Jhep04(2016)011

JHEP04(2016)011 Springer April 4, 2016 March 4, 2016 : March 30, 2016 : January 4, 2016 : : , Revised Published Accepted Received 10.1007/JHEP04(2016)011 doi: Published for SISSA by [email protected] , . 3 1512.08922 The Authors. c Cosmology of Theories beyond the SM, Higgs Physics

We investigate feasibility of efficient baryogenesis at the electroweak scale , [email protected] School of Physics, TheNSW University 2006, of Australia Sydney, E-mail: [email protected] ARC Centre of Excellence for Particle Physics at the Terascale, Open Access Article funded by SCOAP ArXiv ePrint: generated in this framework.constrained This from range the of LHC the Run 2 Higgs data anomalous and couplings be can probed beKeywords: at further high luminosity LHC and beyond. the a strongly first-orderthat phase the transition. anomalous Higgs The vacuumWe distinguished expectation identify feature value a of is range this generallywhere of non-zero transition sphaleron in anomalous is both rates couplings, phases. arein consistent the sufficiently with ‘broken’ fast current phase in experimental and demonstrate data, the that ‘symmetric’ the phase desired and baryon asymmetry are can suppressed indeed be Abstract: within the effective fieldtroweak gauge theory symmetry. framework In based thisscalar framework on field, the a which, LHC therefore, Higgs non-linear admits boson realisation newthe is interactions. eletroweak of described gauge Assuming by the bosons that a elec- are Higgs singlet cubic couplings as coupling with in and the the Standard CP-violating Model, Higgs-top we quark demonstrate anomalous that couplings the alone Higgs may drive Archil Kobakhidze, Lei Wu and Jason Yue Electroweak baryogenesis with anomalous Higgs couplings JHEP04(2016)011 – 14 (1.1) 11 ]. Hence, the observed matter- , ] has important implications for 11 , 11 2 , − 1 10 ] in a more quantitative manner. It 7 10 × ]–[ 3 6 . 14 8 5 ∼ 5 7 – 1 – 5 /s 9 B n := 17 B 12 2 η 13 ]: 13 16 , 12 ] and the standard Cabibbo-Kobayashi-Maskawa CP-violation is not 9 1 , 8 denote respectively the net baryon number and entropy densities] provides s and 70 GeV [ B n ∼ Such new physics can be encoded in an effective field theory (EFT) extension of the 4.1 The electroweak4.2 sphaleron rate CP-violation via4.3 the anomalous Higgs-top The couplings baryon asymmetry 3.1 Higgs vacuum3.2 at zero temperature Higgs vacuum at finite temperature ] and references therein. with a strong hint inand current favour status of of new the physics16 electroweak beyond baryogenesis the readers Standard are Model. referred to For the an reviews overview [ Standard Model which contains a set of nonrenormalizable, gauge invariant operators of sufficient to generate the desiredantimatter baryon asymmetry asymmetry [ [ [here cosmology. Namely, with theelectroweak Higgs phase data transition available, we andhas can baryogenesis been now [ analyse established theModel already nature the well of required the before first-orderthan phase the transition Higgs is discovery not realised that for within the the Higgs boson Standard heavier 1 Introduction The discovery of a 125 GeV Higgs particle at LHC [ A Finite temperature effective potential B Transport equations C The electroweak sphaleron 5 Conclusion and outlook 3 The electroweak phase transition 4 Computing the baryon asymmetry Contents 1 Introduction 2 Description of the model JHEP04(2016)011 ) = (1.2) x ( H (1). For the ], we identify a ]. New physics ∼ O 38 36 ]. , ] and are relatively less R 32 22 10 TeV [ ]–[ Ht L ]–[ ∼ 24 Q ] for supersymmetric models). 17 ) H 35 , † H 34 complex doublet Higgs field ( 2 ¯ th Y t Λ c scattering in the Higgsless Standard Model = U(1) th × t – 2 – ] (see also [ O are the dimensionless constants process at energies WW 33 ¯ th t , → c 3 ) WW H † and WW ]. This shades some doubts on the validity of the EFT → H 6 ( c ¯ t 18 2 t 6 c Λ ] are: = = 125 GeV the successful baryogenesis requires the low ultraviolet 23 6 h O 246 GeV of the SU(2) m ≈ v ]. It admits a tree-level cubic Higgs coupling and CP-violating Higgs-Top 840 GeV or so [ 33 . The purpose of this work is to investigate feasibility of efficient baryogenesis at the Although the current Higgs data is consistent with the Standard Model predictions, Some technicalities are collected in the appendices for the2 reader’s convenience. Description of theThe model electroweak symmetry withinpectation SM value is spontaneously broken via nonzero vacuum ex- quence of thethat model the with Higgs anomalous vacuuminto tree-level expectation account cubic value current constraints Higgs does onparameter not and the range vanish Higgs-Top anomalous at where Higgs-Top couplings couplings high the is proceed [ temperatures. electroweak with phase Taking computing transition the is baryon strongly asymmetry first-order. in We section then 4. We conclude in section 5. troweak gauge symmetry.sector Namely, only we [ consider ainteractions, which model play with a agenesis. significant modified role This Higgs-Yukawa in model thetemperature electroweak is effective phase described potential transition in and and the baryo- analyse next the section. phase In transition. section 3 A we remarkable compute conse- a finite at such high energies maymeasurements escape of the deviations detection frombased at on LHC. SM nonlinear In physics realisation situations parametrized become like imperative. within these, precision the effectiveelectroweak scale theories within the EFT framework based on a non-linear realisation of the elec- (SM). It is expectedpower-law that growth at of high energies scattering newthe amplitudes scale resonances with where show energy new up, in physics whichconsidered. is perturbation unitarise expected theory. rapid, to For emerge example, However, cruciallyunitarity in depends is SM on the with violated specific the for process anomalous the top-Yukawa couplings perturbative one should bear in mindunderstood that yet. the nature of Ingauge the particular, electroweak symmetry symmetry instead can breaking of be isrepresentation the not nonlinearly of fully standard realised the linear with symmetry realisation theNonlinearly group the realised Higgs [ electroweak electroweak boson gauge residinggies, theory in the becomes the famous strongly singlet example interacting being at high ener- Higgs boson of mass cut-off, Λ framework. Several specific renormalizable modelsof of the the LHC electroweak Higgs baryogenesis data in have light been also discussed recently in [ tors, which are relevantconstrained for at the present electroweak [ baryogenesis [ where Λ is a cut-off scale and canonical dimension greater than 4. The lowest dimensional nonrenormalizable opera- JHEP04(2016)011 , is × H ): 0 t † x (2.3) (2.4) (2.5) (2.6) (2.1) (2.2) ( H m vector , X = ∗ 0 2 denoting ρ X i 2 ,Z electroweak σ iσ Y W = ˜ X , with U(1) } 3 × , 2 ., , c 1 . ∈ { + h i ) is fixed to the standard value R , t 4 , ˜ ρ X 2.5 ! L 4 λ . ¯ Q 0 1 , ) + i x 3 2 X ( 0) values only.

ρ µ √ i X 3 κ 246 GeV T D ) ) ρ/ ρ < 2 † x x t + is a complex coupling. ) ( ( i √ Y | ≈ 2 ρ π X iξ v ρ i – 3 – 2 | µ e + 2 t e 2 0 t D invariance, the non-linear realisation of the elec- µ y ) = ( m v , x − 2 = Y h 2 ( ) = ρ t is no longer obliged to form the electroweak doublet = x − H Y ( i ) = h 0) or negative ( ρ U(1) = ρ X (see the next section): h ( ). The standard Higgs doublet then can be identified with × ρ V x ρ > Top 1 ( ) are the three would-be Goldstone bosons spanning SU(2) ρ − x ) we consider modified Higgs potential, which contains tree-level ( i π Higgs 2.3 are the three broken generators for is the third generation left-handed quark doublet, L 3 T i ) field is to be identified with the modulus of the electroweak doublet field, ) δ x L 1 , where one neutral scalar component is the physical Higgs field, while ( coset space. With non-linear realization of SU(2) ρ T , b singlet field − L ) i t Y x σ EM ( 0 = ( = U(1) L i ,H U(1) ) T Q × / x Y ( In addition to eq. ( While maintaining SU(2) We note that if + 1 H as in SM, i.e., it should be restricted to positive ( tation value of the Higgs field The absolute value of thesince vacuum the expectation Higgs value interactions in with ( the electroweak gauge bosons are assumed to be the same cubic term: We assume that the scalar potential has a global minimum for a non-zero vacuum expec- basis of diagonal up-quark Yukawa matrix, looks as: where an additional mass parameter and troweak gauge symmetry allows aA number generic of model new is interactions severely beyond constrainedphysics those by and present the in the electroweak SM. Higgs precision measurements, data.most flavour relevant Therefore, for here the we considerrent electroweak data. only baryogenesis extra and Namely, we interactions are consider which relatively CP-violating are less Higgs-top constrained Yukawa interactions, by which, cur- in the SU(2) the following composite field: where the Pauli matrices and U(1) gauge invariance the Higgsirreducible field representation. We entertain the possibility that the Higgs boson resides in the remaining three components describebosons. longitudinal degrees In of our freedom model, of the later degrees of freedom are combined in nonlinear field JHEP04(2016)011 ) ρ ]). 2.3 42 (2.7) (2.8) (2.9) (2.10) → − S, T, U ρ ]. , established at 37 ] (see also [ 29 41 − . ], we have computed 10 39 × ξ 7 . 8 2 t cos 2 h v < m m t . | y e 2 2 . is the SM top-Yukawa coupling. d | ln √ . Due to the quadratic relation we . Following [ t ). It is then convenient to express SM f t 0 t t − t y d m ξ m m v v m 2.7 2 t 2 y ) explicitly breaks the discrete − √ sin ) (125 GeV) 2 1.2 , and x t t v ( ≈ SM t y y 2 t ρ cm, and y v y · 0 t t e – 4 – 2 = 173 GeV is defined in our model from eq. ( m m ρ

using eq. ( ) = : ξ and − 0 t 27 ≈ x ξ 2 t v − ( V ξ m t 2 h m 10 ∂ m 4 ∂ρ∂ρ · sin 29 sin . and 5 q 4 9 0 . = h covariant derivative. The shifted field 2 2 h ≈ = through Y ≈ m 0 t f 1 2 ¯ e d ¯ m d / = f U(1) ) d × ( 0 t , e.g., in terms of m f 2 ρ v 3 αm | π f 16 Q | is an SU(2) = µ f ¯ d D ]. We take these constraints into account in what follows. Finally, we note that the The Higgs data constraints on CP-violating Higgs-top couplings have been discussed Remarkably, the electroweak precision observables parametrized through the Note that, the cubic interaction term in ( The tree-level top quark mass 38 We found that the most ofwith the the parameter experimental space constraints allowed by on90% the the confidence Higgs electron data level EDM, is using alsoOne polar consistent should thorium also monoxide bear (THO)may in molecules lead mind [ accidental that cancellations possible in anomalous the couplings fermion EDMs. with other SM fermions where in [ new source of CP-violationelectric in dipole the moments Higgs-top (EDMs) of sector chargedthis induces fermions, contribution additional in contribution our to model: the introduced any new particle andrather the than 1-loop Higgs-Yukawa couplings. oblique parametersself-interaction Also, depend couplings the on and current particle data they masses model is are satisfies not unconstrained the sensitive weak at to power present. the counting renormalisability Higgs In as addition, discussed in the [ above symmetry, thus avoiding potentiallyimportantly, dangerous as cosmological will be domain shown wall below,troweak problem. phase it transition plays More a and, crucial together role withing in the in enhancing additional Higgs-top the additional Yukawa first-order interactions, CP-violation elec- appear- leads to the the successfuloblique electroweak baryogenesis. parameters are essentially unaffected in our model. This is because we have not upon expressing the extra mass parameter account an ambiguity in definition of describes the physical excitation associatedsquared: with the Higgs particle with the tree-level mass where JHEP04(2016)011 . ) 2 h ) is m 2.5 (3.3) ( (3.1) (3.2) − 2.5 v . There in terms . One of 2 h . 2 h v λ m vκ > − → − vκ < m v vκ < ) and the anomalous . and = 0. We find that the κ 2.4 i ρ h 2 0 t 2 → − v m κ 0 and, hence, . λ > and the quartic coupling + 3 ln , ) is realised as an absolute minimum 2 0 or, equivalently, ξ vκ µ 0, which, in turn, implies 2 < 2.5 vκ − 2 > 2 h µ + 2 cos m 2 2 h µ ; m ξ 2 2 h – 5 – v 1 m 1 2 2 cos . In this case there are two trivial solutions for the − 2 3 0. t κ λ = = 125 GeV, the Higgs vacuum expectation value π y 0 t 2 − λ ≈ 16 µ m = 2 h ) significantly alter the Higgs vacuum configuration. In this v √ < vκ < m < vκ < 3 2.3 2 h ), 2 h 2 h − m m 3 m κ : − − − κ → = κ vκ ) must be taken into account. At one loop level this can be achieved by 2.3 = 0 ( 2 µ extremum equation, which represent an inflection point of the potential. In this case therealised trivial providing configuration is a local maximum and the minimum ( the local minima inelectroweak this symmetry case breaking is minimum atof ( a the trivial potential configuration if The tachyonic mass parameter, i.e., For The non-tachyonic mass parameter, i.e., 0. In this case the radiative corrections induced by the anomalous Higgs-top Yukawa Within the perturbation theory, one expects that the above tree-level analysis modifies i. ii. iii. ≈ 3.2 Higgs vacuum atThe finite anomalous temperature couplings have eventemperature. more profound To effect illustrate on the the main phase transition point at let finite us first consider 1-loop finite temperature κ interactions ( the substitution: Notice, the symmetry of the above vacuum solutions under insignificantly, except the case when the tree level cubic parameter is vanishingly small, are three cases to consider: The potentially must be bounded from below, that is 3.1 Higgs vacuum atWe find zero convenient temperature to rewriteof the the mass (tree parameter level) Higgsand mass, the cubic coupling The presence of theHiggs-top cubic Yukawa couplings ( term in thesection tree-level we discuss Higgs the potential Higgs ( vacuumelectroweak at phase zero transition. and finite temperatures and the corresponding 3 The electroweak phase transition JHEP04(2016)011 ] . c s T br T T v v 44 , (3.4) (3.5) 43 and , c ρ ) and find s T 2 12 v A T , we will have ρ ξ cos 0 t T m t y ]. From these tables one 2 38 √ . We constraint the variation denotes a critical temperature + 3 1 , c κ 2 t T ξ y + cos + 4 2 0 t ρ λ 1. On the other hand, sphalerons must 2 2 m 1. Here t > T y + 4 < c 2 -dependence on temperature is kept: 2 1 2 t 2 c √ g y /T T . c /T c br + are given in table T – 6 – c br T + 3 . This means that the gauge symmetry is never v s + 4 2 T 2 ξ v ρ v g κ λ 3 = 4 3 and thermal vacuum expectation values, c and + 4 s T t c v 2 1 y T g , ≈ − electroweak gauge couplings. Note that besides the stan- κ + T Y v 2 2 (1) where the thermal effective potential is plotted for various T g 3 1 δV ) + ) + ρ ρ ( ( V V ≈ ) which is linear in field ) = thermal correction to the thermal Higgs potential, we account an additional 3.4 2 according to bounds obtained in our previous work [ are SU(2) and U(1) ρ, T ρ ( ξ 2 1 . Thermal effective potential at various temperatures. Non-zero thermal ground state , T T 2 V g ∼ and We have analysed the full one-loop thermal effective potential (see appendix The above observation has an important implication for the baryogenesis scenario. On t y a parameter area where thevalues above requirements of for the the critical phase transition temperature for are various anomalous satisfied. couplings Some of temperatures. the one hand, we require stronglyto first-order be phase suppressed transition in with the sphaleronbe broken effects effective phase, in [ the i.e. ‘symmetric’ phase,of i.e. the phase transition defined as: which is proportionalThis to is the ilustrated anomalous in couplings figure and is essentially independent of where dard term in eq. ( restored in our framework.non-zero Indeed, expectation even value, for very large temperatures Higgs effective potential where only leading Figure 1 persists at high temperatures. JHEP04(2016)011 - . R a R | ). is i − − b d κ | 2 ss . δn δn B.1 130 (0 ≡ ≡ & a | ≈ B κ D | and reaches | SM y ). κ | Γ and 2 a R B.2 , the right-handed u L t b δn y SM t )]. As the magnitude y δn ≡ 3.3 + a L U = t , see eq. ( , y a L δn , the top quark mass chirality H d y ≡ :Γ δn and Q + , and weak and the QCD sphaleron SM y a L ), since these are generated through h T u h that correspond to a strong first-order , n c δn Q br T decreases too. For very large ≡ v | ≡ c a s H T , in a primordial plasma that undergoes the v decreases with the increase of Q 2 and | c c T ]. The non-equilibrium electroweak phase tran- i s T T – 7 – 7 v µ i g ) are particle (antiparticle) number density of the 1 6 c i n ( ≈ i c i n n , respectively. The strong sphaleron interaction rate Γ ]. These equations describe the evolution of the net parti- ss − 45 i ] produce more baryons than antibaryons. Finally, the net n 44 ≡ , and Γ i 43 ws δn is the statistical factor (2 for bosons and 3 for fermions in thermal equi- i 50 GeV. At the same time, g in our scenario gets modified due to the anomalous Higgs-top Yukawa is the chemical potential. The relevant species initially are the left-handed , the Higgs self interactions at a rate Γ ∼ y i m ) and the neutral Higgs particle ( µ approaches to zero and essentially becomes an inflection point of the effective R t c 6 GeV, the phase transition is not much different from the one in the Standard s T v δn . | 2). These densities are related to the ones of the third generation through eq. ( ≡ κ , | T The rate Γ The quantitative description of the above picture is provided by the solutions of the increases, we observe several values of = 1 κ a as well as for the two( light generation quarks: Therefore, it is suffice to consider the evolution of interactions compared to the Standard Model rate Γ change through the top quark Yukawa interactionflip at at a a rate rate Γ Γ interactions at rates Γ fast enough to maintain chemical equilibrium betweenand left-handed hence and right-handed they quarks, generate net particle number densities for right-handed bottom th particle specie, librium), and top and bottom quarks with nettop particle ( number density the dominant Higgs-top Yukawa interactions. The individual particle asymmetries can broken phase. The sphaleronto transitions avoid must the be wash-out sufficiently of suppressed the inside generated the baryon bubble asymmetry. coupled transport equations [ cle number densities, eletroweak phase transition. Here a bubble wall anddensities these in scatterings front generate of CP theasymmetries wall, (and providing C) diffuse the asymmetries underlying into in theory thesphaleron is particle symmetric CP number transitions non-invariant. phase [ The where CP baryon the charge rapid created outside baryon the number bubble violating wall is swept up by the expanding wall into the Before proceeding to the calculation oflous the Higgs baryon asymmetry couplings, in we the recall modelthe a with so-called the qualitative charge anoma- picture transport of mechanism [ thesition electroweak proceeds baryogenesis through within the nucleationphase and within subsequent the expansion surrounding of plasma bubbles in of the the symmetric broken phase. Plasma particles scatter of values as low as 160 GeV, potential. Hence, the desired first-order phase transition ceases4 to exist for such Computing large the baryon asymmetry small Model. The anomalous Higgs-toporder couplings phase are transition not through the largeof radiative enough contribution either [see to eq. drive ( phase the transition. first- The critical temperature can draw a picture of the phase transition in different areas of parameter space. Namely, for JHEP04(2016)011 ] are taken 38 . Another set of simultaneously. ) for various values ) − . c ( . . . . . − 0 t . . . . ( T 0 t SM SM t t 94 SM t m . y y y . . . . m 4 2 1 9 2 6 2 532 ...... , 213 76 57 . = . . . . − − 8, 174 7, 187 obtained in [ . . 7, 151 3, 190 113 101 128 101 03, 206 83, 238 , 113 . 59 , 221 . . . 82, 200 66, 219 . ξ . = 1 . . ↔ − 187 58 18 9 1 (+) = 0 = 0 5 0 t 6 41 16 3 t t t − − − − − − y m y y (+) and 0 t t m y ...... SM π t 68 60 SM SM π t t . y 5 y y . 25 . 9 63 2 92 and . . . . . 7 81 9 81 0 54 7 58 . . 99 447 152 . . . . , 219 69 52 . c − − − − − − − − . . . . = 0 5, 182 3, 192 2, 209 8, 144 br T 51, 211 . . . 3, 188 = 0 . 113 102 135 107 72, 224 33, 238 . 90, 190 = 0 . ξ v . . 68, 99 80, 219 . | 1 = 0 . . | 159 194 67 23 11 6 1 , = 0 = 0 2 20 ξ 31 4 ξ | t − c | t t − − − − − − 74 y s T y y v in the last table , π κ and the critical temperature . . . . . 5 . . . . . c . . SM t SM SM t t br T y – 8 – y y 4 v 0 55 5 83 4 104 6 . . . . . , 214 9 75 3 83 7 56 0 67 . . . . 77 = 0 , , 224 . . . . . 62 46 . . − − − − − − 9, 187 5, 175 c . . | − − − − − − − − − − − − − − − − − − − − 4, 188 6, 196 0, 212 . . 53 80 98 77 46, 220 44, 206 s . . . T 1, 186 ξ 72 114 110 104 86 58 142 115 77 79, 226 . . . | 666, 238 . v 189 31, 218 5 9 19 61 . ) in the first two tables, as no first-order phase transition 197 = 0 . 73 26 13 7 = 0 = 0 0 − 36 6 t − − − − − t t − − − − y − 2.9 y y 0. For c c c c c c c c c c c c c c c c c c br br br br br br T T T T T T br br br br br br br br br br br br T T T T T T T T T T T T c c c c c c c c c c c c c c c c c c , v , v , v , v , v , v T T T T T T v > , v , v , v , v , v , v , v , v , v , v , v , v T T T T T T T T T T T T c c c c c c c c c c c c c c c c c c s s s s s s T T T T T T s s s s s s s s s s s s T T T T T T T T T T T T v v v v v v from eq. ( v v v v v v v v v v v v (+) 2 h 0 t 2 h 2 h 5 0 5 0 5 1 1 5 0 5 0 5 5 5 ...... m ...... 1 2 3 /m 2 2 1 1 0 0 /m /m | 0 0 1 1 2 2 0 1 2 | | − − − v − − − − − − v v | − − − − − − − − | | and positive κ κ κ ) − ( 0 t corresponding to the strong first-order phase transition. The dash sign indicates that ξ m . The thermal expectation values and t y , κ of the transition is a crossover orinto of the account. second-order. We Constraints use on is found for valiable solutions is obtained by revercing signs of Table 1 JHEP04(2016)011 ] 0 ]. ). in ws h 48 ) is 49 [ is a 3 and, D z (4.3) (4.1) (4.2) | 3 ( z > 2 v /T ]: | g 10 2 + 7 ). It will g = 49 q CPV × t = 1 , S 3 D λ 2.3 ≈ . 48 V 5 √ (9 ) − 1 ≈ 16 ω T , ( # T is ) rot = ) 0 0 ) sph z z ¯ ( E D ( k NV − is given by [ CPV t ) CPV t z T S S ( ) ) in the appendix) evaluated v ) 0 0 exp z z ], while 7 − − B.6 z 26 and (   z CPV 47 t ( ( [ | + − ≈ T is the wall’s velocity. k k Γ T 2 T w v e e dzS | 2 4 tr v w 0 D g 0 π − v N 4 is still fast enough and we employ the 4 0 dz −∞ dz 10 y ) is the effective diffusion constant given Z respectively. The weak sphaleron rate Γ z ∞ × q 1 + in our case is larger than the Standard Model one, −∞ z rot c 4.1 9 D ) λ Z Z diffusion constants as s . ). Ignoring the weak sphaleron rate inside the T s 160 GeV. v h B ∓ − dz dz – 9 – !" NV = 4 D 1 ( 1 30 0 0 ss and − tr ss   −∞ −∞ in eq. ( c Z Z N D | ∼ br T w ¯ D 64Γ 3 κ 4 D v + − v | 2 k k − w w − − = T

v v k k π w 1 for 4 and Higgs − − α − − k we found the rate Γ − q ws | 2 2 7 + − w + + . t ] related to the Higgs-top anomalous interactions ( 5 is the weak isospin fine structure constant; D k k v k k 0 3 . y q q 3Γ | 46 T ∼ − ] (see also appendix | 29 D D − 2 / T + + 45 1 v xω ), the Higgs quartic coupling | 4 λ/g 2 = ≈ g 3.2 √ are the effective Higgs decay rate (cf. eq. ( This readily follow from the fact that sphaleron solution in our case does B π ≈ − n 4 2 / 2 2 ) using the appropriate sph g Γ and Γ B.7 = is a coordinate normal to the bubble wall in the wall’s rest frame, with . Even for small 129. Hence, + . w z 0 03 contains the contributions of the positive modes of the fluctuation operator [ 0) being the broken (symmetric) phase, and α . SM y 0 ≈ According to eq. ( 2 ≈ 4)Γ . SM z < The numerical values for theto above a factors reasonable are estimated accuracy,cubic within serve coupling. the our SM case for as well, especially forλ not too large anomalous Here dynamical pre-factor which is related tofluctuation the operator absolute around value of the the sphaleronrepresent negative solution; normalisation eigenvalue of factors the related tox the translational and rotational zero-modes and The finite temperature electroweakthe sphaleron case rate of within non-zero a Higss thermal thermal volume vacuum expectation value where Γ with eq. ( is discussed in the next section. 4.1 The electroweak sphaleron rate where ( the CP-violating source [ be discussed in moredefined details below. in terms ofThe quark strong sphaleron rate is given by Γ approximation of ref. [ bubble is negligible, the net baryon number density in the broken phase is computed to be: 1 JHEP04(2016)011 . C , we (4.4) (4.5) (4.6) c T 97] for . fixed as 1 = λ ≈ T 0) ) with , λ, 2 ) is comparable or g ( , λ, κ 4.3 B ). The Standard Model 2 7 ) is applicable for both g / = ( 3.2 1 ( B 4.5 λ | SM c | s , T v B | , v ) | | T for which the sphaleron transition v c | , λ, κ for fixed 2 055 37 /T g . at a critical temperature κ | ( s 0 ∼ s T B T v | | dependence of & | | 4 w v T v 26 the the non-exponential factor becomes 2 | α κ | v . g | π | 0 s 4 ) with estimation ( c ∼ s T T sph . v for – 10 – | 97. 4.3 E c . 2 1 SM 7 37 /T | ≈ ) = s − = 0) = s T T 0) ) as a function of v ( T ( 4.4 , λ, sph exp 2 sph g E )( ( E | B c c s T T , λ, κ = v 2 | g ( SM B B parameter. See figure 1. However, for κ . c 160] GeV. Recall that eq. ( , /T ): | ). [6 s s . The factor T 4.4 v ∈ 3.2 ) during the phase transition. By demanding that the rate ( κ The sphaleron energy at finite temperature is computed by scaling the zero temperature 4.3 − To avoid a largerequire exponential suppression ofeven the smaller. left-hand Therefore, we side find of a finite this range of equation we must for broken and symmetric phase.in In ( the symmetric phaselarger we than would the like to Standardobtain: avoid Model suppression rate Γ in eq. ( energy ( is close to thea corresponding wide value range in of the Standard Model [ Figure 2 value for the factor is not differ much fromConsequently, the the zero one temperature in sphaleron the energy SM, as it is explicitly demonstrated in appendix JHEP04(2016)011 ): 3.4 (4.7) (4.8) (4.9) ] and c (4.12) (4.11) (4.10) /T 16 , c /T , thinner is the κ [3 2 ∈ . ξ w 0 L (inside the bubble) and . → ) sin ) at a critical temperature ξ z z , ( , ( | h ) ρ 2 h — larger is 2 z t w vκ . z ]: ( m 0 as → −∞ | y κ t L √ θ 1 z 46 z → ∂ . 1 + ) as + | z c s 2 s ( c br T s T 2 t | CPV t h T v v ) entering the formular for the baryon | ξ S m z m κv ( | w 3 1 + tanh ), ξ . v 7 w . We observe that the width of the wall is ≈ ) cos c | / 2.3 CPV 1, so the shpalerons are ineffective inside the t br z T 1 | v ( ξ c v ) cos T γ S is to ensure that the expanding wall creates non- – 11 – s T h z & 2 | 2 ( w − 2 c | ) are the modulus and phase of the complex top (outside bubble), as it should be. The wall width t 3 π ). We note that the CP-violation is entirely defined c | v h λv s T λ v c y z 2 t ) sin z | √ v ( s T ( c y | z √ /T ∞ θ v s T ( ρ | 3 ≈ + 6 + c v + + h ) and takes the form [ 0 t br ) T t + κ 0 t z v y , c 2 | m ( | → ) 2.3 br m T 055 z . 2 v ( ) and z 0 t = z √ s iθ ( CPV t t w as e ) = S ). In our model it originates from the CP-violating anomalous ) L c m z z s T ( ( ) = ) = ; 4.1 t v ρ z z 2 w ( ( m v t t θ m − ) = 1 tan z as parameters and vary them in the expected ranges 3]. The lower bound on ( p t / √ w of the anomalous interactions ( v / M ]). We do not attempt to perform such calculations in this paper. Instead, we 1 ξ , = 1 52 3 and − w ]–[ γ w 50 [10 L Ignoring the wall curvature, the bubble wall configuration can be analytically computed when the effective potential is approximated by eq. ( ∈ w w keep v equilibrium, while the upper boundThe wall corresponds must move to subsonic the in order soundin the speed front CP asymmetric in of particle relativistic the number densities, plasma. wall, created to diffuse efficiently into the symmetric phase. where in the lastessentially defined step through we the assume wall. anomalous cubic More coupling the precise determination definition of of thewall. wall the The velocity wall typical requires calculations width more ine.g., various careful requires models [ study contain numerical large of theoretical calculations. the uncertainties (see, dynamics of Also, the the symmetric vacuum L has the following simple form: This configuration approaches the broken vacuum in the background of the bubbleby wall a phase where quark mass, 4.2 CP-violation viaNext the we anomalous discuss Higgs-top the couplings number density CP-violating in source eq.Higgs-top ( Yukawa interactions ( In the broken phasebubble. we require is unsuppressed in the symmetric phase: JHEP04(2016)011 ] . 2 , T 38 5 . . w = 0 , where L h 2 /s 100 count- B ]. However, /m 8 5 8 5 6 8 8 n | ∼ SM SM t t − − − − − − − v 38 computed with | ∗ y y = . κ g 10 10 10 10 10 10 10 B 76 57 B 3 . . η × × × × × × × η subject to constraints 41 38 35 01 59 24 21 = 0 = 0 ) and ...... t t t 2 1 3 1 1 1 1 y ). In this case the same y y ) (+) t 0 ) is not satisfied and the − ( 0 t m 4.7 9 5 9 5 6 8 m 10 subject to constraints from [ and figures SM SM t t π − − − − − − − π t y y = 2 5 y 25 10 10 10 10 10 10 . 10 . 69 52 . . × × × × × × (+) × 0 t = 0 = 0 m 41 92 98 00 72 48 = 0 = 0 ξ 71 ...... ξ ). For example, the bubble wall width . t t 3 1 3 1 1 3 5 y y , various values of 1.1 π = +246 GeV and 25 5 6 6 8 . 10 10 12 simultaneously. SM SM 45 is the entropy density with v t t − − − − − − − v / y y 3 10 10 10 10 = 0 10 10 10 – 12 – , various values of 62 46 T . . ξ π × × × × ∗ and × × × 5 g . the asymmetry becomes less dependent on 2 κ 02 18 14 02 2 (with = 0 = 0 28 31 10 . . . . = +246 GeV and π . . . w t t 2 6 8 1 − = 0 we have plotted the dependence of the baryon asymme- 8 9 2 v v y y , ξ 5 3 = 2 . 0 s . The asymmetry increases with the increase of the anoma- 2 h 2 h − ξ and for (i) 0 5 0 5 5 . . . . . c 5 0 ); (ii) /m /m = . . ) 2 0 2 2 1 | | 5 (with 0 2 . − /T v v 2 h − − − − − ( ) and | | 2 0 t κ κ − m /m , = 3 4.1 | , the later being subject to the constraints obtained in [ are treated as free parameters, rather than being defined by studying 2 t v | . A relatively large variation in the asymmetry parameter is observed w y − w κ and phase w L , v /s /s 5 t L . B B y 0 and , n n − κ κ ] and and = = = 01 and . With the increase of 38 . either the first-order phase transition fails, or eq. ( w B B 246 GeV and w 2 h for which a significant baryon asymmetry can be produced. There are some large v η η v − κ κ = 0 . Representative numerical values of baryon asymmetry parameter /m = | on w v v | v B We observe that a significant asymmetry can be produced assuming the Higgs anoma- κ η and its velocity is computed in eq. ( w B which reproduces the observed asymmetry inL eq. ( the dynamics of the wall. Intry figure for small lous couplings for large sphaleron rate in therange of symmetric phase reducestheoretical significantly. uncertainties in Thus, our there calculations and,thorough is therefore, scanning only we do of a not the finite attempt parameter here at area more in order to determine the range of parameters n ing the effective number ofThe relativistic results degrees of of freedom numerical in calculations equilibrium are at presented temperature in table lous couplings results are obtained by reversing signs of 4.3 The baryon asymmetry We have all the ingredients now to compute the asymmetry parameter, Table 2 fixed obtained in [ (with and JHEP04(2016)011 and ) can t ) and y produc- 1.1 3.5 ¯ tt h 3. . We then computed √ ξ / 1 production cross section. on the width of the bubble . B w cross section increases, such η v htj . htj 001 . ). In the Standard Model the ¯ tt h – 13 – ). The notable point of our scenario is that the 1 and observed that the measured asymmetry ( ). B 3 η for which the sphaleron transitions are not suppressed. The c does not vanish in the symmetric phase, see eq. ( s T v c 10 TeV. In the worth case scenario, the underlying physics may cross section decreases, while s T v . The wall velocity is bounded 0 ∼ ¯ tt and figure ) of w h v 2 4.7 and CP-violating Higgs-top Yukawa interactions defined by modulus ) or piar of top-antitop quarks ( κ . The dependence of the baryon asymmetry parameter and its velocity . The model is a restricted case of more general Standard Model with the elec- htj . This has important implications for baryogenesis as one must ensure sufficiently ξ w 1 L The critical experimental test of the proposed scenario comes with the measurements We have found the strong first-order phase transition is realised for a wide range measured at LHC in processesquark of ( the Higgs associated productiontion with cross single section top (antitop) isHowever, an in order the of presencebaryogenesis magnitude of scenario, larger a than significant the pseudoscalar couplings, as it is required by the necessary for baryogenesis CPthe violation baryon asymmetry is parameter governedbe by the comfortably accommodated phase withininteractions the (see allowed table range of parameters of theof anomalous the Higgs-top Yukawa and Higgs cubic couplings. The Higg-top Yukawa coupling can be of the anomalous couplingsHiggs (see expectation table value figure fast baryon number violating sphaleronfound transitions a take finite place range in ( the symmetric phase. We to relatively high scales, even escape the detection atfrom the the LHC. electroweak In precision addition, andLHC the electric data. model dipole is moment relatively measurements less and constrained the current In this paper weasymmetry have studied in the the electroweak Standard phasestrength Model transition with and the computed anomalousphase the Higgs baryon cubic self interactionstroweak symmetry with being non-linearly realised. As an effective theory, the model is valid up Figure 3 wall 5 Conclusion and outlook JHEP04(2016)011 , ], in 76 79 (1) T δV ) LHC with 25- 1 − parameters in the di-Higgs t ]. Thus, accurate measure- y 53 ]). At 1-loop level there are two ]. and 78 68 κ 10 times relative to the SM value [ ]–[ ]–[ − 69 3 57 ∼ The recent studies show that the anomalous 3 ]. – 14 – invariant mass distribution becomes less peaked is the SM value for the cubic coupling] there is a partial 75 2 h | ¯ tt v m | h 3 = SM κ cross sections would provide an important indirect evidence invariant mass distribution exhibits the opposite behaviour. [ cross section becomes larger [ ]. Although promising, the measurements of the anomalous SM htj htj κ ], due to the entanglement of htj 5 54 , increases, the 78 . | ] for recent studies), one requires high luminosity to obtain sizeable and 38 ξ κ ¯ tt 56 1 the . , h can be determined at 14 TeV high-luminosity (3000 fb tan | 55 & κ SM | κ ξ channel [ . As tan | htj htj ). This term comprises of 1-loop zero temperature Coleman-Weinberg potential [ 3.4 With the anomalous couplings favourable by our baryogenesis scenario, the cross sec- The measurement of the Higgs cubic coupling at the LHC is even more challenging. and For positive 3 ], especially for the negative cubic coupling. ¯ tt Finite temperature 1-loop correctionseq. ( to the tree-level potential are encoded in cancellation between triangle and box diagrams [ Liu and Nicholas Manton forAustralian useful Research discussions. Council. This workScience AK was Foundation partially was under supported also the by project supported the No. in DI/12/6-200/13. part byA the Rustaveli National Finite temperature effective potential the related mechanism for baryogenesis,as we the ultimately planned need International to Linear resort Collider to or/and more new powerful colliders, 100 such TeV hadron collider. Acknowledgments We would like to thank Masaki Asano, David Curtin, Manuel Drees, Mark Hindmarsh, Tao tion for di-Higgs production can be78 enhanced by Higgs coupling 50% accuracy at most [ production cross section. Hence, to probe the cosmological eletroweak phase transition and diagrams responsible for thisand process: Higgs triangle cubic diagram couplingscoupling. which involves and Higgs-top Thus, the Yukawa in boxdisentangle order the diagrams contributions to which from probe the involve the triangle only and effects Higgs-top box of Yukawa diagrams. the Higgs cubic coupling, one must to sensitivity to the anomalousanomalous Higg-top Higgs-top interactions Yukawa can couplings. be found Other in [ recentThe studies best on way the toof probe double the Higgs cubic production Higgs (for coupling recent works can see be [ is through the radiative process The CP-violating nature of theby anomalous measuring Higgs-top Yukawa the couplings can angularcays also distributions be in of probed leptons resultingHiggs-top from Yukawa couplings polarised are top ratherSo challenging quark together at de- with LHC further due improvementstions in to (see, experimental the techniques e.g., large and [ backgrounds. theoretical calcula- ments of the total on the validity ofanomalous our Higgs-top scenario. Yukawa couplings Anotherh is observable which the carries three-body the invariantat information mass small on distribution masses, the while for the that for JHEP04(2016)011 ρ 2 T ) we (A.6) (A.7) (A.8) (A.1) (A.2) (A.3) (A.4) (A.5) φ ∼ ( i ) are: m ρ ( 2 i . The 1-loop m T T δV i + , /T ) 4 , ρ / ( 2 2 i CW 0. This infrared divergences ρ m 3 2 ) δV → 2 + 1 2 , g − p p = + √ ) for bosons for fermions . ξ . ) 2 2 − ρ (1) = 0 and do not worry about the sub- v 2 ρ T 2 ( e g . i ( 2 R i , 2 i T s 2 i π µ } δV 2 sin m 2 m 2 ], 12 1) x ) = ( ρ i ). In particular, the linear term ρ − − 2 t 80 J ( 2 ( , [ i ). Note that this linear term is different y ln O DR subtraction scheme, looks as: 2 Z 3 3.4 n , − T and field-dependent masses + + , 6 , A.5 1 4 i ]. 2 | , 2 2 2 δV h / v 6 1 π m | 3 81 { i x ), ( ln λρ ξ x X 64 h,W,Z,t n ]. 2 – 15 – 6 π diverges in the infrared = = } 2 i O 84 A.4 } + 3 cos terms in the above expansions and the Higgs and – − 2 √ 4 + A.7 ρ dp p π 2 t x 82 T κρ x 2 y 2 , m ∞ X 2 12 ρ,W,Z,t π /T 0 24 { π 4 ρ,W,Z,t -particle. In the high-temperature limit + 2 + 2 i { Z / = ) + 0 t i ) = 2 2 n − ρ ρ 4 ρ ( µ m ( is taken to be 4 45 2 i π 2 ) = π T − g ) = 2 360 7 R m − ρ µ δV ( ( term discussed in [ /T = ) ) = ) = ) = ρ ρ ρ ρ ρ x CW 3 ( ( ( ( ) = 2 i 2 t h 2 T x δV ∼ 2 W ( m m m i ( ∼ m J i J ), the bosonic logarithm in φ ( and their respective degrees of freedom are: i 4 m , } runs over the the fields which give the dominant contribution to the potential, i.e., denoting the spin of the 5 T i i , and the one 1-loop thermal potential s The 1-loop finite temperature potential takes the form: We work in the unitary gauge by setting the Goldstone fields For 4 5 CW ρ, W, Z, t tleties related with the gaugefields dependence seems of to the be effective numerically potential. insignificant Anyway, [ the contributionare of dealt Goldstone by resumingeffective the potential. multi-loop We bosonic will contributions not which consider result them in here. the so-called ring-terms in the These expansions have been used inoriginates deriving from eq. the ( top quark field-dependent masses, eqs.from ( the spurious with obtain: where, Note that, due toare the no anomalous longer interactions, proportional the to the Higgs Higgs and expectation top-quark value tree-level masses where { The renormalisation scale Coleman-Weinberg potential computed using the δV JHEP04(2016)011 in ]: T (B.7) (B.3) (B.4) (B.5) (B.6) (B.1) 45 : and H Q only [ H . = 0 (B.2) = 0 = 0 . In fact, the first and y T T H 1 T T T CPV CPV H g g t t Γ g , → ∞ , S S − + Q y ss + − h 1 Γ Q Q . Γ Q Q ,Γ , g g 2 ) ) ss − and / − 6 T T T a O t . = 0 m ∂ Q = B B + + + T Γ m w T k k g q , v , Γ = Q Q − = 6 H ) CPV t h a + h 9( 9( − = H S 30 37 D D , z g H T + − T H ∂ + − g g − + Γ T , T T T 32 + 7 g 16 T T = 3 z = = H + − g g m q CPV t 16 Γ a + B Q ) we obtain the equation solely for H D S − + g Q U H g g 9 7(Γ 7 − H ,T − H Q Q – 16 – Q Q 00 g − B.2 − g g = = 2 2 = 2 = + − Γ = Q y y T D Q 1 DH g g Γ B Γ Q ss Q , CPV t g + − ss − 0 S y 6Γ ss = 2 − 0 1 = Γ Γ 3Γ − H Q H w T y g − − w v 0 Γ v + O Q − − − ,D w Q + 0 00 v T T 20 H H − w h v 00 = 24 37 D Q − h q = 00 D D T Q q D : H denotes differentiation w.r.t. coordinate 0 and where: Plugging this into the third equation of ( These equations are furthertwo simplified equations in are the reduced limit toterms of algebraic of large expressions, Γ which allows us to express Here As a result it is sufficient to consider the transport equations for The most general set of transporting equations are simplifications rather can complicated be to made deal based with.sphaleron on the transitions The physics follow- tend motivated approximations. to Theand populate strong right-handed plasma quarks with of right-handed thesities bottom first are quarks related two and with generations. left- those The of left-handed corresponding top net and particle bottom den- and right-handed top quarks as: B Transport equations JHEP04(2016)011 , of 43 via ss 1 ) = Γ z (B.8) (B.9) (C.5) (C.1) (C.2) (C.3) (C.4) B ]: ( (B.10) (B.11) n L O 44 n , ) with an Ansatz ) 43 0 2.4 , z ( ) ρ ) CPV t ( S V ) symmetric ,T ). Setting the fermion 0 ) z − z T ( − 2.4 z , ρ ( (3) 50 Σ) + ( , i ) is given by ( − c ν O k ρ 2 W , D 1 ( e ) W ( − m z b † ) = 0 µ ! 0 V ) ( z 0 0 1 ( ∞ = W dz Σ)

i L H . Evaluating the leading h U ∞ n w abc ( D ∞ Γ ) z v ( ). y 1 z 2 ∞ Z 2 U Γ ( 2 ss ) − ρ , ig 4.1 ζ Γ ) dU ws ( ss 1 ) + ) z + − h Γ ( 0 ζ a f 3Γ µ a ρ 00 z ( µ i ( 2 W v − H W W ρ∂ √ a q f ν ) i σ ∂ z – 17 – ∂ D CPV t ≈ O 2 ( = = 1 2 g S − i 0 B ) ) φ i 2 z 0 n a ν + gauge fields and and 3 z ( dx 28 w ) contribution we obtain: − a a − i ij T , a W v µ z − µ µ ( ) ss W W − ∂ ∂ W + a Γ a k ij ) ,T e σ ) = z ) + 4 ) := := 2 ( W z T z z 0 g ( ( ( 1 4 µ 00 B i y a 2 L µν ρ 63 dz n ) we find that the net left-handed number density Q D ( n q 2 t W x z D ) can be found by using the Green’s function method: 3 B.4 −∞ m d Z ) = 5 = B.5 z Z ( − m 2 k = 1 Γ Q − ) and ( − sph D + E ) + B.1 k z ( 1 ) = Q z 0, the electroweak sphaleron energy of at zero temperature is given by [ ( H ) + → z ] does not significantly modifies in our model. Since we are assuming the Higgs-gauge ( W the following form: are the SU(2) fieldadditional strengths constant for that adjustThe such static that sphaleron the solution integrand can approaches be zero search for asymptotically. by taking an where interactions are the samecomes from as the in cubic Higgs thefields self interaction SM, and term the in time the only componentθ potential modification ( of to gauge the fields standard to analysis zero and working in the SU(2) limit, i.e., solution to which is given in the main textC in eq. ( The electroweak sphaleron In this appendix44 we demonstrate that the standard electroweak sphaleron solution [ The net left-handed quark numberthe density weak is sphaleron converted processes. into the This net process baryon is density described by the equation: Using eqs. ( Q (requires assumption that Γ The solution to eq. ( JHEP04(2016)011 = κ 2 (C.8) (C.6) )] (C.11) 32 GeV. W f ≈ − − | 2 h functions the . v (1 | m h h 2 5 . f , 0 , f − + [ ) (C.7) h W 2 = f f W 2 f 2 κ h v µ − dζ df − ) for (1 2 h 2 h 2 f f ζ . C.8 v κ 2 . 1 2 4 ζ = 0) with dashed curves. These ! + + c − 3 κ h iy 2 ) and ) are presented in figure 4 for ) and ( + )] + λf r W 2 | W f ) v C.7 C.8 f 2 | h 2 2 2 2 ) = 0) = 1 (C.9) (C.10) ζ g iy z − g h h − vf ( + + z x , f , f 2 (1 := 2 )(1 2 µ x W W W ) ζ ) and ( – 18 – f f − f W . The results are shown on figure [ − ( ( W f 0 κ satisfying (

2 f 3 8 C.7 ζ ) → − h r 1 lim h →∞ ζ − lim ζ + (1 , f vf := (1 2 ( W h W 3 f κ f f ∞ W U + dζ = 2 = 2 df 4 ) ) for various h 2 W h 4 f vf 2 " dζ dζ ( df d must satisfy the following equations of motion: C.11 2 4 λ 2 dζ ζ h ζ = 0) are also plotted with dashed curves. ∞ 4 0 , f κ 2 v d Z dζ 2 ζ 2 W | g f v 2 | + g π . We have also plotted the SM solutions ( 4 | v | . The sphaleron functions is a dimensionless radial distance = / 2 h ζ sph m The numerical solutions to eqs. ( 5 E . 0 − two sets of solutions areHaving very found close numerical to solutions eachthe other for sphaleron and the energy curves electroweak in ( sphaleron, figure we 4 essentially have overlap. then computed sphaleron energy takes the form: that guarantee the finiteness of the sphaleron energy. In terms of with boundary conditions where The functions Figure 4 The SM solutions ( JHEP04(2016)011 ] B ] Nucl. , ]. (1994) Phys. ]. , Ann. Rev. , SPIRE (1986) 364] B 430 IN Nucl. Phys. 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