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Electroweak Phase Transition and Sphaleron

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Electroweak Phase Transition and Sphaleron Electroweak Phase Transition and Sphaleron Eibun Senaha (Natl Taiwan U) April 7, 2017 ACFI Workshop: Making the Electroweak Phase Transition (Theoretically) Strong @Umass-Amherst Outline • part1: EWPT and sphaleron in a complex-extended SM (cxSM) CW Chiang, M. Ramsey-Musolf, E.S., in progress • part2: Band structure effect on sphaleron rate at high-T. K. Funakubo, K. Fuyuto, E.S., arXiv:1612.05431 Introduction - Standard perturbative treatment of EWPT is gauge-dependent. T=TC TC: T at which Veff has degenerate minima vC: minimum of Veff at TC - Gauge-independent methods: vC (1) vC and TC are determined by (2) Patel-Ramsey-Musolf (PRM) scheme [JHEP07(2011)029] vC and TC are determined separately. We analyze EWPT and sphaleron in the cxSM using 2 methods. SM with a complex scalar (cxSM) H: SU(2)-doublet Higgs, S: SU(2)-singlet Higgs 2 We assume all parameters are real. m , λ, δ2, b2, d2, a1, b1 v0, vS0, mH1 , mH2 , ↵, mA, a1 SM with a complex scalar (cxSM) H: SU(2)-doublet Higgs, S: SU(2)-singlet Higgs 2 We assume all parameters are real. m , λ, δ2, b2, d2, a1, b1 v0, vS0, mH1 , mH2 , ↵, mA, a1 246GeV SM with a complex scalar (cxSM) H: SU(2)-doublet Higgs, S: SU(2)-singlet Higgs 2 We assume all parameters are real. m , λ, δ2, b2, d2, a1, b1 v0, vS0, mH1 , mH2 , ↵, mA, a1 246GeV 125GeV SM with a complex scalar (cxSM) H: SU(2)-doublet Higgs, S: SU(2)-singlet Higgs 2 We assume all parameters are real. m , λ, δ2, b2, d2, a1, b1 v0, vS0, mH1 , mH2 , ↵, mA, a1 246GeV 125GeV Patterns of PT Because of 2 fields, there are many patterns of phase transitions. 3 (a) (b) minima at A and B. In this case, the EWPT at T = TC will be first order. Note, however, that an oversized pos- itive δ2 may render B at T = 0 metastable, since the energy di↵erence between A and B phases, which is sup- posed to be positive, can become negative. Therefore, there should be an upper bound on the magnitude of δ2, as will be discussed below. For negative δ2, in contrast, TC is always raised which may preventv ¯C /TC becoming asufficient size. In fact, SFOEWPT cannot be found for (c) (d) δ2 < 0 in our numerical analysis. The situation can be more complex in the presence of non-vanishing a1 and the remaining zero-temperature and thermal loop e↵ects encoded in the one-loop e↵ective potential: Ve↵ ('; T )=V0(')+V1('; T ) (15) where ' =(', 'S); 4 2 T m¯ j FIG. 1: Patterns of symmetry breaking at finite temperature. V ('; T )= n V (¯m2)+ I , 1 j CW j 2⇡2 B,F T 2 j ! X (16) where ' and 'S denote the neutral doublet and singlet background fields, respectively, and and nj count the degrees of freedom for particle species j, andm ¯ j are '-dependent masses. The Coleman-Weinberg 2 2 2 2 λ δ 3g + g y potential VCW and IB,F (a ) are respectively given by [16, ⌃ = + 2 + 2 1 + t T 2 ⌃¯ T 2 , (11) H 8 24 16 4 ⌘ H 17] 2 4 2 T 2 2 m m ⌃ =(δ + d ) ⌃¯ T . (12) VCW(m )= ln c , (17) S 2 2 12 ⌘ S 64⇡2 µ2 − ✓ ◆ For m < 0 and b < 0, T = 0 minima will exist for 2 1 2 px2+a2 2 2 IB,F (a )= dx x ln 1 e− , (18) non-zero v0 and vS0. In the limit of vanishing a1, at suffi- 0 ⌥ Z ⇣ ⌘ ciently high-T the only minimum of the theory occurs at where c =3/2 for scalars and fermions and 5/6 for gauge the origin, denoted by “O”. As T decreases, one generi- bosons, and µ is the renormalization scale. cally expects that a secondary minimum at ' v˜ =0 s S For a1 = 0, the high-T minimum will no longer lie will first appear, since ⌃¯ < ⌃¯ . At a temperature⌘ 6 T , 6 S H 1 at the origin but will be shifted by a1 along the 'S the minimum atv ˜S will become the global minimum in- direction. Alternately, EWSB may occur− directly from dicated by “A” in Fig. 1. As T further decreases, an ad- the origin to point B, as Type (d) in Fig. 1. However, ditional minimum at (' v = 0, ' v = 0) develops, ⌘ 6 S ⌘ S 6 this phase transition is not first order sincev ˜S is zero becoming the global minimum at temperature T2 <T1, as seen from Eq. (13). In what follows, we exclusively corresponding to point “B” in Fig. 1. The universe will explore two cases: Type-(a) EWPT with a1 = 0 and then follow a two-step symmetry-breaking trajectory in 6 Type-(c) EWPT with a1 = 0, i.e., field space indicated Fig. 1, where one may have either S1: mH = 230 GeV, vS0 = 40 GeV, a1 = v˜S(T2) vS(T2) > 0 or < 0. We will henceforth denote T2 2 − (110 GeV)3, as the EWSB critical temperature, TC and the value of ' − at this temperature as vC (TC ). After a straightforward S2: mH2 = 150 GeV, vS0 = 0 GeV, δ2 = 2, d2 = 1, calculation, one finds and mA = mH1 /2 62.5 GeV in both cases. In Fig. 2, temperature evolutions' of the VEVs are plotted in S1 2δ2v˜S(TC ) v¯C v˜S(TC ) vS(TC ) , (13) (Left) and S2 (Right). For the former, it is found that ' λ − r A B transition is first-order, and T2 = TC = 90.4 GeV 2 ! 1 2 v˜S(TC ) andv ¯C = 158.2 GeV. For the latter, on the other hand, TC ¯ m δ2 . (14) ' s2⌃H − − 2 one can see that O A transition is second order while ✓ ◆ A B transition is! first order. We obtain T = 176.8 ! 1 Here, the bar over vC indicates that it has been com- GeV for the former transition and T2 = TC = 85.8 GeV, puted using the high-T e↵ective potential (10). TC and v¯C = 181.4 GeV,v ¯SC = 0 GeV andv ˜SC = 153.8 GeV for v¯C obtained in this way are the leading-order (LO) gauge- the latter transition. Fig. 3 shows contours of the high- invariant results. For positive (negative) δ2 one has T e↵ective potential at T = 180 GeV (Upper Left), 100 v˜S(TC ) larger (smaller) than vS(TC ). In addition, for GeV (Upper Right), TC (Lower Left) and 0 GeV (Lower positive δ2, the potential will have a barrier between the Right) in the case of S2. Patterns of PT Because of 2 fields, there are many patterns of phase transitions. 3 (a) (b) minima at A and B. In this case, the EWPT at T = TC will be first order. Note, however, that an oversized pos- itive δ2 may render B at T = 0 metastable, since the energy di↵erence between A and B phases, which is sup- posed to be positive, can become negative. Therefore, there should be an upper bound on the magnitude of δ2, as will be discussed below. For negative δ2, in contrast, TC is always raised which may preventv ¯C /TC becoming asufficient size. In fact, SFOEWPT cannot be found for (c) (d) δ2 < 0 in our numerical analysis. The situation can be more complex in the presence of non-vanishing a1 and the remaining zero-temperature and thermal loop e↵ects encoded in the one-loop e↵ective potential: Ve↵ ('; T )=V0(')+V1('; T ) (15) where ' =(', 'S); 4 2 T m¯ j FIG. 1: Patterns of symmetry breaking at finite temperature. V ('; T )= n V (¯m2)+ I , We will focus on type (a) PT. 1 j CW j 2⇡2 B,F T 2 j ! X (16) where ' and 'S denote the neutral doublet and singlet background fields, respectively, and and nj count the degrees of freedom for particle species j, andm ¯ j are '-dependent masses. The Coleman-Weinberg 2 2 2 2 λ δ 3g + g y potential VCW and IB,F (a ) are respectively given by [16, ⌃ = + 2 + 2 1 + t T 2 ⌃¯ T 2 , (11) H 8 24 16 4 ⌘ H 17] 2 4 2 T 2 2 m m ⌃ =(δ + d ) ⌃¯ T . (12) VCW(m )= ln c , (17) S 2 2 12 ⌘ S 64⇡2 µ2 − ✓ ◆ For m < 0 and b < 0, T = 0 minima will exist for 2 1 2 px2+a2 2 2 IB,F (a )= dx x ln 1 e− , (18) non-zero v0 and vS0. In the limit of vanishing a1, at suffi- 0 ⌥ Z ⇣ ⌘ ciently high-T the only minimum of the theory occurs at where c =3/2 for scalars and fermions and 5/6 for gauge the origin, denoted by “O”. As T decreases, one generi- bosons, and µ is the renormalization scale. cally expects that a secondary minimum at ' v˜ =0 s S For a1 = 0, the high-T minimum will no longer lie will first appear, since ⌃¯ < ⌃¯ . At a temperature⌘ 6 T , 6 S H 1 at the origin but will be shifted by a1 along the 'S the minimum atv ˜S will become the global minimum in- direction. Alternately, EWSB may occur− directly from dicated by “A” in Fig. 1. As T further decreases, an ad- the origin to point B, as Type (d) in Fig. 1. However, ditional minimum at (' v = 0, ' v = 0) develops, ⌘ 6 S ⌘ S 6 this phase transition is not first order sincev ˜S is zero becoming the global minimum at temperature T2 <T1, as seen from Eq.
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