Temperature Dependence of Standard Model CP Violation

0.18 MB·English

by  Tomas Brauner

Temperature Dependence of Standard Model CP Violation
Toma´ˇs Brauner,1,2 Olli Taanila,1,3 Anders Tranberg,4 and Aleksi Vuorinen1
1Faculty of Physics, University of Bielefeld, D-33615 Bielefeld, Germany
2Department of Theoretical Physics, Nuclear Physics Institute ASCR, 25068 Rˇeˇz, Czech Republic
3Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland
4Niels Bohr International Academy, Niels Bohr Institute and Discovery Center, Blegdamsvej 17, 2100 Copenhagen, Denmark
Weanalyze thetemperaturedependenceof CP violation effectsin theStandardModel bydeter-
miningtheeffectiveactionofitsbosonicfields,obtainedafterintegratingoutthefermionsfromthe
theory and performing a covariant gradient expansion. We find non-vanishing CP violating terms
startingatthesixthorderoftheexpansion,albeitonlyintheCodd/Pevensector,withcoefficients
thatdependon quarkmasses, CKMmatrix elements, temperatureandthemagnitudeoftheHiggs
field. TheCPviolatingeffectsareobservedtodecreaserapidlywithtemperature,whichhasimpor-
2 tant implications for the generation of a matter-antimatter asymmetry in the early Universe. Our
1 results suggest that the cold electroweak baryogenesis scenario may be viable within the Standard
0 Model, provided theelectroweak transition temperature is at most of order 1 GeV.
2
n PACSnumbers: 11.30.Er,98.80.Bp
a
J
Introduction.—One of the most fundamental observa- these works, numerical simulations employing the zero
6
tions in modern cosmology is the existence of an asym- temperature CP violating operator of Ref. [10] demon-

] metry between matter and antimatter in the Universe. strated that one obtains a baryon asymmetry exceeding
h
In order to generate such a state, high energy processes the observed one by four orders of magnitude [13]. Tak-
p
- must have been efficient in breakingthe baryonnumber, ing into account the known suppression of CP violation
p C (chargeconjugation)andCP (combined chargeconju- in the limit of veryhigh temperatures [3], this highlights
e
gation and parity) symmetries in an environment out of the importance of determining the temperature depen-
h
[ thermalequilibrium[1]. Alltheseingredientsarepresent dence of CP violating effects within the SM.

in the Minimal Standard Model (SM), but the precise In the presentLetter, our objective is simple: To eval-
2
mechanismto generatethe correctabundance(if oneex- uate the CP violating part of the SM bosonic effective
v
ists) has so far eluded us [2]. actionasafunctionoftemperature. Theresultingrateof
8
1 temperaturedependenceshould,incombinationwiththe
Within the SM, CPviolationoriginatesfromthe com-
8 results of Ref. [13], give at least a rough estimate of the
plex phase of the Cabibbo-Kobayashi-Maskawa (CKM)
6 temperature range,in whichSMcoldelectroweakbaryo-
matrix. One approach to include its effects in out-of-
.
0 genesismightbeviable. Inthelimitofzerotemperature,
equilibrium field dynamics is to integrate the fermions
1 ourcalculationshouldalsoprovideanindependentcheck
out from the theory and recover CP violating bosonic
1
oftheverynon-trivialpredictionsofRef.[11],addressing
1 operators in the resulting effective action—a technique
the discrepancy between the results of Refs. [10, 11].
: particularly well suited for classical numerical simula-
v Bosonic effective action.—Consider a Euclidean field
tions. Thiswasfirstcarriedoutinaperturbativesetting
i
X by Shaposhnikov et al. [3], who argued that at tempera- theory of chiral fermions, denoted collectively as ψ(x)≡
(ψ (x),ψ (x)), coupled to gaugefields V (x),V (x) and
r tures T abovethe electroweakscale,CP violating opera- L R L R
a tors are suppressed by coefficients proportional to T−12. possibly to additional scalar fields. The dynamics of
the fermions is described by a Euclidean Lagrangian of
The conclusionwasthatthe SM electroweakphasetran-
the Dirac type, ψ(x)D(x)ψ(x), where the Dirac opera-
sition, takingplace atT 100 GeV, cannotconstitute a
≃ tor D(x) in the chiral (left-right) basis has the generic
viable mechanism for baryogenesis (see also Ref. [4]).
form
Somewhatlater,itwasarguedthatelectroweakbaryo-
genesismightstillworkiftherewasamechanismtodelay D(x)= ∂/+V/L(x) mLR(x) , (1)
thetransitionuntilatemperaturescaleofT ≃1–10 GeV, (cid:18) mRL(x) ∂/+V/R(x)(cid:19)
ascenariotypicallyreferredtoascoldelectroweakbaryo-
using the Feynman slash notation, v/ v γ . The func-
genesis [5–8]. Refs. [9–12] subsequently employed a co- ≡ µ µ
tionalintegraloverthefermionicfields canbe performed
variant gradient expansion to determine the leading CP
exactly, resulting in an additionalcontributionto the ef-
violating terms in the bosonic SM effective action at
fective action of the bosonic fields,
zero temperature; however, with contradictory results.
Inparticular,whileRef.[10]claimedtheexistenceofaC Γ[V ,V ,m ,m ]= TrlogD. (2)
L R LR RL
even/Poddoperatoratthe sixthorderoftheexpansion, −
Ref.[11]reportedthatonlytheCodd/Pevensectorgives As demonstrated in Refs. [14], the P even and P odd
a non-vanishing CP violating contribution. Following partsofthiseffectiveaction,Γ+andΓ−,canbeexpressed
2
in the form Here, Z , W± and G are the weak intermediate bo-
µ µ µ
son and gluon fields, respectively, and V is the CKM
1
Γ+ = ReTr(logK), (3) matrix. Furthermore, D = ∂ +(2/3)B and D =
−2 uµ µ µ dµ
∂ (1/3)B , where B is the hypercharge gauge field.
Γ− = 1iImTr(γ logK)+Γ , (4) (Nµo−te that wµe are usingµa “mixed” basis in terms of Z
−2 5 gWZW µ
and the hypercharge field B rather than the commonly
µ
wherethesecondorderdifferentialoperatorK isdefined usedZ andthephotonfieldA .) Theindicated2 2ma-
µ µ
as K m m D/ m−1D/ m . The gauged trix structure of the Dirac operator corresponds t×o weak
≡ LR RL − LL RL RR RL
Wess-Zumino-Witten term Γ encodes the chiral isospin; Dirac, color and family indices are suppressed.
gWZW
anomaly. The left-rightand right-leftcomponents ofthe Dirac op-
The trace-logarithm of the Dirac operator cannot erator are identical, m = m = (φ/v)diag(m ,m ),
LR RL u d
be evaluated in a closed form for general coordinate- where m are the (diagonal) mass matrices for u and
u,d
dependent background fields. Therefore, one usually re- d type quarks. Throughout the calculation, we use the
sortstoacovariantgradientexpansion,thatis,anexpan- unitary gauge, in which the only physical scalar field of
sion in gauge fields and covariant derivatives. A useful theSM—theHiggsfieldφ—fluctuatesarounditsvacuum
toolforthisisthemethodofcovariantsymbols[14,15]— expectation value, v 246 GeV.
≈
a generalmethod for the evaluation of traces of differen- The above expressions for the Dirac operator com-
tial operators of the type f(D,m), where f is an alge- pletely fix our conventions. Our fields are related to the
braicfunction ofits twoarguments,a (matrix covariant) canonically normalized (Euclidean) fields, denoted here
derivativeD anda(matrix)fieldm(x). At nonzerotem- byatilde,viaW± =(g/√2)W˜±,Z =[g/(2cosθ )]Z˜ ,
µ µ µ W µ
perature, the trace of this operator reads B = g′B˜ , and G = (g /2)λ G˜ . Here, as usual,
µ µ µ s a aµ
g,g′,g denoterespectivelytheweakisospin,hypercharge
s
Trf(D,m)= tr f(D,m)11 (5) andstrongcouplingconstants,while θ is the Weinberg
W
∞Z(x,pi)k(cid:2) (cid:3) anTglheeaenldemλeanatsreofththeeGCeKll-MMamnantrmixatarrieceksn.owntoagood
+ −k! tr (D0∇0)kf(D,m)11 , accuracy [19] and are fixed up to independent rephas-
Zx,pk=1
X (cid:2) (cid:3) ings of the quark fields, which must leave all physical
where ∂/∂p , “tr” stands for a trace over observables unchanged. The simplest rephasing invari-
µ µ
∇ ≡
matrix indices, and the bar denotes a conjugation, ant constructed from the CKM matrix is the Jarlskog
v eiD·∇e−ip·xveip·xe−iD·∇ [24]. Finally, the inte- invariant J, defined by Im(VijVj−k1VkℓVℓ−i1) = Jǫikǫjℓ.
≡
gration measures in a d-dimensional Euclidean space- Any CP violating bosonic operator thus has to contain
time have the forms 1/T dx dd−1x and at least four W± fields. As shown by Smit [9], there
x ≡ 0 0 p ≡ is, however, no CP violation even at fourth order of
T dd−1p/(2π)d−1, wherep takesthe values 2nπT
p0 R R 0 R R the gradient expansion at zero temperature—a conclu-
or (2n+1)πT with integer n, depending on whether the
P R sionstraightforwardlygeneralizableto nonzero tempera-
trace is taken over a bosonic or fermionic space.
ture. ThisinparticularimpliesthattheanomalousWess-
A detailed derivation of Eq. (5) will be given else-
Zumino-Witten term need not be considered, as it only
where [17]; here, we simply remark that at zero tem-
contributesatfourthorder[11]. Hence,the CPviolating
perature, the derivative D as well as the field m are
part of the bosonic effective action can be determined
Lorentzcovariant,thefirsttermofthisexpressioniscom-
solely using the Dirac operators (6) and (7) as well as
pletely Lorentz invariant, and the second term vanishes.
Eqs. (3) and (4).
At nonzero temperature this is no longer the case. One
Finally, we note that the above exercise can naturally
should additionally note that due to the presence of ex-
berepeatedfortheleptonsectoroftheSM,assumingthe
plicit free temporal derivatives in the second term, the
neutrinos to have Dirac type masses. Their contribution
individual terms in the sum are not gauge invariant.
is,however,alwayssuppressedby manyordersofmagni-
ApplicationtotheStandardModel.—Weneednotwrite
tude with respect to the quarks, and we have therefore
downthe full SM Lagrangian,but simply displayits Eu-
not considered it in our work.
clidean Dirac operator in order to fix our conventions,
Results.—We have implemented the gradient expan-
following closely the notation of Ref. [11]. Upon diago-
sion and performed the traces indicated in Eqs. (3) and
nalizing the mass matrices, the chiralcomponents of the
(4) using the Mathematica package Feyncalc [18]. At
Dirac operator in the quark sector take the form
theleadingnon-vanishing(sixth)order,theCPviolating
D/ +Z/ +G/ W/+V part of the effective action can be written in the form
D/ = u , (6)
LL W/−V−1 D/d−Z/ +G/! ΓCP-odd = (8)
D/ +G/ 0 i 1/T v 2
D/ = u . (7) N JG κ dx d3x ( + + ),
RR 0 D/ +G/ −2 c F CP 0 φ O0 O1 O2
(cid:18) d (cid:19) Z0 Z (cid:18) (cid:19)
3
where N = 3 is the number of colors, G = 1/(√2v2)
c F
the Fermi coupling, and the zero temperature effective
coupling κ reads
CP
∆ d4p 6 1
κ (p2)3 3.1 102,
CP ≡ G (2π)4 (p2+m2)2 ≈ ×
F Z f=1 f
Y
∆ (m2 m2)(m2 m2)(m2 m2)
≡ u− c u− t c − t
(m2 m2)(m2 m2)(m2 m2). (9)
× d− s d− b s − b
The index n in the functions counts the number of
n
O
Z or ϕ fields where ϕ (∂ φ)/φ. Each of these three
µ µ
≡
termscanbesubsequentlydividedintoPevenandPodd
parts, = ++ −. These functions containnot only
On On On
Lorentzinvariantoperators,butalsoonescontainingone
FIG. 1: The coefficients c1–c13 plotted as functions of the
or several vectors uµ δµ0, specifying the rest frame of effectivetemperatureTeff ≡vT/φ. Theboldlinescorrespond
≡
the thermal bath. The non-invariant terms must natu- tothesmallestandlargestoftheci thatapproachoneatzero
rally vanish in the limit of zero temperature, which we temperature, i.e. c1 and c10.
have indeed verified.
The Lorentz invariant operators in + read explicitly
On
Ref. [19]), we have used m = 2.5 MeV, m = 5 MeV,
u d
+ = c1(W+)2W−W− + 5c2(W+)2W−W− mc = 1.27 GeV, ms = 100 MeV, mt = 172 GeV,
O0 − 3 µµ νν 3 µν µν m =4.2 GeV as well as J =2.9 10−5.
b
×
c 4c
1(W+)2W−W− + 3W+W+W− W− The coefficients ci depend on the quark masses mf as
− 3 µν νµ 3 µ ν µα αν wellasthetemperatureT andtheHiggsfieldφ(x),which
2c1W+W+W−W− 2c W+W+W−W− appearintheparticularcombinationTeff vT/φ. Inthe
− 3 µ ν µα να− 4 µ ν αµ αν zerotemperature limit, c –c approachu≡nity while c –
1 11 12
+4c3W+W+W−W− c.c., (10) c13 tend to zero, reducing our result to that of Ref. [11]
3 µ ν µν αα− and thus independently verifying its conclusions.
8
O1+ = 3(Zµ+ϕµ) c5(W+)2Wµ−Wν−ν asIfnunFcitgi.o1n,swoef Tdisp,laoybttahienebdehthavroiourgohfntuhme ceoriecffialcieevnatlsuac-i
eff
−c6(W+)2W(cid:2)ν−Wµ−ν −c6(W+)2Wν−Wν−µ tionoftheir defining one-loop,multi scalesum-integrals.
−c3(W+·W−)Wµ+Wν−ν ATt low10t–e3m0pMereaVturtehse,ytbheegicn1–tco11falelvroalpveidlsylo.wTlhyeucnti–lcat
+c (W+ W−)W+W− +c W+W+W−W− eff ≃ 12 13
7 · ν µν 7 µ ν α αν on the other hand first exhibit a fast increase, reaching
−c12(W+·W−)Wν+Wν−µ−c12Wµ+Wν+Wα−Wν−α theirmaximumaround20–30MeV,afterwhichthey,too,
+c W−W+W+W− c.c., (11) begin to decrease. The largest coefficient at all temper-
13 µ ν α να − atures is c , which reaches its maximal value of 1.3 at
O2+ = 4(ZµZν +ϕµϕν) (cid:3) Teff 18 M10eV. In addition, we note that at Teff > 10
× c8(W+)2Wµ−Wν−−c8(W−)2Wµ+Wν+ GeV≈, all ci’s are at most of order 10−14, consistent∼with
16 the estimates of Shaposhnikov et al. [3].
(cid:2) (Z ϕ) c (W+ W−)2 2c (W+)2(cid:3)(W−)2
− 3 · 9 · − 6 Fig. 2, on the other hand, demonstrates the depen-
4 (cid:2) (cid:3) dence of our results on the value of the Higgs field φ.
+ (Z ϕ +Z ϕ )
3 µ ν ν µ As is evident from the functional form of Teff = vT/φ,
c (W+)2W−W−+c (W−)2W+W+ larger values of φ ameliorate the thermal suppression of
× 10 µ ν 10 µ ν CPviolatingeffects. Thisobservationhasimportantim-
−2(cid:2)c11(W+·W−)(Wµ+Wν−+Wν+Wµ−) , (12) plications for cold electroweak baryogenesis simulations,
(cid:3) highlightingthenecessityofdeterminingthedistribution
while we find that
oftheHiggsfieldduringthecoldspinodaltransition[20].
− = − = − =0. (13) Finally, we want to stress that in the present work we
O0 O1 O2
havecompletely neglected the effects of the strong inter-
Here, “c.c.” stands for complex conjugation, acting on action, an issue of increasing severity as one approaches
the fields as W± W∓, Z Z, and ϕ ϕ. In ad- the deconfinement transition (see also the discussion in
→− →− →
dition,wehavedenotedthehyperchargecovariantderiva- Ref. [12]). For an extended discussion of this issue as
tives of W± by W± (∂ B )W±. For the numerical well as of the details of our computation (including the
µν ≡ µ± µ ν
valuesofthequarkmassesandtheJarlskoginvariant(see Lorentz breaking operators present at nonzero tempera-
4
violating operatorsin such anenvironment. As this task
is,however,technicallyextremelydemanding(unlessone
includes the fermions themselves in the dynamics, which
hasbecomeviable onlyrecently[22]), ourcalculationas-
sumedthermalequilibriumataneffectivetemperatureT.
As a rough estimate of the temperature range, in which
cold electroweak baryogenesis might be viable, we ob-
serve that for φ=v, the largestc coefficient c reaches
i 10
the value 10−5 at T 1 GeV. It should be noted, how-
≃
ever,that the effective actionofEqs.(8)–(12)consistsof
a variety of operators with coefficients of different mag-
nitude, which may either enhance each others’ effects or
lead to surprising cancelations.
Theresultwehaveobtainedshouldbecontrastedwith
FIG.2: ThedependenceoftheciontheHiggsfieldφ,plotted the energy scales typically associated with cold elec-
for three different temperatures, T = 1/3 GeV, 1 GeV and troweak baryogenesis. If the relevant momentum scale
3 GeV. Ineachcase,thegreybandisspannedbyc1 andc10. of the dynamics is identified with the inverse time scale
of the electroweak transition, lattice simulations using
a hypothetical order four operator indicate an effective
ture), we refer the reader to Ref. [17]. temperature of T τ−1 5–10 GeV, where τ is the
Conclusions and outlook.—In this Letter, we have de- ≃ Q ≃ Q
maximal time scale of a transition sufficiently out of
termined the leading CP violating operators in the co-
equilibrium to generate a large enough baryon asymme-
variant gradient expansion of the bosonic effective ac-
try[23]. Thesenumbersaresomewhat,butnotdramati-
tion of the Standard Model, displayed in Eqs. (8)–(12)
cally higher than the approximative1 GeV we obtained.
above. In the zero temperature limit, we independently
To further quantify the result, classical numerical simu-
confirmed the result of Ref. [11]. While this agreement
lationsofthebosoniceffectiveaction,includingallofthe
itself does not resolve the conflict between the results of
operators we derived, are clearly required.
Refs. [10, 11], it suggests that the source of the discrep-
An important technical note in this context is that,
ancy must lie either in the details of the computation
sinceinthebosonicsimulationstheasymmetryisquanti-
performed in Ref. [10], or in a subtlety in separating Γ−
fiedviatheChern-Simonsnumberratherthanthebaryon
into an anomalous and a regular part, cf. Eq. (4).
numberitself,the dynamicshavetobe explicitly CPbut
Thegradientexpansionweemployinvolvessettingthe
also P violating to generate an asymmetry. Given that
externalmomentatozero. Oneshouldbearinmindthat
the CP violating terms found here conserve P, one must
this may limit the range of applicability of the result.
inadditionincludetheCodd/PoddbutCPevenopera-
In agreement with Refs. [10, 11], we expect the gradi-
tors. TheseareinherentinthefermionicsectoroftheSM
ent expansion to be valid up to externalmomenta of the
andappearatlowerordersinthederivativeexpansionof
order of m . This is based on the observation that the
c the bosonic effective action than the CP violating op-
result is insensitive to the three smallest quark masses.
erators. We will further address this important point in
In Ref. [17], we address the issue of convergence by cal-
Ref.[17]. Beyondthis,improvingtheaccuracyofthepre-
culating the next order in the expansion.
dictions would require not integrating out the fermions
Nonzerotemperaturewasobservedtoaffectthevalues
but rather incorporating them into the (quantum) sim-
of the couplings of the operators already present at zero
ulations. We plan to return to these questions in future
temperature, as well as to give rise to altogether new
work.
operators. Still, similarly to the zero temperature limit,
We are grateful to Lorenzo L. Salcedo for his in-
all CP violating operators turned out to be P even/C
sightful comments, and to Ju¨rgen Berges, Thomas Kon-
odd. Another important feature of our result was that
standin, Mikko Laine and Jan Smit for useful discus-
the temperature entered only through the combination
sions. T.B., O.T. and A.V. were supported by the Sofja
T vT/φ, involving the local Higgs field φ and its
eff
≡ Kovalevskaja program of the Alexander von Humboldt
vacuum expectation value v.
Foundation, and A.T. by the Carlsberg Foundation.
DirectnumericalsimulationsusingtheSMeffectiveac-
tionwithaP-oddoperator(ofthe −-typeinourclassi-
O1
fication) suggest that the cold electroweak baryogenesis
scenario is able to produce the observed baryon asym-
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