Research Article Renormalization of QED Near Decoupling Temperature

Research ArticleRenormalization of QED Near Decoupling Temperature

Samina S Masood

Physics Department University of Houston Clear Lake Houston TX 77058 USA

Correspondence should be addressed to Samina S Masood masooduhcledu

Received 26 December 2013 Revised 28 April 2014 Accepted 7 May 2014 Published 24 June 2014

Academic Editor Ali Hussain Reshak

Copyright copy 2014 Samina S Masood This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study the effective parameters of QED near decoupling temperatures and show that the QED perturbative series is convergentat temperatures below the decoupling temperatureThe renormalization constant of QED acquires different values if a system coolsdown from a hotter system to the electron mass temperature or heats up from a cooler system to the same temperature At 119879 = 119898the first order contribution to the electron self-mass 120575119898119898 is 00076 for a heating system and 00115 for a cooling system and thedifference between two values is equal to 13 of the low temperature value and 12 of the high temperature value around 119879sim119898 Thisdifference is a measure of hot fermion background at high temperatures With the increase in release of more fermions at hottertemperatures the fermion background contribution dominates and weak interactions have to be incorporated to understand thebackground effects

1 Introduction

Renormalization techniques of perturbation theory are usedto calculate temperature dependence of renormalizationconstants of QED (quantum electrodynamics) at finite tem-peratures [1ndash11] The values of electron mass charge andwavefunction at a given temperature represent the effectiveparameters of QED at those temperatures The magneticmoment of electrons dynamically generated mass of pho-tons and QED coupling constants are estimated as functionsof temperature However thermal contributions to electricpermittivity magnetic permeability and dielectric constantof the medium are derived from the vacuum polarizationSome of the important parameters of QED plasma such asDebye shielding length plasma frequency and the phasetransitions can be obtained from the properties of themedium itself

In this paper we reexamine the analytical results of tem-perature dependent renormalization constants and prove thatQED can safely be renormalized at finite temperatures usingthe perturbation theory in a vacuum below the neutrinodecoupling temperature We use the renormalization schemeof QED in real-time formalism [1ndash8] to calculate the electronmass wavefunction and charge of electron as renormaliza-tion constants of QED [9ndash11] It is now well known that

the existing first order calculations of the renormalizationconstants in the real-time formalism give the quadraticdependence of QED parameters on temperature119879 expressedin units of electron mass 119898 Renormalization constants ofQED using the perturbation theory give effective parametersof QED in a hot and dense medium and are very usefulto understand the physics of the universe However therenormalization scheme is fully reliable below the decouplingtemperature only It is explicitly checked that in the existingscheme of calculations the theory remains renormalizable at119879 le 4119898 sim 2MeV However the perturbative correctionswill exceed the original values of QED parameters at highertemperatures and hard thermal loops have to be dealt withusing already developed methods [12ndash14] However belowthe neutrino decoupling temperature the real part of thepropagators is enough to describe the perturbative behaviorof the system and doubling of the field is not required

We discuss here the distinct behavior of QED belowthe neutrino decoupling temperatures At these tempera-tures QED coupling starts to play its role in modifyingQED parameters for nucleosynthesis With the help of theseeffective parameters of QED the abundance of helium inthe early universe can be estimated [12] precisely at a giventemperature The temperature dependent QED correctionsto the nucleosynthesis parameters improve the results of the

Hindawi Publishing CorporationPhysics Research InternationalVolume 2014 Article ID 489163 9 pageshttpdxdoiorg1011552014489163

2 Physics Research International

standard big bang model of cosmology [13ndash16] and theyhelp to relate the observational data for example WMAP(Wilkinson Microwave Anisotropy Probe [15]) with the bigbang theory The same techniques can even be used tocalculate perturbative effects in QCD [17] and electroweakprocesses [18] at low temperatures and estimate their contri-butions at high temperatures alsoWe assume that the changein the properties of neutrinos [19ndash23] does not affect thedecoupling temperature significantly

Without giving the calculational details we have to brieflyoverview the existing formofQEDrenormalization constantsin real-time formalismThe Feynman rules of vacuum theoryare used with the statistically corrected propagators given as

119863120573 (119896) =119894

1198962 minus 1198982 + 119894120576+

2120587

119890120573119864119896 minus 1120575 (1198962minus 1198982) (1)

for bosons and

119878119865 (119901) =119894

119901 minus 119898 + 119894120576minus2120587 (119901 + 119898)

119890120573119864119901 + 1

120575 (1199012minus 1198982) (2)

for fermions in a hot medium Since the temperature cor-rections appear as additive contributions to the fermion andboson propagators in (1) and (2) temperature dependentterms can be handled independently of vacuum terms at theone-loop level We restrict ourselves up to the two-loop levelto show explicitly that second order thermal corrections arefinite and smaller than the first order corrections [9] belowthe neutrino decoupling

In the next section we give the calculations of thermalcorrections to the mass renormalization constant 120575119898119898 ofQEDand the physicalmass of electrons at finite temperaturesSection 3 is devoted to the calculations of electron wavefunc-tion at finite temperature Section 4 gives the calculationsof the electron charge and the QED coupling constant upto the two-loop level whereas Section 5 the last section isdevoted to the discussions of the results of all the calculationsto explicitly prove the renormalizability of the theory up tothe two-loop level below the decoupling temperature

2 Self-Mass of Electron

The renormalized mass of electrons119898119877 can be represented asa physical mass119898phys of electrons and is defined in a hot anddense medium as

119898119877 equiv 119898phys = 119898 + 120575119898 (119879 = 0) + 120575119898 (119879) (3)

where119898 is the electronmass at zero temperature and 120575119898(119879 =

0) represents the radiative corrections from a vacuum and120575119898(119879) are the contributions from the thermal background atnonzero temperature 119879 The physical mass of electrons canthen be represented as a perturbative series in 120572 and can bewritten as

119898phys cong 119898 + 120575119898(1)+ 120575119898(2)+ sdot sdot sdot (4)

where 1205751198981 and 1205751198982 are the shifts in electron mass in the

first and second order in 120572 respectively The physical mass

is deduced by locating the pole of the fermion propagator(119894(119901 + 119898)(119901

2minus 1198982+ 119894120576)) in thermal background For

this purpose we sum over all of the same order diagramsRenormalization is established by demonstrating the order-by-order cancellation of singularities All the finite termsfrom the same order in 120572 are combined together to evaluatethe same order contribution to the physical mass given in (4)The physical mass in thermal backgrounds up to order 1205722[7 8] is calculated using the renormalization techniques ofQED Self-mass of electron is expressed as

Σ (119901) = 119860 (119901) 119864120574119900 minus 119861 (119901) sdot 120574 minus 119862 (119901) (5)

where 119860(119901) 119861(119901) and 119862(119901) are the relevant coefficientsthat are functions of electron momentum only We take theinverse of the propagator with momentum and mass termsseparated as

119878minus1(119901) = (1 minus 119860) 119864120574

119900minus (1 minus 119861) 119901 sdot 120574 minus (119898 minus 119862) (6)

The temperature dependent radiative corrections to theelectron mass up to the first order in 120572 are obtained from thetemperature dependent propagator as

1198982phys equiv 119864

2minus |p|2 = 1198982 [1 minus 6120572

120587119887 (119898120573)]

+4120572

120587[119898119879119886 (119898120573) +

2

31205721205871198792minus

6

1205872119888 (119898120573)]

(7)

giving

120575119898

119898≃

1

21198982(1198982phys minus 119898

2)

≃1205721205871198792

31198982[1 minus

6

1205872119888 (119898120573)] +

2120572

120587

119879

119898119886 (119898120573) minus

3120572

120587119887 (119898120573)

(8)

where 120575119898119898 is the relative shift in electron mass due to finitetemperatures which was determined in [1 9] with

119886 (119898120573) = ln (1 + 119890minus119898120573) (9a)

119887 (119898120573) =

infin

sum

119899=1

(minus1)119899119864119894 (minus119899119898120573) (9b)

119888 (119898120573) =

infin

sum

119899=1

(minus1)119899 119890minus119899119898120573

1198992 (9c)

The convergence of (4) can be ensured for 119879 lt 2MeVas 120575119898 119898 is always smaller than unity within this limit Thisscheme of calculations will not work for higher temperaturesand the first order corrections may exceed the original valuesofQEDparameters after 5MeVAt the low temperatures119879 lt

119898 the functions 119886(119898120573) 119887(119898120573) and 119888(119898120573) fall off in powersof 119890minus119898120573 in comparison with (119879119898)2 and can be neglected inthe low-temperature limit giving

120575119898

119898

119879⋖119898997888997888997888rarr

1205721205871198792

31198982 (10)

Physics Research International 3

In the high-temperature limit 119886(119898120573) and 119887(119898120573) arevanishingly small and the total fermion contribution comesfrom 119888(119898120573) rarr minus120587

212 yielding

120575119898

119898

119879⋗119898997888997888997888rarr

1205721205871198792

21198982 (11)

The above equations give 120575119898 119898 = 7647 times 10minus3(11987921198982)

for the low temperature and 120575119898119898 = 1147times10minus211987921198982 for

the high temperature showing that the rate of change of mass120575119898 119898 is larger at 119879 gt 119898 as compared to 119879 lt 119898 Subtracting(10) from (11) the change in 120575119898 119898 between low- and high-temperature ranges can be written as

Δ(120575119898

119898) = plusmn

1205721205871198792

61198982= plusmn38 times 10

minus3 1198792

1198982 (12)

showing that the Δ(120575119898 119898) = 38 times 10minus3 at 119879 = 119898

120575119898 119898 = 00076 for a heating system and 120575119898 119898 = 00115

for a cooling system and the difference between two valuesΔ(120575119898 119898) = 00038 such that Δ(120575119898 119898) is equal to 13 ofthe low-temperature value and 12 of the high-temperaturevalue at 119879 = 119898 This difference is due to the photon back-ground contributions at low temperatures and additional hotfermionic background at high temperatures Therefore theabsence of hot fermion background contributes to a 50decrease in self-mass as compared to cooling universe corres-ponding 119879 value The high 119879 behavior will give 33 moreself-mass as compared to the low 119879 behavior Since Δ(120575119898

119898) quadratically grows with temperature the fermion back-ground contribution dominates over the hot boson back-ground after the nucleosynthesis

The temperature dependence of QED parameters is alittle more complicated and significant during nucleosyn-thesis because of the change in matter composition duringnucleosynthesis Therefore (7) and (8) are required for the119879 sim 119898 region and help to estimate the change in QEDstatistical behavior due to the change in composition Thisdifference becomesmore visible when we plot (12) and (13) ofself-mass at low temperature and high temperature showingthat both values start to give a disconnected region near119879 sim 119898 that is the nucleosynthesis temperature Figure 1shows that the slope of both graphs (corresponding to (10)and (11)) never meet at the common point 119879 = 119898 The dis-connected region around 119879 sim 119898 indicates a change inthermal properties for a heating and a cooling system In aheating system fermions start to produce around 119879 sim 119898whereas in a cooling system fermions are eliminating aroundthese temperatures Gap between two curves around 119879 sim 119898

is a measure of background fermion contributionThat modification in the electron mass behavior in the

range 119879 sim 119898 is estimated by (10) It is also clear from(11) that after around 5MeV the temperature dependencecorrection term (120575119898119898) approaches unity or even bigger forhigher temperatures even at the one-loop level Higher order

0

002

004

006

008

01

012

0 05 1 15 2 25 3 35

Disconnected region

Low T

High T

T (m)

Elec

tron

mas

sFigure 1 A graph of self-mass of electron near 119879 = 119898 for a coolingsystem from higher temperature and a heating system from a lowertemperature Both high119879 and low119879 curves have different slopes andneither value coincides

corrections [7 8] will also grow rapidly at high temperaturesgiving

119898phys = 119898[1 +120575119898

119898+1

2(120575119898

119898)

2

+ sdot sdot sdot ] asymp 119898 exp(120575119898119898

)

(13)

A change occurs around 119879 sim 119898 and is clearly related tonucleosynthesis where during the cooling of the universeright after decoupling the beta decay processes involvingthe electron mass change the composition of matter andelectrons pick up thermal mass from the hot fermion loopAt 119879 gt 4MeV the renormalization scheme of perturbativeQEDmay not be a very good theory as beta decay contributesthrough weak interactions which may cause hard thermalloops and the associated singularities inQEDTherefore all ofour discussions in the following sections are referred to belowdecoupling temperature

3 Second Order Contribution

Second order thermal corrections to the electron mass comefrom the two-loop diagrams The overlapping diagrams giveoverlapping hot terms with divergent cold terms and thecalculations become really cumbersome However in thelimiting cases simpler expressions can be obtained both forlow 119879 and for high 119879 limits In the limit 119879 lt 119898 the secondorder thermal contribution to electron mass [9 10] is

1205751198982

119898≃ (

1205751198981

119898)

2

+212057221198792

31198982

times [33

2+1

V(8119898

119864minus 1) + (

5

Vminus1

2minus4119898

119864V2) ln 1 + V

1 minus V


minus 1 +1

2(1 +

1

V) ln 1 + V

1 minus V

times[120574 minus ln 2 + 6

1205872

infin

sum

119903=1

1

1199032ln(119903119898

119879)]]

(14)

This equation shows that the low-temperature expressionfor thermal contribution is very complicated due to theoverlapping hot and cold terms at the two-loop level ascompared to the thermal one-loop contribution which isjust 12057212058711987921198982 However the leading order low-temperaturesecond order contribution is simply

1205751198982

119898≃ (

1205721205871198792

31198982)

2

+1012057221198792

31198982 (15)

whereas the first term indicates the contribution from thedisconnected graph which is usually expected from theiterationmethodThe second term in this expression is clearlydominant for 119879 lt 119898 In the limit 119879 gt 119898 the electron masscontribution is a long expression and can be found in [10 11]in detail Numerical evaluation is not simple andwe postponeit for now However the leading order contribution at 119879 gt 119898

can be written as

(120575119898(1)

119898)

2

≃ 1205722[M1(

119879

119898)

4

+M2(119879

119898)

3

+M3(119879

119898)

2

+M4 (119879

119898) +M5]

(16)

The coefficients Mrsquos in (16) are complicated functionsof electron mass energy and velocity of electron giving theleading order contribution as

(120575119898(1)

119898)

2

≃ 1205722(119879

119898)

5

(17)

The second order contribution in the above equations isjust a leading order contribution to prove that the secondorder contribution cannot be higher than the first ordercontribution below the decoupling temperature only Higherorder terms can blow up even at the lower temperatures

4 Wavefunction Renormalization

The electron wavefunction in QED is related to the self-massof electron through the Ward identity The factor (1 minus 119860) isrequired for renormalization because then the propagatorcan also be renormalized by replacing

1

119901 minus 119898 + 119894120576997888rarr

119885minus12

119901 minus 119898 + 119894120576 (18)

Thus for Lorentz invariant self-energy the wavefunctionrenormalization constant can equivalently be expressed as

119885minus12 = 1 minus 119860 = 1 minus

120597Σ (119901)

120597119901 (19)

The fermion wavefunction renormalization in the finitetemperature field theory can be obtained in a similar wayas discussed in vacuum theories However the Lorentzinvariance in the finite temperature theory is imposed bysetting 119860 = 119861 in (5) Thus using (14) and (5) one obtains [9]

119885minus12 (119898120573) = 119885

minus12 (119879 = 0) minus

2120572

120587int

infin

0

119889119896

119896119899119861 (119896) minus

3120572

120587119887 (119898120573)

+1205721198792

120587V1198642ln 1 + V1 minus V

1205872

6+ 119898120573119886 (119898120573) minus 119888 (119898120573)

(20)

giving the low-temperature values as

119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896)

+1205721205871198792

61198642

1

Vln 1 minus V1 + V

(21)

and high-temperature value as

119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896)

+1205721205871198792

41198642

1

Vln 1 minus V1 + V

(22)

For small values of the electron velocity V thermalcontributions to the wavefunction renormalization constantcan be determined from (21) and (22) as

119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896) minus

1205721205871198792

31198642 (23)

for low temperature and

119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896) minus

1205721205871198792

21198642 (24)

for high temperatureThe finite part of (21) and (22) is equal to 120575119898119898 at

119864 = 10119898 in the relevant temperature range These terms aresuppressed at large values of electron energy 119864 and they aresuppressed by a factor 11987921198982 However the calculated valueat that temperature is significantly differentThe difference inthe thermal contribution can easily be found to be about 50of the low-temperature value and around 33 of the high-temperature value just as in 120575119898119898 This difference can bementioned as

Δ (119885minus12 ) asymp

1205721205871198792

61198642= 38 times 10

minus31198792

1198642 (25)

The finite part of the wavefunction renormalization con-stant can be obtained by finding a ratio of temperature withthe Lorentz energy 119864 The minimum value of this energy isequal to mass Following (13) the higher order contributionsto the wavefunction can then be written as

119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896) + exp(120572120587119879

2

21198642) (26)


Equation (26) indicates that thermal contributions to thewavefunction are always smaller than 120575119898119898 because 119864 isalways greater than119898 At119879 = 119898 and even at the temperatureshigher than nucleosynthesis convergence of the series canbe established as 119879 lt 119864 at those temperatures So the twointeresting physical limits give smaller thermal contributionin electron wavefunction as the relevant temperature limitscan be defined as 119898 lt 119879 lt 119864 and 119879 lt 119898 lt 119864 whichensures the renormalizability ofQED at comparatively highertemperature as compared to self-mass

5 Second Order Thermal Corrections tothe Electron Wavefunction

The renormalization of the wavefunction is directly relatedto the self-mass of electrons and the guaranteed finiteness ofelectron mass at finite temperatures below the decouplingtemperature ensures the finiteness of the wavefunction auto-matically However the detailed expression for the wavefunc-tion renormalization constant can be found in [10] and can begiven as

119885minus12

119879lt119898997888997888997888rarr 1 +

120572

4120587(4 minus

3

120576) minus

120572

41205872(119868119860 minus

1198680

119864)

minus1205722

41205872(3 +

1

120576) 119868119860 +

212057221198792

312058721198982

(27)

Similarly the high-temperature limit for the wavefunc-tion renormalization constant gives

119885minus12

119879gt119898997888997888997888rarr 1 minus 120572[

2119868119860

120587+

1

4120587(3

120576minus 4) +

41205871198792

3]

minus 1205722[

1

412058733

120576(119868119860 + 119869119860) + (3119868119860 + 5119869119860) minus

81198792

31198982

+1

8

infin

sum

119899119903119904=1

(minus1)119903119879

times 119890minus119903120573119864

[119891+ (119904 119903)120574p1198642V2

minus119868119861119868119862

641205872

+ ℎ (119901 120574)119891minus (119899 119903)119868119862

8120587minus 119891minus (119904 119903)

+ 119891+ (119899 119903)120574p1198642V2

119868119861

8120587

+ 2

119898ℎ (119901 120574) minus

120574p1198642V2

119891minus (119904 119903) ]]

(28)

in terms of the one-loop integrals 119868rsquos and 119869rsquos which can befound in the original papers [9 10]

6 Photon Self-Energy andQED Coupling Constant

The self-energy of photons and the electron charge alsobehave differently for a cooling and a heating system around

119879 = 119898 It is well known that the electron charge andthe coupling constant do not show significant temperaturedependence for 119879 lt 119898 However they have significantthermal contributions at high temperatures (119879 gt 119898)Differences in the behavior of a cooling and a heating QEDsystem start again near 119879 sim 119898 from the lower side and nearthe decoupling temperature from a high side This differencein the coupling constant looks more natural due to the betadecay processes during nucleosynthesis

Calculations of the vacuum polarization tensor showthat the real part of the [9 10] longitudinal and transversecomponents of the polarization tensors can be evaluated inthe limit 120596 rarr 0 as

ReΠ120573119871 (119896 0) =4120587120572

31198792+

1198962

21205872ln 119898

119879+ sdot sdot sdot (29a)

ReΠ120573119879 (119896 0) =2120572

31205871198962 ln 119898

119879+ sdot sdot sdot (29b)

giving the interaction potential in the rest frame of thecharged particle as [8]

119881 (119896) equiv 11989021198771205751205830 [

119906120583119906]

1198962 minus (41205871205723) 1198792 + (119896221205872) ln (119898119879)

+

119892120583] minus 119906120583119906]

1198962 minus (21205723120587) 1198962 ln (119898119879)]

119881 (119896) = 11989021198771205751205830Δ 120583]120575]0

(30)

where 119890119877 is the renormalized charge 119881(119896) can be written atlow temperature as

119881 (119896) equiv 1198902119877 (1 +

2120572

3120587ln 119879

119898)[

11990620

1198962 + (412058712057211987923)+1 minus 11990620

1198962]

(31)

The constant in the longitudinal propagator is the plasmascreening mass therefore the outside factor correspondsto the charge renormalization and in turn to the couplingconstant We may then write the coupling constant at lowtemperatures as

120572 (119879) = 120572 (119879 = 0) (1 +2120572

3120587ln 119879

119898)

= 120572 (119879 = 0) (1 + 155 times 10minus3 ln 119879

119898)

(32)

The factor 155times103 ln(119879119898) is a slowly varying functionof temperature and does not give any significant contributionnear the decoupling temperature and remains insignificantfor a large range of temperature due to the absence ofsignificant numbers of hot electrons in the backgroundTherefore the coupling constant is not modified at 119879 lt 119898

at all The temperature dependent factor in the longitudinalpropagator (412057212058711987921198982) is the plasma screening frequencies


or self-mass of photons that contribute to the QED couplingconstant at finite temperatures For generalized temperaturesthe charge renormalization constant 1198853 can be written as [11]

1198853 = 1 minus21198902

1205872

times 119888 (119898120573)

1205732minus119898119886 (119898120573)

120573minus1

4(1198982minus1205962

3) 119887 (119898120573)

(33)

Also the electric permittivity is

120576 (119870) ≃ 1 +41198902

12058721198702(1 minus

1205962

1198962)

times (1 minus120596

2119896ln 120596 + 119896120596 minus 119896

)(119888 (119898120573)

1205732minus119898119886 (119898120573)

120573)

minus1

4(21198982minus 1205962+111198962+ 37120596

2

72) 119887 (119898120573)

(34)

and the magnetic permeability is

1

120583 (119870)

≃ 1 +21198902

120587211989621198702

times [12059621 minus

1205962

1198962minus (1 +

1198962

1205962)(1 minus

1205962

1198962)120596

2119896ln 120596 + 119896120596 minus 119896

times (119888 (119898120573)

1205732minus119898119886 (119898120573)

120573)

minus1

8(61198982minus 1205962+1291205962minus 109119896

2

72) 119887 (119898120573)]

(35)

In the limit 119879 gt 119898 the wavefunction renormalizationconstant can be written as

1198853 = 1 +1205721198792

61198982 (36)

giving the renormalized coupling constant as

120572 =1198902 (ℎ119888)41205871205980

(1 +1205721198792

61198982) =

12058301198902119888

2ℎ(1 +

1205721198792

61198982) (37)

Equation (36) gives 1198853 = 1 + 12 times 10311987921198982 and leaves

the perturbation series valid for at least 119879 le 4119898It is clear from (32)ndash(36) that the coupling constant is

basically changed due to the hot fermion loop contributionsHot bosons do not change the coupling constant as thevacuum fluctuations occur due to fermion loops at the firstorder in 120572 However the situation is different for higher ordercontributions

At119879 gt 4119898 (the decoupling temperature) thermal contri-butions are significant enough to grow the coupling constantto the level where it can create a problem for the convergenceof the perturbative series of QED In that case we need to usenonperturbative methods to establish the renormalization ofQED at high temperatures However thermal corrections tothe coupling constant are significant until the temperatureis of the order of electron mass When it is lower than theelectron mass and the primordial nucleosynthesis almoststops 120572 attains the constant value (1137) It happens becausethe constant thermal contribution from fermion background(119888(119898120573) = minus120587

212) at 119879 gt 119898 becomes negligible as soon as

the universe cools down to 119879 lt 119898

7 Second Order Correction tothe QED Coupling Constant

The first order thermal corrections do not contribute to thecoupling constant at119879 lt 119898 but they do not vanish at the two-loop level because of the overlap of the vacuum term and thethermal term The low-temperature second order correctionto coupling constant can be given as

1198853 = 1 +1205722119879

61198982

2

(38)

and the high-temperature contributions are given as

120572119877

= 120572 (119879 = 0)

+8120572

1205871198982[119898119886 (119898120573)

120573minus119888 (119898120573)

1205732

+119887 (119898120573)

4(1198982+1

31205962)]

+1205722

1198982[1198792

6+

infin

sum

119899119903119904=1

(minus1)119904+119903119890minus119904120573119864 119879

(119899 + 119904)

times 24119879

(119903 + 119904)minus 12119898

2

times [119890minus119898120573(119904+119903)

119898minus (119903 + 119904)

times 120573119864119894 minus119898120573 (119903 + 119904)]]

(39)

Since the contribution to the coupling constant is alwaysproportional to 11987921198982 and is sufficiently smaller than 120572 thatensures the renormalizability at the temperatures below thedecoupling temperature

In the next section we will summarize these results anddiscuss the behavior of QED in thermal background belowthe decoupling temperature


8 Results and Discussion

A quantitative study of the QED renormalization constantsat finite temperatures shows that all the renormalizationconstants are finite at119879 le 4119898 Also the higher order radiativecorrections are smaller than the lower order perturbativecorrections in real-time formalism In this range of temper-ature the largest thermal contributions come from the massrenormalization constantThewavefunction renormalizationconstant is suppressed especially at low temperatures and rel-ativistic energies There is no low-temperature contributionto the electron charge and QED coupling constant as the hotfermion loop contributions are ignorable at low temperatures(119879 lt 119898) However a rapid growth in the renormalizationconstants at large temperatures indicates increasing ther-mal corrections with the increase of hot electrons in thebackground A comparison between the one-loop and two-loop contributions is shown to prove renormalizability of thetheory including thermal corrections What happens around119879 ge 119898 is expressed in terms of 119886(119898120573) 119887(119898120573) and 119888(119898120573)

functions at the first loop level These functions give vanish-ing contributions at low temperatures as they arise from theintegration of the hot fermion propagator Contributions of119888(119898120573) vanish at low 119879 and it sums up to (minus120587212) for large 119879values However at the two-loop level thermal contributionsoverlap with the vacuum terms and are expressed in a muchmore complicated overlapping series as well as the same119886(119898120573) 119887(119898120573) and 119888(119898120573) functions The problem ariseswhen diverging vacuum terms overlap with the temperaturedependent terms However they are renormalizable belowdecoupling temperatures The low-temperature and high-temperature contributions are derivable from these generalexpressions

We plot thermal contributions to electron mass (120575119898119898)electron wavefunction renormalization constant (119885minus12 +

(2120572120587) int(119889119896119896)119899119861(119896)) and the electron charge (1198853 minus 1) thatcan be derived for 119879 lt 119898 and 119879 gt 119898 ranges from thesame (8) (19) and (33) respectively at the one-loop levelSince we are dealing with the exponential functions in thisstudy even less than an order of magnitude difference isa safe limit to use low-temperature and high-temperaturelimits Just to prove the renormalizability we present theplots of low-temperature and high-temperature terms ofrenormalization constants as they are derived from generalexpressions We give a comparison between a heated and acooled QED system around the common temperature of theorder of electron mass (see Figure 1) The difference betweenthermal background contributions of a heating and a coolingsystem changes due to the difference of background duringthe heating and cooling process A cooling system such as theearly universe starts off with more fermions and keeps losingthem during the cooling process whereas a heating systemwill start with a minimum number of fermions and they willbe created during the heating process We just quantitativelydiscuss the self-mass contribution as it changes the physicallymeasurable mass a very important parameter of the theory

Figure 1 indicates the difference between the self-massesof electron due to the low- and high-temperature first ordercorrections It is found that the difference between two values

000200400600801

01201401601802

minus1 0 1 2 3 4 5T (m)

Elec

tron

mas

s

Two-loop

One-loop

Figure 2 Comparison of self-mass contribution from the hotbackground at the one-loop (solid line) and at the two-loop (brokenline) level below the decoupling temperature

00002000400060008

0010012001400160018

002

Elec

tron

char

ge

0 1 2 3 4 5T (m)

Two-loop

One-loop

Figure 3 Comparison of QED coupling constant contribution fromthe hot background at the one-loop (solid line) and at the two-loop(broken line) level below the decoupling temperature

is equal to 13 of the low-temperature value and 12 of thehigh-temperature value at 119879 = 119898 Similarly the electronwavefunction and charge for a system that approaches fromlower temperature to 119879 = 119898 will be different from theone reaching to the same temperature by cooling of a hottersystem It is understandable as the photons do not couplewitheach other directlyThey only couple to the charged fermionsand the presence of charge fermion is required to affect theQED couplingThe fermion distribution function (2) actuallybrings in the fermion loop contributions at high119879whenmorefermions are generated during the nucleosynthesis and arenot ignorable any more in the medium

We compare thermal contribution to the first order [9]and the second order [10] corrections to the self-mass ofelectron in Figure 2 The solid line corresponds to the firstorder corrections and the broken lines correspond to thesecond order corrections Figure 3 gives a similar graph forthe renormalized coupling constant of QED in a hotmediumA plot of these renormalization constants shows that thetemperature corrections do not affect the renormalizability ofthe theory for119879 sufficiently smaller than119898up to the two-loop


Table 1 The values of the self-mass of electron and QED coupling small constant below the decoupling temperature

119879119898Electron mass Charge Wavefunction for 119864 = 10m

Low 119879 High 119879 High 119879 Low 119879 High 119879002 306119864 minus 06 minus713948119864 minus 15

02 0000306 minus713948119864 minus 11

1 0007647 001147 16119864 minus 07 minus446218119864 minus 08 minus15059119864 minus 07

2 004588 256119864 minus 06 minus24094119864 minus 06



5 028675 00001 minus94116119864 minus 05

6 041292 0000207 minus0000195167 056203 0000384 minus0000361568 073408 0000656 minus000061689 092907 000105 minus00009879910 1147 00016 minus000150585

level The distinction in behavior starts near 119879 sim 119898 withoutaffecting the renormalizability Due to the small contributionthe behavior of renormalization constants for both orders iscomparable at low temperatures indicated by Figures 2 and3This difference is significant at high temperatures The plotof the QED coupling constant as a function of temperatureshows the very small effect at low temperatures at both loopsIt only becomes significant for 119879 gt 119898 In the approximationsused in this paper one-loop thermal contribution is zeroat the first loop level at low temperature However at hightemperatures the coupling constant grows quadratically withtemperature expressed in units of electronmassWith a largecoupling constant the renormalizability of the theory cannotbe guaranteed and nonperturbative methods have to be usedto treat the hard thermal loops However it is not neededunder the decoupling temperatures at all

Table 1 shows the leading order thermal contributionsto the numerical values of the renormalization constantsfor electron mass and the QED coupling constant near thedecoupling temperatures It also shows that the thermal con-tribution of the renormalization constants at temperaturesaround the decoupling temperature becomes significant Itis a table of the leading order contributions to prove therenormalizability So we consider 119864 le 4119898 just to get anapproximate thermal contribution to electron wavefunction

However it is explicitly shown in the above figures andTable 1 that the renormalization scheme of QEDworks belowthe neutrino decoupling temperature Above the decou-pling temperature a large number of hot electrons in thebackground lead to the failure of the QED renormalizationscheme at larger temperatures and the electroweak theory hasto be incorporated

The cooling universe of the standard big bang modelbehaves differently after the neutrino decoupling Nucleosyn-thesis starts right [17 18] after the neutrino decoupling andthe helium synthesis takes place when the temperature ofthe universe is cooled down to the temperature of elec-tron mass This is actually a temperature where the finitetemperature corrections to QED parameters are significantbut complicated enough to evaluate it numerically However

the temperature dependent QED parameters are needed todescribe the observations of WMAP data After the nucleo-synthesis is complete the temperature dependence is back tothe quadratic dependence on temperature though it is notexactly the same as at the low temperature Fermion back-ground contribution can easily be seen

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S S Masood QED at Finite Temperature and Density LambertAcademic Publication 2012

[2] J F Donoghue and B R Holstein ldquoRenormalization and radia-tive corrections at finite temperaturerdquo Physical Review D vol28 no 2 pp 340ndash348 1983

[3] J F Donoghue and B R Holstein ldquoErratum to ldquoRenormaliza-tion and radiative corrections at finite temperaturerdquordquo PhysicalReview D vol 29 no 12 p 3004 1984

[4] J F Donoghue B R Holstein and R W Robinett ldquoQuantumelectrodynamics at finite temperaturerdquo Annals of Physics vol164 no 2 pp 233ndash276 1985

[5] A E I Johansson G Peressutti and B-S Skagerstam ldquoQuan-tum field theory at finite temperature renormalization andradiative correctionsrdquoNuclear Physics B vol 278 no 2 pp 324ndash342 1986

[6] G Peressutti and B-S Skagerstam ldquoFinite temperature effectsin quantum field theoryrdquo Physics Letters B vol 110 no 5 pp406ndash410 1982

[7] H AWeldon ldquoCovariant calculations at finite temperature therelativistic plasmardquo Physical Review D vol 26 no 6 pp 1394ndash1407 1982

[8] N P Landsman and C G van Weert ldquoReal- and imaginary-time field theory at finite temperature and densityrdquo PhysicsReports vol 145 no 3-4 pp 141ndash249 1987

[9] S S Masood ldquoRenormalization of QED in superdense mediardquoPhysical Review D vol 47 no 2 pp 648ndash652 1993


[10] M Q Haseeb and S S Masood ldquoSecond order thermal correc-tions to electron wavefunctionrdquo Physics Letters B vol 704 no1-2 pp 66ndash73 2011

[11] S S Masood and M Q Haseeb ldquoSecond-order corrections tothe magnetic moment of electron at finite temperaturerdquo Inter-national Journal of Modern Physics A vol 27 no 32 Article ID1250188 9 pages 2012

[12] Y Fueki H Nakkagawa H Yokota and K Yoshida ldquoChi-ral phase transitions in QED at finite temperaturemdashDyson-Schwinger equation analysis in the real time hard-thermal-loopapproximationrdquo Progress of Theoretical Physics vol 110 no 4pp 777ndash789 2003

[13] N Fornengo CW Kim and J Song ldquoFinite temperature effectson the neutrino decoupling in the early Universerdquo PhysicalReview D vol 56 p 5123 1997

[14] N Su ldquoA brief overview ofhard-thermal-loop perturbationtheoryrdquoCommunications inTheoretical Physics vol 57 no 3 pp409ndash421 2012

[15] E Komatsu K M Smith J Dunkley et al ldquoSeven-yearwilkinson microwave anisotropy probe (WMAP) observationscosmological interpretationrdquo Astrophysical Journal SupplementSeries vol 192 no 2 p 18 2011

[16] G Steigman IAU Symposium No 265 2009[17] S S Masood and M Q Haseeb ldquoGluon polarization at finite

temperature and densityrdquo Astroparticle Physics vol 3 no 4 pp405ndash412 1995

[18] S S Masood ldquoNucleosynthesis at finite temperature and den-sityrdquo httparxivorgabs13103608

[19] S S Masood ldquoNeutrino physics in hot and dense mediardquo Phy-sical Review D vol 48 no 7 pp 3250ndash3258 1993

[20] A Perez-Martınez S S Masood H Perez Rojas R Gaitan andS Rodriguez-Romo ldquoEffective magnetic moment of neutrinosin strong magnetic fieldsrdquo Revista Mexicana de Fisica vol 48no 6 pp 501ndash503 2002

[21] M Chaichian S S Masood C Montonen A Perez Martınezand H Perez Rojas ldquoQuantum magnetic collapserdquo PhysicalReview Letters vol 84 no 23 pp 5261ndash5264 2000

[22] S S Masood ldquoMagnetic moment of neutrinos in the statisticalbackgroundrdquo Astroparticle Physics vol 4 no 2 pp 189ndash1941996

[23] S S Masood ldquoThermodynamics of nucleosynthesisrdquo httparxivorgabs13103608

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



Advances in

Astronomy

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


standard big bang model of cosmology [13ndash16] and theyhelp to relate the observational data for example WMAP(Wilkinson Microwave Anisotropy Probe [15]) with the bigbang theory The same techniques can even be used tocalculate perturbative effects in QCD [17] and electroweakprocesses [18] at low temperatures and estimate their contri-butions at high temperatures alsoWe assume that the changein the properties of neutrinos [19ndash23] does not affect thedecoupling temperature significantly

Without giving the calculational details we have to brieflyoverview the existing formofQEDrenormalization constantsin real-time formalismThe Feynman rules of vacuum theoryare used with the statistically corrected propagators given as

119863120573 (119896) =119894

1198962 minus 1198982 + 119894120576+

2120587

119890120573119864119896 minus 1120575 (1198962minus 1198982) (1)

for bosons and

119878119865 (119901) =119894

119901 minus 119898 + 119894120576minus2120587 (119901 + 119898)

119890120573119864119901 + 1

120575 (1199012minus 1198982) (2)

for fermions in a hot medium Since the temperature cor-rections appear as additive contributions to the fermion andboson propagators in (1) and (2) temperature dependentterms can be handled independently of vacuum terms at theone-loop level We restrict ourselves up to the two-loop levelto show explicitly that second order thermal corrections arefinite and smaller than the first order corrections [9] belowthe neutrino decoupling

In the next section we give the calculations of thermalcorrections to the mass renormalization constant 120575119898119898 ofQEDand the physicalmass of electrons at finite temperaturesSection 3 is devoted to the calculations of electron wavefunc-tion at finite temperature Section 4 gives the calculationsof the electron charge and the QED coupling constant upto the two-loop level whereas Section 5 the last section isdevoted to the discussions of the results of all the calculationsto explicitly prove the renormalizability of the theory up tothe two-loop level below the decoupling temperature

2 Self-Mass of Electron

The renormalized mass of electrons119898119877 can be represented asa physical mass119898phys of electrons and is defined in a hot anddense medium as

119898119877 equiv 119898phys = 119898 + 120575119898 (119879 = 0) + 120575119898 (119879) (3)

where119898 is the electronmass at zero temperature and 120575119898(119879 =

0) represents the radiative corrections from a vacuum and120575119898(119879) are the contributions from the thermal background atnonzero temperature 119879 The physical mass of electrons canthen be represented as a perturbative series in 120572 and can bewritten as

119898phys cong 119898 + 120575119898(1)+ 120575119898(2)+ sdot sdot sdot (4)

where 1205751198981 and 1205751198982 are the shifts in electron mass in the

first and second order in 120572 respectively The physical mass

is deduced by locating the pole of the fermion propagator(119894(119901 + 119898)(119901

2minus 1198982+ 119894120576)) in thermal background For

this purpose we sum over all of the same order diagramsRenormalization is established by demonstrating the order-by-order cancellation of singularities All the finite termsfrom the same order in 120572 are combined together to evaluatethe same order contribution to the physical mass given in (4)The physical mass in thermal backgrounds up to order 1205722[7 8] is calculated using the renormalization techniques ofQED Self-mass of electron is expressed as

Σ (119901) = 119860 (119901) 119864120574119900 minus 119861 (119901) sdot 120574 minus 119862 (119901) (5)

where 119860(119901) 119861(119901) and 119862(119901) are the relevant coefficientsthat are functions of electron momentum only We take theinverse of the propagator with momentum and mass termsseparated as

119878minus1(119901) = (1 minus 119860) 119864120574

119900minus (1 minus 119861) 119901 sdot 120574 minus (119898 minus 119862) (6)

The temperature dependent radiative corrections to theelectron mass up to the first order in 120572 are obtained from thetemperature dependent propagator as

1198982phys equiv 119864

2minus |p|2 = 1198982 [1 minus 6120572

120587119887 (119898120573)]

+4120572

120587[119898119879119886 (119898120573) +

2

31205721205871198792minus

6

1205872119888 (119898120573)]

(7)

giving

120575119898

119898≃

1

21198982(1198982phys minus 119898

2)

≃1205721205871198792

31198982[1 minus

6

1205872119888 (119898120573)] +

2120572

120587

119879

119898119886 (119898120573) minus

3120572

120587119887 (119898120573)

(8)

where 120575119898119898 is the relative shift in electron mass due to finitetemperatures which was determined in [1 9] with

119886 (119898120573) = ln (1 + 119890minus119898120573) (9a)

119887 (119898120573) =

infin

sum

119899=1

(minus1)119899119864119894 (minus119899119898120573) (9b)

119888 (119898120573) =

infin

sum

119899=1

(minus1)119899 119890minus119899119898120573

1198992 (9c)

The convergence of (4) can be ensured for 119879 lt 2MeVas 120575119898 119898 is always smaller than unity within this limit Thisscheme of calculations will not work for higher temperaturesand the first order corrections may exceed the original valuesofQEDparameters after 5MeVAt the low temperatures119879 lt

119898 the functions 119886(119898120573) 119887(119898120573) and 119888(119898120573) fall off in powersof 119890minus119898120573 in comparison with (119879119898)2 and can be neglected inthe low-temperature limit giving

120575119898

119898

119879⋖119898997888997888997888rarr

1205721205871198792

31198982 (10)



212 yielding

120575119898

119898

119879⋗119898997888997888997888rarr

1205721205871198792

21198982 (11)




Δ(120575119898

119898) = plusmn

1205721205871198792


minus3 1198792

1198982 (12)








0

002

004

006

008

01

012

0 05 1 15 2 25 3 35

Disconnected region

Low T

High T

T (m)

Elec

tron

mas



119898phys = 119898[1 +120575119898

119898+1

2(120575119898

119898)

2


)

(13)




1205751198982

119898≃ (

1205751198981

119898)

2

+212057221198792

31198982

times [33

2+1

V(8119898

119864minus 1) + (

5

Vminus1

2minus4119898

119864V2) ln 1 + V

1 minus V


minus 1 +1

2(1 +

1

V) ln 1 + V

1 minus V


1205872

infin

sum

119903=1

1

1199032ln(119903119898

119879)]]

(14)


1205751198982

119898≃ (

1205721205871198792

31198982)

2

+1012057221198792

31198982 (15)


can be written as

(120575119898(1)

119898)

2

≃ 1205722[M1(

119879

119898)

4

+M2(119879

119898)

3

+M3(119879

119898)

2

+M4 (119879

119898) +M5]

(16)


(120575119898(1)

119898)

2

≃ 1205722(119879

119898)

5

(17)




1

119901 minus 119898 + 119894120576997888rarr

119885minus12

119901 minus 119898 + 119894120576 (18)



120597Σ (119901)

120597119901 (19)


119885minus12 (119898120573) = 119885

minus12 (119879 = 0) minus

2120572

120587int

infin

0

119889119896

119896119899119861 (119896) minus

3120572

120587119887 (119898120573)

+1205721198792

120587V1198642ln 1 + V1 minus V

1205872

6+ 119898120573119886 (119898120573) minus 119888 (119898120573)

(20)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896)

+1205721205871198792

61198642

1

Vln 1 minus V1 + V

(21)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896)

+1205721205871198792

41198642

1

Vln 1 minus V1 + V

(22)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896) minus

1205721205871198792

31198642 (23)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896) minus

1205721205871198792

21198642 (24)




1205721205871198792

61198642= 38 times 10

minus31198792

1198642 (25)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896) + exp(120572120587119879

2

21198642) (26)





119885minus12

119879lt119898997888997888997888rarr 1 +

120572

4120587(4 minus

3

120576) minus

120572

41205872(119868119860 minus

1198680

119864)

minus1205722

41205872(3 +

1

120576) 119868119860 +

212057221198792

312058721198982

(27)


119885minus12

119879gt119898997888997888997888rarr 1 minus 120572[

2119868119860

120587+

1

4120587(3

120576minus 4) +

41205871198792

3]

minus 1205722[

1

412058733

120576(119868119860 + 119869119860) + (3119868119860 + 5119869119860) minus

81198792

31198982

+1

8

infin

sum

119899119903119904=1

(minus1)119903119879

times 119890minus119903120573119864

[119891+ (119904 119903)120574p1198642V2

minus119868119861119868119862

641205872

+ ℎ (119901 120574)119891minus (119899 119903)119868119862

8120587minus 119891minus (119904 119903)

+ 119891+ (119899 119903)120574p1198642V2

119868119861

8120587

+ 2

119898ℎ (119901 120574) minus

120574p1198642V2

119891minus (119904 119903) ]]

(28)






ReΠ120573119871 (119896 0) =4120587120572

31198792+

1198962

21205872ln 119898


ReΠ120573119879 (119896 0) =2120572

31205871198962 ln 119898



119881 (119896) equiv 11989021198771205751205830 [

119906120583119906]

1198962 minus (41205871205723) 1198792 + (119896221205872) ln (119898119879)

+

119892120583] minus 119906120583119906]

1198962 minus (21205723120587) 1198962 ln (119898119879)]

119881 (119896) = 11989021198771205751205830Δ 120583]120575]0

(30)


119881 (119896) equiv 1198902119877 (1 +

2120572

3120587ln 119879

119898)[

11990620

1198962 + (412058712057211987923)+1 minus 11990620

1198962]

(31)


120572 (119879) = 120572 (119879 = 0) (1 +2120572

3120587ln 119879

119898)

= 120572 (119879 = 0) (1 + 155 times 10minus3 ln 119879

119898)

(32)





1198853 = 1 minus21198902

1205872

times 119888 (119898120573)

1205732minus119898119886 (119898120573)

120573minus1

4(1198982minus1205962

3) 119887 (119898120573)

(33)


120576 (119870) ≃ 1 +41198902

12058721198702(1 minus

1205962

1198962)


2119896ln 120596 + 119896120596 minus 119896

)(119888 (119898120573)

1205732minus119898119886 (119898120573)

120573)

minus1

4(21198982minus 1205962+111198962+ 37120596

2

72) 119887 (119898120573)

(34)


1

120583 (119870)

≃ 1 +21198902

120587211989621198702


1205962

1198962minus (1 +

1198962

1205962)(1 minus

1205962

1198962)120596

2119896ln 120596 + 119896120596 minus 119896

times (119888 (119898120573)

1205732minus119898119886 (119898120573)

120573)

minus1

8(61198982minus 1205962+1291205962minus 109119896

2

72) 119887 (119898120573)]

(35)


1198853 = 1 +1205721198792

61198982 (36)


120572 =1198902 (ℎ119888)41205871205980

(1 +1205721198792

61198982) =

12058301198902119888

2ℎ(1 +

1205721198792

61198982) (37)









1198853 = 1 +1205722119879

61198982

2

(38)


120572119877

= 120572 (119879 = 0)

+8120572

1205871198982[119898119886 (119898120573)

120573minus119888 (119898120573)

1205732

+119887 (119898120573)

4(1198982+1

31205962)]

+1205722

1198982[1198792

6+

infin

sum

119899119903119904=1

(minus1)119904+119903119890minus119904120573119864 119879

(119899 + 119904)

times 24119879

(119903 + 119904)minus 12119898

2

times [119890minus119898120573(119904+119903)

119898minus (119903 + 119904)

times 120573119864119894 minus119898120573 (119903 + 119904)]]

(39)










000200400600801

01201401601802

minus1 0 1 2 3 4 5T (m)

Elec

tron

mas

s

Two-loop

One-loop


00002000400060008

0010012001400160018

002

Elec

tron

char

ge

0 1 2 3 4 5T (m)

Two-loop

One-loop








02 0000306 minus713948119864 minus 11





5 028675 00001 minus94116119864 minus 05









References






























FluidsJournal of


Journal of







Advances in

Astronomy



Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





212 yielding

120575119898

119898

119879⋗119898997888997888997888rarr

1205721205871198792

21198982 (11)




Δ(120575119898

119898) = plusmn

1205721205871198792


minus3 1198792

1198982 (12)








0

002

004

006

008

01

012

0 05 1 15 2 25 3 35

Disconnected region

Low T

High T

T (m)

Elec

tron

mas



119898phys = 119898[1 +120575119898

119898+1

2(120575119898

119898)

2


)

(13)




1205751198982

119898≃ (

1205751198981

119898)

2

+212057221198792

31198982

times [33

2+1

V(8119898

119864minus 1) + (

5

Vminus1

2minus4119898

119864V2) ln 1 + V

1 minus V


minus 1 +1

2(1 +

1

V) ln 1 + V

1 minus V


1205872

infin

sum

119903=1

1

1199032ln(119903119898

119879)]]

(14)


1205751198982

119898≃ (

1205721205871198792

31198982)

2

+1012057221198792

31198982 (15)


can be written as

(120575119898(1)

119898)

2

≃ 1205722[M1(

119879

119898)

4

+M2(119879

119898)

3

+M3(119879

119898)

2

+M4 (119879

119898) +M5]

(16)


(120575119898(1)

119898)

2

≃ 1205722(119879

119898)

5

(17)




1

119901 minus 119898 + 119894120576997888rarr

119885minus12

119901 minus 119898 + 119894120576 (18)



120597Σ (119901)

120597119901 (19)


119885minus12 (119898120573) = 119885

minus12 (119879 = 0) minus

2120572

120587int

infin

0

119889119896

119896119899119861 (119896) minus

3120572

120587119887 (119898120573)

+1205721198792

120587V1198642ln 1 + V1 minus V

1205872

6+ 119898120573119886 (119898120573) minus 119888 (119898120573)

(20)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896)

+1205721205871198792

61198642

1

Vln 1 minus V1 + V

(21)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896)

+1205721205871198792

41198642

1

Vln 1 minus V1 + V

(22)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896) minus

1205721205871198792

31198642 (23)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896) minus

1205721205871198792

21198642 (24)




1205721205871198792

61198642= 38 times 10

minus31198792

1198642 (25)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896) + exp(120572120587119879

2

21198642) (26)





119885minus12

119879lt119898997888997888997888rarr 1 +

120572

4120587(4 minus

3

120576) minus

120572

41205872(119868119860 minus

1198680

119864)

minus1205722

41205872(3 +

1

120576) 119868119860 +

212057221198792

312058721198982

(27)


119885minus12

119879gt119898997888997888997888rarr 1 minus 120572[

2119868119860

120587+

1

4120587(3

120576minus 4) +

41205871198792

3]

minus 1205722[

1

412058733

120576(119868119860 + 119869119860) + (3119868119860 + 5119869119860) minus

81198792

31198982

+1

8

infin

sum

119899119903119904=1

(minus1)119903119879

times 119890minus119903120573119864

[119891+ (119904 119903)120574p1198642V2

minus119868119861119868119862

641205872

+ ℎ (119901 120574)119891minus (119899 119903)119868119862

8120587minus 119891minus (119904 119903)

+ 119891+ (119899 119903)120574p1198642V2

119868119861

8120587

+ 2

119898ℎ (119901 120574) minus

120574p1198642V2

119891minus (119904 119903) ]]

(28)






ReΠ120573119871 (119896 0) =4120587120572

31198792+

1198962

21205872ln 119898


ReΠ120573119879 (119896 0) =2120572

31205871198962 ln 119898



119881 (119896) equiv 11989021198771205751205830 [

119906120583119906]

1198962 minus (41205871205723) 1198792 + (119896221205872) ln (119898119879)

+

119892120583] minus 119906120583119906]

1198962 minus (21205723120587) 1198962 ln (119898119879)]

119881 (119896) = 11989021198771205751205830Δ 120583]120575]0

(30)


119881 (119896) equiv 1198902119877 (1 +

2120572

3120587ln 119879

119898)[

11990620

1198962 + (412058712057211987923)+1 minus 11990620

1198962]

(31)


120572 (119879) = 120572 (119879 = 0) (1 +2120572

3120587ln 119879

119898)

= 120572 (119879 = 0) (1 + 155 times 10minus3 ln 119879

119898)

(32)





1198853 = 1 minus21198902

1205872

times 119888 (119898120573)

1205732minus119898119886 (119898120573)

120573minus1

4(1198982minus1205962

3) 119887 (119898120573)

(33)


120576 (119870) ≃ 1 +41198902

12058721198702(1 minus

1205962

1198962)


2119896ln 120596 + 119896120596 minus 119896

)(119888 (119898120573)

1205732minus119898119886 (119898120573)

120573)

minus1

4(21198982minus 1205962+111198962+ 37120596

2

72) 119887 (119898120573)

(34)


1

120583 (119870)

≃ 1 +21198902

120587211989621198702


1205962

1198962minus (1 +

1198962

1205962)(1 minus

1205962

1198962)120596

2119896ln 120596 + 119896120596 minus 119896

times (119888 (119898120573)

1205732minus119898119886 (119898120573)

120573)

minus1

8(61198982minus 1205962+1291205962minus 109119896

2

72) 119887 (119898120573)]

(35)


1198853 = 1 +1205721198792

61198982 (36)


120572 =1198902 (ℎ119888)41205871205980

(1 +1205721198792

61198982) =

12058301198902119888

2ℎ(1 +

1205721198792

61198982) (37)









1198853 = 1 +1205722119879

61198982

2

(38)


120572119877

= 120572 (119879 = 0)

+8120572

1205871198982[119898119886 (119898120573)

120573minus119888 (119898120573)

1205732

+119887 (119898120573)

4(1198982+1

31205962)]

+1205722

1198982[1198792

6+

infin

sum

119899119903119904=1

(minus1)119904+119903119890minus119904120573119864 119879

(119899 + 119904)

times 24119879

(119903 + 119904)minus 12119898

2

times [119890minus119898120573(119904+119903)

119898minus (119903 + 119904)

times 120573119864119894 minus119898120573 (119903 + 119904)]]

(39)










000200400600801

01201401601802

minus1 0 1 2 3 4 5T (m)

Elec

tron

mas

s

Two-loop

One-loop


00002000400060008

0010012001400160018

002

Elec

tron

char

ge

0 1 2 3 4 5T (m)

Two-loop

One-loop








02 0000306 minus713948119864 minus 11





5 028675 00001 minus94116119864 minus 05









References






























FluidsJournal of


Journal of







Advances in

Astronomy



Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus 1 +1

2(1 +

1

V) ln 1 + V

1 minus V


1205872

infin

sum

119903=1

1

1199032ln(119903119898

119879)]]

(14)


1205751198982

119898≃ (

1205721205871198792

31198982)

2

+1012057221198792

31198982 (15)


can be written as

(120575119898(1)

119898)

2

≃ 1205722[M1(

119879

119898)

4

+M2(119879

119898)

3

+M3(119879

119898)

2

+M4 (119879

119898) +M5]

(16)


(120575119898(1)

119898)

2

≃ 1205722(119879

119898)

5

(17)




1

119901 minus 119898 + 119894120576997888rarr

119885minus12

119901 minus 119898 + 119894120576 (18)



120597Σ (119901)

120597119901 (19)


119885minus12 (119898120573) = 119885

minus12 (119879 = 0) minus

2120572

120587int

infin

0

119889119896

119896119899119861 (119896) minus

3120572

120587119887 (119898120573)

+1205721198792

120587V1198642ln 1 + V1 minus V

1205872

6+ 119898120573119886 (119898120573) minus 119888 (119898120573)

(20)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896)

+1205721205871198792

61198642

1

Vln 1 minus V1 + V

(21)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896)

+1205721205871198792

41198642

1

Vln 1 minus V1 + V

(22)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896) minus

1205721205871198792

31198642 (23)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896) minus

1205721205871198792

21198642 (24)




1205721205871198792

61198642= 38 times 10

minus31198792

1198642 (25)


119885minus12 = 119885

minus12 (119879 = 0) minus

2120572

120587int119889119896

119896119899119861 (119896) + exp(120572120587119879

2

21198642) (26)





119885minus12

119879lt119898997888997888997888rarr 1 +

120572

4120587(4 minus

3

120576) minus

120572

41205872(119868119860 minus

1198680

119864)

minus1205722

41205872(3 +

1

120576) 119868119860 +

212057221198792

312058721198982

(27)


119885minus12

119879gt119898997888997888997888rarr 1 minus 120572[

2119868119860

120587+

1

4120587(3

120576minus 4) +

41205871198792

3]

minus 1205722[

1

412058733

120576(119868119860 + 119869119860) + (3119868119860 + 5119869119860) minus

81198792

31198982

+1

8

infin

sum

119899119903119904=1

(minus1)119903119879

times 119890minus119903120573119864

[119891+ (119904 119903)120574p1198642V2

minus119868119861119868119862

641205872

+ ℎ (119901 120574)119891minus (119899 119903)119868119862

8120587minus 119891minus (119904 119903)

+ 119891+ (119899 119903)120574p1198642V2

119868119861

8120587

+ 2

119898ℎ (119901 120574) minus

120574p1198642V2

119891minus (119904 119903) ]]

(28)






ReΠ120573119871 (119896 0) =4120587120572

31198792+

1198962

21205872ln 119898


ReΠ120573119879 (119896 0) =2120572

31205871198962 ln 119898



119881 (119896) equiv 11989021198771205751205830 [

119906120583119906]

1198962 minus (41205871205723) 1198792 + (119896221205872) ln (119898119879)

+

119892120583] minus 119906120583119906]

1198962 minus (21205723120587) 1198962 ln (119898119879)]

119881 (119896) = 11989021198771205751205830Δ 120583]120575]0

(30)


119881 (119896) equiv 1198902119877 (1 +

2120572

3120587ln 119879

119898)[

11990620

1198962 + (412058712057211987923)+1 minus 11990620

1198962]

(31)


120572 (119879) = 120572 (119879 = 0) (1 +2120572

3120587ln 119879

119898)

= 120572 (119879 = 0) (1 + 155 times 10minus3 ln 119879

119898)

(32)





1198853 = 1 minus21198902

1205872

times 119888 (119898120573)

1205732minus119898119886 (119898120573)

120573minus1

4(1198982minus1205962

3) 119887 (119898120573)

(33)


120576 (119870) ≃ 1 +41198902

12058721198702(1 minus

1205962

1198962)


2119896ln 120596 + 119896120596 minus 119896

)(119888 (119898120573)

1205732minus119898119886 (119898120573)

120573)

minus1

4(21198982minus 1205962+111198962+ 37120596

2

72) 119887 (119898120573)

(34)


1

120583 (119870)

≃ 1 +21198902

120587211989621198702


1205962

1198962minus (1 +

1198962

1205962)(1 minus

1205962

1198962)120596

2119896ln 120596 + 119896120596 minus 119896

times (119888 (119898120573)

1205732minus119898119886 (119898120573)

120573)

minus1

8(61198982minus 1205962+1291205962minus 109119896

2

72) 119887 (119898120573)]

(35)


1198853 = 1 +1205721198792

61198982 (36)


120572 =1198902 (ℎ119888)41205871205980

(1 +1205721198792

61198982) =

12058301198902119888

2ℎ(1 +

1205721198792

61198982) (37)









1198853 = 1 +1205722119879

61198982

2

(38)


120572119877

= 120572 (119879 = 0)

+8120572

1205871198982[119898119886 (119898120573)

120573minus119888 (119898120573)

1205732

+119887 (119898120573)

4(1198982+1

31205962)]

+1205722

1198982[1198792

6+

infin

sum

119899119903119904=1

(minus1)119904+119903119890minus119904120573119864 119879

(119899 + 119904)

times 24119879

(119903 + 119904)minus 12119898

2

times [119890minus119898120573(119904+119903)

119898minus (119903 + 119904)

times 120573119864119894 minus119898120573 (119903 + 119904)]]

(39)










000200400600801

01201401601802

minus1 0 1 2 3 4 5T (m)

Elec

tron

mas

s

Two-loop

One-loop


00002000400060008

0010012001400160018

002

Elec

tron

char

ge

0 1 2 3 4 5T (m)

Two-loop

One-loop








02 0000306 minus713948119864 minus 11





5 028675 00001 minus94116119864 minus 05









References






























FluidsJournal of


Journal of







Advances in

Astronomy



Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119885minus12

119879lt119898997888997888997888rarr 1 +

120572

4120587(4 minus

3

120576) minus

120572

41205872(119868119860 minus

1198680

119864)

minus1205722

41205872(3 +

1

120576) 119868119860 +

212057221198792

312058721198982

(27)


119885minus12

119879gt119898997888997888997888rarr 1 minus 120572[

2119868119860

120587+

1

4120587(3

120576minus 4) +

41205871198792

3]

minus 1205722[

1

412058733

120576(119868119860 + 119869119860) + (3119868119860 + 5119869119860) minus

81198792

31198982

+1

8

infin

sum

119899119903119904=1

(minus1)119903119879

times 119890minus119903120573119864

[119891+ (119904 119903)120574p1198642V2

minus119868119861119868119862

641205872

+ ℎ (119901 120574)119891minus (119899 119903)119868119862

8120587minus 119891minus (119904 119903)

+ 119891+ (119899 119903)120574p1198642V2

119868119861

8120587

+ 2

119898ℎ (119901 120574) minus

120574p1198642V2

119891minus (119904 119903) ]]

(28)






ReΠ120573119871 (119896 0) =4120587120572

31198792+

1198962

21205872ln 119898


ReΠ120573119879 (119896 0) =2120572

31205871198962 ln 119898



119881 (119896) equiv 11989021198771205751205830 [

119906120583119906]

1198962 minus (41205871205723) 1198792 + (119896221205872) ln (119898119879)

+

119892120583] minus 119906120583119906]

1198962 minus (21205723120587) 1198962 ln (119898119879)]

119881 (119896) = 11989021198771205751205830Δ 120583]120575]0

(30)


119881 (119896) equiv 1198902119877 (1 +

2120572

3120587ln 119879

119898)[

11990620

1198962 + (412058712057211987923)+1 minus 11990620

1198962]

(31)


120572 (119879) = 120572 (119879 = 0) (1 +2120572

3120587ln 119879

119898)

= 120572 (119879 = 0) (1 + 155 times 10minus3 ln 119879

119898)

(32)





1198853 = 1 minus21198902

1205872

times 119888 (119898120573)

1205732minus119898119886 (119898120573)

120573minus1

4(1198982minus1205962

3) 119887 (119898120573)

(33)


120576 (119870) ≃ 1 +41198902

12058721198702(1 minus

1205962

1198962)


2119896ln 120596 + 119896120596 minus 119896

)(119888 (119898120573)

1205732minus119898119886 (119898120573)

120573)

minus1

4(21198982minus 1205962+111198962+ 37120596

2

72) 119887 (119898120573)

(34)


1

120583 (119870)

≃ 1 +21198902

120587211989621198702


1205962

1198962minus (1 +

1198962

1205962)(1 minus

1205962

1198962)120596

2119896ln 120596 + 119896120596 minus 119896

times (119888 (119898120573)

1205732minus119898119886 (119898120573)

120573)

minus1

8(61198982minus 1205962+1291205962minus 109119896

2

72) 119887 (119898120573)]

(35)


1198853 = 1 +1205721198792

61198982 (36)


120572 =1198902 (ℎ119888)41205871205980

(1 +1205721198792

61198982) =

12058301198902119888

2ℎ(1 +

1205721198792

61198982) (37)









1198853 = 1 +1205722119879

61198982

2

(38)


120572119877

= 120572 (119879 = 0)

+8120572

1205871198982[119898119886 (119898120573)

120573minus119888 (119898120573)

1205732

+119887 (119898120573)

4(1198982+1

31205962)]

+1205722

1198982[1198792

6+

infin

sum

119899119903119904=1

(minus1)119904+119903119890minus119904120573119864 119879

(119899 + 119904)

times 24119879

(119903 + 119904)minus 12119898

2

times [119890minus119898120573(119904+119903)

119898minus (119903 + 119904)

times 120573119864119894 minus119898120573 (119903 + 119904)]]

(39)










000200400600801

01201401601802

minus1 0 1 2 3 4 5T (m)

Elec

tron

mas

s

Two-loop

One-loop


00002000400060008

0010012001400160018

002

Elec

tron

char

ge

0 1 2 3 4 5T (m)

Two-loop

One-loop








02 0000306 minus713948119864 minus 11





5 028675 00001 minus94116119864 minus 05









References






























FluidsJournal of


Journal of







Advances in

Astronomy



Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1198853 = 1 minus21198902

1205872

times 119888 (119898120573)

1205732minus119898119886 (119898120573)

120573minus1

4(1198982minus1205962

3) 119887 (119898120573)

(33)


120576 (119870) ≃ 1 +41198902

12058721198702(1 minus

1205962

1198962)


2119896ln 120596 + 119896120596 minus 119896

)(119888 (119898120573)

1205732minus119898119886 (119898120573)

120573)

minus1

4(21198982minus 1205962+111198962+ 37120596

2

72) 119887 (119898120573)

(34)


1

120583 (119870)

≃ 1 +21198902

120587211989621198702


1205962

1198962minus (1 +

1198962

1205962)(1 minus

1205962

1198962)120596

2119896ln 120596 + 119896120596 minus 119896

times (119888 (119898120573)

1205732minus119898119886 (119898120573)

120573)

minus1

8(61198982minus 1205962+1291205962minus 109119896

2

72) 119887 (119898120573)]

(35)


1198853 = 1 +1205721198792

61198982 (36)


120572 =1198902 (ℎ119888)41205871205980

(1 +1205721198792

61198982) =

12058301198902119888

2ℎ(1 +

1205721198792

61198982) (37)









1198853 = 1 +1205722119879

61198982

2

(38)


120572119877

= 120572 (119879 = 0)

+8120572

1205871198982[119898119886 (119898120573)

120573minus119888 (119898120573)

1205732

+119887 (119898120573)

4(1198982+1

31205962)]

+1205722

1198982[1198792

6+

infin

sum

119899119903119904=1

(minus1)119904+119903119890minus119904120573119864 119879

(119899 + 119904)

times 24119879

(119903 + 119904)minus 12119898

2

times [119890minus119898120573(119904+119903)

119898minus (119903 + 119904)

times 120573119864119894 minus119898120573 (119903 + 119904)]]

(39)










000200400600801

01201401601802

minus1 0 1 2 3 4 5T (m)

Elec

tron

mas

s

Two-loop

One-loop


00002000400060008

0010012001400160018

002

Elec

tron

char

ge

0 1 2 3 4 5T (m)

Two-loop

One-loop








02 0000306 minus713948119864 minus 11





5 028675 00001 minus94116119864 minus 05









References






























FluidsJournal of


Journal of







Advances in

Astronomy



Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










000200400600801

01201401601802

minus1 0 1 2 3 4 5T (m)

Elec

tron

mas

s

Two-loop

One-loop


00002000400060008

0010012001400160018

002

Elec

tron

char

ge

0 1 2 3 4 5T (m)

Two-loop

One-loop








02 0000306 minus713948119864 minus 11





5 028675 00001 minus94116119864 minus 05









References






























FluidsJournal of


Journal of







Advances in

Astronomy



Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







02 0000306 minus713948119864 minus 11





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References






























FluidsJournal of


Journal of







Advances in

Astronomy



Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics























FluidsJournal of


Journal of







Advances in

Astronomy



Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Research Article Renormalization of QED Near Decoupling Temperature

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Transcript of Research Article Renormalization of QED Near Decoupling Temperature