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Electroweak interaction

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Unified description of electromagnetism and the weak interaction
Standard Model ofparticle physics
Elementary particles of the Standard Model

Inparticle physics, theelectroweak interaction orelectroweak force is theunified description of two of thefundamental interactions of nature:electromagnetism (electromagnetic interaction) and theweak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different aspects of the same force. Above theunification energy, on the order of 246 GeV,[a] they would merge into a single force. Thus, if the temperature is high enough – approximately 1015 K – then the electromagnetic force and weak force merge into a combined electroweak force.

During thequark epoch (shortly after theBig Bang), the electroweak force split into the electromagnetic andweak force. It is thought that the required temperature of 1015 K hasnot been seen widely throughout the universe since before the quark epoch, and currently the highest human-made temperature in thermal equilibrium is around5.5×1012 K (from theLarge Hadron Collider).

Sheldon Glashow,[1]Abdus Salam,[2] andSteven Weinberg[3] were awarded the 1979Nobel Prize in Physics for their contributions to the unification of the weak and electromagnetic interaction betweenelementary particles, known as theWeinberg–Salam theory.[4][5] The existence of the electroweak interactions was experimentally established in two stages, the first being the discovery ofneutral currents in neutrino scattering by theGargamelle collaboration in 1973, and the second in 1983 by theUA1 and theUA2 collaborations that involved the discovery of theW and Zgauge bosons in proton–antiproton collisions at the convertedSuper Proton Synchrotron. In 1999,Gerardus 't Hooft andMartinus Veltman were awarded the Nobel prize for showing that the electroweak theory isrenormalizable.

History

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After theWu experiment in 1956 discoveredparity violation in theweak interaction, a search began for a way to relate theweak andelectromagnetic interactions. Extending hisdoctoral advisorJulian Schwinger's work,Sheldon Glashow first experimented with introducing two different symmetries, onechiral and one achiral, and combined them such that their overall symmetry was unbroken. This did not yield arenormalizabletheory, and its gauge symmetry had to be broken by hand as nospontaneous mechanism was known, but it predicted a new particle, theZ boson. This received little notice, as it matched no experimental finding.

In 1964,Salam andJohn Clive Ward[6] had the same idea, but predicted a masslessphoton and three massivegauge bosons with a manually broken symmetry. Later around 1967, while investigatingspontaneous symmetry breaking, Weinberg found a set of symmetries predicting a massless, neutralgauge boson. Initially rejecting such a particle as useless, he later realized his symmetries produced the electroweak force, and he proceeded to predict rough masses for theW and Z bosons. Significantly, he suggested this new theory was renormalizable.[3] In 1971,Gerard 't Hooft proved that spontaneously broken gauge symmetries are renormalizable even with massive gauge bosons.

Formulation

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Main article:Mathematical formulation of the Standard Model
Weinberg's weak mixing angleθW, and relation between coupling constantsg, g′, ande. Adapted from Lee (1981).[7]
The pattern ofweak isospin,T3, andweak hypercharge,YW, of the known elementary particles, showing the electric charge,Q, along theweak mixing angle. The neutral Higgs field (circled) breaks the electroweak symmetry and interacts with other particles to give them mass. Three components of the Higgs field become part of the massiveW andZ bosons.

Mathematically, electromagnetism is unified with the weak interactions as aYang–Mills field with anSU(2) ×U(1)gauge group, which describes the formal operations that can be applied to the electroweak gauge fields without changing the dynamics of the system. These fields are the weak isospin fieldsW1,W2, andW3, and the weak hypercharge fieldB.This invariance is known aselectroweak symmetry.

Thegenerators ofSU(2) andU(1) are given the nameweak isospin (labeledT) andweak hypercharge (labeledY) respectively. These then give rise to the gauge bosons that mediate the electroweak interactions – the threeW bosons of weak isospin (W1,W2, andW3), and theB boson of weak hypercharge, respectively, all of which are "initially" massless. These are not physical fields yet, beforespontaneous symmetry breaking and the associatedHiggs mechanism.

In theStandard Model, the observed physical particles, theW±
 andZ0
 bosons, and thephoton, are produced through thespontaneous symmetry breaking of the electroweak symmetry SU(2) × U(1)Y to U(1)em,[b] effected by theHiggs mechanism (see alsoHiggs boson), an elaborate quantum-field-theoretic phenomenon that "spontaneously" alters the realization of the symmetry and rearranges degrees of freedom.[8][9][10][11]

The electric charge arises as the particular linear combination (nontrivial) ofYW (weak hypercharge) and theT3 component of weak isospin (Q=T3+12YW{\displaystyle Q=T_{3}+{\tfrac {1}{2}}\,Y_{\mathrm {W} }}) that doesnot couple to theHiggs boson. That is to say: the Higgs and the electromagnetic field have no effect on each other, at the level of the fundamental forces ("tree level"), while anyother combination of the hypercharge and the weak isospin must interact with the Higgs. This causes an apparent separation between the weak force, which interacts with the Higgs, and electromagnetism, which does not. Mathematically, the electric charge is a specific combination of the hypercharge andT3 outlined in the figure.

U(1)em (the symmetry group of electromagnetism only) is defined to be the group generated by this special linear combination, and the symmetry described by theU(1)em group is unbroken, since it does notdirectly interact with the Higgs.[c]

The above spontaneous symmetry breaking makes theW3 andB bosons coalesce into two different physical bosons with different masses – theZ0
 boson, and the photon (γ),

(γZ0)=(cos⁡θWsin⁡θW−sin⁡θWcos⁡θW)(BW3),{\displaystyle {\begin{pmatrix}\gamma \\Z^{0}\end{pmatrix}}={\begin{pmatrix}\cos \theta _{\text{W}}&\sin \theta _{\text{W}}\\-\sin \theta _{\text{W}}&\cos \theta _{\text{W}}\end{pmatrix}}{\begin{pmatrix}B\\W_{3}\end{pmatrix}},}

whereθW is theweak mixing angle. The axes representing the particles have essentially just been rotated, in the (W3,B) plane, by the angleθW. This also introduces a mismatch between the mass of theZ0
 and the mass of theW±
 particles (denoted asmZ andmW, respectively),

mZ=mWcos⁡θW .{\displaystyle m_{\text{Z}}={\frac {m_{\text{W}}}{\,\cos \theta _{\text{W}}\,}}~.}

TheW1 andW2 bosons, in turn, combine to produce the charged massive bosonsW±
:[12]

W±=12(W1∓iW2) .{\displaystyle W^{\pm }={\frac {1}{\sqrt {2\,}}}\,{\bigl (}\,W_{1}\mp iW_{2}\,{\bigr )}~.}

Lagrangian

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Before electroweak symmetry breaking

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TheLagrangian for the electroweak interactions is divided into four parts beforeelectroweak symmetry breaking manifests,

LEW=Lg+Lf+Lh+Ly .{\displaystyle {\mathcal {L}}_{\mathrm {EW} }={\mathcal {L}}_{g}+{\mathcal {L}}_{f}+{\mathcal {L}}_{h}+{\mathcal {L}}_{y}~.}

TheLg{\displaystyle {\mathcal {L}}_{g}} term describes the interaction between the threeW vector bosons and theBvector boson,

Lg=−14WaμνWμνa−14BμνBμν,{\displaystyle {\mathcal {L}}_{g}=-{\tfrac {1}{4}}W_{a}^{\mu \nu }W_{\mu \nu }^{a}-{\tfrac {1}{4}}B^{\mu \nu }B_{\mu \nu },}

whereWaμν{\displaystyle W_{a}^{\mu \nu }} (a=1,2,3{\displaystyle a=1,2,3}) andBμν{\displaystyle B^{\mu \nu }} are thefield strength tensors for the weak isospin and weak hypercharge gauge fields.

Lf{\displaystyle {\mathcal {L}}_{f}} is thekinetic term for the Standard Model fermions. The interaction of the gauge bosons and the fermions are through thegauge covariant derivative,

Lf=Q¯jiD/Qj+u¯jiD/uj+d¯jiD/dj+L¯jiD/Lj+e¯jiD/ej,{\displaystyle {\mathcal {L}}_{f}={\overline {Q}}_{j}iD\!\!\!\!/\;Q_{j}+{\overline {u}}_{j}iD\!\!\!\!/\;u_{j}+{\overline {d}}_{j}iD\!\!\!\!/\;d_{j}+{\overline {L}}_{j}iD\!\!\!\!/\;L_{j}+{\overline {e}}_{j}iD\!\!\!\!/\;e_{j},}

where the subscriptj sums over the three generations of fermions;Q,u, andd are the left-handed doublet, right-handed singlet up, and right handed singlet down quark fields; andL ande are the left-handed doublet and right-handed singlet electron fields.TheFeynman slashD/{\displaystyle D\!\!\!\!/} means the contraction of the 4-gradient with theDirac matrices, defined as

D/≡γμ Dμ,{\displaystyle D\!\!\!\!/\equiv \gamma ^{\mu }\ D_{\mu },}

and the covariant derivative (excluding thegluon gauge field for thestrong interaction) is defined as

 Dμ≡∂μ−i g′2 Y Bμ−i g2 Tj Wμj.{\displaystyle \ D_{\mu }\equiv \partial _{\mu }-i\ {\frac {g'}{2}}\ Y\ B_{\mu }-i\ {\frac {g}{2}}\ T_{j}\ W_{\mu }^{j}.}

Here Y {\displaystyle \ Y\ } is the weak hypercharge and the Tj {\displaystyle \ T_{j}\ } are the components of the weak isospin.

TheLh{\displaystyle {\mathcal {L}}_{h}} term describes theHiggs fieldh{\displaystyle h} and its interactions with itself and the gauge bosons,

Lh=|Dμh|2−λ(|h|2−v22)2 ,{\displaystyle {\mathcal {L}}_{h}=|D_{\mu }h|^{2}-\lambda \left(|h|^{2}-{\frac {v^{2}}{2}}\right)^{2}\ ,}

wherev{\displaystyle v} is thevacuum expectation value.

The Ly {\displaystyle \ {\mathcal {L}}_{y}\ } term describes theYukawa interaction with the fermions,

Ly=−yuijϵab hb† Q¯iaujc−ydij h Q¯idjc−yeij h L¯iejc+h.c. ,{\displaystyle {\mathcal {L}}_{y}=-y_{u}^{ij}\epsilon ^{ab}\ h_{b}^{\dagger }\ {\overline {Q}}_{ia}u_{j}^{c}-y_{d}^{ij}\ h\ {\overline {Q}}_{i}d_{j}^{c}-y_{e}^{ij}\ h\ {\overline {L}}_{i}e_{j}^{c}+\mathrm {h.c.} ~,}

and generates their masses, manifest when the Higgs field acquires a nonzero vacuum expectation value, discussed next. The ykij ,{\displaystyle \ y_{k}^{ij}\ ,} for k∈{u,d,e} ,{\displaystyle \ k\in \{\mathrm {u,d,e} \}\ ,} are matrices of Yukawa couplings.

After electroweak symmetry breaking

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The Lagrangian reorganizes itself as the Higgs field acquires a non-vanishing vacuum expectation value dictated by the potential of the previous section. As a result of this rewriting, the symmetry breaking becomes manifest. In the history of the universe, this is believed to have happened shortly after the hot big bang, when the universe was at a temperature159.5±1.5 GeV[13](assuming the Standard Model of particle physics).

Due to its complexity, this Lagrangian is best described by breaking it up into several parts as follows.

LEW=LK+LN+LC+LH+LHV+LWWV+LWWVV+LY .{\displaystyle {\mathcal {L}}_{\mathrm {EW} }={\mathcal {L}}_{\mathrm {K} }+{\mathcal {L}}_{\mathrm {N} }+{\mathcal {L}}_{\mathrm {C} }+{\mathcal {L}}_{\mathrm {H} }+{\mathcal {L}}_{\mathrm {HV} }+{\mathcal {L}}_{\mathrm {WWV} }+{\mathcal {L}}_{\mathrm {WWVV} }+{\mathcal {L}}_{\mathrm {Y} }~.}

The kinetic termLK{\displaystyle {\mathcal {L}}_{K}} contains all the quadratic terms of the Lagrangian, which include the dynamic terms (the partial derivatives) and the mass terms (conspicuously absent from the Lagrangian before symmetry breaking)

LK=∑ff¯(i∂/−mf) f−14 Aμν Aμν−12 Wμν+ W−μν+mW2 Wμ+ W−μ−14 ZμνZμν+12 mZ2 Zμ Zμ+12 (∂μ H)(∂μ H)−12 mH2 H2 ,{\displaystyle {\begin{aligned}{\mathcal {L}}_{\mathrm {K} }=\sum _{f}{\overline {f}}(i\partial \!\!\!/\!\;-m_{f})\ f-{\frac {1}{4}}\ A_{\mu \nu }\ A^{\mu \nu }-{\frac {1}{2}}\ W_{\mu \nu }^{+}\ W^{-\mu \nu }+m_{W}^{2}\ W_{\mu }^{+}\ W^{-\mu }\\\qquad -{\frac {1}{4}}\ Z_{\mu \nu }Z^{\mu \nu }+{\frac {1}{2}}\ m_{Z}^{2}\ Z_{\mu }\ Z^{\mu }+{\frac {1}{2}}\ (\partial ^{\mu }\ H)(\partial _{\mu }\ H)-{\frac {1}{2}}\ m_{H}^{2}\ H^{2}~,\end{aligned}}}

where the sum runs over all the fermions of the theory (quarks and leptons), and the fields Aμν ,{\displaystyle \ A_{\mu \nu }\ ,} Zμν ,{\displaystyle \ Z_{\mu \nu }\ ,} Wμν− ,{\displaystyle \ W_{\mu \nu }^{-}\ ,} and Wμν+≡(Wμν−)† {\displaystyle \ W_{\mu \nu }^{+}\equiv (W_{\mu \nu }^{-})^{\dagger }\ } are given as

Xμνa=∂μXνa−∂νXμa+gfabcXμbXνc ,{\displaystyle X_{\mu \nu }^{a}=\partial _{\mu }X_{\nu }^{a}-\partial _{\nu }X_{\mu }^{a}+gf^{abc}X_{\mu }^{b}X_{\nu }^{c}~,}

withX{\displaystyle X} to be replaced by the relevant field (A,{\displaystyle A,}Z,{\displaystyle Z,}W±{\displaystyle W^{\pm }}) andf abc by the structure constants of the appropriate gauge group.

The neutral current LN {\displaystyle \ {\mathcal {L}}_{\mathrm {N} }\ } and charged current LC {\displaystyle \ {\mathcal {L}}_{\mathrm {C} }\ } components of the Lagrangian contain the interactions between the fermions and gauge bosons,

LN=e Jμem Aμ+g cos⁡θW  ( Jμ3−sin2⁡θW Jμem ) Zμ ,{\displaystyle {\mathcal {L}}_{\mathrm {N} }=e\ J_{\mu }^{\mathrm {em} }\ A^{\mu }+{\frac {g}{\ \cos \theta _{W}\ }}\ (\ J_{\mu }^{3}-\sin ^{2}\theta _{W}\ J_{\mu }^{\mathrm {em} }\ )\ Z^{\mu }~,}

where e=g sin⁡θW=g′ cos⁡θW .{\displaystyle ~e=g\ \sin \theta _{\mathrm {W} }=g'\ \cos \theta _{\mathrm {W} }~.} The electromagnetic currentJμem{\displaystyle \;J_{\mu }^{\mathrm {em} }\;} is

Jμem=∑f qf f¯ γμ f ,{\displaystyle J_{\mu }^{\mathrm {em} }=\sum _{f}\ q_{f}\ {\overline {f}}\ \gamma _{\mu }\ f~,}

where qf {\displaystyle \ q_{f}\ } is the fermions' electric charges. The neutral weak current Jμ3 {\displaystyle \ J_{\mu }^{3}\ } is

Jμ3=∑f Tf3 f¯ γμ  1−γ5 2 f ,{\displaystyle J_{\mu }^{3}=\sum _{f}\ T_{f}^{3}\ {\overline {f}}\ \gamma _{\mu }\ {\frac {\ 1-\gamma ^{5}\ }{2}}\ f~,}

whereTf3{\displaystyle T_{f}^{3}} is the fermions' weak isospin.[d]

The charged current part of the Lagrangian is given by

LC=−g 2  [ u¯i γμ  1−γ5 2MijCKM dj+ν¯i γμ 1−γ5 2ei ] Wμ++h.c. ,{\displaystyle {\mathcal {L}}_{\mathrm {C} }=-{\frac {g}{\ {\sqrt {2\;}}\ }}\ \left[\ {\overline {u}}_{i}\ \gamma ^{\mu }\ {\frac {\ 1-\gamma ^{5}\ }{2}}\;M_{ij}^{\mathrm {CKM} }\ d_{j}+{\overline {\nu }}_{i}\ \gamma ^{\mu }\;{\frac {\ 1-\gamma ^{5}\ }{2}}\;e_{i}\ \right]\ W_{\mu }^{+}+\mathrm {h.c.} ~,}

where ν {\displaystyle \ \nu \ } is the right-handed singlet neutrino field, and theCKM matrixMijCKM{\displaystyle M_{ij}^{\mathrm {CKM} }} determines the mixing between mass and weak eigenstates of the quarks.[d]

LH{\displaystyle {\mathcal {L}}_{\mathrm {H} }} contains the Higgs three-point and four-point self interaction terms,

LH=− g mH2 4 mW H3− g2 mH2 32 mW2H4 .{\displaystyle {\mathcal {L}}_{\mathrm {H} }=-{\frac {\ g\ m_{\mathrm {H} }^{2}\,}{\ 4\ m_{\mathrm {W} }\ }}\;H^{3}-{\frac {\ g^{2}\ m_{\mathrm {H} }^{2}\ }{32\ m_{\mathrm {W} }^{2}}}\;H^{4}~.}

LHV{\displaystyle {\mathcal {L}}_{\mathrm {HV} }} contains the Higgs interactions with gauge vector bosons,

LHV=( g mHV+ g2 4H2 )( Wμ+ W−μ+1 2 cos2⁡ θW Zμ Zμ ) .{\displaystyle {\mathcal {L}}_{\mathrm {HV} }=\left(\ g\ m_{\mathrm {HV} }+{\frac {\ g^{2}\ }{4}}\;H^{2}\ \right)\left(\ W_{\mu }^{+}\ W^{-\mu }+{\frac {1}{\ 2\ \cos ^{2}\ \theta _{\mathrm {W} }\ }}\;Z_{\mu }\ Z^{\mu }\ \right)~.}

LWWV{\displaystyle {\mathcal {L}}_{\mathrm {WWV} }} contains the gauge three-point self interactions,

LWWV=−i g [( Wμν+ W−μ−W+μ Wμν− )( Aν sin⁡θW−Zν cos⁡θW )+Wν− Wμ+ ( Aμν sin⁡θW−Zμν cos⁡θW )] .{\displaystyle {\mathcal {L}}_{\mathrm {WWV} }=-i\ g\ \left[\;\left(\ W_{\mu \nu }^{+}\ W^{-\mu }-W^{+\mu }\ W_{\mu \nu }^{-}\ \right)\left(\ A^{\nu }\ \sin \theta _{\mathrm {W} }-Z^{\nu }\ \cos \theta _{\mathrm {W} }\ \right)+W_{\nu }^{-}\ W_{\mu }^{+}\ \left(\ A^{\mu \nu }\ \sin \theta _{\mathrm {W} }-Z^{\mu \nu }\ \cos \theta _{\mathrm {W} }\ \right)\;\right]~.}

LWWVV{\displaystyle {\mathcal {L}}_{\mathrm {WWVV} }} contains the gauge four-point self interactions,

LWWVV=− g2 4 { [ 2 Wμ+ W−μ+( Aμ sin⁡θW−Zμ cos⁡θW )2 ]2−[ Wμ+ Wν−+Wν+ Wμ−+( Aμ sin⁡θW−Zμ cos⁡θW )( Aν sin⁡θW−Zν cos⁡θW ) ]2} .{\displaystyle {\begin{aligned}{\mathcal {L}}_{\mathrm {WWVV} }=-{\frac {\ g^{2}\ }{4}}\ {\Biggl \{}\ &{\Bigl [}\ 2\ W_{\mu }^{+}\ W^{-\mu }+(\ A_{\mu }\ \sin \theta _{\mathrm {W} }-Z_{\mu }\ \cos \theta _{\mathrm {W} }\ )^{2}\ {\Bigr ]}^{2}\\&-{\Bigl [}\ W_{\mu }^{+}\ W_{\nu }^{-}+W_{\nu }^{+}\ W_{\mu }^{-}+\left(\ A_{\mu }\ \sin \theta _{\mathrm {W} }-Z_{\mu }\ \cos \theta _{\mathrm {W} }\ \right)\left(\ A_{\nu }\ \sin \theta _{\mathrm {W} }-Z_{\nu }\ \cos \theta _{\mathrm {W} }\ \right)\ {\Bigr ]}^{2}\,{\Biggr \}}~.\end{aligned}}}

 LY {\displaystyle \ {\mathcal {L}}_{\mathrm {Y} }\ } contains the Yukawa interactions between the fermions and the Higgs field,

LY=−∑f  g mf 2 mWf¯ f H .{\displaystyle {\mathcal {L}}_{\mathrm {Y} }=-\sum _{f}\ {\frac {\ g\ m_{f}\ }{2\ m_{\mathrm {W} }}}\;{\overline {f}}\ f\ H~.}

See also

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Notes

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  1. ^The particular number 246 GeV is taken to be thevacuum expectation valuev=(GF2)−1/2{\displaystyle v=(G_{\text{F}}{\sqrt {2}})^{-1/2}} of theHiggs field (whereGF{\displaystyle G_{\text{F}}} is theFermi coupling constant).
  2. ^Note thatU(1)Y andU(1)em are distinct instances of genericU(1): Each of the two forces gets its own, independent copy of the unitary group.
  3. ^Although electromagnetism – e.g. the photon – does notdirectly interact with theHiggs boson, it does interactindirectly, throughquantum fluctuations.
  4. ^ab Note the factors 12 (1−γ5) {\displaystyle ~{\tfrac {1}{2}}\ (1-\gamma ^{5})~} in the weak coupling formulas: These factors are deliberately inserted to expunge any left-chiral components of the spinor fields. This is why electroweak theory is said to be a 'chiral theory'.

References

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  1. ^Glashow, S. (1959). "The renormalizability of vector meson interactions."Nucl. Phys.10, 107.
  2. ^Salam, A.; Ward, J. C. (1959). "Weak and electromagnetic interactions".Nuovo Cimento.11 (4):568–577.Bibcode:1959NCim...11..568S.doi:10.1007/BF02726525.S2CID 15889731.
  3. ^abWeinberg, S (1967)."A Model of Leptons"(PDF).Phys. Rev. Lett.19 (21):1264–66.Bibcode:1967PhRvL..19.1264W.doi:10.1103/PhysRevLett.19.1264. Archived fromthe original(PDF) on 2012-01-12.
  4. ^S. Bais (2005).The Equations: Icons of knowledge. p. 84.ISBN 0-674-01967-9.
  5. ^"The Nobel Prize in Physics 1979".The Nobel Foundation. Retrieved2008-12-16.
  6. ^Salam, A.; Ward, J.C. (November 1964)."Electromagnetic and weak interactions".Physics Letters.13 (2):168–171.Bibcode:1964PhL....13..168S.doi:10.1016/0031-9163(64)90711-5.
  7. ^Lee, T.D. (1981). "Particle Physics and Introduction to Field Theory".Physics Today.34 (12): 55.Bibcode:1981PhT....34l..55L.doi:10.1063/1.2914386.
  8. ^Englert, F.; Brout, R. (1964)."Broken symmetry and the mass of gauge vector mesons".Physical Review Letters.13 (9):321–323.Bibcode:1964PhRvL..13..321E.doi:10.1103/PhysRevLett.13.321.
  9. ^Higgs, P.W. (1964)."Broken symmetries and the masses of gauge bosons".Physical Review Letters.13 (16):508–509.Bibcode:1964PhRvL..13..508H.doi:10.1103/PhysRevLett.13.508.
  10. ^Guralnik, G.S.; Hagen, C.R.; Kibble, T.W.B. (1964)."Global conservation laws and massless particles".Physical Review Letters.13 (20):585–587.Bibcode:1964PhRvL..13..585G.doi:10.1103/PhysRevLett.13.585.
  11. ^Guralnik, G.S. (2009). "The history of the Guralnik, Hagen, and Kibble development of the theory of spontaneous symmetry breaking and gauge particles".International Journal of Modern Physics A.24 (14):2601–2627.arXiv:0907.3466.Bibcode:2009IJMPA..24.2601G.doi:10.1142/S0217751X09045431.S2CID 16298371.
  12. ^D. J. Griffiths (1987).Introduction to Elementary Particles. John Wiley & Sons.ISBN 0-471-60386-4.
  13. ^D'Onofrio, Michela; Rummukainen, Kari (2016). "Standard model cross-over on the lattice".Phys. Rev. D.93 (2) 025003.arXiv:1508.07161.Bibcode:2016PhRvD..93b5003D.doi:10.1103/PhysRevD.93.025003.hdl:10138/159845.S2CID 119261776.

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