Simulations of Cold Electroweak Baryogenesis: hypercharge U(1) and the creation of helical magnetic fields

Journal of High Energy Physics, Jun 2017

We perform numerical simulations of Cold Electroweak Baryogenesis, including for the first time in the Bosonic sector the full electroweak gauge group SU(2) × U(1) and CP-violation. We find that the maximum generated baryon asymmetry is reduced by a factor of three relative to the SU(2)-only model of [1], but that the quench time dependence is very similar. In addition, we compute the magnitude of the helical magnetic fields, and find that it is proportional to the strength of CP-violation and dependent on quench time, but is not proportional to the magnitude of the baryon asymmetry as proposed in [2, 3]. Astrophysical signatures of primordial magnetic helicity can therefore not in general be used as evidence that electroweak baryogenesis has taken place.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP06%282017%29075.pdf

Simulations of Cold Electroweak Baryogenesis: hypercharge U(1) and the creation of helical magnetic fields

Received: May Simulations of Cold Electroweak Baryogenesis: hypercharge U(1) and the creation of helical magnetic elds Nottingham 0 U.K. 0 Zong-Gang Mou 0 1 Paul M. Sa n 0 2 Anders Tranberg 0 1 CP-violation. We 0 Open Access 0 c The Authors. 0 Theory, Nonperturbative E ects 0 4036 Stavanger , Norway 1 Faculty of Science and Technology, University of Stavanger 2 School of Physics and Astronomy, University of Nottingham for the rst time in the Bosonic sector the full electroweak gauge group SU(2) U(1) and nd that the maximum generated baryon asymmetry is reduced by a factor of three relative to the SU(2)-only model of [1], but that the quench time dependence is very similar. In addition, we compute the magnitude of the helical magnetic elds, and nd that it is proportional to the strength of CP-violation and dependent on quench time, but is not proportional to the magnitude of the baryon asymmetry as proposed in [2, 3]. Astrophysical signatures of primordial magnetic helicity can therefore not in general be used as evidence that electroweak baryogenesis has taken place. magnetic; elds; Cosmology of Theories beyond the SM; CP violation; Lattice Quantum Field Contents 1 Introduction 2 3 3.1 Conclusion A Lattice implementation B Initial condition C CP-violation, Introduction Variance of observables CP-odd: the baryon asymmetry and helicity Short and long time evolution Hypercharge impact on asymmetry Helical magnetic eld and baryogenesis Sphaleron unravelling The SU(2) U(1)-Higgs model with CP-violation quench time [1, 8]. rst-principles considered hypercharge and the generation of magnetic elds [12, 13], or have included percharge on the tion value, and dependence of the baryon asymmetry itself. The SU(2) U(1)-Higgs model with CP-violation SU(2) and U(1) gauge elds, with the classical action S = dt d3x + e2 (t) y + ( y )2 + y Tr W The eld strength tensors are W for SU(2) and B for U(1), with the dual de ned by e2 (t) = 2t= q) The covariant derivative D is given by = 12 around T = 1 GeV. number, de ned as an integral over time or equivalently in a spatial representation Ncs;2(t) Ncs;2(0) = dt d3x Tr W Ncs;2(t) = WiaWjak hNB(t) NB(0)i = 3h[Ncs;2(t) Ncs;2(0)]i: (g0)2 Z hypercharge Chern-Simons number Ncs;1(t) = d3x ijkBiBjk; 2 = These are naturally generalised to outside of the unitary gauge, where we have de ned = W 3 cos + B sin ; = W 3 sin B cos : = naW a cos + B sin ; = naW a sin B cos ; Nw = dx3 ijkTr[U y@iU U y@j U U y@kU ]; U(x) = hNB(t) NB(0)i = 3h[(Ncs;2 Ncs;1)(t) (Ncs;2 Ncs;1)(0)]i; sition involving true fermion production always has Ncs;1 = 0. elds, respectively, EW , EB, E . na = ' = to 'y@ ' a manifestly gauge covariant way as 'y(D ') sin = @ (naW a) @ (naW a) value for the mixing angle , the eld A is the photon eld throughout, but only at the end of the transition is it massless [19]. We will include in our list of observables the component, being composed of part of EB and part of EW . ! (0; 0; 1) everywhere, mH = 2 2 = p 2 v = 125 GeV; mW = mZ = = 0; gv = 77:5 GeV; = 88:4 GeV; with our choices of parameters1 v = 246 GeV ; sin2 = 0:231 ; e = g = = 0:63 ; g0 = = 0:35 : In the A 1 Z Nh = d3x ijkAiFjk: while our CP-even (P-even, C-even) observables are 2, all the energy components EB, The gauge impact on the nal results. appendix A. ip occurs. The the evolution from the CP-conjugated initial data. and energy is transferred to the gauge elds as particles are created. At rst, only the modes k ensemble composed of such pairs. a certain fraction ip with 1 or even in rare cases 2. Averaging over these instances of 0; 1; 2 gives our ensemble averages. We nd, that when CP-violation is not present, the vanishing of the CP-odd observables is not a result of the cancelling of positive and negative ips. No ips occur at all. Energy written as E = dt d3x y (x; t); temperature is also smaller by a factor Tslowquench / (Einitial E)1=4 ' 0:7 Tfastquench. motion only, and not the gauge elds themselves. Variance of observables CP-odd quantities w2 = hNw2 i h2 = hNh2i Chern-Simons number have been rescaled to t inside the plot. cs;2 = cs;1 = w = h = the volume), describing the (mH t)1:4 and number as hNcs;2i(t) = cs;2(t) Te (t) cs;2(t)dt; their di usion rate, and bias coe cient. P violating term in the action is dt cs;2(t)Ncs; cs;2(t) = up the di usion rate, and simply nd hNcs;2i(t) = CP-odd: the baryon asymmetry and helicity Short and long time evolution two quench-times in the gure4 0:294 exp( 0:003 mH t); 0:273 exp( 0:002 mH t): minimum occurs at 0:71mH q + 10. Ncs;2 ! Nw also produces good ts. From this mH t ' 300{500. Moreover, we see from gure 7 that the winding number stays put after { 10 { cp = 6:83. 400, and infer the nal asymmetry from the value of Nw. In this gure, we show the accumulated errors from the time integration. Hypercharge impact on asymmetry metry itself. { 12 { taking cp = 6:83. strongly correlated with the nal asymmetry (this was also observed in [8]). But we see very little impact of including the hypercharge U(1). Noting from gure 10 (left) Nw(t ! 1) = 1:39 the CP-symmetric ensemble. { 13 { with mH q = 16. Helical magnetic eld and baryogenesis hNcs;1i ' 1:56 that the dependence on cp is close to linear, and we nd hNhi ' 1:66 cp; We show in extrapolated to an asymptotic value as for Ncs;2 in gure 7. We see that for both observ{ 15 { the magnetic eld, also generating helicity. In [2], the prediction is that hNhi ' 300hNcs;2i for the decay of Sphalerons. By order Sphaleron unravelling elds, with Nh the equations of motion. 7In the plot we multiply Ncs by 3 to match the convention used in [3]. { 17 { Chern-Simons number is one, the helicity uctuates around Conclusion timescale of Higgs eld at minima of its mean value. violation and quench time we discovered that, for a xed quench time, the Chern-Simons of the Higgs eld, be used as a proxy for baryogenesis. { 18 { there as well. This is currently under investigation [27]. Acknowledgments Lattice implementation L = X Tr U0i 1 X Tr h(Di )y(Di ) Tr Uij where cQ = 2 t Q and L2 is the CP-violating term, momentum _ = [ (x + 0) (x) = U (x) (x + )Z (x) and the gauge covariant derivative U (x) Z (x) = V (x) V (x) gdx W a(x) ; V (x) g0dx B (x) ; of (A.1). Eia(x) = the Higgs eld: electric elds on the lattice as, Tr hi aUi(x+0)Uiy(x) gdtdxi ; Ei(x) = Vi(x+0)Viy(x) Vi(x)Viy(x+0) ig0dtdxi (x+0) 2 (x)+ (x 0) Tr[ y(x) (x)] (x) + X 1 @ L2 k ; where k is one 2 2 matrix in (1; i 1; i 2; i 3), so that = P k , and (x + 0) = (x) + dt _ (x): Vi(x + 0) = 4 dtdxiEi(x) i dtdxiEi(x)5 Vi(x): [Vij(x) Vji(x)+Vji(x j) Vij(x j)] ji ; Tr hi aUi(x)Uj(x + i)Uiy(x + j)Ujy(x) Tr hi aUjy(x j)Ui(x j)Uj(x + i j)Uiy(x) For the U(1) gauge eld, we nd: Ei(x) Ei(x 0) For the SU(2) gauge elds: Eia(x) = X ji = J i;a = Ui(x + 0) = dtdxiEia(x) + X i a g dtdxiEia(x) Ui(x): Tr h y(x)Ui(x) (x + i)Zi(x)i 3i ; Tr h y(x)i aUi(x) (x + i)Zi(x)i : { 20 { X Ei(x) Ei(x i) X Eia(x) = 0 = a + Eia(x) = 1 X Tr hi aUiy(x)i bUi(x)i Eib(x); 0 = g0 1 = g 2 = g 3 = g the Higgs charge densities, (x) = 0; = 0; 1; 2; 3: invariant combination, A = 2 sin 1 dx g 2 U (x) (x + ) Z (x) dx g0 cos Im [V (x)] : Initial condition Modes and initial charge. The Higgs eld panded on the lattice as and its canonical momenta _ are ex(x) = p (x) = p p2!p np + ; h p p yi = 2; h p p yi = 2; are complex following three criteria: (1) q1 < (N1 (2) q1 = (N1 (3) q1 = (N1 q1)%N1; q1)%N1; q2 < (N2 q2)%N2; q1)%N1; q2 = (N2 q2)%N2; q3 < (N3 q3)%N3; whereas a \corner" mode obeys Therefore, the elds can also be expanded into: q1 = (N1 q1)%N1; q2 = (N2 q2)%N2; q3 = (N3 q3)%N3: (x) = p (x) = p with real numbers ap , bp , cp , dp satisfying p2!p eipxr !p hcp +idp i r ap + ibp +h:c: + p We only initialise unstable modes, whose momenta ful l 2 X cos(pidxi) H = minimise H to zero, ; for x = ap ; bp ; cp ; dp = 0; 1; 2; 3: Under the parity inversion, (x) ! Ui(x) ! U y( x i We assume the origin point is located at x = (N1 s1; N2 on the lattice. So for s2; N3 s3). The charge conjugation operation is (x) ! Ui(x) ! Ui (x); Vi(x) ! Vi (x); explicitly C-even ensemble of C-conjugate pairs. CP-violation, The CP-violating term is On the lattice, we adopt L2 = 4 2d4x Tr [I01I23 + I02I31 + I03I12] ; + U y(x + U y(x 1 hU (x)U (x + )U y(x + )U y(x) + U (x)U y(x + )U y(x )U (x )U (x )U (x )U (x + )U y(x) )U (x Here we list its derivatives with respect to di erent elds, @@WLia2 = 3 CPgdxi 32 2m3W d4x (K1[jk] K1[kj] + Tr hi aUi(x)Uj(x + i) 0k(x + i + j)Uiy(x + j)Ujy(x)i + Tr hi aUi(x)Uj(x + i)Uiy(x + j) 0k(x + j)Ujy(x)i + Tr hi aUi(x)Uj(x + i)Uiy(x + j)Ujy(x) 0k(x)i j)Uj(x Tr hi aUi(x)Ujy(x + i j)Uiy(x Tr hi aUi(x)Ujy(x + i j)Uiy(x j)Uj(x j)Uj(x j)Uj(x + Tr hi aUi(x)Uiy(x + 0) jk(x + 0) + Tr hi aUi(x)Uiy(x + 0) jk(x)i Tr hi aUi(x) jk(x + i)Uiy(x Tr hi aUi(x) jk(x + i 0)Uiy(x Tr hi aUi(x)Uiy(x Tr hi aUi(x)Uiy(x 0) jk(x (x) = j (x)j2I (x): WL0a2 = 3 CPgdt 32 2m2W d4x (K3[ijk] + K3[jki] + K3[kij]); + Tr hi aUi(x + 0) jk(x + i + 0)Uiy(x)i + Tr hi aUi(x + 0) jk(x + i)Uiy(x)i + Tr hi aUi(x + 0)Uiy(x) jk(x)i 1 @ L2 k = 251302 [astro-ph/0101261] [INSPIRE]. [INSPIRE]. [hep-ph/9902449] [INSPIRE]. JHEP 07 (2012) 087 [arXiv:1203.5012] [INSPIRE]. rst principles in the [INSPIRE]. quench due to an extra singlet, in progress. [1] Z.-G. Mou , P.M. Sa n and A. Tranberg , Simulations of Cold Electroweak Baryogenesis: [2] T. Vachaspati , Estimate of the primordial magnetic eld helicity , Phys. Rev. Lett . 87 ( 2001 ) [3] C.J. Copi , F. Ferrer , T. Vachaspati and A. Achucarro , Helical Magnetic Fields from Sphaleron Decay and Baryogenesis, Phys. Rev. Lett . 101 ( 2008 ) 171302 [arXiv:0801.3653] [4] L.M. Krauss and M. Trodden , Baryogenesis below the electroweak scale , Phys. Rev. Lett. 83 [5] J. Garc a-Bellido , D. Yu . Grigoriev, A. Kusenko and M.E. Shaposhnikov , Nonequilibrium electroweak baryogenesis from preheating after in ation , Phys. Rev. D 60 (1999) 123504 [6] E.J. Copeland , D. Lyth , A. Rajantie and M. Trodden , Hybrid in ation and baryogenesis at the TeV scale , Phys. Rev. D 64 ( 2001 ) 043506 [hep-ph/0103231] [INSPIRE]. [7] A. Tranberg and J. Smit , Baryon asymmetry from electroweak tachyonic preheating , JHEP [8] A. Tranberg , J. Smit and M. Hindmarsh , Simulations of cold electroweak baryogenesis: [9] A. Tranberg and B. Wu , On using Cold Baryogenesis to constrain the Two-Higgs Doublet [11] Z.-G. Mou , P.M. Sa n and A. Tranberg , Cold Baryogenesis from [12] A. Diaz-Gil , J. Garc a-Bellido , M. Garc a Perez and A. Gonzalez-Arroyo , Magnetic eld production during preheating at the electroweak scale , Phys. Rev. Lett . 100 ( 2008 ) 241301 [13] A. Diaz-Gil , J. Garc a-Bellido , M. Garc a Perez and A. Gonzalez-Arroyo , Primordial [15] D. Grasso and H.R. Rubinstein , Magnetic elds in the early universe , Phys. Rept . 348 ( 2001 ) [16] A. Tranberg and J. Smit , Simulations of cold electroweak baryogenesis: Dependence on Higgs mass and strength of CP-violation , JHEP 08 ( 2006 ) 012 [hep-ph/0604263] [INSPIRE]. [17] T. Brauner , O. Taanila , A. Tranberg and A. Vuorinen , Temperature Dependence of Standard Model CP-violation , Phys. Rev. Lett . 108 ( 2012 ) 041601 [arXiv:1110.6818] [INSPIRE]. [18] G. 't Hooft, Magnetic Monopoles in Uni ed Gauge Theories, Nucl. Phys . B 79 ( 1974 ) 276 [21] J. Garc a-Bellido , M. Garc a Perez and A. Gonzalez-Arroyo , Symmetry breaking and false vacuum decay after hybrid in ation , Phys. Rev. D 67 ( 2003 ) 103501 [hep-ph/0208228] [19] M. D'Onofrio , K. Rummukainen and A. Tranberg , Sphaleron Rate in the Minimal Standard Model , Phys. Rev. Lett . 113 ( 2014 ) 141602 [arXiv:1404.3565] [INSPIRE]. [20] A. Rajantie , P.M. Sa n and E.J. Copeland, Electroweak preheating on a lattice , Phys. Rev. [22] J. Smit and A. Tranberg , Chern-Simons number asymmetry from CP-violation at electroweak [23] J.-I. Skullerud , J. Smit and A. Tranberg , W and Higgs particle distributions during electroweak tachyonic preheating , JHEP 08 ( 2003 ) 045 [hep-ph/0307094] [INSPIRE]. [24] K. Enqvist , P. Stephens , O. Taanila and A. Tranberg , Fast Electroweak Symmetry Breaking and Cold Electroweak Baryogenesis , JCAP 09 ( 2010 ) 019 [arXiv:1005.0752] [INSPIRE]. [25] B. J .W. van Tent , J. Smit and A. Tranberg , Electroweak scale in ation, in aton Higgs mixing and the scalar spectral index , JCAP 07 ( 2004 ) 003 [hep-ph/0404128] [INSPIRE]. [26] T. Konstandin and G. Servant , Natural Cold Baryogenesis from Strongly Interacting Electroweak Symmetry Breaking , JCAP 07 ( 2011 ) 024 [arXiv:1104.4793] [INSPIRE]. [27] Z.G. Mou , P.M. Sa n and A. Tranberg , Simulations of Cold Electroweak Baryogenesis: [28] S. Yu . Khlebnikov and M.E. Shaposhnikov , The Statistical Theory of Anomalous Fermion Number Nonconservation, Nucl . Phys . B 308 ( 1988 ) 885 [INSPIRE]. [29] Y. Burnier , M. Laine and M. Shaposhnikov , Baryon and lepton number violation rates across the electroweak crossover , JCAP 02 ( 2006 ) 007 [hep-ph/0511246] [INSPIRE].


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP06%282017%29075.pdf

Zong-Gang Mou, Paul M. Saffin, Anders Tranberg. Simulations of Cold Electroweak Baryogenesis: hypercharge U(1) and the creation of helical magnetic fields, Journal of High Energy Physics, 2017, 1-27, DOI: 10.1007/JHEP06(2017)075