Phase transitions in the electroweak theory at finite temperature and density

PHYSICS LETTERS B

Volume 185, number 3,4

PHASE TRANSITIONS

IN THE ELECTROWEAK

AT FINITE TEMPERATURE

19 February 1987

THEORY

AND DENSITY

E J FERRER ‘, V DE LA INCERA’

and A E SHABAD

Lebedev Phyncal Institute, II 7924 Moscow, USSR Received 8 November 1986

It is shown that the critxal leptomc charge density necessary for the condensation of W-mesons 111the electroweak theory IS increased by heating the system The phase-transltlon diagrams between the dtiferent condensed phases (*gs- and W*-condensate) are found, and the relation between the crltlcal temperatures and the leptomc charge den&y are given m the high-temperature hmlt

In recent years the Bose-Einstein condensation (BE) m gauge theories at fimte temperature T and/or density, as well as Its lmpllcatlons to cosmology and astrophysics, has been mvestlgated m many papers [l--7]. In this context the BE of the Hlggs field m the electroweak model at lllgh temperature and fernuon density [3], or at high temperature but wth external charge, instead of the fernuon density [4], have been studed. Some remarks on the quantization of this kmd of system urlth external charge were made [5], and It was concluded that m this case an extra degree of freedom appears that cannot be physically interpreted. On the other hand, at T = 0 It was proved that when the density of fermlons m an electroweak system is larger than some crrtlcal value, the usual ground state (W*> = 0 IS unstable, and the system falls m a homogeneous condensate of vector particles (W’> # 0 [6] Nevertheless, the properties of the electroweak plasma at fimte temperature and density, with both condensates and their mutual mfluence taken mto account, remam unknown, and only recently this problem was attacked under the supposItion that the chermcal potential related to the neutral-weak charge 1s equal to zero [7], Here we consider the leptomc sector of the Wemberg-Salam model m the presence of a nonzero leptomc charge density and we study how the W- and Hlggs-condensates behave at fimte temperatures, From the equatlons obtained, It can be seen that for both condensates the system undergoes second-class phase transItIons at cntical temperatures that depend on the leptomc charge density The phase translhon curves, obtained formally for sufficiently high temperatures, can be extrapolated to lower temperatures down to T = 0. In our picture the chermcal potentials associated to the commutmg conserved charges (j+ to the electromag netic and ,uz to the neutral-weak one) are defined as functions of the temperature and fernuomc density through equations that descnbe the eqmhbrmm state Let us consider the effective lagrangan of the SU(2) X U(1) Wemberg-Salam model wlthout baryons m the Feynman gauge (CY= I, a’ = 1) and euchdean metric.

tp

hl($L@=R + “R@+$~)- U4 +m2P

+ (1/201)@,Q2

t (1/2~~‘)(a,B,)~

(aCIGaCtgeabc~~)arSc+ga2~,

(1)

’ On leave of absence from IMACC, Saence Academy of Cuba, Havana, Cuba 0370-2693187/$ (North-Holland

03 50 0 Elsevler Science Publishers Physics Pubhshmg Dlvlslon)

BV

407

Volume 185, number 3,4

PHYSICS LETTERS B

19 February 1987

where

Dla=~-llgwagT"a-llg'Bla, ~L = g a ( l + ~ ' 5 )

e

,

eR

(-D~=~ta--llgW~7"a +alg'B~ +ld3~4# , (1--~,5)e

~-a and ~ are the ghosts, the other notatmns are standard. The term P364, reside the covanant derlvatave ~ , m (1) originates from the term p3 N added to the harmltoman densaty, where N is the conserved leptomc charge densxty. 1-

N = e74 e + ~ v74(1 + 75)v

(2)

One should consader the e q m h b r m m equatmns

0 V~/O~ = O,

0 V~/ap3 = ] 0 ,

(3)

where V~ ~s the effecUve potential at fimte temperature T = ~ - 1 , ~ox stands for every average f e l d of the system and 3"0 as the average of the leptomc charge density Jo = which we shaU take, generally, different from zero Lmde [6] considered the problem ( 1 ) - ( 3 ) at zero temperature taking for V ~° the tree approxamation with respect to bosons and one.loop approxmaation ~ t h respect to ferrmons (filled Ferma-spheres). He found that, apart from the usual nonzero average <¢> = (o/0 ~), there appears a homogeneous condensate of W-bosons with a nonzero + >1 = ICt I, where~ xt /= 1,2, 3. SlmuRaneously, the average value of W43 and B 4 are also nonzero. average value I = - I ( U l - P 2 c°s20)/g,


(4)

where 0 as the Wemberg angle, we may see that after the shaft by , , the parameters Pl and P2 become coefficaents of the electric charge density Ne = e3 b c wc G 4by + ~- l [(D 4 ¢)? (r 3 + / ) ¢ - ¢? (r 3 + / ) D 4 ¢] - le 74 e

(5)

and the weak-neutral charge density

N w = e 3bc wbG~v cos20 -- x l [(D4¢)? (r 3 cos20 - I sin20)¢ - ~b+(r3 cos20 - I sin20)D4~b ] 1

--

-- ~ l~L74 (7-3 COS20 + 1 sln20) ~L -- 21-leR')'4 sin20 eR

(6)

m the hamlltoman, respectively Therefore Pl and/22 are the chermcal potentials for these charges Eqs (3) define the constants ~, (Cz)2, Pl, P2,/23 as functions o f J 0 and the temperature, if nonzero. Note, that eqs (3) with % = ( W3 >, imply that the average values of the charge densities (5), (6) are kept zero, as is needed for the thermodynamlcal eqmhbrmm. Our mare purpose is to obtmn the phase diagram for the system under conslderataon m the (9"0, T)-plane To thas end it is necessary to calculate the effecUve potential V~ [8] as a funcUon of C 2, ~2 and the chermcal potentials P l , / / 2 , P3. The condensates found within the tree approximation, at T = 0 [6], can be affected by temperature-dependent radiative corrections If we consider X >> e 4 , all the interesting effects are expected to take place at hagh temperature, when T>> me, mw, C 2, P l , P2, P3 In this case the leading contribution to Vt3 comes from the most divergent &agrams, thas means - apart from the T4-term that depends neither on the p's nor on C 2, ~2, and is hence of no interest as far as the phase transition is concerned - those m which all loops beyond the lowest order are quadratlcaUy divergent [9] Thas leading T2-contnbutmn to the effective potential is proportional to quadratic combinations of Pl,2,3, C, ~ and can be presented m the following form

408

Volume 185, number 3,4 Vf =

--

PHYSICS LETTERS B

19 February 1987

/m3\2n33 \ ' v 4 ' ~x44(0 ) _ (B4)2II44(0) -- Qb)2II(0)_ (wtl)2ii~tl(0)_ /23 2 II(p3 )(u3 )(0) - 2/23B4II(p 3 )(B4)(0),

(7) where IIuz,ab (0), IIuu (0), II (0) are the finite parts of the temperature-dependent polanzaUon operators for the fields Wa, Bp, ¢, respectwely, taken at zero values of/21,2,3, C, ~ m the zero-frequency k 4 = 0, zero-momentum k -+ 0 hmat, II(~t3 )(/.t3 ), II(g 3 )(B4 ) are the fimte part of the polanzatmn operators for the "/23-field" II(~t3)(p3 ) = _ ~(02Vl~/3p2),

II(p3)(B4) = 02VO/Op3 OB4 ,

taken m the same hnut The quantity IInaa (0) disappears due to gauge mvanance. Consequently, the W-boson condensate does not contribute to the leading T 2-behavmur, tins conclusion being approxamaUon-mdependent within the perturbaUve expansion. In the one-loop approxamatlon of the temperature Green functaons expansion the polanzataon operators for Yang-Mdls and abehan vector fields in the zero momentum and ingh temperature hnuts can be found in refs [10,11 ] The other terms in (7) can be easily calculated within the same approxlmatmn Finally, we obtain for the effective potentaal 1 2 V# = - (/21 -/22 c°s20) C2 + (e2C2/4 sin20 - g/22 + ~- X$2 - ~-m2) ~2

+ (5/12/32)(gl -/22 cos20) 2 - (5/12/32)(/21 +/22 sin20) 2 -/22/2/32

+ [(e2/sm220)(1 + 2 cos20) + 2~.] ~2/3/32 -- (2/3/32)(pl +/22 sin20) P 3 ,

(8)

where the first two terms are the tree approxamaUon and the T4-term is omitted. A phase translUon occurs when at least one of the condensates, ~ or C, vanishes for some temperature, and simultaneously a 2 Va/a$ 2 or a 2 Va/ac 2 turns to zero. Taking into account that Va depends only on $2 and C2 , we see that the phase translUon con&Uon lmphes that av~/a~ 2 = 0 for ~ = 0, or OVa/aC 2 = 0 for C = 0 Moreover, when one of the condensates, C or ~, is different from zero then the corresponding equahbnum equations ava/aa = o, o = ~ or C, reduces to ava/ao 2 = o. Therefore, in order to obtain the phase transmon we must start from the following set of equations, vnth either ~ or C equal to zero.

av#/a~2 = (e2/4 sin20 ) C 2 - g1 P22 + ½ X~2 - ½m2 + (1/3/32) [(e2/sm220)( 1 + 2 cos20) + 2X] = o ,

(9a)

~V~/8C2 = - ( P l -/'t2 c°s20) 2 + @2/4 sin20) ~2 = 0 ,

(9b)

~V#/OPl = -2(/21 -/22 c°s20) C2 + (5/6132)(pl -/22 c°s20) - (7/8/32)(/21 +/22 sin20) + ~J0 = 0 ,

(9c)

'OV[1/OP2 = 2(/21 -/22 c°s20) C2 c°s20 - (5/6/32)(/21 -/22 c°s20) c°s20 - 41-P2~ 2 2

+ g J0 sin20 - (7/18/32)(/21 + P2 sin20) sin20 = O.

(9d)

Here we have already used the equation for the leptomc charge eqmhbrmm

aV#/ala3 = -(2/3/32)(pl + P2 sin20) -/23//32 =J0 .

(10)

In these equations the electron masses are neglected throughout Note that (9c) and (9d) express the electrical and neutral-weak neutrahty of the system, respectwely. Eqs (9) define, at ~ = 0 or C = 0 cnttcal temperatures as functions of J0, T = Te(J0), e g the phase transition curves The phase transition curve, found as a solution to eqs. (9) with C = 0, ~ :/= 0 separates the Hlggs- and W-condensed phase C 4= 0, ~ ¢ 0 from the Hlggs-condensed phase C = 0, ~ =~ 0 (the curve "a" in fig. 1) One can find 409

Volume 185, number 3,4

PHYSICS LETTERS B

C:~o

/a,

19 February 1987

b

JL

.1-~ Fig 1 Phase dmgmms for W-+-and Hlggs-condensates The curve "a" (C = 0, ~ ~ 0) marks the boundary of the W-condensed phase, wlule the curve "b" (C = ~ = 0) separates the phases with and without the Hlggs condensate JL = (1/6~r2)(e ~/2 sm 0) 3 is the crltlcal density at T = 0 found by Lmde [6] The point T2c = ~-m2/R is the usual ermcal temperature m the model without leptomc charge H ere cur yes "a" and "b" behave a sympt o tlcally as J~ ~ 1 81 R T~cand 4 ~ 0 9 R Tc6,respeetlvely that it does not lie in the region j 2 < 5 18a 2 T 6, and for large J0 >> 1 it is gtven b y the asymptotic equation Tc6 0 55j2/R. In the above formulae a =~- e sm 0, a n d R = (e2/sm220)(1 + 2 cos20) + 2X These results are rehable in the region of Ingh temperatures F o r lower temperatures the phase transltton curve should be extrapolated to the Llnde point [6] T = O, Jo =JL = (1/6n2)(e~/2 sm 0) 3 separating the same phases at zero temperature The other phase transition curve, that IS a solution o f e q s ( 9 a ) - ( 9 c ) with ~ = C = 0, is the boundary between the Hlggs-con densate C = 0, ~ 4 : 0 and the fully s y m m e m c phase C = g = 0 Tins curve is marked by " b " in fig 1. It starts at the point Jo = O, T c2 = 3~ m 2 /R, which is the well-known symmetry restoration temperature for J0 = 0, and behaves asymptoUcally as 1;6 ~ 1.10j2/R for large T and J0. The phase transition curve " b ' hes entarely m the region o f high temperatures and is therefore rehable m the sense that it is consistent with the high-temperature approyamation adopted above One can show that in between the curves " a " and " b " the phase C -- 0, ~ 4 : 0 IS stable. There are also soluUons o f eqs. (9) with C v~ 0, ~ = 0 In them C remains arbitrary, since eqs (9c) and (gd) be, come dependent The corresponchng farmly of curves would he in fig 1 between the curves " a " and " b " These curves do not, however, describe the phase transltaon (~ :# 0) -+ (~ = 0) since they relate to the case C v~ 0, whereas the state we deal with between the curves " a " and " b " is characterized b y the value C = 0. Observe that the W-condensate which appears at T = 0 for J0 > JL tends to &sappear with the increase o f temperature This fact can be easily understood ff we remember [6] that at fixed J0 > JL = (1/6n2)( e~/2 sin 0) 3 an electric charge appears In the ferrnlonlc part of the system, winch only can be compensated by a nonzero electrically charged condensate, i e., by a W-condensate When T is increased many charged particles can be excited and at some sufficiently large T (which depends on the fixed value of the leptonlc charge density J 0 ) they are so many that the neutrahty is reached w~th C = 0 Another mteresUng fact described in fig 1 is linked to the neutrality with respect to the neutral-weak charge When J0 :/: 0 (even if J0 < J L ) there exists some nonzero neutral-weak charge density belonging to the fernnonic sector The thermodynanucal equlhbnum also reqtares that tins charge be compensated and b y this reason we must consider eq (9d) v, Ith/~2 to be determined from the system of all equations The compensating charge is concentrated at zero temperature In the g and C condensates When the temperature Increases the C-condensate evaporates first, as we chscussed above. After that the bosomc component of the weak-neutral charge is concentrated in the G-condensate and m the overcondensate gas Finally, for higher temperatures tins condensate also evaporates, the symmetry is completely retained and the ferrmomc and bosomc components of the gas possess mutually compensating neutral-weak charges 410

Volume 185, number 3,4

PHYSICS LETTERS B

19 February 1987

Our phase diagrams are not m agreement with those of ref [7] In tins paper the authors have found that an increment m the temperature gives rise to the emergence of the W-condensate, at leptomc charge density lower than JL Tins is a chrect consequence of the fact that they set/.t2 = 0 instead of solving eq. (9d). Tins consideration is eqtuvalent to dlsregarchng the conservation of the average neutral-weak charge (9d), and it may be related with a noneqtnhbrlum s~tuatlon The results of our paper nught have some interest for cosmology and astrophysics, where the external conchtmns of Ingh temperature and ferrmomc density could be reahzed. We are indebted to Professor E S Fradkm and Dr A.D. Lmde for many suggestions and helpful dascusslons We also should ltke to thank Dr O.K. Kalashmkov and Dr. H Perez Rojas for comments

References [1] D A Ktrzhmts and A D Lmde, Phys Lett B 42 (1972) 471, Ann Phys 101 (1976) 195 [2] H E Haber and H A Weldon, Phys Rev D 25 (1982) 502, V A Rubakov and A N Tavkhehdze, Phys Lett B 165 (1985) 109, V A Rubakov, Progr Theor Phys 75 (1986)366 [3] A D Lmde, Phys Rev D 14 (1976) 3345, Rep Progr Phys 42 (1979) 389 [4] J I Kapusta, Phys Rev D 24 (1981) 426 [5 ] V de la Incera, E J Ferret and A E Shabad, m Group theoretical methods in physics, Proc 3rd Seminar (Yurmala), eds M A Markov (Nauka, Moscow, 1985), E J Ferret and V de la Incera, Rev Cubana de Fls IV (1984) 51, E J Ferret, V de la Incera and A E Shabad, in prepaxatlon [6] A D Lmde, Phys Lett B86 (1979)39, I V Knve, Yad Flz 31 (1980)1279 [7] O K Kalastuukov and H Perez Rolas, Kxatk Soob po FlZ 2 (1986) 23 [English translation Soy Phys Lebedev Inst Reports, AUerton Press] [8] L Dolan and R Jacklw, Phys Rev D 9 (1974)3320 [9] S Wemberg Phvs Rev D 9 (1974)3357 [10] O K Kalashnlkov and V V Khmov, Yad Flz 31 (1980) 1357, 33 (1981) 848 [11] E S Fradkm, Proc Lebedev Phys Inst 29 (1975) 7

411