Lidated within the high temperature regime (T0 = 2 -1 ) for different valuesLidated within

Lidated within the high temperature regime (T0 = 2 -1 ) for different values
Lidated within the high temperature regime (T0 = 2 -1 ) for many values of . The agreement at = 1 is great for both k = 0 and k = two, though deviations may be observed in the vicinity of your boundary when is decreased. Ultimately, panel (c) shows the SC as a function in the inverse from the distance to the boundary, (1 – 2r/ )-1 , at k = 0 and || 1 (0 and 0.9 are considered), represented making use of a logarithmic scale. It could be observed that as the temperature is improved, the validity of your asymptotic worth 1/4 two is extended to smaller distances from the boundary.0.03 1 T = 0.five, = 0 = 0.9 =1 1/4 two 0.02 0.1 T = two, = 0 = 0.9 =(b) ( (= /2) T0 = 2)0.SC0.SC0.(k (0.0.005 = 0) = /2) 0.2 0.four 0.(a)k = 0, = 0 = 0.9 =1 k = two, = 0 = 0.9 =3 SCan0 0 0.80.001 0 0.2 0.four 0.6 0.82r/2r/0.T0 = 1, = 0 T0 = 1, = 0.9 T0 = ten, =T0 = ten, = 0.9 T0 = 100, = 0 T0 = 100, = 0.9 1/40.0.SC0.015 three 0.01 0.( (c)= /2)(k= 0)0(1 – 2)-1 rFigure two. Profiles from the scalar condensate (SC) in the equatorial plane ( = /2). (a) Massless (k = 0) quanta at low (T0 = 0.5 -1 ) and higher (T0 = 2 -1 ) temperatures, for various values of . (b) High-temperature outcomes for k = 0 and k = 2 at a variety of . (c) Validity in the vanishing mass high-temperature lead to 20(S)-Hydroxycholesterol manufacturer Equation (115) as a function of the radial coordinate, as T0 and are increased. The black dotted lines represent the high-temperature lead to Equation (119), which reduces to Equation (115) when k = 0.We further validate the result in Equation (119) inside the limit of crucial rotation ( = 1), when the SC is continuous inside the equatorial plane. Figure three shows the SC as a function of temperature for numerous values of k. In panel (a), it may be seen that the numerical resultsSymmetry 2021, 13,23 ofapproach the asymptotic form in Equation (119) about T0 1. Panel (b) displays the difference SC – SCan , exactly where SCan refers towards the asymptotic GYY4137 Biological Activity expression in Equation (119). At T0 1, all curves give essentially zero, hence validating the contributions of order 0 O( T0 ). At T0 1, the asymptotic expression loses its validity because of the terms which are logarithmic in or independent of T0 . Because the SC vanishes as T0 0, these terms will dominate SC – SCan , as confirmed by the dotted black lines.1 k=0 0.25 1/ 3 0.75 2/SCan0.06 k=0 0.25 1/ three 1 2/-SCan0.3 (SC – SC ) an0.SC0.-0.( (a)= /2)(( (b)= /2)(= 1)-0.04 10-= 1)0.01 10-10-TTFigure three. (a) Log-log plot with the SC in the equatorial plane ( = /2) as a function of T0 for = 1 and a variety of values of k. (b) Linear-log plot of the distinction SC – SCan involving the numerical result and also the higher temperature analytical expression in Equation (119).The O( T 0 ) term appearing in Equation (119) could originate from the truth that we have only taken into account the thermal part of the expectation value of your SC. We now Had take into consideration its complete expectation worth by adding the vacuum contribution SCvac to the result in Equation (119). According to Ref. [42], Hadamard regularisation givesHad SCvac =1 41-=-1 4k – k2 k3 2k(1 – k2 ) (k) C – ln(Had two) six k 1 – k2 – k3 – 2k (1 – k2 ) (1 k ) C – ln(Had two), (120)exactly where Had is definitely an arbitrary (unspecified) regularisation mass scale and also the relation (A13) was employed to go from the 1st to the second line. Adding with each other the thermal as well as the vacuum contributions givesHad Had SCtotal SC SCvac =MT two M R – two M2 six 12lnHadTM R 32 – a2 two 24- O( T -1 ). (121) 0, we obtain(122)Imposing agreement using the Minkowski value (117) in the limit1 M SC Had = eC- two ,exactly where M = k -1 is the mass on the fi.