGG), at one loop and ignoring mO/mH effects, 2 singles out a particular choice of HO given the mass mO: 2 16mO HO = . (17) 25v2 2 2 Taking into account the (small) mH /mO corrections, we plot the required choice of HO as a function 2 of mO in Figure 6 along with the physical mass of the octet. Notice that, unless the new octet state is very light, the coupling quickly becomes extremely large. In particular, once the physical octet mass reaches about 400 GeV,the coupling is nonperturbatively large. Hence, this scenario is only viable with relatively light states. In fact, the quartic part of the potential becomes unbounded below unless the condition 2 HO O O;min (18) ⌘ 4H

is satisﬁed. We have also plotted O;min in Fig. 6. It becomes nonperturbatively large already when mO 300 GeV, a point at which the physical mass is only about 180 GeV. Of course, a potential that is ⇡ 10 Tree-Level Potential TeV4 Tree-Level Potential TeV4 20 10 0.25 0 3.0 -0.01 @ D @ D 0.20 0.001 2.5 0 -0.005 D D 2.0 0.15

TeV 15 TeV @ -5 @ R R -0.003 t t é 1.5 é = 5 = 0.10 L L t t é é 1.0 0 -0.000115 0.05 -0.00025 0.5

0.0 0.00 -1 0 1 2 3 0.15 0.20 0.25 0.30 0.35 0.40 h TeV h TeV

@ D @ D Figure 4: Tree-level potential V (h, ˜t L, ˜tR) along the subspace ˜t L = ˜tR. We have ﬁxed mQ = mU = 800 GeV and adjusted X t to produce A(hGG)= ASM(hGG). The right-hand plot zooms in near the good EWSB vacuum where h 246 GeV and the stops have no VEV. A much deeper minimum is located near the D-ﬂat direction where theh i⇡ Higgs and stop VEVs are all equal. The barrier separating the two minima is shallow.

> a bounce action S0 400, necessary for a sufﬁciently long-lived metastable vacuum to describe our universe, occurs only⇠ for a light stop mass eigenstate below 70 GeV. Such a light stop is excluded by ˜ 0 LEP,even in the case of smallNEWt1 ˜1 mass SCALARS splitting [59, 60]. WITH 3.2 Inverting hGG with chargedQUARTICS scalar color octets Here we will consider a different possibility that does not involve large mixing effects. If we drop the assumption of supersymmetry, we can consider charged scalar octets that have a mass that decreases with increasingAnother Higgs mass, option, e.g. Manohar-Wise scalar octet O:

2 † † 2 2 † † † 2 V = µ H H + H H H + mO HOH H O O + O O O , (16) Ä ä Ä ä Ä ä with HO > 0.Increasing This is a simplified Higgs subset diphoton of the interactions means large that arise, negative for example, for the Manohar- Wise scalar in the 8, 2 representation of the Standard Model gauge group 61 . Other interactions ( )1/2 4 [ ] contract the SU(2)HHOO indices quartic, of H with requiring those of O. Therethe O is no quartic principled to reason be even to ignore them, but we restrict to a low-dimensionallarger to prevent parameter a space runaway for ease negative of plotting potential. the results and because we expect it will capture the qualitative story of the interplay between vacuum stability and Higgs corrections. Quantitatively, it could be worthwhile to explore the full set of operators, but this is beyond the scope of this paper. Again, tree-level vacuum stability problem. The Manohar-Wise representation contains both a neutral scalar O0 and a charged scalar O+; assuming they have the same mass, as they do with this simplified set of interactions with the Higgs, one finds that they affect the Higgs decay widths as shown by the dashed purple curve in Figure 2, vent a minimum-finding step from skipping over a shallow minimum and falling into a deep one.

9 Collider bound Bounce Action 10 400 As you would expect, in all 100 8 but a tiny sliver of the

200 parameter space, having a 6 O l potential that’s unbounded

4 from below leads to very quick vacuum decay. 2 600 120 140 160 180 200 220 240 260 Physical Octet Mass GeV

Figure 7: The bounce action for tunneling away from the metastable minimum in the scalar octet case, as a Unbounded below@ D 2 2 function of the physical octet mass mO HOv and the octet quartic O. The region below the dashed orange curve has a potential that is unbounded from below. Nonetheless, the tunneling calculation shows that a portion of this region is metastable enoughp to provide a viable vacuum. The vertical red dotted line is an estimate of the collider bound, showing that any surviving parameter space is at masses near 200 GeV and strong coupling > O 4, or must decay in a manner that evades the ATLAS paired dijet search. Kinks in the curves are from the parameter⇠ grid of the numerical scan, not physics. to the Higgs contribute terms d 3 y4 in the RGE for the Higgs quartic. These corrections drive dt = 8⇡2 negative at relatively low energies, leading to yet another vacuum instability. Of course, there is a way out: if the new colored fields come in complete supermultiplets, the scalars contribute an opposite contribution to the running of and the quartic can be saved from turning negative. Thus, one per- spective on this correction is that it gives a bound on the size of the allowed splitting between fermions and scalars in the new multiplet; this is essentially the naturalness point of view discussed in Ref. [37]. The first observation relates to fermionic top partners. In particular, suppose we have new fields T, T in the (3, 1) 2/3 representations of the Standard Model gauge group. We can add both a vectorlike mass for these fields± and a mixing term with the SM left-handed quarks, ¯ † MTT + yT HQT + yT H QT. (19) Such top partners contribute a correction to the hGG amplitude:

A(hGG) vyT yT = 1 . (20) ASM(hGG) Myt vyT yT If we wish this to equal 1, we must take y y 2 y M . If the new colored states are to be heavier T T = 3 t v than the top quark, this requires large Yukawas. Furthermore, these states are highly mixed with the top, and require that we signiﬁcantly alter yt from its Standard Model value. This is an awkward solution that will be difﬁcult to reconcile with experimental bounds. A safer approach is to add a pair of vectorlike fermions, as in Section 2.2, which are not mixed with the SM top. To be concrete, we will take these states to have the same quantum numbers as the SM Q

12 The minimal version of split SUSY cannot give a big enough e⌅ect – indeed, the only source for enhancement is the same chargino loop as in natural SUSY. Thus a large enhancement of 1.5 - 2 immediately rules out this version of split SUSY. We can however certainly imagine extra fermions near the TeV scale; a collection of fermions can have their masses protected by a common chiral symmetry and set by the same scale. In what follows we ask whether the recent LHC data can be explained in a framework of this sort. We show that restricting to un-natural models with only new fermions immediately leads us to a very narrow set-up with sharp theoretical and experimental implications: (1) new, vector-like, un-colored fermions with electroweak quantum numbers must exist and be very light, within the range 100 150 GeV; (2) the cut-o⌅ scale of the theory where additional bosonic degrees of freedom must kick in, cannot be high and is in fact bounded by ⇤ < 1 10 TeV. The cut-o⌅ can be somewhat UV increased but only at the expanse of signiﬁcant model-building⇥ gymnastics, which further destroys any hope of perturbative gauge coupling uniﬁcation.

2 The diphoton rate and includingVACUUMcopies with identical couplings, STABILITY the relevant RGEs read [48 , AND49] N A fermionic loop contribution enhancing the Higgs-diphoton coupling requires vector-like represen- dy 3 9g2 9g2 tations16⇤2 and= y largey2 Yukaway2 + couplingsy2 + yc2 to+ they2 + Higgsyc2 +3 boson.y2 This2 has1 important, ramifications for the dt 2 n N n n t 4 4 consistency of⇧ the theory at highFERMIONS scale. To see this, note that in the⌃ presence of a new fermion f with dy 3 ⇥ ⇥ 9g2 9g2 16electric⇤2 n charge= y Q,they2 hy2 + partialy2 + yc width2 + y2 reads+ yc24 +3y2 2 1 , dt n 2 n N n n t 4 20 ⇧ ⌅ ⌃ ⇥ 2 ⇥ 2 2 2 Arkani-Hamed,2 dyt 2 c2 Blum,2 c2 D’Agnolo,9yt 2 9 g2Fan17 g1207.44821 2 16⇤ = y y +(hy +y )+ y + 1 8g 4 log mf , 7 m dt t N ⌅ n n 1+ 2 Q23 4 20 1+ h , (2.1) ⇧ (h ) ⇥ 3 log v ⌃ 120 m2 2SM ⇤ ⇤⇥ SM f ⇤ 2 d 2⌅ 9g1 2⇤ A 2 c2⌃ 2 c2 ⌥ 4 c4 ⇤ 4 c4 4 16⇤ = 24 9g2 + 12yt⇤ +4 yn + yn + y + y 2 y + y ⇤+ yn + yn 6yt dt 52 3 ⇤ N2 N ⇤ ⇧ GF mh ⇤⇥ 5 ⇥ ⌃ ⇤ with (h )SM = 3 2 SM and SM = 6.49.⇥ Constructive interference⇥ between the ⌅ 3 128⇤32g⇤2 |A | A SM and the+ new2 fermiong4 + g2 amplitude+ 1 requires. electroweak symmetry breaking to contribute negatively(A.1) to 8 2 ⌅ 2 5 ⇧ the mass of the⌥ new fermion.⇧ Thus⌃ f must be part of a vector-like representation with an electroweak- c c Theconserving RGEs for sourcey and ofy mass.n are similar to that for y and yn. The gauge beta functions are YukawasThe basic buildingfrom blocka mass is41 then6 matrix the charged19 vector-like2 fermion mass matrix, b = + N ,b= + N ,b= 7. (A.2) 1 10 5 2 6 3 3 m yv ⌅ Q Vector doublets + triplet (“wino-higgsino”).+Q +Q For⇧ ⇤ our2 “wino-higgsino” scenario, including M = ⌅ ⇤ c + cc, (2.2) L ⌦ y v ↵ Q N copies with identical couplings and allowing for an additional⇤ m⌅ singlet ⇤n, the relevant RGEs read [50] ⇥ 2 2 dy c c 5 2 1 2 1 c2 2 c2 2 c2 2 9 2 33 2 with16⇤ the Higgs= y yn VEVyn + y giveny by+ Hyn = v/y ↵+2N = 1743y +3 GeV.y Eq.+ yn (2.2+ y)n contains+3yt oneg1 physicalg2 phase,, ⇥ = drivedt Higgs quartic4 4 negative 2 2 fast (unless superpartners 20 4 c ⇧ ⌃ arg mdy⇧⇥ m⌅⇥ yy , that cannot3 be3 rotated3 away by field redefinitions.⇥ It is straightforward9 9 to show that 16⇤2 n =3yc yyc + y y2 + yc2 + y2 + N 3y2 +3yc2 + y2 + yc2 +3y2 g2 g2 , ⇥are=⌅ 0dt maximizesnearby)n⇧ the e⌅nect4 wen are2 after,n 4 making2 ⇥ = 0 an un-illuminatingn n complicationt 20 1 4 for2 our current ⇧ ⇧ ⌃ purpose.dy Hence for simplicity we assume ⇥9= 0 in what17 follows.9 We⇥ are then allowed to take all of 16⇤2 t = y N 3y2 +3yc2 + y2 + yc2 + y2 8g2 g2 g2 , the parametersdt t in2 Eq. (2.2) to ben realn and positive.2 t 3 The 20 two1 Dirac4 2 mass eigenvalues are split by an ⇧ ⌃ amount2 d 2 c2 2 ⇥ c2 2 4 c4 4 c4 16⇤ = 24 +2 3y +3y + yn + yn + 12yt N 5y +5y + yn + yn dt N c 2 2 c 2 2 2 c 2 c 2 2 2 c2 c22 22yy v 4 2 (y y ) v 2 (m⇧ m⌅) m = m 21+ynyn⇥yy + ⇥ (+y ⇥+ yn ,)(y⇥ +=y⇥n) 6yt,⇥ ⇥ = , ⇥ =⇥ . (2.3) 2 1 N v Ny m v 2 y 2 m 2 2 m1 2m1 m1 ⌅ g1 2 27 4 ⇧ 9 2 2 9 4 9 + g2 + g1 + g2g1 + g2. (A.3) 4At leading-log plus5 leading finite-mass200 correction;20 see8 e.g. [4]forarecentdiscussion. ⇧ ⌃ 5At leading-log, the SM amplitude is given by the top quark and W boson contributions to the QED beta function, = b + b =+(4/3)2 7. Finite mass corrections modify this prediction slightly to = 6.49. SM leading log t W 41 2 19 SM A b = + N ,b= +2 ,b= 7. A(A.4) 1 10 5 2 6 N 3 We take as initial conditions, at a scale µ = 100 GeV,

2 mh g1 =0.36 5/3,g2 =0.65,g3 =1.2,yt =0.99, = 2 =0.129. (A.5) –3– 2v

The vacuum stability cuto⇥ scale UV is determined by [51]

2 2⇤ UV (UV )= = 0.065 1 0.02 log10 , (A.6) 3 log H 100 GeV UV ⇧ ⇧ ⌃⌃

⇤ ⌅ 42 with the Hubble constant H = 70 km/s/Mpc = 1.5 10 GeV. We comment that for the problem · under study, Landau poles of the Yukawa couplings appear at much higher scales, beyond the scale where the vacuum instability sets in, posing no additional constraint.

– 13 – parameter space. Here, for simplicity, we consider a few decay modes in single exclusive channels. If we take, for example, BR(L Z + l) 100%, and assume a flat 70% eciency times acceptance for 1 ! ⇡ each of the four leptons10, then we find that the region m (100 120) GeV can be excluded already Significant fermion loop effects require veryL1 ⇢ low cutoff: in the “vector-like lepton” model, while the limit extends to 180 GeV in the “wino-higgsino” case. ⇡ This estimate was made using a standard CLs technique described in App. B from a single channel 4l =1 =0 =1 =0 m m m m with MET < 50 GeV and HT < 200 GeV and a Drell-Yan lepton pair from a Z decay. The limits are 700 700 slightly weaker for L1 Z + ⌧. In this case we are not yet sensitive to the “vector-like lepton” model, µ =2 µ =1.75 µ =1.5 ! µ =2 µ =1.75 µ =1.5 650 while we are sensitive to masses up650 to 130 GeV in the “wino-higgsino”µ =1.5 one. Again this estimate µ =1.5 ⇡ 11 c c was obtained by looking at a single channel: 3l +1⌧h with(y=2y MET) < 50 GeV and HT < 200 GeV and (y=2y ) =1 TeV =1 600 UV m 600 a Drell-Yan lepton pair from a Z decay. It is clear that the rest of the relevant parameter space =1 can =1 TeV m easily be covered by the end of the year and that, combining di↵erent channelsUV and possibly results 550 =0 550 m from the two experiments, the sensitivity would be increased, covering also more generic scenarios =0 in =10 TeV m UV 500 which the branching ratio to these500 final states is not exactly one. =10 TeV UV

[GeV] In summary, for = 1, an L decaying to SM leptons is either already ruled out or within reach =10 TeV [GeV] 1 UV

c h L2 450 (y=2y ) N 450 =10 TeV

m UV m of the 8 TeV LHC. If, instead, L W ⇤n (⌫) dominates, the relevantc final state is WW+ MET from 1 ! 1 (y=2y ) L L production, which is still unconstrained for the “vector-like lepton” model, but also within reach 400 1 1 400 of the 8 TeV LHC. In the worst case, when L and n are nearly degenerate in masses, the monojet vector doublet + singlet 1 1 vector doublet + triplet N=1 350 µ =1.25 searches will be able to probe the350 relevant parameter space at the 14N=1 TeV LHC. In the latter case, µ =1.25 other interesting channels, especially for > 1, wouldwino be−higgsino the WWW+MET and, to a lesser extent, 300 Z 300 N WZ+MET final states arising + from the production of L1N as depicted in the right panel of Fig. 5. p + L1 250 Dedicated analyses,Z, beyond the scope250 of this paper, would improve the current sensitivities for some 100 110 120 130 140 150 160 170 180 190 12 200 100 110 120 130 140 150 160 170 180 190 200 m [GeV] of the channels . For our purpose here it suces to show thatm if[GeV] the enhancement of the rate will L1 l be confirmedp(¯p) and an un-naturalL1 theory is responsible for it, then we expect the new fermions involved to be detected in the nextZ few years, or even months.

Z W ( ) ( ) Figure 2. Left: “vector-like lepton” model. Right: “wino-higgsino” model. The horizontal andW vertical axes Viable region would n + 1 p + n1 correspond to the light and heavy mass eigenvalues,p L respectively.+ Pink bandsL1 denote thep diphoton+ enhancement 1 Z, L1 Z, ( ) c W c µ. Graybe bandsprobed denote directly: the vacuum instability cut-o↵⇤UV . Dark is for y = y ; pale is for y =2y .The p(¯p) L1 p(¯p) L1 p(¯p) N n + n width of the bands (for both µ and ⇤UV ) correspond1 to varying the electroweak-conserving massL1 1 splitting

( ) Z ( ) term m (see Eq. (2.3)) from zero to one. Green dashedW band, on the right, denotes the SUSYW wino-higgsinoW scenario. 1 ( ) g˜ q˜ ˜0 W q q Figure 5. Feynman diagrams for new fermion production and decay. n =1 =0 =1 =0 1 m m m m p L+ Z , 1 700 Finally, in addition to direct searches,700 light non-singlet fermions are constrainedq q q indirectly by electroweakµ =2 precision tests (EWPTs), especially so given the need for a large=2 electroweak1 breaking 650p(¯p) L1 q˜µ g˜ q˜ µ =1.75˜0 650 µ =1.75 n1 13 mass to a↵ect µ. Indeed, specializing to the “vector-like lepton” example , in the minimal field g 600 600 ( ) 10From [40]wegetaneciency of the kinematical cuts 0.87.W Taking into account the finite acceptance (somewhat ⇠ q S g optimistically) we obtain the=1 final 0.7 [41]. Notice that this is a huge simplification of the experimental set-up that does 550 =1 TeV m 550 =1 not evenUV distinguish between electrons and muons, and is thus only intended to giveq an˜ orderB˜ ofS˜ magnitudeG˜ estimate.m =1 TeV 11 UV Assuming an hadronic tau identification eciency, for the HPS algorithm used in the CMS paper, ✏⌧h =0.35 [41–44] 500 =0 500 and the same 0.7eciencym as before for any extra lepton. =0 12 g˜ m

[GeV] 1

=10 TeV [GeV] It isUV sucient to think about possible three-lepton resonance searches or monojet searches with the additional h =10 TeV L2 450 450 UVg

requirement of soft leptons in the ﬁnalm state [36]. m 13We expect similar results to hold for the “wino-higgsino” model, as can be deduced e.g.s˜ from [45]. 400 400 s G˜ g vector doublet + singlet vector doublet + triplet 350 N=2 350 N=2 g µ =1.5 µ =1.5 300 300 µ =1.25 µ =1.25 2 – 10 – 1 250 250 100 110 120 130 140 150 160 170 180 190 200 100 110 120 130 140 150 160 170 180 190 200 m [GeV] m [GeV] L1 l

Figure 3.SameasFig.2, but for = 2 copies of vector like fermions. N

3 Collider signals and electroweak constraints

The light charged fermions discussed in the previous section are produced through electroweak pro- cesses with appreciable rates at hadron colliders. In this section we consider constraints and detection prospects from current and upcoming searches, assessing charchteristic detection channels and provid- ing rough estimates of the experimental sensitivity. We stress that our analysis is simplistic, and can by no means replace a full-fledged collider study. Nevertheless, our estimates provide solid motivation and concrete guidelines for a more dedicated study in the future, should the diphoton enhancment be precision constraints on this field content, in the context of modified Higgs couplings. 8 c See Eqs. (2.2-2.3)andthediscussionbetweenthemforthedefinitionofy, y , m and .

–7– 3 Correlation between CP-odd and CP-even observables

3.1 Operators and corresponding observables We will assume that new charged matter does not interact or mix with the SM fermions at tree level and they do not contribute to the EDMs of the SM fermions at one-loop order. (Thus, we will also not discuss the discrepancy in the measured muon g 2, which would typically require new physics with leptonic quantum numbers exerting a one-loop e↵ect.) The one-loop EDM is generically ruled 2 out unless the new CP violating phases are tuned to be small (< 10 ). However, new charged particles could still contribute at the two-loop order to the SM fermion⇠ EDMs through Barr-Zee type diagrams [88]. To see the correlations between the CP-odd observables (including EDMs) and CP- even observables, it will be useful to perform an operator analysis first, which strictly speaking is only valid in the limit when the charged matter is heavy and could be integrated out. We will use operator analysis to clarify the correlations of the observables, whereas for the numerical evaluations we will use the full-fledged loop calculations. Charged matter with physical phases contributes to 6 CP violating dimension-six operators builtHIGGS out of the Higgs AND and the SM CP gauge fields. Among them, two involve the SU(3) color field strength, one of which is the famous Weinberg operator [89]. We only inspect couplings (neglecting SM gauge coupling e↵ects) are extracted from the general formulas [81–84]: the four operators generated by loops of colorless particles. They and their corresponding CP-even Operators2 modifying the2 Higgs2 to2 diphoton rate have CP- operators with similar16 structures⇡ (g⇠)=g are,⇠ 2 g following⇠ + g the+4 notationsg in [90], (2.27) violating cousins: | | | | | | 2 ⇣ 2 2 2⌘ 2 c 2 c 16⇡ (g )=g 3a⌫g b+ g⇠cµ+4 g + y +˜ya⌫ b4g† yycµ (2.28) W = ✏abcWµ| W| ⌫ W| | , | ˜| =|✏|abcW| µ| W⌫ W O ⇣ OW ⌘ 2 a 2 aµ⌫ 2 2 1 2 ac 2 aµ⌫ c 16⇡ (g=)=Hg†HW5 g W+ g⇠, +2 g ˜+= Hy†H+W˜y W 2g † yy (2.29) OhW |µ⌫| | | |Oh|W 2 | | | µ⌫| ✓ µ⌫ ⇣ ˜ ⌘µ◆⌫ hB2 = H†1HBµ⌫2B , 2 2 hB˜ =cH2 †HBµc ⌫ B 16O⇡ (y)= y g + g +5 yO+2 y 2y †g g (2.30) 2 a| | a| | µ⌫ | | | | a a µ⌫ WB =(H† H)W B , ˜ =(H† H)W˜ B , (3.1) 2 c 1 c⇣ 2 µ⌫ 2 c WB2 2⌘ µ⌫ 16O⇡ (y )= y g + g +5 yO +2 y 2y†g g (2.31) 2 | | | | | | | | where a denotes the three Pauli matrices⇣ and a is the isospin⌘ index. The operators have coecients We will evolve these RGEs to examine the extent to which annihilating through resonant enhancement a bounded by the interval [ 1/⇤2 , 1/⇤2 ], where ⇤ is some high scale.2 i relievesCP-violating the Landau pole phases problemneg of in the masspos all-scalar matrix model. of fermions running in Now weThe specify modified theh observables rate, in these the limit operators of large m contribute, is determined to. It by is the well low known energy that theoremsWB gives ! , O the S parameterto bethe [3, 61 in,loop62 the, 70 electroweak]:generate precisionthese operators tests (EWPT), and affect the rate:

2 2 1 2 2 @ 2 2 2 @ µ = 1+ Q log4 detsW cW† v aWB+ Q arg det , (2.32) 3 @ logSv= M M , @ log v M SM SM A ↵ A µ⌫ where the first term arises from the modified CP-even hF µ⌫ F vertex and the second term from where sWthe CP-oddsin ✓WhF, cWF˜µ⌫ costerm.✓W Here, and v == 2466.49 GeV. represents the SM amplitude. As in the scalar case, ⌘ µ⌫ ⌘ ASM Amongthe result all the is modified operators, by familiarW , WB loop( functions˜ , that˜ , correct˜ )modifyCP-evenTGCs(CP-oddTCGs). for finite mass, which are given in the O O OW OhW OWB More concretely,CP-even and the CP-odd general case triple respectively gauge by couplings [85] could be parametrized as [93],

V + µ 3 ⌫ +2 1 µ⌫ iV + ⌫ µ WWV /gWWV = ig AW1/2(⌧Wf )= V⌧f 1+(1h.c. +⌧if)V arcsinW W V + W W (2.33)V 1 µ⌫ 2 µ ⌫⌧f 2 µ⌫ L s ! mW ˜ 1 ˜ + µ⌫ i2 V + ⌫ µ +i˜ AW1/2(⌧Wf )=V˜⌧f arcsin+ W, W V˜ + , (2.34) (3.2) V µ ⌫ 2s⌧f µ⌫ mW ··· (with ⌧ =4m2 /m2 ) and asymptote to 1 when 2m m . (Note that there is a minor error in where V is eitherf Z for h and g = e cot ✓ ,gf h= e. The ﬁrst line of Eq. 3.2 contains Ref. [86], which assumes theWWZ same loop function forW theWW scalar and pseudoscalar decay modes.) CP-even TGCsWe choose while two the representative second line points contains in parameter CP-odd space TGCs. that Theﬁt a 50% dots enhanced representh C-violating rate. In TGCs V! arising fromboth cases, operators we arrange at high for the orders light in mass the eigenstate SM e↵ective to be at theory. 140 GeV, In such the that SM, itg is1 near= 1, theV DM= 1 and mass but slightly too heavy for the dark matter to annihilate into two of our new charged fermions. 2In some literature [91, 92], more CP-odd operators were listed. As we show in Appendix B,thoseadditionaloperators can be written inModerate terms of phase: the operatorsm = m in= Eq. 3463.1 GeV,usingy =1 the.37 equationse0.2⇡i, yc of=1 motion..37. The mass eigenvalues are 140 • and 577 GeV. As we will see in the next section, this point is excluded by the nonobservation 27 of an electron EDM: it predicts d /e =9.3 10 cm. However, this exclusion can be avoided e ⇥ if the EDM is canceled by another contribution which must be tuned to about the 10% level. (And note that our phase is already somewhat small; maximal CP violation would require the EDM to be tuned at a few-percent level.)

– 160.02 – ⇡i c Small phase: m = m = 333 GeV, y =1.11e , y =1.11. The mass eigenvalues are 140 and • 28 526 GeV. This point is currently safe from EDM constraints, predicting d /e =7.3 10 cm, e ⇥ but would likely be detected with next-generation electron EDM measurements [63].

In both cases, we will also consider equal couplings g g⇠ = g = g of the pseudoscalar resonance, 27 ⌘3 chosen to achieve a gamma-ray line rate of 1.0 10 cm /s. These turn out to be relatively insensitive ⇥

– 13 – CP VIOLATION

Could try to probe this with CPV Higgs decays (Voloshin 1208.4303), but also: RGE mixes

˜µ µ H†HFµ F and LHµ eF¯ ⇥˜0 W +

⇥˜ Generically, new physics that alters Higgs decays with

⇥˜0 W fermions will produce a nonzero electron (or neutron) EDM. “Higgs CP problem”Figure 4: Some annihilation modes

Hu ⇥ “Barr-Zee” type diagrams: X + may be familiar, e.g. split SUSY Z, h

eL eR

Figure 5: Two-loop EDMs in supersymmetric theories. The one-loop diagram in the dashed box is a “CPV-EWPT" term.

References

[1] R. Essig, “Direct Detection of Non-Chiral Dark Matter,” Phys. Rev. D 78, 015004 (2008) arXiv:0710.1668 [hep-ph].

[2] M. Farina, M. Kadastik, D. Pappadopulo, J. Pata, M. Raidal and A. Strumia, “Implica- tions of XENON100 and LHC results for Dark Matter models,” Nucl. Phys. B 853, 607 (2011) arXiv:1104.3572 [hep-ph].

[3] C. P.Burgess, M. Pospelov and T. ter Veldhuis, “The Minimal model of nonbaryonic dark matter: A Singlet scalar,” Nucl. Phys. B 619, 709 (2001) hep-ph/0011335.

[4] J. Giedt, A. W. Thomas and R. D. Young, “Dark matter, the CMSSM and lattice QCD,” Phys. Rev. Lett. 103, 201802 (2009) arXiv:0907.4177 [hep-ph].

[5] M. Ackermann et al. [Fermi-LAT Collaboration], “Constraining Dark Matter Models from a Combined Analysis of Milky Way Satellites with the Fermi Large Area Tele- scope,” Phys. Rev. Lett. 107, 241302 (2011) arXiv:1108.3546 [astro-ph.HE].

6 Doublets + Singlets N=1, f=0.2p Doublets + Singlets N=1, f=0.03p 600 600 2 2 1.25 1.75 550 H L 550 H L

1.75 500 500 L L 450 1.5 450 GeV GeV

H H 1.5

H 400 1.1 H 400 m m

350 350

Constraints300 from EDMs: green 300contours are Higgs to 1.25 250 250 diphoton100 120enhancement140 160 180 200 100 120 140 160 180 200 mL GeV mL GeV Doublets + Triplet N=1, f=0.2p Doublets + Triplet N=1, f=0.05p 600 600 2 2 1.25 1.75 H L H L 550 H L 550 H L

1.75 500 500 L L 450 1.5 450 GeV GeV

H H 1.5

H 400 1.1 H 400 m m

350 350

300 300

1.25 250 250 100 120 140 160 180 200 100 120 140 160 180 200

mL GeV mL GeV -27 Solid purple: de/eH =L 1.05 10 cm; H L c Figure 10. Upper: “vector-like lepton” model; Lower: “wino-Higgsino” model. N =1,m = m,y = y in c -28 all these plots. =arg(yy m ⇤ m⇤ ). The horizontal and vertical axes correspond to the light and heavy mass dashed purple: de/e = 10 cm 27 eigenvalues. The solid purple line is the current EDM constraint de/e =1.05 10 cm with the grey region 28 ⇥ excluded; the dashed purple line is the projected constraint de/e =10 cm. The green lines denote the diphotonJ. Fan enhancement and MR,µ. 1301.2597 calculation, it can also be understood as a consequence of the fact that the arg det coupling arises M from an anomalous rotation of fermion ﬁelds, whereas scalars have no anomalies. However, if there is a pseudoscalar particle in the spectrum that can run in the two-loop EDM diagram in place of the Higgs, or if CP-violation leads the Higgs to have a small pseudoscalar-like coupling to the electron (e.g. by mixing with a pseudoscalar), there will still be a two-loop EDM [107]. Thus, in the case of charged scalars, the Higgs CP problem would be less robust: if all pseudoscalars are heavy, the EDMs can be rather small. (There are other diculties for such an interpretation of an increased h ! rate, as new charged scalars typically have vacuum stability problems [40, 46], although there is still viable parameter space for quite light scalars [55].)

– 20 – Harvard University Department of Physics Colloquium

April 29, 2013, Monday in Jefferson 250 @ 4:15 pm

Tea will be served in Jefferson 450 @ 3:30 p.m.

“Particle Physics with Cold Molecules: The ACME Experiment”

The ACME collaboration (DeMille/Doyle/Gabrielse) claims to be able toHarvard improve University the EDM limit by an order Department of Physics Colloquium of magnitude (or measure it). Latest news at colloquium April 29, 2013, Monday in Jefferson 250 @ 4:15 pm on Monday (April Tea29) will be atserved Harvard? in Jefferson 450 @ 3:30 p.m. “Particle Physics with Cold Molecules: The ACME Experiment” John Doyle Harvard Physics

Measurement of a non-zero electric dipole moment (EDM) of the electron within a few orders of magnitude of the current best limit[1] of |de|<1.05*10 -27 e*cm would be an indication of CP violation beyond the Standard Model, probably an indication of SUSY and possibly indicating the T violation necessary for explaining the matter-antimatter asymmetry of the universe. The ACME Collaboration is searching for an electron EDM by performing a precision measurement of electron spin precession signals from cold thorium monoxide (ThO). I will discuss the current status of the experiment. Based on a data set acquired from 65 hours of running time, we have achieved a one-sigma statistical uncertainty in the value of de of 1*10-

[1] JJ Hudson et al., "Improved measurement of the shape of the electron." Nature 473, 493 (2011) John Doyle Harvard Physics

[1] JJ Hudson et al., "Improved measurement of the shape of the electron." Nature 473, 493 (2011) MORAL

Loop-level corrections to Higgs properties are small in almost any well-behaved model. Still worth looking! But don’t expect effects larger than ~20% without running into vacuum instabilities that are likely to kill or severely constrain the model.

Many reasonable people who thought about the diphoton excess concluded it would go away and, at CMS, it has. MORAL

A more plausible scenario for modifying Higgs couplings is through an extended Higgs sector (e.g. 2 2 2HDM mZ /mA effects in the MSSM). Not surprising: it’s tree-level.

Then, expect correlated changes in diphoton and ZZ, WW.

Important to look for other Higgses as well! STANDARD MODEL VACUUM STABILITY

Aside from neutrino masses and gravity modifying physics at high energies, and dark matter with no deﬁnite connection to the SM, we currently have no compelling beyond Standard Model signals.

This raises the question: does the SM Higgs potential give us any reason to suspect new physics at some scale? VACUUM STABILITY IN THE SM The Higgs quartic coupling runs according to: 1 () 122 +6y2 3y4 ⇡ 8⇡2 t t The top Yukawa has: 1 9 (y )= y3 4y g2 t 8⇡2 4 t t s ✓ ◆ Experimentally, we now know 0.13,yt 1. ⇡ ⇡ So, the leading effect is that the Higgs quartic decreases at high scales due to the top Yukawa, and top Yukawa decreases due to the strong gauge coupling. SHAPE OF THE HIGGS POTENTIAL

To good accuracy, the running quartic ( µ ) describes the behavior of the potential at large ﬁeld values H: V (H) ( H ) H 4 at H v. ⇡ | | | | | | This is due to the RG improvement of the Coleman- H 2 Weinberg potential, which involves log | | terms. µ2

The leading µ dependence cancels between logs in the Coleman-Weinberg potential and the logs in the running couplings. TUNNELING There is a simple approximation for the bounce action of the tunneling solution (Isidori, Ridolfi, Strumia [IRS] hep- ph/0104016): 1 2 µ H = h(r),r = xµx p2 where d2h 3 dh + + V 0(h)=0 dr2 r dr solution with h = v at infinity and h’(0) = 0: 2 2R 8⇡2 h(r)= ,S= r2 + R2 0 3 s| | | | Here is taken fixed and negative. and including copies with identical couplings, the relevant RGEs read [48, 49] N dy 3 9g2 9g2 16⇡2 = y y2 y2 + y2 + yc2 + y2 + yc2 +3y2 2 1 , dt 2 n N n n t 4 4 ✓ ◆ dy 3 9g2 9g2 16⇡2 n = y y2 y2 + y2 + yc2 + y2 + yc2 +3y2 2 1 , dt n 2 n N n n t 4 20 ✓ ◆ dy 9y2 9g2 17g2 16⇡2 t = y y2 + yc2 + y2 + yc2 + t 8g2 2 1 , dt t N n n 2 3 4 20 ✓ ◆ d 9g2 16⇡2 = 24 9g2 1 + 12y2 +4 y2 + yc2 + y2 + yc2 2 y4 + yc4 + y4 + yc4 6y4 dt 2 5 t N n n N n n t ✓ ◆ 2 3 3g2 + 2g4 + g2 + 1 . (A.1) 8 2 2 5 ✓ ◆ ! c c The RGEs for y and yn are similar to that for y and yn. The gauge beta functions are 41 6 19 2 b = + N ,b= + N ,b= 7. (A.2) 1 10 5 2 6 3 3 Vector doublets + triplet (“wino-higgsino”). For our “wino-higgsino” scenario, including N copies with identical couplings and allowing for an additional singlet n, the relevant RGEs read [50] dy 5 1 1 9 33 16⇡2 = ycy yc + y y2 + y2 yc2 + N 3y2 +3yc2 + y2 + yc2 +3y2 g2 g2 , dt n n 4 4 n 2 2 n n t 20 1 4 2 ✓ ◆ dy 3 3 3 9 9 16⇡2 n =3yc yyc + y y2 + yc2 + y2 + N 3y2 +3yc2 + y2 + yc2 +3y2 g2 g2 , dt n n 4 n 2 n 4 2 n n t 20 1 4 2 ✓ ◆ dy 9 17 9 16⇡2 t = y N 3y2 +3yc2 + y2 + yc2 + y2 8g2 g2 g2 , dt t 2 n n 2 t 3 20 1 4 2 ✓ ◆ d 16⇡2 = 24 +2 3y2 +3yc2 + y2 + yc2 + 12y2 N 5y4 +5yc4 + y4 + yc4 dt N TUNNELINGn n t 2 n n 2 y yc yyc (y2 + yc2)(yc2 + y2 ) 6y4 N n n N n n t g2 27 9 9 9 1 + g2 + g4 + g2g2 + g4. (A.3) Our toy solution5 2 has200 1an 20arbitrary2 1 8 2size modulus R, and it’s ✓ ◆ unclear what scale to evaluate the quartic at. IRS do a 41 2 19 one-loop calculationb1 = + includingN ,b2 = the +2functional,b3 = 7determinant. (A.4) 10 5 6 N to pin them down. Should take: R 1 /µ with µ near We take as initial conditions, at a scale µ = 100 GeV,⇠ the scale where the potential goes unstable. m2 g =0.36 5/3,g =0.65,g =1.2,y =0.99, = h =0.129. (A.5) 1 2 3 t 2v2 p TheScale vacuum of stability the instability: cuto↵ scale ⇤UV is determined by [51] 2 2⇡ ⇤UV (⇤UV )= = 0.065 1 0.02 log10 , (A.6) 3 log H 100 GeV ⇤UV ✓ ✓ ◆◆

⇣ ⌘ 42 with the Hubble constant H = 70 km/s/Mpc = 1.5 10 GeV. We comment that for the problem · under study, Landau poles of the Yukawa couplings appear at much higher scales, beyond the scale where the vacuum instability sets in, posing no additional constraint.

– 13 – RUNNING QUARTIC

Recently: 1205.6497 by Degrassi, Di Vita, Elias-Miró, Espinosa, Giudice, Isidori, Strumia. NNLO.

1.0 0.10

g Mh = 125 GeV yt s 0.08 0.8 3s bands in M = 173.1 ± 0.7 GeV L t m H 0.06 a M = 0.1184 ± 0.0007 l s Z 0.6 g 0.04

coupling H L g1 couplings 0.4 0.02 SM quartic Mt = 171.0 GeV 0.00 Higgs 0.2 as MZ = 0.1205 -0.02 a M = 0.1163 l s Z 0.0 yb MtH = 175.3L GeV -0.04 102 104 106 108 1010 1012 1014 1016 1018 1020 102 104 106 108 1010 1012 10H14 10L 16 1018 1020 RGE scale m in GeV RGE scale m in GeV

MetastableFigure 1: Left: potential SM RG evolution of the, instability gauge couplings nearg1 = 5/the3g0,g 2intermediate= g, g3 = gs,ofthe scale. top and bottom Yukawa couplings (y ,y ), and of the Higgs quartic coupling . All couplings are t b p deﬁned in the MS scheme. The thickness indicates the 1 uncertainty. Right: RG evolution of ± varying M and ↵ by 3. t s ±

We stress that both these two-loop terms are needed to match the sizable two-loop scale dependence of around the weak scale, caused by the 32y4g2 +30y6 terms in its beta t s t function. As a result of this improved determination of (µ), we are able to obtain a

signiﬁcant reduction of the theoretical error on Mh compared to previous works.

Putting all the NNLO ingredients together, we estimate an overall theory error on Mh of 1.0GeV(seesection 3). Our ﬁnal results for the condition of absolute stability up to the ± Planck scale is M [GeV] 173.1 ↵ (M ) 0.1184 M [GeV] > 129.4+1.4 t 0.5 s Z 1.0 . (2) h 0.7 0.0007 ± th ✓ ◆ ✓ ◆

Combining in quadrature the theoretical uncertainty with the experimental errors on Mt and

↵s we get M > 129.4 1.8GeV. (3) h ± From this result we conclude that vacuum stability of the SM up to the Planck scale is

excluded at 2 (98% C.L. one sided) for Mh < 126 GeV.

Although the central values of Higgs and top masses do not favor a scenario with a

vanishing Higgs self coupling at the Planck scale (MPl)—apossibilityoriginallyproposed

2 VACUUM INSTABILITY

180 7 10 1010 200 Instability InstabilityInstability Meta-stability GeV Non GeV 150 stability in 175 - t - in M perturbativity t Meta 1,2,3 s M 100 Stability mass mass 170 top 9 Top 10 12 50 10 Stability Pole

0 165 0 50 100 150 200 115 120 125 130 135

Higgs mass Mh in GeV Higgs mass Mh in GeV

FigureDegrassi 5: Regions et of al. absolute ﬁnd stability, that meta-stabilitythe measured and instability values of the SMfall vacuum in a inregion the Mt– Mh plane. Right: Zoom in the region of the preferred experimental range of Mh and Mt (the graywhere areas denote the thequartic allowed regionruns at negative, 1, 2, and 3). but The threethe boundaries lifetime lines is correspondlong to ↵ (M )=0.1184 0.0007, and the grading of the colors indicates the size of the theoretical error. enoughs Z that± it could be our universe (metastable region). The dotted contour-lines show the instability scale ⇤ in GeV assuming ↵s(MZ )=0.1184.

3.3 Phase diagram of the SM

The ﬁnal result for the condition of absolute stability is presented in eq. (2). The central value of the stability bound at NNLO on Mh is shifted with respect to NLO computations

(where the matching scale is ﬁxed at µ = Mt)byabout+0.5GeV,whosemaincontributions can be decomposed as follows: +0.6GeVduetotheQCDthresholdcorrectionsto (in agreement with [14]); +0.2GeV due to the Yukawa threshold corrections to ; 0.2GeVfromRGequationat3 loops(from[12,13]); 0.1GeVfromthee↵ective potential at 2 loops. As a result of these corrections, the instability scale is lowered by a factor 2, for M 125 ⇠ h ⇠ GeV, after including NNLO e↵ects. The value of the instability scale is shown in ﬁg. 4.

The phase diagram of the SM Higgs potential is shown in fig. 5 in the Mt–Mh plane, taking into account the values for Mh favored by ATLAS and CMS data [1, 2]. The left plot illustrates the remarkable coincidence for which the SM appears to live right at the border between the stability and instability regions. As can be inferred from the right plot, which zooms into the relevant region, there is significant preference for meta-stability of the SM potential. By taking into account all uncertainties, we find that the stability region is disfavored by present data by 2.ForMh < 126 GeV, stability up to the Planck mass is excluded at 98% C.L. (one sided).

17 ANOTHER VIEW

1209.0393 by Isabella Masina: emphasizes uncertainty10 in top mass (is what’s measured precisely the pole mass?) 166 176

165 D 164 174 METASTABILITY D GeV @ GeV L 163 a t

3 @ t

m 0.1213 a3 mZ = H m t 162 m 172 =0.1196 l m a3 mZ H L 0.1179 161 a3 mZ = H L 160 STABILITY 170 H L 124.5 125.0 125.5 126.0 126.5 127.0 mH GeV Becomes important to reduce uncertainties in the top FIG. 5: The solid (black) line marks the points in the plane [mH , mt(mt)] where a second sector vacuum,in order degenerate to with make the electroweak a one,deﬁnite is@ obtainedD just claim. below the Planck scale. The (red) diagonal arrow shows the e↵ect of varying ↵ (m )=0.1196 0.0017 [19]; the (blue) horizontal 3 Z ± one shows the e↵ect of varying µ (the matching scale of ) from mZ up to 2mH . The shaded (yellow) vertical region is the 2 ATLAS [1] and CMS [2] combined range, m = 125.65 0.85 H ± GeV; the shaded (green) horizontal region is the range m (m ) = 163.3 2.7 GeV, equivalent to t t ± m = 173.3 2.8 GeV [15]. t ±

the input parameters and the one associated to the matching procedure. To illustrate this,

we consider in particular the point on the transition line associated to the value mH =126

GeV; for such point, and both vanish at a certain scale µ (see ﬁg. 4). The arrows show how, if some inputs or the matching scale are changed, the position of this point

have to change in order to keep having, at the same scale µ, a vacuum degenerate with the electroweak one. The diagonal arrow is obtained by varying the strong coupling in its allowed range, ↵ (m )=0.1196 0.0017 [19]; the short (long) dashed line shows how 3 Z ± the solid line would move if ↵3(mZ )wereequaltoitsminimum(maximum)presently allowed value. Notice that the error on ↵3(mZ ) induces an uncertainty in both the Higgs and top masses of about 0.7 GeV. In ref.[14] the impact of the variation of ↵ (m )on ± 3 Z m was estimated to be 0.5 GeV (see their table 1). The two results are in substantial H ± agreement, considering that in our analysis ↵ (m )=0.1196 0.0017 at 1 [19], while 3 Z ± ref.[14] considers a smaller error, ↵ (m )=0.1184 0.0007 at 1. Since the variation 3 Z ± of the other input parameters in eq.(4) induces much smaller e↵ects then the one due WHAT DOES IT MEAN?

•Bad news for supersplit supersymmetry. MSSM boundary conditions at a high scale lead to larger Higgs masses, ~ 140 GeV.

•Some are using it as propaganda for “asymptotic safety,” an idea that seems to me to be obviously inconsistent with black hole physics & semiclassical GR.

•Many others are adopting a (related?) view that quadratic divergences are not real. But quadratic sensitivity to large mass scales is physical, the SM is not UV complete, and gravity is real, so I can’t make sense of this viewpoint. WHAT DOES IT MEAN? INSIGHTS FROM TWITTER CONCLUSIONS •Vacuum instabilities plague most attempts to signiﬁcantly modify the loop-level Higgs couplings to photons and to gluons. Tree-level alterations (mixing) typically safer.

•If large loop corrections observed, possible “Higgs CP problem.”

•Standard Model: probably metastable, with quartic running negative at intermediate scale. Lots of room for unknown new physics to modify the story.

•I’m skeptical of any deep meaning attached to small lambda at high scale.