Download pdf - Electroweak baryogenesis with a pseudo-Goldstone Higgs boson

Electroweak baryogenesis with a pseudo-Goldstone Higgs boson

Benjamın Grinstein* and Michael Trott+

Department of Physics, University of California at San Diego, La Jolla, California 92093, USA(Received 1 July 2008; published 22 October 2008)

We examine the nature of electroweak Baryogenesis when the Higgs boson’s properties are modified by

the effects of new physics. We utilize the effective potential to one loop (ring improving the finite

temperature perturbative expansion), while retaining parametrically enhanced dimension six operators of

Oðv2=f2Þ in the Higgs sector. These parametrically enhanced operators would be present if the Higgs is a

pseudo-Goldstone boson of a new physics sector with a characteristic mass scale �� TeV, a coupling

constant 4� � g � 1, and a strong decay constant scale f ¼ �=g. We find that generically the effect of

new physics of this form allows a sufficiently first order electroweak phase transition so that the produced

Baryon number can avoid washing out, and has enhanced effects due to new sources of CP violation. We

also improve the description of the electroweak phase transition in perturbation theory by determining the

thermal mass eigenstate basis of the standard model gauge boson fields. This improves the calculation of

the finite temperature effects through incorporating mixing in the determination of the vector boson

thermal masses of the standard model. These effects are essential to determining the nature of the phase

transition in the standard model and are of interest in our pseudo-Goldstone Baryogenesis scenario.

DOI: 10.1103/PhysRevD.78.075022 PACS numbers: 14.80.Cp, 11.10.Wx

I. INTRODUCTION

The standard model (SM) of electroweak interactionswith a single Higgs field responsible for spontaneous sym-metry breaking is fully compatible with precision data(EWPD). The theory predicts the existence of a new par-ticle, the Higgs boson, which has not been observed.However, the current bound on its mass, mh >114:4 GeV [1], is not in conflict with precision tests.

While this is the case, the issues of the hierarchy prob-lem and triviality problems of the Higgs s ector of the SMstill strongly motivate theorists to think that new physics(NP) will be discovered by LHC at the TeV scale. EWPD,flavor constraints, and the desire for a lack of fine-tuninggenerically pushes the scale of possible NP degrees offreedom to �TeV. EWPD also favors a light Higgs mh &v. If the Higgs is part of a NP sector that addresses thehierarchy and triviality problems with a mass scale � �v * mh, then the model class where the Higgs is a pseudo-Goldstone boson of this NP sector [2–4] is an interestingpossibility to consider.1

Pseudo-Goldstone Higgs (PGH) models would be morecompelling if the PGH scenario were to address anotherproblem of the SM not by explicit construction but as anatural consequence of the structure of theory. The purposeof this paper is to critically examine recent claims thatPGH models can naturally accommodate a very desirablelow energy effect, namely, the generation of the observed

baryon-antibaryon asymmetry of the Universe at the elec-troweak phase transition (EWPT).For � � v it is appropriate to examine the effect of

these PGH models on the EWPT using an effective fieldtheory. This is appropriate if the scale � and the details ofthe new sector are such that all of the NP effects can bedescribed by local operators modifying the SM. We areinterested in parameter choices of the various modelswhere this is the case. We perform a general effective fieldtheory analysis in this paper and are not wedded to anyparticular PGH model. Some modern examples of modelsof this form are little Higgs models [6–8] possibly includ-ing a custodial symmetry [9]; and holographic compositeHiggs models [10] possibly including a custodial symme-try [11,12]. Generally speaking, when one imposes custo-dial symmetry (SUCð2Þ) the models can be in accordancewith EWPD with a relatively low � scale �� TeV.For this reason, the effective theory we use when inves-

tigating the low energy effect of PGH models is the SMsupplemented with SUCð2Þ invariant dimension six opera-tors that are SUð3Þ � SUð2Þ � Uð1Þ invariant and built outof SM fields. In this paper, we will use Buchmuller andWyler’s [13] version of this higher dimension operatorbasis, although we note that it is overcomplete [14]. Thisapproach allows us to calculate in a relatively modelindependent manner the lower energy effects of this entiremodel class.An important point emphasized in [15] when consider-

ing dimension six operators induced from a PGH model isthat operators that only involve the PGH and derivativesare suppressed by the decay constant f, not the scale �.These scales are related by f ¼ �=g, where gsm � g �4� is the coupling constant of the new sector. This para-metric enhancement of the effects of NP in the Higgs sector

*[email protected][email protected] Higgs is generally an exact Goldstone boson in the new

sector and receives its mass and self-couplings from SM correc-tions in interesting models of this form. For a recent review ofthe physics of pseudo-Goldstone Higgs models see [5].

PHYSICAL REVIEW D 78, 075022 (2008)

1550-7998=2008=78(7)=075022(28) 075022-1 � 2008 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.78.075022

(when g > 1) make the properties of the Higgs a veryimportant probe of such models.

This parametric enhancement is important when consid-ering electroweak Baryogenesis (EWB), as many of theconstraints on EWB are constraints on the self-couplings ofthe Higgs. In the SM, these directly translate into a con-straint on the Higgs mass. In [16], the significant effect ofNP (such as PGH models) on the relationship between theHiggs self-coupling and the Higgs mass was studied indetail. This turns out to be crucial in the pseudo-GoldstoneBaryogenesis (PGBG) scenario we examine in this paper,as it will allow the phase transition to be sufficiently firstorder for the produced baryon-antibaryon asymmetry toavoid washing out while the Higgs mass is greater than114.4 GeV.2 If the PGHmodel also contains new sources ofCP violation, these sources of CP violation are also para-metrically enhanced and PGBG could occur for a scale f inthe range 500 GeV & f & 1 TeV.

Our results agree with some aspects of a recent study byWells et al. [17], although we do find some disagreementsand improve upon the analysis in a number of ways (seeSec. ). The most important improvements are that wedetermine and use a thermal mass eigenstate basis of thegauge boson fields of the SM that distinguishes betweenthe transverse and longitudinal masses; and we also use acanonically normalized Lagrangian incorporating the ef-fects of the parametrically enhanced kinetic sector opera-tors. This latter improvement expands the allowedparameter space for PGBG considerably.

The outline of this paper is as follows. In Sec. II, wereview the deficiencies of the SM with regards to EWB. InSec. III, we state the effective theory that we will use toinvestigate the low energy effects of PGH models. InSec. IV, we review some PGH models that can induce theWilson coefficients of interest. In Sec. V, we derive theeffective potential including finite temperature effects andimprove the SM calculation of finite temperature masses.In Sec. VI, we examine the EWPT in the low energyeffective theory and determine when it can be first order.In Sec. VII, we discuss the necessary conditions on the lowenergy realization of PGH models so that the producedbaryon asymmetry will not wash out and briefly discuss thelow energy effect on bubble nucleation. In Sec. VIII, weconclude.

II. SM BARYOGENESIS AT THE EW SCALE

Following the initial realization that anomalous electro-weak (EW) baryon number violation is large at high tem-peratures [18,19], the possibility of EWB [20] proceedingthrough a first order EWPTwas suggested. This suggestion

grew into a promising theory [21], see [22] and referencestherein for a summary. However, this elegant theoreticalmechanism is now ruled out for the SM with one Higgsdoublet. A brief review of the problems of EWB in the SMis appropriate before we reexamine EWB in the context ofNP. In the SM, baryogenesis at the EW scale has a numberof serious problems. For baryogenesis to take place, theSakharov conditions [23](i) C and CP violation,(ii) baryon number violation, and(iii) a departure from thermal equilibrium,

must be satisfied. All three of these conditions are quali-tatively present in the SM with a light Higgs. C [24,25] andCP violation are present [26]. A departure from thermalequilibrium and baryon number violation was potentiallypresent [20] in the EWPT if the Higgs self-coupling wassufficiently small. In the SM, this requires a small Higgsmass, mh & 70 GeV.In part due to the Higgs mass bound mh > 114:4 GeV

[1], it is known that the SM alone, quantitatively, does notgenerate the correct baryon asymmetry observed in theUniverse. The current value of the baryon asymmetry at95% C.L. is determined from big bang nucleosynthesiscosmology [27] and 3 yr WMAP data [28], to be

YB ¼ �B

s¼ ð6:7; 9:2Þ � 10�11 BBN;

ð8:1; 9:2Þ � 10�11 WMAP;(1)

where �B is the baryon number density of the Universe,and s is the entropy density of the Universe. The number tocompare for the SM baryon asymmetry due to EWB iszero. This is due to the following problems:(P1) The EWPT in the SM is not first order as the Higgs

mass exceeds 70 GeV. This statement is based on latticesimulations [29–32], and similar conclusions are reachedin perturbative studies. Therefore, the required departurefrom thermal equilibrium is not present in the SM.(P2) Even if the EWPT was weakly first order, EWB

could not occur. The EWPT must be strongly first order forthe resulting YB to not wash out, and this requires a Higgsmass that is quite small. Washout can occur as thermalBoltzmann fluctuations can erase a generated YB. Thesefluctuations depend on the energy of the EW vacuumbarrier field configurations [33,34] as expð�Esph=TcÞ(where Esph is the sphaleron energy) and translate into

the following bounds on the expectation value of Higgsfield at the critical temperature of the phase transition Tc

h�ðTcÞiTc

* b: (2)

Here, b is a numerical constant estimated to be in the range1:0 & b & 1:3 [35] from the uncertainty in the calculationof the functional determinant associated with the staticsaddle point solution of the Yang-Mills Higgs equations,the sphaleron. This solution translates into a constraint on

2We restrict our study to Higgs masses 114:4 GeV & mh &160 GeV. The upper bound on the Higgs mass is dictated byinsisting the perturbative methods employed are under control.This issue is discussed in Appendix A.

BENJAMIN GRINSTEIN AND MICHAEL TROTT PHYSICAL REVIEW D 78, 075022 (2008)

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the Higgs mass in the SM as [36]

h�ðTcÞiTc

� 4Ev2

m2h

; (3)

where E ¼ ð4m3w þ 2m3

zÞ=ð12�v3Þ, which gives

mh & v

ffiffiffiffiffiffi4E

b

s; (4)

yielding bounds of mh & 35 GeV for b ¼ 1:3 and mh &39 GeV for b ¼ 1:0.3

(P3) Finally, CP violation in the SM is far too small forEWB [38–40].

A. New mechanism or Higgs effective field theory?

Comparing these problems with our baryon-antibaryonasymmetric existence, one could conclude that a totallynew mechanism such as leptogenesis [41] or perhaps su-persymmetry [42] is involved in the generation of YB.Leptogenesis is also an appealing mechanism and willbecome much more so if neutrinoless double beta decay[43] is unambiguously established in future experiments.The window for the minimal supersymmetric standardmodel to allow EWB is now quite constrained [44].Minimal supersymmetric standard model baryogenesis re-quires a Higgs mass mh & 115 GeV.

Another solution to problems P1, P2 could be that theHiggs would have properties that deviate from the SM. Ifthe Higgs couples to a NP sector with a mass scale ��TeV, then the properties of the Higgs are naturally ex-pected to deviate from the SM below the scale �. Thequestion in detail becomes to what degree must the prop-erties of the Higgs effective field theory (HEFT) Higgsdeviate from the SM Higgs for EWB to occur, and hownatural is the required deviation in a NP setting?

The naive expectation that the properties of the effectivetheory Higgs will deviate insignificantly from the SMHiggs fails in many model extensions of the SM, see[14,16,45–51]. In particular, in PGH models with a newstrong interaction at a TeV, there is a parametric enhance-ment of the NP effects on the properties of the Higgs [15].When this is the case, the properties of the Higgs in theeffective theory can change dramatically, and the problemsof EWB can be addressed as follows:

(P1) The relationship between the Higgs mass and theHiggs self-coupling is significantly relaxed in the effectivetheory [16]. A first order EWPT is possible due to a smalleffective Higgs self-coupling, while mh > 114:4 GeV; wewill show this by determining the required conditions on

the dimension six operator Wilson coefficients so that theEWPT is first order. To accomplish this we utilize ringimproved finite temperature perturbation theory and ana-lytically study the EWPT in our effective theory. We alsodetermine the relevant thermal mass eigenstates for thescalar and gauge boson masses that are crucial to determin-ing the nature of the phase transition. This improves theSM calculation of the EWPTand is of interest in our HEFT.(P2) We also determine the constraint on the coefficients

of dimension six operators from the washout condition inour effective theory. There are several effects that modifythe determined value of h�ðTcÞi in the effective theory. Weinclude Higgs self-energy loops, which are large and ne-glected in the expression E ¼ ð4m3

w þ 2m3zÞ=ð12�v3Þ.

Constraints on the Higgs mass are significantly effectedas NP changes the order of the polynomial of the effectivepotential, and the thermal mass basis we derive signifi-cantly modifies the determined value.(P3) We are assuming that the Higgs is a pseudo-

Goldstone boson of a new sector that can have the requirednew sources of CP violation. The low energy expression ofthe new CP violation is through operators that are alsoparametrically enhanced. The required scale suppressingthe NP operators for the SM to be supplemented withenough CP violation is in the range 500 GeV &f & 1 TeV, according to recent independent studies[52,53]. This does not contradict the current bounds onnon-SM CP violation from electric dipole moment (EDM)experiments, see [52,53].One would expect 500 GeV & f & 1 TeV in PGH

models with a new strong interaction at �TeV. When fis in this range, the effective Higgs self-coupling can besmall enough for a strong EWPT, while mh > 114:4 GeV.With this outline of our approach in mind, we first deter-mine the operator basis that expresses the low energyeffects of a new strong interaction at a TeV in the nextsection.

III. THE LAGRANGIAN DENSITY WITH D ¼ 6HIGGS OPERATORS

We now construct the Lagrangian density of a HEFT dueto integrating out the degrees of freedom with massesgreater than v of a PGH model. The SM Lagrangiandensity is given by

L 4� ¼ ðD��ÞyðD��Þ � Vð�Þ; (5)

where � is the Higgs scalar doublet. The covariant deriva-tive of the � field is given by

D� ¼ 1@� � ig12B� � ig2

�I

2WI

�; (6)

where �I are the Pauli matrices,WI�, B� are the SU(2) and

U(1) SM gauge bosons, and the hypercharge of 1=2 hasbeen assigned to the Higgs. The Higgs potential at treelevel is given by

3The more optimistic end of this estimated bound �ðTcÞ=Tc *1 is frequently used in the literature. We will treat the constraintas exact and consider both ends of the bound when we considerthe effects of NP. The effects of NP inducing the ð�y�Þ3operator on the sphaleron bound were studied in [37] and foundto be negligible, so we retain the SM bound.

ELECTROWEAK BARYOGENESIS WITH A PSEUDO- . . . PHYSICAL REVIEW D 78, 075022 (2008)

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Vð�Þ ¼ �m2�y�þ �1

2ð�y�Þ2: (7)

We expand the real field hðxÞ around a real constantbackground field value ’ in Landau gauge introducingthree real Goldstone boson fields �iðxÞ

�ðxÞ ¼ 1ffiffiffi2

p �1ðxÞ þ i�2ðxÞ’þ hðxÞ þ i�3ðxÞ

� �: (8)

When the tree level masses are fixed by a minimization ofthe SM potential, they are given by

m2hð’Þ ¼

�1

2ð3’2 � v2Þ; m2

�ð’Þ ¼ �1

2ð’2 � v2Þ;

m2Wð’Þ ¼

g22’2

4; m2

Zð’Þ ¼ðg21 þ g22Þ’2

4;

m2i ð’Þ ¼

f2i ’2

2:

(9)

In the SM, one has ’ ¼ v � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m2=�1

p.

We utilize Landau gauge, as this gauge choice allows usto avoid subtleties that occur in unitary gauge in finitetemperature field theory, see [54,55]. The gauge fixing isperformed by taking � ! 0 for the Lagrangian term

Lgauge ¼ � 1

2�ð@�Wi

� � �

2g2’�

iÞ2

� 1

2�ð@�B� � �

2g1’ðhþ i�3ÞÞ2: (10)

We now turn to the low energy effect of PGH modelsinducing parametrically enhanced higher dimension opera-tors. The effective Lagrangian density with operators thatcontain Higgs doublets is

L � ¼ L4� þL6

�

�2þO

�v4

�4

�; (11)

with the dimension six Lagrangian density (recall g is thecoupling constant of the new sector and gsm � g � 4�)

L6� ¼ g2C�@

�ð�y�Þ@�ð�y�Þ � g2�2

3!ð�y�Þ3

þ ChG

2ð�y�ÞG�G

� þ Ch ~G

2ð�y�ÞG�

~G�

þ ChW

2ð�y�ÞW�W

� þ Ch ~W

2ð�y�ÞW�

~W�

þ ChB

2ð�y�ÞB�B

� þ Ch ~B

2ð�y�ÞB�

~B� þ � � � ;(12)

where the Wilson coefficients are independent of g. Wehave written the custodial symmetry [56,57] (SUð2ÞC)preserving terms involving only Higgs doublets and fieldstrengths. As mentioned, approximate custodial symmetryis favored as it will suppress the T-parameter operatorð�yD��Þ2 that contributes to the � parameter [15,16].

The PDG quotes �0 ¼ 1:0002þ0:0007�0:0004 for the global fit

[27] of EWPD, and this operator is only suppressed bythe decay constant scale. The coefficient of this operatorhas been determined [16,58] to be C< 4� 10�3 forf ¼ 1 TeV.4

Thus, if SUCð2Þ is not approximately preserved in ex-tensions of the SM of the form we are discussing, thisoperator would have to be suppressed by fine-tuning, or thedecay constant scale would have to be quite high. With anapproximately SUCð2Þ invariant NP sector however, thestrong decay constant could be as low as f ¼ �=g�500 GeV. In this case, the effects on the Higgs sectorself-couplings are very significant [16]. This is the scenariowe are interested in. We do not consider this to be a strongassumption, as SUCð2Þ is also approximately preserved inthe SM.In the SM, SUCð2Þ is only an approximate symmetry as

custodial symmetry is violated by the U(1) and Yukawainteractions. The dimension six operators involving mod-ifications of the Yukawa sector of the SM provide a furthersource of CP violation required for EWB, see [52,53].These operators and their Hermitian conjugates are alsosuppressed by f and are given by

Oe� ¼ g2ð�y�Þð �‘e�Þ; (13)

Ou� ¼ g2ð�y�Þð �qu�Þ; (14)

Od� ¼ g2ð�y�Þð �qd�Þ: (15)

So long as f is in the range 500 GeV & f & 1 TeV, theseoperators can supply the extra CP violation for EW baryo-genesis in our PGBG scenario.Further distinctions can be made on the L6

� operator

basis. All of the operators of interest can come from under-lying tree-level topologies [60]; thus, their Wilson coeffi-cients need not be suppressed by factors of 16�2 in navedimensional analysis [61]. One expects the field strengthoperators that must be induced by loops to be significantlysuppressed compared with the parametrically enhancedoperators in the Higgs sector for this reason.5

We also note that an important aspect of PGH modelsthat has been neglected in some of the literature on theEWPT [17,37,47,62] is that operator extensions of the SMinduce a noncanonical effective Lagrangian. This is due tothe presence of dimension six kinetic operators. As in[15,16,63], we normalize the kinetic term of the resultingLagrangian for h to 1=2. We use the field redefinition

4The need for a SUCð2Þ symmetry in models of this form (andmany other SM extensions) has been appreciated for quite sometime, and many PGH models, such as [9,12,59] incorporate thissymmetry by construction.

5For PGH theories, phenomenological signals involving fieldstrength operators have been examined in [15].


075022-4

hðxÞ ! h0ðxÞð1þ 2 ’2

f2C�Þ1=2

: (16)

This gives the potential, before minimization fixes ’, theform

VCðh0;�0iÞ ¼�m2

2

� Xi¼1;3

�2i þ

�’þ h0

ð1þ 2’2

f2C�Þ1=2

�2�

þ�1

8

� Xi¼1;3

�2i þ

�’þ h0

ð1þ 2’2

f2C�Þ1=2

�2�2

þ �2

48f2

� Xi¼1;3

�2i þ

�’þ h0

ð1þ 2’2

f2C�Þ1=2

�2�3:

(17)

Neglecting constant terms and expanding in the f ! 1limit, while retaining only 1=f2 terms, one has for h

VCðh0Þ ¼ ahð’Þð’3Þh0 þm2hð’Þ2

h02 þ ’�eff3

3!h03 þ �eff

4

4!h04

þ 15�2

5!f2’h05 þ 15�2

6!f2h06; (18)

where the parameters ahð’Þ, m2hð’Þ, and the effective

couplings are given by

ahð’Þ ¼ �1

2

�1� v2

’2

��1� ’2

f2C�

�þ �2

8f2

�’2 � v4

’2

�

m2hð’Þ ¼

�1

2ð3’2 � v2Þ

�1� 2

’2

f2C�

�þ �2

8f2ð5’4 � v4Þ

�eff3 ð’Þ ¼ 3�1

�1� 3

’2

f2C�

�þ 5

2�2

’2

f2

�eff4 ð’Þ ¼ 3�1

�1� 4

’2

f2C�

�þ 15

2�2

’2

f2:

Note that we have eliminated m with Eq. (34). We use thezero temperature minimization condition that determinesm2, even though we are interested in the inclusion of finitetemperature effects. The inclusion of finite temperatureeffects shifts the mass for the Higgs field but does notchange the minimization condition that determines m2, asthe UV subtraction is defined when T ¼ 0. Also, we ne-glect the higher order Coleman-Weinberg terms in theminimization condition. Similarly, for each �i field one has

VCð�0iÞ ¼

m2�ð’Þ2

ð�0iÞ2 þ ð�0

iÞ4��1

8þ �2’

2

16f2

�þ ð�0

iÞ6�2

48f2;

where we have

m2�ð’Þ ¼ �1

2ð’2 � v2Þ þ �2

8f2ð’4 � v4Þ; (19)

and there are many cross terms in the potential that we havenot written here for brevity.

We suppress the primes on the redefined fields for theremainder of the paper. For the background field ’, wehave

VCð’Þ ¼ �m2

2’2 þ �1

8’4 þ �2

48f2’6: (20)

Once we obtain the full effective potential in Sec. VI, itwill be clear that neglecting to reduce the effectiveLagrangian to a canonical form will effect the h’i.Canonically normalizing the effective Lagrangian alsoeffects the crucial relationship between the Higgs massand the Higgs self-couplings. As both the effects of C�

and �2 are enhanced by the same parameter (g2), oneshould not neglect the effects of canonically normalizingthe HEFT Lagrangian when examining PGBG.With this effective theory, we can investigate the low

energy effects of PGH models on the EWPT and thewashout condition. Firstly, we examine some examplesof integrating out degrees of freedomwith mass scales�>v and matching to induce the HEFT we are discussing

IV. MODEL PARTICULARS

As we have explained above, our analysis is based on aLagrangian with precisely the same field content as that ofthe SM. It has been supplemented with additional terms,irrelevant operators characterized by the dimensionfullparameter f. The advantage of this approach is that onecan study the conditions for successful baryogenesis with-out specifying a specific ‘‘ultraviolet completion,’’ that is,without committing to one specific model of interactionsbeyond the standard model. All that is required is then thatthe model includes a light Higgs and that the parameters ofthe resulting low energy effective Lagrangian fall in acertain range, as will be shown in Figs. 7–10.It is easy to display simple UV completions of the

effective Lagrangian under study. A minimalistic exampleconsists of the SM supplemented by a neutral, real scalarfield S and additional terms in the Lagrangian

�L ¼ 1

2@�S@�S� 1

2f2S2 � 1

3!1fS

3 � 2fS�y�:

(21)

A quartic self-interaction for S can be added to make thepotential bounded from below, but is irrelevant for ourpurposes. Integrating out the scalar S at energies belowits mass f, one obtains an effective potential of the form ofEq. (12) with the couplings

�1 ¼ �� 22

2; (22)

g2�2

�2¼ �1

32

f2; (23)


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g2C�

�2¼ 2

2

2f2; (24)

where � is the quartic Higgs self-coupling before theeffects of the S field are included. Hence, we see that thissimple model produces, at lowest order, only the terms inthe effective Lagrangian that play a significant role in ouranalysis of baryogenesis but does not give any other termsincluding notably those which could be significantly con-strained by precision tests of the EW sector. This model ofcourse does not address the hierarchy problem. Models thatinvolve a new strong interaction can address the hierarchyproblem and can also generate the EW scale throughdimensional transmutation and are more compelling.

The possibility that a light Higgs is a composite particlewhose constituents are bound by a new interaction thatgoes strong at a scale �1 TeV has been extensivelystudied; for a review see [5]. In most of these models, theHiggs mass remains small compared with the scale of thenew strong interactions because it is a pseudo-Goldstoneboson of a global symmetry broken only weakly (typicallyby the EW and Yukawa interactions in the SM). We candetermine which of these models have succesful EW bar-yogenesis by determining their effective Lagrangian. Inparticular, we need to know the magnitude of f2, �2, andC�. As we shall see, models in which the precision EW

constraints are evaded by adopting a large-scale, f�10 TeV, require unacceptably large coefficients of �2

and C�. However, since the precision EW constraints are

most severe for the � (or T) parameter, the scale f can betaken significantly smaller in models with an SUCð2Þ sym-metry that automatically suppresses corrections to �. Letus consider some examples of strongly coupled pseudo-Goldstone Higgs theories with custodial symmetry.

The littlest Higgs with custodial symmetry [64] is atheory with flavor symmetry SO(9) in which it is assumedthat techni-strong interactions induce a condensate thatbreaks flavor to SOð5Þ � SOð4Þ. An SUð2Þ3 � Uð1Þ sub-group of the flavor group is gauged weakly, but some ofthese gauged symmetries are spontaneously broken at thescale of the condensates so that, in fact, only the SM gaugegroup remains unbroken and, of the original Goldstonebosons, only the Higgs doublet remains light. At lowenergies, this model is of the type we are investigating,with the couplings

g2�2

�2’ �1

f2; (25)

g2C�

�2¼ 1

4f2: (26)

In the above, the exact expression for �2 has not beencomputed; the expression above satisfies the counting rulesof [15].

The holographc Higgs model [10–12] is a warped 5Dtheory with 4D-boundaries. A bulk SOð5Þ � Uð1Þ � SUð3Þgauge symmetry is broken to Oð4Þ � Uð1Þ � SUð3Þ on theUV boundary and to the SM on the IR one. Matching to thelow energy 4D effective theory gives [15]

g2�2

�2¼ c

�1

f2; (27)

g2C�

�2¼ 1

2f2; (28)

where c ¼ 0, 1 in the models of Refs. [11,12],respectively.6

In these examples of matching, we find that, firstly, thesymmetry breaking that induces �2 is proportional to �1;and secondly, C� is positive. However, our small number

of examples are in no way exhaustive of all PGH models.In particular, models of the little Higgs form, by construc-tion can have a symmetry breaking structure that is quitesurprising due to collective symmetry breaking, see [5].Thus, we will perform our effective theory analysis in twocases.C1: In the first case, we will retain the maximum model

independence that can be afforded in the PGH structureand allow �2 to be independent of �1.C2: In the second case, we will impose that �2 is

proportional to �1 and determine a constraint on ~�2 ¼�2=�1 and C�.

We now turn to the construction of our effective poten-tial for the low energy effective theory of PGH models.

V. EFFECTIVE POTENTIAL

We now calculate the effective potential to one loop todetermine the leading quantum corrections to the classicaltree level potential. Studies of this form were inauguratedby Coleman and Weinberg in [67], and several reviews ofthe application of the effective potential in studies of theelectroweak phase transition exist [22,55]. As well as theone loop temperature independent terms, there are also oneloop finite temperature terms determined using thermalfield theory, see [68–70]. First, we consider the tempera-ture independent effective potential.

A. One loop effective potential

The effective potential is determined as the sum of 1PIdiagrams with arbitrary numbers of external legs and zeroexternal momenta as shown in Fig. 1. We will renormalizethe one loop contributions to the effective potential term by

term using dim reg with d ¼ 4� 2� and MS.

6Models of this form can also supply a dark matter candidate[65] and can also increase the strength of the phase transitionthrough other 5D effects in gauge-Higgs unification [66].


075022-6

We neglect terms due to higher dimension operatorinsertions in the above loop diagrams when determiningthe zero temperature effective potential. In the next sectionand in the appendices, we do include the effects of NP inthermal loops. We do this as the latter are significantlynumerically enhanced and have an important thermalscreening effect on the one loop effective potential. Thisreduces the problem with the imaginary part of the effec-tive potential as we will show. These NP effects in thermalloops and the effects of NP that change the relationshipbetween the Higgs mass and the self-couplings in theHEFT are the dominant effects of NP that we areinvestigating.

B. Scalar contributions

The effective potential is determined in terms of theclassical background field ’. The contributions of theHiggs self-interactions to the one loop effective potentialare given by

VeffS;hð’Þ ¼

�4�d

2

Z d4kEð2�Þ4 lnðk2E þ V00

Cð0ÞÞ; (29)

where we have introduced the renormalization scale �.Note that we have rotated to Euclidean space. From theprevious section, we have V 00

Cð0Þ ¼ m2hð’Þ. We perform the

integral to obtain

VeffS;hð’Þ ¼ �m4

hð’Þ32�2

4�ð2� d=2Þdðd� 2Þ

�m2

hð’Þ4��2

�d=2�2

;

where V 00C indicates two derivatives with respect to the

dynamical field h. We find

VeffS;hð’Þ ¼

m4hð’Þ

64�2

�log

�m2

hð’Þ�2

�� 3

2� Cuv

�;

where

Cuv ¼ 1

�� E þ logð4�Þ: (30)

There are also contributions from the three �i fields for thescalar contribution to the effective potential. Each �i fieldgives a contribution

VeffS;�i

ð’Þ ¼ m4�ð’Þ

64�2

�log

�m2

�ð’Þ�2

�� 3

2� Cuv

�:

C. Vector bosons and fermions

The one loop effects due to the spinors that receive theirmass from the vacuum expectation value of the Higgs arewell known, see [22] for a review. The one loop results aregiven by

VeffF ð’Þ ¼ �X

i

3m4i ð’Þ

16�2

�log

�m2

i ð’Þ�2

�� 3

2� Cuv

�:

We neglect all but the top quark contributions. Because ofthe operator Ot�, there are also 1=f2 corrections of the

form �Re½Ct�’2=f2 to the mass, see [14] for a recent

study of these operator effects. As these contributions tothe potential are suppressed by ’2=ð16�2f2Þ, we neglectthem. For the W and Z fields one obtains

VeffV ð’Þ ¼ 3m4

Zð’Þ64�2

�log

�m2

Zð’Þ�2

�� 5

6� Cuv

�;þ 3m4

Wð’Þ32�2

��log

�m2

Wð’Þ�2

�� 5

6� Cuv

�:

The one loop contribution to the effective potential forour low energy theory is thus

Veffð’Þ ¼ VCð’Þ þ VeffS;hð’Þ þ 3Veff

S;�1ð’Þ þ Veff

V ð’Þþ Veff

F ð’Þ:

D. Renormalization

We are using MS and dimensional regularization todefine the UV subtraction in the T ! 0 limit. The UVcounter terms are given by

L c:t: ¼ �þ m2’2 þ �1’4; (31)

where the counterterm parameters are given by

� ¼ m4

16�2Cuv m2 ¼ � 3�1m

2

64�2Cuv

�1 ¼ 3Cuv

64�2

��21 � f4t þ g42

8þ ðg21 þ g22Þ2

16

�:

(32)

The first renormalization condition defines the vacuumexpectation value of ’. Although we will retain an unfixed’ when examining the electroweak phase transition, as acheck of our results so far we can minimize the potential inthe T ! 0 limit, while neglecting the higher order effectsof the one loop effective potential terms. We interpret theeffective potential as a function of � ¼ hþ ’ and mini-mize with respect to � and take h�ii ¼ 0, hhi ¼ 0, andh�i ¼ v. Solving for m2 in the minimization condition upto neglectedOð’4=f4; ’2=ðf216�2ÞÞ terms in the one loopeffective potential one finds

m2

’2¼ �1

2þ �2’

2

8f2: (33)

iW, Z t

FIG. 1. One loop diagrams that contribute to the effectivepotential.


075022-7

The solution of this equation for the classical minimum is

’2 ¼ v2 � 2f2

�2

½��1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�21 þ 2�2m

2=f2q

: (34)

With this definition, we find for the h field

VCðhÞ ¼ m2h

2h2 þ v�eff

3

3!h3 þ �eff

4

4!h4 þ 15�2

5!f2vh5

þ 15�2

6!f2h6; (35)

where the parameters are

m2hðvÞv2

¼ �1

�1� 2C�

v2

f2

�þ �2

2

v2

f2; (36)

�eff3 ðvÞ ¼ 3�1

�1� 3C�

v2

f2

�þ 5

2�2

v2

f2; (37)

�eff4 ðvÞ ¼ 3�1

�1� 4C�

v2

f2

�þ 15

2�2

v2

f2: (38)

For the �i fields, the minimized potential causes themass of the fields to vanish, as expected.

In Appendix A, we derive a range of values for j�1j. Thelargest values that j�1j can take on while the loop expan-sion is under control are j�1j � g32. We now examine one

aspect of how this power counting affects the computationof the effective potential. The effective potential and henceits derivatives correspond to Green functions with vanish-ing external momenta (P2 ¼ 0). Conversely, the physicalparameters are defined at the scale m2

h. Thus, formally, we

have

d2Veff

d�2¼ m2

h � ��d3Veff

d�3¼ �eff

3 � ��3

d4Veff

d�4

¼ �eff4 ��4;

where we are following and extending the convention laiddown in [17], and hats denote physical parameters, forexample, mh is the pole of the Higgs propagator. Wehave introduced the shifts for the 1PI 2, 3, and 4 pointfunctions

�� ¼ �ðP2 ¼ m2hÞ ��ðP2 ¼ 0Þ;

��3 ¼ �3ðP2 ¼ m2hÞ � �3ðP2 ¼ 0Þ;

��4 ¼ �4ðP2 ¼ m2hÞ � �4ðP2 ¼ 0Þ

(39)

to denote this discrete running of the parameters.Note that if j�1j & g32 holds, we should neglect the small

effects due to this shift in the parameters as

�ðP2 ¼ m2hÞ ��ðP2 ¼ 0Þ � g21;2p

2;�g21;2m2h; (40)

� g21;2v2

��1 þ �2v

2

2f2

�; (41)

and similarly,

�3ðP2 ¼ m2hÞ � �3ðP2 ¼ 0Þ � g21;2v�

eff3 ; (42)

�4ðP2 ¼ m2hÞ � �4ðP2 ¼ 0Þ � g21;2�

eff4 : (43)

Clearly, if j�1j & g32, we can neglect g21;2�1 effects. We

also must neglect g21;2�2 effects as these are loop sup-

pressed and suppressed by f2; we have neglected manysuch effects in the effective potential and consistencydemands that we drop these terms. If one chooses to retainthese terms because �1 is not small, then these effects canbe significant. However, at the same time the convergenceof the loop expansion will be poor, and perturbative inves-tigations will be limited in the reliability of their conclu-sions as shown in Appendix A.As advocated in [17,71], it can be important to deter-

mine the running of the parameters in the Higgs sector toformally cancel the IR divergence that occurs when ’ !v, T ! 0. The correct description of the T ! 0 physics ofthe system after the EW phase transition is completeshould cancel this IR divergence. However, as this is notour focus in this paper, we neglect this higher order effect

and renormalize in the standard manner using MS anddimensional regularization.A much more significant effect when j�1j & g32 is dis-

tinguishing between the transverse and longitudinal massesof the gauge bosons and determining the thermal massbasis for the SM appropriate for ring resummation. Thisis an Oðg21;2mTÞ effect that imposes significant physical

constraints when extensions to the SM still have �1, m2 >

0 and is numerically important in our HEFT. We now turnto finite temperature effects and determining the transverseand longitudinal thermal mass basis in the SM.

E. Finite temperature effects

The finite temperature effects are calculated using fieldswith (anti)periodic boundary conditions for the (fermion)boson fields on the time interval � ¼ 1=T [72]. Theseboundary conditions allow one to decompose the Bose(�) and fermion (�) fields in Fourier modes [70]

�ðx; �Þ ¼ X1n¼�1

�nðxÞ expði!Bn�Þ;

�ðx; �Þ ¼ X1n¼�1

c nðxÞ expði!Fn�Þ;

where we have !Bn ¼ 2n�T and !F

n ¼ ð2nþ 1Þ�T. Theone loop functions J are obtained [54,73] by using resi-dues to transform the sum over Fourier modes into the sumof the usual T ¼ 0 loop contributions to propagators(which are renormalized in the standard way) and addi-


075022-8

tionally finite temperature contributions that have correc-tion factors for the Fermi-Dirac and Bose-Einstein particledistributions. The temperature dependent contributions arewritten in terms of the integrals [54]

Jðy2i Þ �Z 1

0dxx2 log

�1� expð�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2i

qÞ�; (44)

where y2i ¼ m2i =T

2. The temperature dependent one loopterms are given by

VTð’; TÞ ¼ T4

2�2

��X

F

gfJ�ðy2i Þ þXB

gBJþðy2i Þ�; (45)

where the sums are taken over all degrees of Boson (B) andFermion (F) freedom. The number of degrees of freedomgi for the W

, Z, t, h, �i fields are

gt ¼ 12; gW ¼ 6; gZ ¼ 3;

gh ¼ 1; g� ¼ 3: (46)

The Fermi-Dirac and Bose-Einstein particle distributioncorrection factors modify the loop expansion parameter.The finite temperature loop expansion is an expansion in

g2SMexpð�E�Þ 1

; (47)

where E is the typical energy scale of a process and one hasa þð�Þ sign for Fermi-Dirac (Bose-Einstein) particle dis-tributions [74]. As T � E, the effective expansion parame-ter for the Fermions is still given by g2SM. However, for thebosonic degrees of freedom, as T � E, the expansionparameter is given by

g2SMT

E: (48)

Thus, at high temperature, perturbation theory begins tobreak down in the bosonic loops. This fact is essential tothe phenomena of high temperature symmetry restoration.Otherwise, perturbative corrections (for all T) would neverrestore EW gauge symmetry at high temperatures.

The IR divergence T � E driven breakdown of finitetemperature field theory is decidedly inconvenient in per-turbative studies of the EW phase transition. A mathemati-cal sign of this breakdown is the presence of am3ð’ÞT termin the high temperature expansion of the finite temperatureintegral for the bosons. We resum a class of higher orderdiagrams that act to introduce a thermal mass / T2, whichscreens the IR divergence in the bosonic propagators [75].This ring resummation improves the nature of the thermalperturbative expansion and can be formally justified by apower counting analysis [22,73,76,77], which considersthe tadpole diagrams calculated at finite temperatureshown in Fig. 2. A scalar tadpole, at leading order inM=T gives a finite temperature contribution

�1

T2

4: (49)

Consider adding n quadratically divergent subdiagrams toa tadpole as in Fig. 2. This diagram will scale as�

�1T2

M2

�n�1TM ¼ ð�1Þ2 T

3

M

��1T

2

M2

�n�1

: (50)

For temperatures where �1T2 �M2, this class of ring

diagrams should be resummed for a reliable perturbativeexpansion. In fact, in our HEFT, the temperature scales ofthe EWPT are such that this factor is typically less thanone. However, ring resummation is still an important im-provement on the naive thermal perturbative expansion asemploying ring resummation improves the convergence ofthe loop expansion [73,78] and reduces the imaginary partof the effective potential.The imaginary part of the effective potential is of con-

cern as our description of the phase transition assumes thatthe field is sufficiently stable for the transition to be de-scribed by bubble nucleation. So long as ImðVeffÞ �ReðVeffÞ, the imaginary part can be interpreted following[79] as the decay rate per unit volume of a state, see also[55]. The imaginary part of the effective potential canpotentially come from two sources: the logarithms of theColeman-Weinberg terms when a mass squared is negativeand the cubic mass terms that appear in the expansion ofthe Jþ. The logarithmic dependence on the mass cancelswhen finite temperature effects are included as the finitetemperature integrals J are the Coleman-Weinberg termsregulated with a finite temperature cutoff [22], and ringresummation cures the remaining imaginary part in thefollowing manner: Consider the Higgs mass

m2hð’Þ ¼

�1

2ð3’2 � v2Þ; (51)

whenm2 is eliminated. Typically, ’ � v until far after thephase transition has occurred, and this term is negativebefore thermal corrections are taken into account. Whenperforming a ring resummation, we rewrite the potential asthe standard one loop finite temperature contributionsVTð’; TÞ and then an extra term that includes the shift inthe mass due to the ring resummation thermal corrections�hðTÞ ¼ �1T

2=4þ T2BT , where

BT ¼ 4f2t þ 3g22 þ g2116

: (52)

See Appendix B for details. For the Higgs, we have

1

2

3

n

FIG. 2. The n tadpole loop contribution to the diasy diagram ofthe Higgs propagator.


075022-9

VRð’; TÞ ¼ T

4�2

Z 1

0dkk2 log

�1þ �hðTÞ

k2 þm2hð’Þ

�: (53)

This integral can be directly evaluated using the Leibnizrule for m2

h, integrating over k, and subsequently integrat-

ing over m2h; one finds

VRð’; TÞ ¼ T

12�½m3

hð’Þ � ðm2hð’Þ þ�hðTÞÞ3=2: (54)

The first term cancels against an equivalent cubic massterm in the high temperature expansion of Jþ for the Higgs.The remaining term is crucial in determining the nature ofthe EWPT. The expression m2

hð’Þ þ�hðTÞ is an example

of a thermal eigenstate mass. Thus, we see that ring re-summation introduces a thermal mass term so that

m2hð’; TÞ ¼

�1

2ð3’2 � v2Þ þ �1T

2

4þ T2BT (55)

for the cubic mass dependence. As we are studying thephase transition as T decreases from values T � v thesethermal mass terms introduce a positive real contributionfor the mass that makes m2

hð’Þ positive for the range of ’,T of interest in our study of the effective potential. Asimilar argument holds for all bosonic degrees of freedom[75]. The class of diagrams suppressed by a single power of�1 compared with the diagrams of the ring resummationare the ‘‘setting sun’’ diagrams, see Fig. 3. These diagramsare not included in the resummation, which is justified solong as j�1j is small.

The key difference in this part of our analysis comparedwith the past literature is taking thermal contributions tothe gauge boson polarization tensor self-consistently intoaccount and defining a thermal mass eigenstate basis forthe gauge boson fields. Past calculations have neglectedthis subtlety in defining the mass eigenstates in the contextof thermal corrections. This improvement is of numericalimportance as the phase transition occurs when all theterms in the effective potential are approximately thesame size and are canceling against one another. Thecritical value of the vacuum expectation value h’ðTcÞiand the critical temperature Tc that determine the washoutcriteria are sensitive to small changes in these mass terms.We discuss at length the calculation of the gauge bosonmass eigenstate basis in Appendix B 2. We find the follow-ing longitudinal vector boson masses:

ðmLWð’; TÞÞ2 ¼ g22

�11T2

6þ ’2

4

�;

ðmLAð’; TÞÞ2 ¼

11T2

6ðg21cos2ð�ðTÞÞ þ g22sin

2ð�ðTÞÞÞ

þ ’2

4ðg1 cosð�ðTÞÞ � g2 sinð�ðTÞÞÞ2;

ðmLZð’; TÞÞ2 ¼

11T2


2ð�ðTÞÞÞ

þ ’2

4ðg1 sinð�ðTÞÞ þ g2 cosð�ðTÞÞÞ2; (56)

where we have introduced a thermal Weinberg angle �ðTÞthat characterizes the degree of mixing in the longitudinalvector boson masses. For the transverse masses (deter-mined again in Appendix B 2), we incorporate the effectsof mixing and introduce a second thermal Weinberg angle�0ðTÞ characterizing the degree of mixing in the transversevector boson masses. We also introduce a parameter � thatsignifies a nonperturbative magnetic mass term that isimportant, as it screens the transverse mass of the W, Zfields in the ’ ! 0 limit. We will use the value � ¼ 4:2,which has been determined for the deconfined hot SU(2)gauge theory [80] in Landau gauge. We expect this to be agood approximation to the � of the SM. We find thefollowing transverse masses:

ðmTWð’; TÞÞ2 ¼

�2g429�2

T2 þ g22’2

4;

ðmTAð’; TÞÞ2 ¼

g21T2cos2½�0ðTÞ

24

þ ’2ðg2 sin½�0ðTÞ � g1 cos½�0ðTÞÞ24

;

ðmTZð’; TÞÞ2 ¼

g22mTWð’; TÞTcos2½�0ðTÞ

3�

þ g21T2sin2½�0ðTÞ

24

þ ’2ðg2 cos½�0ðTÞ þ g1 sin½�0ðTÞÞ24

:

(57)

Note that both of our thermal Weinberg angles �ðTÞ, �0ðTÞreduce to �W in the T ! 0 limit.In our HEFT, we also have to deal with the effects of NP

on the ring resummation for the Higgs and the would beGoldstone-boson fields �i. Again, examining the Higgsmass we have

m2hð’Þ ¼

�1

2ð3’2 � v2Þ

�1� 2

’2

f2C�

�þ �2

8f2ð5’4 � v4Þ:

Thus, NP can make matters worse in a number of ways. If�2 and �1 are independent, the T2 thermal mass term issmall compared with the term proportional to ��2v

4=f2

(�1 is at best �g32 whereas, when �1 and �2 are indepen-

32

1 n

FIG. 3. The n tadpole loop contribution to the sunset diagramof the Higgs propagator.


075022-10

dent, we allow �2 �Oð1Þ). So the mass is negative, and theeffective potential is not dominated by its real part near theEWPT. Further, for large regions of parameter space inPGBG �1 < 0, so that the quadratic thermal correctionsmake the situation worse. When this occurs, one must have�2 > 0 to stabilize the potential. Both of these problemsare solved (and the loop expansion is improved) if one alsoincorporates the ring diagrams proportional to �2. Thisintroduces terms of the form �2T

4=f2 and �2T2’2=f2

that act to ensure that the Higgs and would beGoldstone-boson fields have a positive mass for the ’, Tof interest. We perform this calculation in Appendix B 1.

We find the following expressions for the masses appro-priate for the ring resummed effective potential for the hand �i:

m2hð’; TÞ ¼ m2

hð’Þ þT2�1

4

�1� 3C1

�

’2

f2

�þ ’2T2

2f2�2

þ T2BT

�1� 4C1

�

’2

f2

�þ 3T4�2

4f2;

m2�ð’; TÞ ¼ m2

�ð’Þ þ T2�1

4

�1� C1

�

’2

f2

�þ ’2T2

2f2�2

þ T2BT þ 3T4�2

4f2: (58)

Note that we neglect the Oðg3SMTÞ and Oð�1gSMTÞ loopsuppressed contributions from the one loop gap equations[81] for the scalars.

We note that the ring resummation utilizes the result ofthe high temperature expansion of the Jðy2i Þ, and�hðTÞ isapproximated by its leading T2 term. In our effectivetheory the critical temperature Tc at which the minima ofthe potential become degenerate can be significantly lessthan the EW scale v. Further, the effects of supercoolingdue to the expansion of the Universe delaying the onset of

the phase transition lead to the physically relevant nuclea-tion temperature Tn < Tc being, in some cases, Tn & mh,mW , mZ [17]. As this is the case, one might question thegeneral use of the high temperature expansion in thisanalysis and others. In particular, one might doubt theconvergence of the expansion used in Appendix B todetermine the thermal mass basis.However, this approach is under control7 for the bosonic

fields, when we expand the Jþ integral as

Jþðy2Þ ¼ �2y2

12� �ðy2Þ3=2

6� y4

32log

�y2

ab

�; (59)

where logab ¼ 5:408, and logaf ¼ 2:635. Taking into ac-

count the effects of the expansion of the Universe, we arerestricted to the situation where ’c � Tc and Tn is not toofar below Tc for most of the ð�2; C�Þ parameter space of

interest. Lower temperatures lead to metastable vacuumsolutions, and the EWPT does not occur, see [17]. UsingEq. (59) is sufficiently accurate, so long as mi=T < 2�.The neglected higher order terms are a numerically sup-pressed expansion given by

� 2�7=2X1‘¼1

ð�1Þ‘ �ð2‘þ 1Þð‘þ 1Þ! �ð‘þ 1=2Þ

�m2

ð4�2T2Þ�‘þ2

:

(60)

Thus, Eq. (59) is clearly sufficient for all known masses asthe lowest physically interesting temperatures are T �20 GeV. For the unknown Higgs mass, we restrict our-selves to considering low Higgs massesmh & 160 GeV forthis reason.

F. The effective potential in the PGH effective theory

We find the following effective potential

Vringeff ð’; TÞ ¼

a

2ðT2 � T2

bÞ’2 þ �2

48f2’6 þ ��1ðT; fÞ’4 � T

12�½m3

hð’; TÞ þ 3m3�ð’; TÞ

� T

12�½ðmL

AÞ3ð’; TÞ þ ðmLZÞ3ð’; TÞ �

T

12�½2ðmL

WÞ3ð’; TÞ þ 2ðmTAÞ3ð’; TÞ

� T

12�½2ðmT

ZÞ3ð’; TÞ þ 4ðmTWÞ3ð’; TÞ þ Lð’; TÞ þO

�g4sm; �

21; f

4t

’2

16�2f2

�: (61)

We have used the condition

@Veffðv; 0Þ@’

� 0 (62)

to fix m2 in defining the potential and have adopted thenotation

a ¼ BT þ �1

4

�1þ v2

3f2C�

�; (63)

T2b ¼ m2=a;¼

��1v

2

2þ �2v

4

8f2

��a; (64)

7Once again, this statement holds so long as j�1j is small.When j�1j � 0:2, generally the temperatures Tn, Tc are toosmall for a reliable high temperature expansion. This is anothersign of the lack of a consistent perturbative treatment for largenegative �1.


075022-11

�� 1ðT; fÞ ¼ �1

8þ �2T

2

24f2� �1C�

T2

8f2; (65)

and the logarithmic terms are given by

Lð’; TÞ ¼ � 3f2t64�2

’4

�log

�afT

2

�2

�� 3

2

�

þ 3m4�ð’; 0Þ64�2

�log

�abT

2

�2

�� 3

2

�

þm4hð’; 0Þ64�2

�log

�abT

2

�2

�� 3

2

�

þ 3m4Wð’; 0Þ32�2

�log

�abT

2

�2

�� 5

6

�

þ 3m4Zð’; 0Þ64�2

�log

�abT

2

�2

�� 5

6

�: (66)

We choose the renormalization scale � ¼ MZ. The tem-perature Tb sets the temperature scale at which the phasetransition occurs and dictates the convergence of the hightemperature expansion. Tb is a function of the NP parame-ters and mh. We find the following for TbðmhÞ:T2bð115 GeVÞð130 GeVÞ2 ’ 1þ ð320 GeVÞ2

f2C� � ð220 GeVÞ2

f2�2;

T2bð130 GeVÞð150 GeVÞ2 ’ 1þ ð310 GeVÞ2


f2�2;

T2bð160 GeVÞð170 GeVÞ2 ’ 1þ ð300 GeVÞ2


f2�2;

where we have rounded to two significant digits and usedthe zero temperature relationship between m2

h and �1. Wewill consider Tb � 100 GeV in what follows. Note that forthe region of parameter space, where �2 �Oð1Þ is orderone and positive (while and C� �Oð1Þ is order one andnegative) the effects of NP can significantly reduce T2

b andcan even in principle cause the sign of T2

b to change.However, as we will show, when this occurs the EWPT isnot sufficiently first order for the washout condition to besatisfied. In fact, one can use the above approximate ex-pressions as a quick check of the low energy expression ofa PGH model to see if the washout condition is potentiallypassed.

Relation to previous work

Our final potential agrees with some aspects of paststudies [17,37,47,62,82], although we do find some dis-agreements, and our results extend previous investigationsin a number of ways. The origin of the disagreements andimprovements are the following.

We reiterate that our demand for a reliable perturbativestudy imposed the power counting �1 & g32; thus, we donot retain the higher order effects of running our parame-

ters. We also neglect temperature independent terms in theeffective potential that are suppressed by ’2=ð16�2f2Þ.The temperature dependent one loop effects of NP areretained consistently because this class of terms leads tothe critical thermal screening that suppresses the imaginarypart of the effective potential. We have introduced a ther-mal screening due to NP effects that is required when thering resummation is employed with the �2 operator. Wenote that these thermal effects are significantly numericallyenhanced compared twith ’2=ð16�2f2Þ effects.We also reemphasize that we have determined the po-

tential in a canonical low energy effective theory; i.e. werescale the Higgs field to remove the dimension six kineticterms, which introduces the dependence on C� in our

effective potential. (The dependence on this operator willturn out to be critical when �2 / �1.) Further, we havedetermined the longitudinal and transverse thermal massesof the gauge boson fields and have used them in oureffective potential.We have also emphasized that an important feature of

the low energy description of PGH models is that therelationship between the Higgs mass and the Higgs self-coupling is significantly relaxed in the effective theory, asemphasized in [16]. This effect is essential for perturbativestudies of PGBG to be reliable, and is a generic low energysignal of a new strong interaction at a TeV with a PGH. Wenow turn to determining the condition on the NP parame-ters and the Higgs mass that allow a first order phasetransition to occur while our perturbative study is reliable.

VI. ON THE POSSIBILITY OF PSEUDO-GOLDSTONE BARYOGENESIS

Before turning to the possibility of PGBG, we firstreview the condition on the parameters in the potential inthe SM for there to be a first order EWPT. This may seemesoteric as if the washout condition is passed one knowsthat the phase transition is first order. However, deriving ananalytic constraint on our Wilson coefficients is useful as itreduces the subsequent region of Wilson coefficients to testfor satisfying the washout condition. In addition, our ap-proach in this section (and Appendix A) will establish theregion of Wilson coefficients where our perturbative re-sults will be reliable and illustrate how �1 < 0 avoids thefirst order phase transition constraint. When this is the case,we derive a further constraint that will ensure our analysisavoids unreliably concluding PGBG could occur by pass-ing the washout condition when �1 < 0 and the loopexpansion is nonperturbative.

A. The first order phase transition condition: SM

We will emphasize the limitations that the non-Abelianmagnetic mass discussed in [81,83] place on the Higgsself-coupling in this approach. This approach determines alimit on the Higgs mass in the SM for a first order EWPTthat reasonably approximates the mass limit determined in


075022-12

lattice investigations. Consider the potential of the form of

Vringeff ð’; TÞ, where one has taken �2, C� ! 0. We consider

the limitations arising from the gauge boson masses. Thescalar sector of the theory is known to always give a secondorder phase transition for all values of �1 [78,81].

One can obtain a necessary condition on the existence ofa first order phase transition following [81] by first takingthe potential in the simplified form

Vð’; TÞ ¼ a

2ðT2 � T2

bÞ’2

�Xi

biT

3ðc2i T2 þ ’2Þ3=2;þ�1

8’4; (67)

where the constants a, bi, ci are all positive. Note that interms of our perturbative couplings gSM, we have bi /g3SM, a / g0SM, cA / g0SM. However, it is important to note

that for the longitudinal mass of the W, Z fields theparameters are cLW , c

LZ / g0SM, whereas for the transverse

mass cTW , cTZ / gSM, when ’ ! 0. Insisting that the tem-

perature is high enough that there is a minimum at theorigin, one imposes d2Vð0; TÞ=d’2 � 0, and one finds thatfor temperatures

T2 � ~T2b �

T2b

1� ðPi biciÞ=a; (68)

there is a minimum at the origin. For there to be a first orderphase transition we also require that for some’ above zerowe have another minimum with a potential barrier inbetween the two minima. For field values just above ’ ¼0 and for temperatures above ~T2

b the potential increases. Atlarge ’ values the potential is dominated by the ’4 termand therefore also grows with ’. For there to be a secondminimum away from ’ ¼ 0, there must be a maximumtoo. The condition that there is a second minimum awayfrom the origin is weakest when the height of the barrierapproaches zero. In this case, the location of the maximumof the barrier moves toward ’ ¼ 0. Hence, a necessarycondition for the first order phase transition is that as T !~Tb the derivative of the potential vanishes at some infini-tesimal value of ’. Taylor expanding the derivative, wefind the condition

0 ¼ a

2

� Pi biciT

2b

a�Pi bici

�� ’2

4

�Xi

bici

� �1

�: (69)

Recall that a, bi, ci, di are all positive, a�Pibici > 0 and

T2b is positive in the SM. Thus, for this equation to have a

solution for ’> 0 one must have

�1 <Xi

bici; (70)

which is the first order phase transition condition for theSM of [81]. The condition is dominated by the contributionto the constraint for the transverse W and Z masses, whichgives for

Pibi=ci

3g2224�

þ 3ðg2 cos½�0ðTbÞ þ g1 sin½�0ðTbÞÞ424�ð3�2g21sin

2½�0ðTbÞ=2þ 4g42�cos2½�0ðTbÞÞð1=2Þ

:

Using our approximate results for thermal Weinberg angle,tree level results for g1, g2, and � ¼ 4:2 [80], we find thephase transition is first order in the SM for a Higgs mass

mh < 58 GeV: (71)

This is consistent with general expectations that the tran-sition is first order in the SM if mh & mW and qualitativelyagrees with lattice simulations [29–32]. For example, [32]finds a first order EWPT for the SM for mH < 72:41:7 GeV. Thus, we consider this condition in the contextof new physics to analytically study the relaxation of thisbound in PGBG.

B. First order phase transition condition: PGBG

In our PGBG scenario,our effective theory introducesthe following changes in the potential:(i) The constant a is changed by �1v

2C�=ð12f2Þ. Notethat a is still positive for the range of NP models wewill consider as we expanded in C�, which required

2v2C�=f2 < 1.

(ii) The barrier temperature Tb is changed through thechange in a and the term �2v

4=ð8f2Þ. Demandingthe NP effects are such that m2 is still positive gives

�1v2

2þ �2v

4

8f2> 0: (72)

(iii) The coefficient of ’4 obtains temperature depen-dence and the effecting coupling that will bebounded is shifted by the NP Wilson coefficients.

(iv) The potential now has a ’6 term.(v) When one relates the Lagrangian density parameters

ðm2; �1; �2Þ in terms of the physical parametersðv;m2

h; �effi Þ, one must introduce the dependence

on C� that comes from canonically normalizing

the physical Higgs field.The simplified form of the potential is now

Vð’; TÞ ¼ a

2ðT2 � T2

bÞ’2 �Xi

biT

3ðc2i T2 þ ’2Þ3=2;

þ ��1ðT; fÞ’4 þ �2

48f2’6: (73)

For there to be a second minima for ’> 0, we now havethe condition

m2h

v2

�1þ 2C�

v2

f2

�� 2

v2

2f2þ

��2

3f2� �1C�

f2

�

��

2m2hv

2

m2h þ 4BTv

2

�<

Xi

bici; (74)

where we use the zero temperature result for �1, and we


075022-13

again neglect the effects of running this parameter to ~Tb, asit is a higher order effect.

Let us examine the constraints on the NP Wilson coef-ficients. Numerically, the sum is

Xi

bici

’ 5:6� 10�2 (75)

for � ¼ 4:2. When �1 < 0, which can happen in our HEFT,the first order phase transition condition of the SM isevaded. However, we will still require j�1j & g32 (which

we conservatively take to be j�1j & 0:2) so that our per-turbative investigation has a loop expansion that is undercontrol, see Appendix A. This condition and the first orderphase transition condition become the important constraint

� 0:2 &m2

h

v2

�1þ 2C�

v2

f2

�� 2

v2

2f2& 5:6� 10�2:

(76)

Thus, we have two inequalities and four unknownsmh,C�,

�2, f. As shown in Fig. 4, for mh & 160 GeV a first orderphase transition can be present if the SM is modified withparametrically enhanced dimension six operators. We plota number of cases where the strong decay scale is in therange dictated by the requirement of enough CP violationfor EWB to occur, i.e. 500 GeV & f & 1000 GeV.

Recall our cases defined in Sec. IV. In C1, the requiredWilson coefficient for �2 is Oð1Þ, with �2 > 0 required.One would also expect �2 > 0 for the potential to bestabilized in the presence of these NP terms. For C�, the

Wilson coefficient can vanish or beOð1Þ and of either sign.InC1, there are large regions of parameter space, where thephase transition is first order when �1 is positive or nega-tive as we show in Fig. 4.

In C2, the important region of constraint for ~�2 and C�

is given by

� 0:2 &m2

h

v2

�1þ 2C�

v2

f2� ~�2

v2

2f2

�& 5:6� 10�2:

(77)

In C2, for almost all of the parameter space where thephase transition is first order, �1 is positive in our HEFT, aswe show in Fig. 5. When �1 > 0 in our HEFT, we find that

so long as ~�2 is positive and C� is Oð1Þ and negative, the

phase transition can be first order. We do note however, thatthe models discussed in Sec. IV all have C� Wilson co-

efficients that are Oð1Þ and positive. We now turn todetermining the critical washout condition in our PGBGscenario.

FIG. 4 (color online). Case 1 where �2 and �1 are treated as independent. The green (0< ��1ðf; TÞ< 5:6� 10�2) and light blue(� 0:2< �1 < 0) regions satisfy the first order phase transition (and small �1) conditions for Higgs masses of 115 GeV (top) and130 GeV (bottom). Also plotted is the condition that 2v2C�=f

2 < 1, which is the region between the horizontal dashed lines, Eq. (81),

which is satisfied below the short dashed line and the ascending solid line above which T2b is positive. For each Higgs mass, we plot the

region of allowed Wilson coefficients for a strong decay constant of f ¼ 500, 750, 1000, 1250 GeV (left to right). The region that ourcalculation is self-consistent, with a perturbative loop expansion that is under control, and has the signs of �1 andm

2 the same as in theSM is the small region in the green band bounded between the ascending solid and short dashed lines. For almost all of the viableparameter space the nature of the EW phase transition is different than in the SM. For the blue region the potential must be stabilizedby the �2 operator.


075022-14

VII. WASHOUT CONDITION

When considering the washout condition it is best to

have a picture of the phase transition in mind. We plot Vringeff

when the parameters in the Lagrangian density are suchthat a first order phase transition is possible for mh ¼120 GeV in the cases where ��1ðT; fÞ< 0 and ��1ðT; fÞ>0 and for the SM in Fig. 6.

A first order phase transition proceeds through the nu-cleation of bubbles where ’> 0 inside the bubble, ’ ¼ 0outside the bubble, and the expectation value of the Higgschanges rapidly as one goes through the bubble wall. Asthe Universe cools down, when the phase transition occurseventually a critical temperature Tc is reached where thehigh temperature minima at the origin and the minima at’c are degenerate. The conditions defining ’c, Tc are

Veffð’c; TcÞ ¼ Veffð0; TcÞ; @Veffð’c; TcÞ@’

¼ 0; (78)

and correspond to the blue line (that returns to 0) in Fig. 6.Wewish to solve for’c and Tc, as the washout condition

must be satisfied for the phase transition to be sufficientlyfirst order. Sufficiently, first order is defined as the condi-tion discussed in Sec. II,

’c=Tc � b; (79)

with 1 & b & 1:3. Satisfying the washout condition [35]guarantees that once baryogenesis has taken place outsidethe bubble wall, as the bubble expands and envelops theproduced baryon number, the remaining sphaleron inducedBþ L violating Boltzman fluctuations inside the bubble donot erase the produced baryon number. The sensitivity of

the right-hand side of Eq. (79) to the’6 term was examinedin [37] and found to be a percent level effect that weneglect.As our effective potential is quite complicated, we solve

for ’c, Tc numerically. The procedure we use is to first

translate Vringeff ½v; �1; �2; C�; f;T;’ to

Vringeff ½v;m2

h; �2; C�; f;T;’; (80)

using our zero temperature definition of �1, while neglect-ing the effects of running. We then choose a m2

h, f and

numerically solve for ’c and Tc by scanning the allowedregion of �2,C� parameter space determined in Figs. 4 and

5. Our results are reported in Figs. 7–10.We find that T2

b > 0 when the washout condition is

passed. We note that there is a region where T2b > 0 is

roughly parallel to the T2b ¼ 0 line, where the washout

condition is not passed. We find empirically that the fol-lowing constraint equation determines this region wherethe washout condition is not passed and T2

b > 0

��aT2b �

Xi

biTb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2i T

2b þ v2

qþ T2

bv2

f2

��2

6� 4�1C�

9

�

þ �2v4

8f2

��<��1v

2

2

�� (81)

This equation is inspired by the fact that it is known thatthe SM with a Higgs mass in the region we consider114:4 GeV & mh & 160 GeV does not pass the washoutcondition; thus, the relationship between the m2 and �1

parameters need to be significantly effected in order tosatisfy our washout condition. The equation characterizes

FIG. 5 (color online). Case 2 where we plot ~�2 ¼ �2=�1. As in Fig. 4, the green (0< ��1ðf; TÞ< 5:6� 10�2) and light blue(� 0:2< �1 < 0) regions satisfy the first order phase transition (and small �1) conditions for Higgs masses of 115 GeV (top) and130 GeV (bottom). The lines are the same as in Fig. 4. For almost all of the parameter space, �1 is positive; however, the nature of theEW phase transition is still quite different than in the SM, as we discuss in Sec. VII.


075022-15

the relationship between �1 and m2 as T ! Tb. When thewashout condition is not passed and this equation is sat-isfied, the critical signs of m2 and �1 are the same(although both are negative in C1, unlike in the SM). Inthe region of parameter space dictated by this equation andthe T2

b > 0 condition, the washout condition is not passed.

This condition can be understood by the following ap-proximation in case 1: Recall the zero temperature mini-mization condition, Eq. (33) when ’ ¼ v. When m2 ¼ 0,this equation dictates

�1 ¼ � v2

4f2�2: (82)

Substituting this result in Eq. (36), we obtain the constraint

C� ¼ �2

8

v2

m2h

� f2

2v2; (83)

which reasonably approximates the plotted Eq. (81). When

the washout condition is satisfied in PGBG, T2b > 0, and

the relationship between m2 and �1 must be significantlyeffected in the sense that Eq. (81) is not satisfied. There is afurther constraint on PGBG due to the effect that anexpanding universe has on the possibility of the bubbleformation. The results of [17] indicate that for the caseC� ¼ 0, the supercooling effect due to the expansion of the

Universe is a small shift in Tc for most of the relevantparameter space. The temperatures for most of the parame-ter space above are �100 GeV.

VIII. CONCLUSIONS

We have shown how an effective theory of the SMHiggsthat would be the low energy description of a PGH canaddress all of the problems of EW scale SM baryogenesis.Our results indicate that PGH models with Wilson coef-ficients �2 and C� that areOð1Þ and a strong decay scale fin the range (500 Gev, 1 TeV) may successfully account for

FIG. 6 (color online). The temperature dependence of the EW phase transition in a number of cases when mh ¼ 120 GeV. As thedashes get shorter and as the color decends in hue, the Universe is cooling down. Comparison of the three graphs clearly illustrates thesensitivity of the EW phase transition to the low energy expression of a new strong interaction with a TeV mass scale and the influenceof the C� operator. The potentials are normalized to zero at the origin. Top left: The decay constant is f ¼ 700 GeV and �2 ¼ 4,

C� ¼ 0. In this case, �1 < 0. Temperatures plotted ¼ ð115; 110; 105; 103; Tc ¼ 102:2; 101; 100Þ GeV and ’c ¼ 113:4 GeV so that

’c=Tc ¼ 1:11. Top right: The decay constant is f ¼ 700 GeV and �2 ¼ 2, C� ¼ �2. In this case, �1 > 0. Temperatures plotted ¼ð80; 75; 74; 73; Tc ¼ 72:5; 72; 71:5Þ GeV and ’c ¼ 118:1 GeV so that ’c=Tc ¼ 1:63. Bottom: The SM for comparison. In this case,�1 > 0. Temperatures plotted ¼ ð150; 145; 141; 139; Tc ¼ 136:9; 135; 133Þ GeV and ’c ¼ 0:28 GeV so that ’c=Tc ¼ 2:1� 10�3.Note that as ’ ! 0 formally the loop expansion breaks down and thus the behavior of the graphs as ’ ! 0 is not reliably determinedin perturbation theory but formally the normalized potential must vanish.


075022-16

the origin of the baryon-antibaryon asymmetry of theUniverse.

The PGBG scenario is falsifiable and should be ruled inor out as the possible origin of the baryon asymmetry of theUniverse in the next few years of experiments. Let usconsider the experimental path that could find evidencefor PGBG being the origin of the baryon-antibaryon asy-metry of the Universe.

If the Higgs self-coupling can be determined through theprocess gg ! hh [16,84] at LHC, and it deviates from theSM value dictated by the determined Higgs mass, ourresults indicate that one should start to seriously considerPGBG. A large effect on the relationship between �1 andmh in the effective theory is absolutely required. If this isestablished and ideally new resonances of a new stronginteraction were discovered, then the possibility is seri-ously raised that PGBG may be the origin of a significantamount of baryon-antibaryon asymmetry in the Universe.Unfortunately, the limited kinematic reach of LHC means

that new strong interaction states could easily be elusive atLHC. Indirect signals of a new strong interaction such as agrowth in the longitudinal gauge boson scattering ampli-tudes despite the presence of a light Higgs [15] are possiblythe best that can be achieved experimentally. If stronginteraction states avoid detection due to LHC’s limitedreach, the large effects of NP in the Higgs sector in thisscenario allows one to have some reasonable hope ofinteresting signals of NP in the properties of the Higgs.In conjuction to these LHC results, PGBG also requires

that EDM experiments also find evidence for non-SM CPviolation. If such a set of discoveries are made, one willactually be able to conclude that PGBG is the likely sourceof the baryon-antibaryon asymmetry in the Universe.8

FIG. 7 (color online). The overlay of the washout condition and our phase transition condition when �2 and �1 are independent. Thelines are defined in Fig. 4, and the decay constant of scale is f ¼ 500 GeV (left) and 750 GeV (right). The black square indicates thatthe stronger washout condition ’c=Tc � 1:3 is passed; the gray triangle indicates that only the weaker washout condition ’c=Tc � 1:0is passed.

8Leptogenesis with new sources of CP violation in the leptonsector would not induce such large effects on EDMs. EDMs donot violate lepton number and de �G2

fmem2 [52,53].


075022-17

This scenario has a number of interesting features thatincrease its viability. The SUCð2Þ operators that are para-metrically enhanced in the Higgs sector (and only exactlythese operators) are exactly the operators that need to besizable in our HEFT for PGBG to occur. These operatorsare not constrained by EWPD to be small. An interestingfeature of PGBG is the coincidence in the required strongdecay scale f. The same range of scales is required for theSM to be supplemented with enough CP violating effectsand the EW phase transition to be sufficiently first order.

If the Higgs is found at LHC and if it is a pseudo-Goldstone Higgs, experiment could soon inform us ifEW pseudo-Goldstone baryogenesis is the origin of thebaryon-antibaryon asymmetry of the Universe.

ACKNOWLEDGMENTS

This work was supported in part by the U.S. Departmentof Energy under Contract No. DE-FG03-97ER40546.

APPENDIX A: CONSTRAINTS FOR RELIABLEPERTURBATIVE STUDIES OF NP AND THE EW

PHASE TRANSITION

We have emphasized the need to have a loop expansionunder control in thermal field theory calculations of theEWPT in our HEFT. We digress for a moment to give somemore detail on why this consideration is so important. Theconcern about the convergence of perturbation theory ismore urgent in investigations of the EWPT. As discussed inSec. VE, finite temperature effects are known to cause theloop expansion to break down for sufficiently high tem-peratures leading to high temperature symmetry restora-tion. In the SM, even with ring improvement, the loopexpansion is still a poor expansion if the scalar doubletquartic self-coupling �1 is large [78,81,85]. Once we em-ploy ring resummation to absorb the thermal mass termsthat scale asOð�T2Þ andOðg2T2Þ, the remaining loops aredominated by momenta of the order of their mass scale,

FIG. 8 (color online). The overlay of the washout condition and our phase transition condition when �2 and �1 are independent. Thelines are defined in Fig. 4, and the decay constant of scale is f ¼ 1000 GeV (left) and 1250 GeV (right). The black square indicatesthat the stronger washout condition ’c=Tc � 1:3 is passed; the gray triangle indicates that only the weaker washout condition’c=Tc � 1:0 is passed. As the scale f grows, the size of the required Wilson coefficient for �2 grows rapidly.


075022-18

and the loop expansion parameters are dictated by�1T=meff and g2SMT=meff [73,78]. These loop expansion

parameters place a constraint on �1 for perturbative studiesto be reliable.

As an example to clarify the issue, consider the ringimproved potential of a pure scalar theory. This potentialappears to give a first order phase transition at leadingorder in the ring improved loop expansion. However, thisconclusion is incorrect. A pure scalar theory is well knownto undergo only a second order phase transition. Thisincorrect conclusion is reached as the loop expansionparameter is order one near the phase transition [78]. Forthe pure scalar theory the loop expansion parameter is�T=meff and meff � �T. This clearly illustrates the needto insist that perturbative studies take note of the nature ofthe expansion parameter and ensure that it is less thanone.

Now consider the (lower order) simplified classicalpotential inspired by our effective potential of the form[78]

Vð’; TÞ ¼ 1

2ðag2T2 �m2Þ’2 þ �1

8’4;

�Xi

biT

3ðc2i T2 þ ’2Þ3=2: (A1)

The phase transition occurs when all terms in the potentialare approximately the same size. When this occurs, onefinds

’c � g3

�1

Tc; ðag2T2c �m2Þ � g6

�1

T2c : (A2)

For this potential, the transverse vector loop expansionparameter (subloops with MT

Wð’; TÞ vectors running inthem) is given by

g2Tc

MTWð’; TÞ

� �1

g22: (A3)

This is why we insist that for the loop expansion tobe under control one must have �1 � g32 [78,81,85].

FIG. 9 (color online). The overlay of the washout condition and our phase transition condition when �2 is proportional to �1. Thelines are defined in Fig. 4 and the symbols in Fig. 8. The decay constant scale is f ¼ 500 GeV (left) and 750 GeV (right).


075022-19

Perturbative studies that do not take this constraint intoaccount run the risk of obtaining unreliable conclusions.For Higgs masses abovemh > 115 GeV, perturbative stud-ies of the EWPT of the SM are unreliable for this reason.

In our perturbative investigation of our PGBG scenario,we must insist that the values of mh, f and the Wilsoncoefficients �2, C� dictate that the loop expansion is under

control. This is not a significant fine tuning for mh &160 GeV. Because of the parametrically enhanced NPeffects on the relationship between the Higgs mass andthe Lagrangian parameter �1 in our effective theory. ForPGH models in case C1, this suppression of the Higgs self-coupling naturally occurs when �2 �Oð1Þ and positiveand C� �Oð1Þ. This suppression of the Higgs self-

coupling in our HEFT also tends to make the phase tran-sition first order, while improving the justification of per-turbative studies. �2 > 0 is also desired so that NPstabilizes the Higgs potential when �1 < 0.9

Some past studies have allowed the Higgs mass to bemh � 160 GeV when considering the effect of NP[17,37,62] and have not taken this constraint on �1 intoaccount. As the Higgs mass increases, the �2 Wilsoncoefficient must become rather large for �1 to remainsmall. One should also note that when Higgs masses areabove 160 GeV, one has a poorly behaved high temperatureexpansion. For these reasons, we restrict our investigationto mh & 160 GeV.This reasoning also gives a constraint on �2 for a reliable

perturbative investigation. Consider the nonrenormalizablepotential of the form

Vringeff ð’; TÞ ¼

a

2ðT2 � T2

bÞ’2 þ �2

48f2’6

þ ��1ðT; fÞ’4;�Xi

biT

3ðc2i T2 þ ’2Þ3=2:

(A4)

Again, the phase transition occurs when all terms in thepotential are approximately the same size. When this oc-curs one again finds the constraint �1 � g32 is appropriate,

FIG. 10 (color online). The overlay of the washout condition and our phase transition condition when �2 is proportional to �1. Thelines are defined in Fig. 4 and the symbols in Fig. 8. The decay constant scale is f ¼ 1000 GeV (left) and 1250 GeV (right).

9Lattice simulations could relax this constraint on �1, whileinvestigating the nature of the electroweak phase transition inPGH scenarios.


075022-20

and we have the additional condition

�2 � �1

f2

T2c

: (A5)

For the decay constant scale f and critical temperatures Tc

of interest, one finds that

�2 � g32f2

T2c

�Oð1Þ; (A6)

which is consistent with the values of �2, which we find arerequired for a first order phase transition in the context ofNP.

When �1 < 0 in our low energy PGH Lagrangian, thefirst order phase transition condition of the SM is evaded.However, we will still require j�1j & g32 (which we con-servatively take to be j�1j & 0:2) so that our perturbativeinvestigation has a loop expansion that is under control.This condition and the first order phase transition conditionbecome the important constraint

� 0:2 &m2

h

v2

�1þ 2C�

v2

f2

�� 2

v2

2f2& 5:6� 10�2:

(A7)

This equation can be satisfied for large regions of C�, �2

parameter space when mh � 115 GeV. The upper boundon this constraint equation has some finite temperatureeffects that we discuss in Sec. VI. The requirement ofj�1j � g32 that is appropriate for �1 < 0 is purely a require-ment for a loop expansion that is under control and inde-pendent of temperature.

APPENDIX B: HIGH TEMPERATUREEXPANSIONS

In calculating the W, Z, A thermal mass terms thatinfluence the nature of the EWPT, we employ high tem-perature expansions. All temperature dependent loop inte-grals can be decomposed in terms of a basic integral

Jðy2Þ �Z 1

0dxx2 log½1� expð�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

qÞ; (B1)

where y2i ¼ m2i =T

2. The derivatives of this integral wedenote as

Iðy2Þ ¼ 2

�dJðy2Þdy2

�; (B2)

Kðy2Þ ¼ dIðy2Þdy2

: (B3)

Expressing our results in terms of these integrals allowsone to easily improve the propagators self-consistentlywith the determined thermal masses. We employ hightemperature expansions for these integrals obtained bytaking derivatives of the following expansions:

Jþðy2Þ ¼ �2y2

12� �ðy2Þ3=2

6� y4

32log

�y2

ab

�þO

�y3þn

2n�n

�;

J�ðy2Þ ¼ ��2y2

24� y4

32log

�y2

af

�þO

�y3þn

2n�n

�; (B4)

where n ¼ 2; 3 . . . and logab ¼ 5:408, logaf ¼ 2:635. We

also find it convenient to define the following functions,which are the results of common loop integrals

F1ðy2a; y2bÞ ¼1

2

ðy2aIþðy2aÞ � y2bIþðy2bÞÞy2a � y2b

þ F6ðy2a; y2bÞ

F2ðy2aÞ ¼ 1

2

�y2aKþðy2aÞ � 1

2Iþðy2aÞ

�

F3ðy2a; y2bÞ ¼ F1ðy2a; y2bÞ þ F2ðy2aÞ;F4ðy2a; y2bÞ ¼ Iþðy2aÞ þ 2Iþðy2bÞ þ

3

y2aðJþð0Þ � Jþðy2aÞÞ;

F5ðy2a; y2bÞ ¼3’2

2�2ðy2a � y2bÞ�Jþðy2aÞy2a

� Jþðy2bÞy2b

�

F6ðy2a; y2bÞ ¼ � 3

2

�ðJþðy2aÞ � Jþðy2bÞÞy2a � y2b

�

F7ðy2a; y2bÞ ¼ Iþðy2aÞ þ 2Iþðy2bÞ þ3

2y2aðJþðy2aÞ � Jþð0ÞÞ;

F8ðy2a; y2bÞ ¼’2

3�2ðy2a � y2bÞðIþðy2aÞ � Iþðy2bÞÞ: (B5)

We express our results in terms of these functions to allowour results to be used, if desired, when the high tempera-ture expansion is not employed.

1. Thermal masses: Scalars

As a simple example of the techniques employed todetermine the thermal mass basis for the gauge bosons,we now obtain the thermal masses of the Higgs andGoldstone-boson fields in the high temperature limit. Wealso improve on past results by using the reasoning behindthe one loop ring resummation to consistently employ ringresummation when a �2 operator is present. This willreduce the imaginary part of the effective potential due tothe scalar masses. To obtain thermal mass contributionsappropriate to shift the mass in ring resummation, one setsthe external momenta ðk0;kÞ to zero by setting k0 ¼ 0 andtaking the limit k2 ! 0. The diagrams to determine arethermal loops given by Fig. 11. We calculate in the WI, Bbasis for the gauge bosons, as we are interested in leadingorder T2 effects for the scalars to illustrate the modificationof the results due to the presence of NP.10

10Note that we use the results of the one loop gap calculationsfor the scalars (neglecting NP) in [81] when we consider the oneloop gap equation results of the vector bosons to be consistent inthe vector boson section.


075022-21

The results for the finite temperature contributions to thescalar self energies for the Higgs are

�1 ¼ 3�1T2

4�2

�Iþðy2hÞ

�1� 4C1

�

’2

f2

��;

þ 3�1T2

4�2

�Iþðy2�Þ

�1� 2C1

�

’2

f2

��;

�2 ¼ � 6m2t T

2

’2�2ðy2t K�ðy2t Þ þ I�ðy2t ÞÞ

�1� 2C1

�

’2

f2

�;

�3 ¼ 3T2

8�2ðg21Iþðy2BÞ þ 3g22Iþðy2WÞÞ

�1� 2C1

�

’2

f2

�:

�NP1 ¼ �2

f2

�3�1T

2

2�2ðIþðy2hÞ þ Iþðy2�ÞÞ

�2;

�NP2 ¼ 9�1T

2

2�2ðIþðy2hÞ þ Iþðy2�ÞÞ

�’2�2

f2

�; (B6)

where we have included the necessary rescalings of thekinetic sector to have a canonical low energy theory.Utilizing the high temperature expansion to expand ineach case to leading order, we find

�hð’; TÞ ¼ T2�1

4

�1� 3C1

�

’2

f2

�þ ’2T2

2f2�2;

þ T2BT

�1� 4C1

�

’2

f2

�þ 3T4�2

4f2;

��ð’; TÞ ¼ T2�1

4

�1� C1

�

’2

f2

�þ ’2T2

2f2�2;

þ T2BT þ 3T4�2

4f2: (B7)

2. Thermal masses: Vector bosons

When employing ring resummation in the SM, a veryimportant effect is the limit the magnetic mass places onthe Higgs self-coupling for a first order EWPT to occur.The magnetic mass is a nonperturbative contribution to thetransverse mass in thermal field theory, the inverse ofwhich corresponds to the magnetic screening length forthe SU(2) sector of theory. Although we cannot calculatethe magnetic mass, we can estimate its effects in perturba-tion theory when one calculates the gauge polarizationtensor in one loop gap equations. The magnetic mass stillimposes a very important constraint on the phase transitioneven in our HEFT. As we are examining deviations fromthe SM in PGH models, it is appropriate to have the SMcalculation of the gauge polarization tensor in thermal fieldtheory performed as accurately as possible. Thus, we de-termine the one loop gap equations for the gauge bosondegrees of freedom. The requisite diagrams to calculate aregiven in Fig. 12, when the propagators are full propagatorswhose masses are dictated by the self-consistent solution ofthe one loop gap equations.Again, we seek to obtain thermal mass contributions

appropriate to shift the mass in ring resummation and setthe external momenta ðk0;kÞ to zero by setting k0 ¼ 0 andtaking the limit k2 ! 0. The complications involved inconsidering mixing originate from the asymmetry betweenthe temporal and spatial components in thermal field the-ory. Because of this asymmetry, the longitudinal (tempo-ral) and transverse (spatial) modes of the gauge fielddevelop different effective masses at finite temperature.

i

f

1

2

3

W , B

21

NP

2

2NP

FIG. 11. One loop diagrams that contribute to the scalar ther-mal masses.

i

4

5

i

B

B B

B

W W

W

W W

W

W , B

6

W W

i

i

W , B W WW , B

B BW , B

7 8

9

W W

1 0

1 1

1 2

1 3

1 4

f f

FIG. 12. One loop diagrams that contribute to the vector bosonthermal masses. f indicates a sum over all spin 1=2 particles.


075022-22

Thus, we decompose the propagator in Landau gauge11 as

iD�i ðkÞ ¼ P�

L

k2 �m2i ��LðkÞ

þ P�T

k2 �m2i ��TðkÞ

;

(B8)

where the transverse and longitudinal projectors are

P�T ¼ g�i

� ij � kikj

k2

�gj ; (B9)

P�L ¼ k�k

k2� g� � P�

T ; (B10)

and m2i is the tree level mass. One can determine the

transverse and longitudinal corrections to the mass via�Lð0Þ ¼ ��0

0 and �Tð0Þ ¼ ��iið0Þ=3. The difficulty is

that once a mass eigenstate basis is known then the cor-rections to the massive gauge bosons are easy to determine;however, what exactly the mass eigenstate basis is dependson thermal corrections. We circumvent this difficulty byfirst calculating the diagrams with no internal massivegauge bosons in the B�, W

I� basis. For the longitudinal

mass the results of the diagrams�4,�5,�7,�8,�9,�10,�11 are

hWaWbi1L ¼ ð�7 þ�9 þ�10 þ�11ÞL;

¼ �g22T2 ab

�2

�F3ðy2�; y2hÞ þ

Iþð0Þ2

�;

þ g22T2 ab

�2

�Iþðy2hÞ þ 3Iþðy2�Þ

8

�;

� 12g22T2 ab

�2½I�ðy2t Þ � y2t K�ðy2t Þ=2; (B11)

hBBi1L ¼ ð�4 þ�5 þ�8ÞL;

¼ � g21T2 ab

�2½F3ðy2�; y2hÞ;

þ g21T2 ab

�2


8

�;

� 20g21T2 ab

�2½I�ðy2t Þ � y2t K�ðy2t Þ=2; (B12)

hW3Bi1L ¼ g1g2T2

8�2½Iþðy2�Þ � Iþðy2hÞ: (B13)

We then rotate these contributions to the two point func-tions by assuming that an angle exists for any T to diago-nalize the external W3, B fields. This defines a thermalbasis of the bosonic fields with

Z ¼ cosð�ðTÞÞW3 � sinð�ðTÞÞB;A ¼ sinð�ðTÞÞW3 þ cosð�ðTÞÞB; (B14)

andW is related toW1;2 in the usual manner. This thermalangle will limit to the Weinberg angle as T ! 0. Theremaining contributions to the projected two point func-tions are obtained from the diagrams �6, �12, �13, �14.One finds

hWþW�i2L ¼ � 8g22sin2ð�ðTÞÞT2

�2F1½ðyTWÞ2; ðyTAÞ2;�

4g22sin2ð�ðTÞÞT2

�2F1½ðyLWÞ2; ðyLAÞ2;

� 8g22cos2ð�ðTÞÞT2

�2F1½ðyTWÞ2; ðyTZÞ2;�

4g22cos2ð�ðTÞÞT2

�2F1½ðyLWÞ2; ðyLZÞ2;

þ g22T2sin2ð�ðTÞÞ2�2

F4½ðyLAÞ2; ðyTAÞ2;þg22T

2cos2ð�ðTÞÞ2�2

F4½ðyLZÞ2; ðyTZÞ2;þg22T

2

2�2F4½ðyLWÞ2; ðyTWÞ2;

þ g424F5½ðyLWÞ2; ðy�Þ2;þ

g21g22

4cos2ð�ðTÞÞF5½ðyLAÞ2; ðy�Þ2;þ

g21g22

4sin2ð�ðTÞÞF5½ðyLZÞ2; ðy�Þ2: (B15)

The high temperature expansion of ðmLWðT;’ÞÞ2 is given by

ðmLWðT;’ÞÞ2 ¼

11g22T2

6þ g22’

2

4� g42’

2T

16�ðmh þmLWÞ

;� g22T

16�ðmh þ 3m� þ 4mL

Z þ 8mLW þ 4mL

AÞ;

þ g22TðmLA �mL

ZÞ cosð2�ðTÞÞ4�

� g21g22v

2T

16�

�sin2ð�ðTÞÞmL

Z þm�

þ cos2ð�ðTÞÞmL

A þm�

�; (B16)

11As this decomposition is gauge dependent, the resultant thermal masses and thermal Weinberg angles will be gauge dependent. Infact, the effective potential itself is gauge dependent as well; however, all physical quantities derived from the effective potential willbe gauge independent. Note there exists a subtlety in the decomposition that leads to a factor of 2=3 [81].


075022-23

which reproduces the known answer for the case of vanishing U(1) charge [81] in the g1 ! 0, � ! 0 limit. Note that wehave added the usual EW term g22’

2=4 to this expression. For thermal photon and Z fields, one finds the following:

hAAi2L ¼ � 8g22sin2ð�ðTÞÞT2

�2F2½ðyTWÞ2;�

4g22sin2ð�ðTÞÞT2

�2F2½ðyLWÞ2;þ

g22T2sin2ð�ðTÞÞ�2

F4½ðyLWÞ2; ðyTWÞ2;

þ g21g22

4cos2ð�ðTÞÞF5½ðyLWÞ2; ðy�Þ2;þ

ðg1 cosð�ðTÞÞ � g2 sinð�ðTÞÞÞ48

F5½ðyLAÞ2; ðyhÞ2; (B17)

hZZi2L ¼ � 8g22cos2ð�ðTÞÞT2


4g22cos2ð�ðTÞÞT2

�2F2½ðyLWÞ2;þ

g22T2cos2ð�ðTÞÞ�2


þ g21g22

4sin2ð�ðTÞÞF5½ðyLWÞ2; ðy�Þ2;þ

ðg1 cosð�ðTÞÞ þ g2 sinð�ðTÞÞÞ48

F5½ðyLZÞ2; ðyhÞ2; (B18)

hAZi2L ¼ � 8g22 cosð�ðTÞÞ sinð�ðTÞÞT2


4g22 cosð�ðTÞÞ sinð�ðTÞÞT2

�2F2½ðyTWÞ2;

þ g22T2 sinð�ðTÞÞ cosð�ðTÞÞ

�2F4½ðyLWÞ2; ðyTWÞ2;þ

g21g22

4cosð�ðTÞÞ sinð�ðTÞÞF5½ðyLWÞ2; ðy�Þ2: (B19)

The thermal Weinberg angle �ðTÞ for the longitudinal massis defined by demanding that

ðmLAZÞ2 ¼ sinð�ðTÞÞ cosð�ðTÞÞðhW3W3i1L

� hBBi1LÞ;þðcos2ð�ðTÞÞ � sin2ð�ðTÞÞÞhW3Bi1Lþ hAZi2L

vanish for a specific T. These expressions are rather daunt-ing. Let us first consider the case of vanishingly smalltemperature. Adding the usual tree level EW terms theexpression ðmL

AZÞ2 reduces to

ðmLAZÞ2T!0 ¼ �’2

8ð2g1g2 cosð2�ðTÞÞÞ

� ’2

8ððg21 � g22Þ sinð2�ðTÞÞÞ; (B20)

demanding that ðmLAZÞ2 ¼ 0, the solution is

sinð�ð0ÞÞ ¼ g1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig21 þ g22

q ; (B21)

which establishes that in the limit T ! 0, the thermalWeinberg angle reduces to the usual �W . In the oppositelimit as T � ’, one finds the leading term

ðmLAZÞ2’!0 ¼

11T2

12ðg22 � g21Þ sinð2�ðTÞÞ; (B22)

and ðmLAZÞ2 for sinð�ðTÞÞ ¼ 0 at high temperature. As ex-

pected, EW symmetry is restored and the diagonal basis isthe basis of the unbroken electroweak theory WI

�, B�.Although not unexpected, this is entertaining.

Employing the high temperature expansion, we canperturbatively solve as the temperature decreases. Thethermal angle will be sinð�ðTÞÞ ¼ 0þOð1=TÞ, and wefind that the temperature dependence of thermalWeinberg angle is

sinð�ðTÞÞT>’ ¼ 3g1g2ðmhð’; TÞ �m�ð’; TÞÞ88ðg22 � g21Þ�T

; (B23)

so that as the temperature lowers, thermal Weinberg anglerises toward �W , and the correct basis changes over to thebasis in the broken electroweak theory. Note that the scalarmasses are the results of the one loop gap equation scalarmasses that are not imaginary for small ’.However, we are interested in mass effects when the

temperature is eventually approaching the temperature Tb.We retain terms of Oðg2sm’2Þ and Oðg2smT2Þ, when solvingthe equation. Note that terms of order Oðg2sm’2Þ from thehigh temperature expansion are loop suppressed comparedwith the tree level electroweak terms and are dropped. Thetemperature scale is set by Tb � 100 GeV for the phasetransition and ’ & Tb < v, as the minima is not yetreached in the potential. Thus, one can see that the hightemperature expansion is properly thought of as a pertur-bative expansion in gsm for the temperatures of interestabout the phase transition. We solve ðmL

AZÞ2 perturbativelyin gSM for this reason. The resulting expression is stillrather daunting. However, the physical dependence canbe deduced by using g22 ¼ 4M2

WðvÞ=v2 and g21 ¼4ðM2

ZðvÞ �M2WðvÞÞ=v2 and expanding in ðT � ’Þ. One

finds

sinð�ðTÞÞ 0:09� 0:15ðT � ’ÞT

þ 0:03ðT � ’Þ2T2

:

With these insights we can state the correct longitudinalthermal masses for the bosonic fields relevant for studies ofthe EWPT to be, to Oðg2SMÞ


075022-24

ðmLWð’; TÞÞ2 ¼ g22

�11T2

6þ ’2

4

�;

ðmLAð’; TÞÞ2 ¼

11T2


2ð�ðTÞÞÞ

þ ’2

4ðg1 cosð�ðTÞÞ � g2 sinð�ðTÞÞÞ2;

ðmLZð’; TÞÞ2 ¼

11T2


2ð�ðTÞÞÞ

þ ’2

4ðg1 sinð�ðTÞÞ þ g2 cosð�ðTÞÞÞ2:

(B24)

The appropriate approximation for thermal Weinbergangle in studies of the electroweak phase transition issinð�ðTbÞÞ. However, our formalism can be used for nu-merical studies not using this approximation if desired.

The calculations for the transverse mass are similar. Forthe transverse mass the results for the diagrams �4, �5,�7, �8, �9, �10, �11 are

hWaWbi1T ¼ ð�7 þ�9 þ�10 þ�11ÞL;

¼ � g22T2 ab

4�2

�Iþðy2�Þ þ

2ðJþðy2�Þ � Jþðy2hÞÞy2� � y2h

�;

þ g22T2 ab

�2


8

�;

þ 6g22T2 ab

�2½y2t K�ðy2t Þ

þ g22T2 ab

2�2Iþð0Þ; (B25)

hBBi1T ¼ ð�4 þ�5 þ�8ÞT;

¼ �g21T2 ab

4�2

�Iþðy2�Þ þ

2ðJþðy2�Þ � Jþðy2hÞÞy2� � y2h

�;

þ g21T2 ab

�2


8

�;

þ 20g21T2 ab

�2½y2t K�ðy2t Þ; (B26)

hW3Bi1T ¼ g1g2T2

8�2½Iþðy2�Þ � Iþðy2hÞ: (B27)

Again, we rotate these contributions to the two pointfunctions by assuming that an angle exists to diagonalizethe external W3, B fields. Note however that this angle isnot the same as in the longitudinal case, although it re-mains true that this second thermal angle will limit to theWeinberg angle as T ! 0. Again, this defines a thermalbasis of fields with

Z ¼ cosð�0ðTÞÞW3 � sinð�0ðTÞÞB;A ¼ sinð�0ðTÞÞW3 þ cosð�0ðTÞÞB; (B28)

and W related to W1;2 in the usual manner. For thediagrams �6, �12, �13, �14, one finds for the transversemass

hWþW�i2T ¼ 8g22sin2ð�0ðTÞÞT2

3�2F6½ðyTWÞ2; ðyTAÞ2;þ

4g22sin2ð�0ðTÞÞT2

3�2F6½ðyLWÞ2; ðyLAÞ2;

þ 8g22cos2ð�0ðTÞÞT2

3�2F6½ðyTWÞ2; ðyTZÞ2;þ

4g22cos2ð�0ðTÞÞT2

3�2F6½ðyLWÞ2; ðyLZÞ2;

þ g22T2sin2ð�0ðTÞÞ3�2

F7½ðyLAÞ2; ðyTAÞ2;þg22T

2cos2ð�0ðTÞÞ3�2

F7½ðyLZÞ2; ðyTZÞ2;

þ g22T2

3�2F7½ðyLWÞ2; ðyTWÞ2 þ

g424F8½ðyLWÞ2; ðy�Þ2;þ

g21g22

4cos2ð�0ðTÞÞF8½ðyLAÞ2; ðy�Þ2;

þ g21g22

4sin2ð�0ðTÞÞF8½ðyLZÞ2; ðy�Þ2: (B29)

So that the high temperature expansion of hWþW�i1T þ hWþW�i2T is given by

ðmTWÞ2 ¼

g22mTWT

3�þ g22m

LWT

12�þ g22’

2

4;þ g22T5ðmT

Zcos2½�0ðTÞ þmT

Asin2½�0ðTÞÞ

12�� g21g

22’

2Tcos2½�0ðTÞ24�ðmL

A þm�Þ

� g21g22’

2Tsin2½�0ðTÞ24�ðmL

Z þm�Þ � 2g22mTWm

TZTcos

2½�0ðTÞ3�ðmT

W þmTZÞ

þ g22Tðmh �m�Þ248�ðmh þm�Þ �

2g22mTWm

TATsin

2½�0ðTÞ3�ðmT

W þmTAÞ

� g42’2T

24�ðmh þmLWÞ

� g22mLWm

LZTcos

2½�0ðTÞ3�ðmL

W þmLZÞ

� g22mLWm

LATsin

2½�0ðTÞ3�ðmL

W þmLAÞ

; (B30)


075022-25

which also reproduces the known answer for the case of vanishing U(1) charge [81] in the g1 ! 0, � ! 0 limit. Thetransverse masses of thermal photon and Z fields are deduced from the addition of the rotated contributions and thefollowing

hAAi2T ¼ � g22sin2ð�0ðTÞÞT2

�2ðIþ½ðyLWÞ2 þ 2Iþ½ðyTWÞ2Þ;þ

2g22T2sin2ð�0ðTÞÞ3�2


þ g21g22

4cos2ð�ðTÞÞF8½ðyLWÞ2; ðy�Þ2;þ

ðg1 cosð�ðTÞÞ � g2 sinð�ðTÞÞÞ48

F8½ðyLAÞ2; ðyhÞ2; (B31)

hZZi2T ¼ �g22cos2ð�0ðTÞÞT2

�2ðIþ½ðyLWÞ2 þ 2Iþ½ðyTWÞ2Þ;þ

2g22T2cos2ð�0ðTÞÞ3�2


þ g21g22

4sin2ð�ðTÞÞF8½ðyLWÞ2; ðy�Þ2;þ

ðg1 cosð�ðTÞÞ þ g2 sinð�ðTÞÞÞ48

F8½ðyLZÞ2; ðyhÞ2; (B32)

hAZi2T ¼ �g22 cosð�0ðTÞÞ sinð�0ðTÞÞT2

�2Iþ½ðyLWÞ2 �

2g22 cosð�0ðTÞÞ sinð�0ðTÞÞT2

�2Iþ½ðyTWÞ2;

þ 2g22T2 sinð�0ðTÞÞ cosð�0ðTÞÞ

3�2F7½ðyLWÞ2; ðyTWÞ2;þ

g21g22

4cosð�0ðTÞÞ sinð�0ðTÞÞF8½ðyLWÞ2; ðy�Þ2:

As in the longitudinal case, thermal Weinberg angle �0ðTÞfor the transverse mass is defined by demanding that

ðmTAZÞ2 ¼ sinð�0ðTÞÞ cosð�0ðTÞÞðhW3W3i1T

� hBBi1TÞ;þðcos2ð�0ðTÞÞ � sin2ð�0ðTÞÞÞhW3Bi1Tþ hAZi2T

vanish for a given T. Again, at small temperature we find

ðmTAZÞ2T!0 ¼ �’2

8ð2g1g2 cosð2�0ðTÞÞÞ

� ’2

8ððg21 � g22Þ sinð2�0ðTÞÞÞ; (B33)

whereas, at high temperature we now have

ðmTAZÞ2’!0 ¼ �T2

48g21 sinð2�0ðTÞÞ; (B34)

and again we have that ðmTAZÞ2 ¼ 0 for sinð�0ðTÞÞ ¼ 0 at

high temperature. Numerically approximating the solutionas before and expanding in ðT � ’Þ. One finds

sinð�0ðTÞÞ 0:49þ 0:03ðT � ’ÞT

þ 0:04ðT � ’Þ2T2

from which we see that at ’ & Tb the transverse masseshave already mixed further into the thermal mass basisfrom the initial EW basis. The transverse thermal massesfor the bosonic fields relevant for studies of the EWPT toOðg3SMÞ are

ðmTWð’; TÞÞ2 ¼ ðmT

WÞ2ð’; TÞ; (B35)

ðmTAð’; TÞÞ2 ¼

g22mTWTsin

2½�0ðTÞ3�

þ g21T2cos2½�0ðTÞ

24;

þ ’2ðg2 sin½�0ðTÞ � g1 cos½�0ðTÞÞ24

;

þ Fþðmhð’Þ; m�ð’ÞÞ;

ðmTZð’; TÞÞ2 ¼

g22mTWTcos

2½�0ðTÞ3�

þ g21T2sin2½�0ðTÞ

24;

þ ’2ðg2 cos½�0ðTÞ þ g1 sin½�0ðTÞÞ24

;

þ F�ðmhð’Þ; m�ð’ÞÞ: (B36)

We have defined the following functions of the scalarmasses:

Fðmhð’Þ; m�ð’ÞÞ ¼ A1’ðg21 þ g22Þ;

A1’ðg21 � g22Þ cos½2�0ðTÞ

6A2’g1g2 sin½2�0ðTÞ; (B37)

A1’ ¼ T

96�

ðmhð’Þ �m�ð’ÞÞðmhð’Þ þm�ð’ÞÞ ðmhð’Þ �m�ð’ÞÞ;

A2’ ¼ T

96�

ðmhð’Þ �m�ð’ÞÞðmhð’Þ þm�ð’ÞÞ ðmhð’Þ þm�ð’ÞÞ:

We have not in fact solved for the transverse masses as yetdue to the appearance of mT

W on both sides of Eq. (B35).


075022-26

This is an important feature of the transverse mass thatleads to the inclusion of a nonperturbative magneticmass term. Consider ’ ! 0, then cosð�0ðTÞÞ ¼ 1,sinð�0ðTÞÞ ¼ 0, and mhð0Þ ¼ m�ð0Þ. In this case, the ex-pression for mT

W is given by

mTWð0; TÞ2 ¼

g22T

3�mT

Wð0; TÞ: (B38)

The physical solution [83] is

mTWð0; TÞ ¼

g22T

3�: (B39)

A non-Abelian gauge theory is expected to have a term ofthis form as a nonperturbative feature [74,86]. Of course,we cannot calculate a nonperturbative result in perturba-tion theory. We retain this term as it plays an important roleas ’ ! 0 in determining the nature of the phase transition.We multiply the magnetic mass term by an unknown �factor to signify its nonperturbative origin in our Vring

eff .Lattice simulations have determined mmðTÞ ¼0:456ð6Þg22ðTÞT [80], which one expects to be a goodapproximation of the magnetic mass of the SU(2) sectorof the SM, giving � ¼ 4:2.

The magnetic mass for mTWð0; TÞ contributes a magnetic

mass term to mTZð0; TÞ and mT

Að0; TÞ. For mTAð0; TÞ, there is

a g1 term that is ’ independent. As ’ ! 0, the magneticmass does not screen thermal photon field. Thus, themagnetic mass effects on the photon field can be dropped.

For mTZð’; TÞ, in the ’ ! 0 limit the magnetic mass

does screen the Z field and is retained. The Oðg2SMÞ trans-verse masses, including only the important ’ independentmagnetic mass terms that are higher order are given by

ðmTWð’; TÞÞ2 ¼

�2g429�2

T2 þ g22’2

4; (B40)

ðmTAð’;TÞÞ2 ¼

g21T2cos2½�0ðTÞ

24;

þ’2ðg2 sin½�0ðTÞ�g1 cos½�0ðTÞÞ24

:

ðmTZð’;TÞÞ2 ¼

g22mTWð’;TÞTcos2½�0ðTÞ

3�þg21T

2sin2½�0ðTÞ24

;

þ’2ðg2 cos½�0ðTÞþ g1 sin½�0ðTÞÞ24

;

where we have neglected the functions F that areOðg2SM�1Þ and suppressed by loop factors.

3. Oð�y�Þ3 finite temperature terms

As we are considering the effects of the operatorð�y�Þ3, one should note that matching corrections tothis operator are obtained by expanding the one loop finitetemperature contributions given in Eq. (45).Expanding Eq. (45) to higher order, it is easy to retain

the m6 term. For the bosons, this gives a contribution

VB6 ðTÞ ¼

ð�y�Þ3�2

�ð3Þ�2

6ð4�2Þ2T2

�m6

h

v6þ 6

m6W

v6þ 3

m6Z

v6

�

to the potential. For the fermions, we similarly have

VF6 ðTÞ ¼

ð�y�Þ3�2

8ð7�ð3Þ � 8Þ�2

ð4�2Þ2T2

m6t

v6: (B41)

These matching corrections are small for the temperaturesof interest. For mh ¼ 120 GeV, v ¼ 246 Gev and thePGD values [27] for the known masses mW ¼ 80:4 GeV,mZ ¼ 91:2 GeV, mt ¼ 172:5 GeV the sum of the match-ing corrections above gives a correction of size 2:6�10�4=T2, to the ð�y�Þ3 operator. Because of its negligiblecoefficient we neglect this matching correction.

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