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KEK-TH-870

On the thermal sunset diagram for scalar ?eld theories

Tetsuo NISHIKAWA,? Osamu MORIMATSU,? and Yoshimasa HIDAKA? Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization, 1-1, Ooho, Tsukuba, Ibaraki, 305-0801, Japan

(Dated: February 1, 2008)

arXiv:hep-ph/0302098v1 13 Feb 2003

1

Abstract

We study the so-called “sunset diagram”, which is one of two-loop self-energy diagrams, for scalar ?eld theories at ?nite temperature. For this purpose, we ?rst ?nd the complete expression of the bubble diagram, the one-loop subdiagram of the sunset diagram, for arbitrary momentum. The temperature-dependent discontinuous part as well as the well-known temperature independent part is obtained analytically. The continuous part is reduced to a one-dimensional integral which one can easily evaluate numerically. We calculate the temperature independent part and dependent part of the sunset diagram separately. For the former, we obtain the discontinuous part ?rst and the ?nite continuous part next using a twice-subtracted dispersion relation. For the latter, we express it as a one-dimensional integral in terms of the bubble diagram. We also study the structure of the discontinuous part of the sunset diagram. Physical processes, which are responsible for it, are identi?ed. Processes due to the scattering with particles in the heat bath exist only at ?nite temperature and generate discontinuity for arbitrary momentum, which is a remarkable feature of the two-loop diagrams at ?nite temperature. As an application of our result, we study the e?ect of the diagram on the spectral function of the sigma meson at ?nite temperature in the linear sigma model, which was obtained at one-loop order previously. At high temperature where the decay σ → ππ is forbidden, sigma acquires a ?nite width of the order of 10 MeV while within the one-loop calculation its width vanishes. At low temperature, the spectrum does not deviate much from that at one-loop order. Possible consequences with including other two-loop diagrams are discussed.

PACS numbers: 11.10.Wx, 12.40.-y, 14.40.Aq, 14.40.Cs, 25.75.-q Keywords: sigma meson, two-loop self-energy, spectral function, ?nite temperature, linear sigma model

? ? ?

Electronic address: nishi@post.kek.jp Electronic address: osamu.morimatsu@kek.jp Electronic address: hidaka@post.kek.jp

2

I.

INTRODUCTION

The so-called sunset diagram, which is depicted in Fig.1, is one of two-loop self-energy sigma model, φ4 theory, and so on. diagrams. The sunset diagram appears in theories with 4-point vertices such as O(4) linear At ?nite temperature, two-loop self-energy diagrams have remarkable features that are not seen at zero temperature. Consider the discontinuity of the self-energy. At zero temperature, the discontinuity of one-loop diagrams is due to two particle intermediate states and that of two-loop diagrams comes from three particle states in addition to two particle ones. In general, in higher loop diagrams new processes appear. However, they contribute to the discontinuity only at higher energies. On the other hand, at ?nite temperature, even though the number of particles participating in the process at a given order of loops is the same as that at zero temperature, new processes appear even at low energy. This is because at ?nite temperature some of particles participating in the process can be particles in the heat bath. Furthermore, as will be shown in this paper, there exist processes which are possible at arbitrary energy. Accordingly, the discontinuity of two-loop self-energies is non-vanishing in all the energy region. Therefore, in some cases at ?nite temperature, to extend calculations to two-loop order has a meaning more than just making more precision. One of such cases is the spectral function for the sigma meson in the linear sigma model. The spectral function of the sigma meson at ?nite temperature was studied by Chiku and Hatsuda at one-loop level [1]. The sigma meson at zero temperature has a large width due to the strong coupling with two pions. However, they found that at ?nite temperature the spectral function near σ → ππ threshold is enhanced as a typical signal of chiral phase transition. This is because, as the temperature increases, the mass of σ decreases while that

? ? ?

FIG. 1: “Sunset” diagram. We label the external particle by “Φ” and internal particles with mass mi by φi .

?

¨

3

of π increases due to partial restoration of chiral symmetry and, accordingly, the phase space available for the σ → ππ decay is squeezed to zero. At ?nite temperature, however, there exist processes, collision with or absorption of a thermal particle in the heat bath, which contributes to the discontinuity at arbitrary energy for σ , as discussed above. If we include them, the structure of the spectral function might be signi?cantly modi?ed. Their e?ects on the spectral function cannot be taken into account till we extend the calculation including two-loop self-energies. In this paper, we take a step forward in the calculation of two-loop self-energy diagrams, i.e. we evaluate the thermal sunset diagram for scalar ?eld theories. After giving a brief review of the real time formalism [2-14] in the second section, which we use for the calculation of diagrams, we ?rst examine the bubble diagram, a one-loop diagram which appears as a subdiagram of the sunset diagram, at ?nite temperature in the third section. Then, we discuss the structure of the discontinuous part of the thermal sunset diagram in the fourth section and explain how we calculate the sunset diagram in the ?fth section. Using our result for this diagram, we study the e?ects of the diagram on the spectral function of σ at ?nite temperature in O(4) linear sigma model in the sixth section. Finally we summarize the paper in the seventh section.

II.

BRIEF REVIEW OF THE REAL TIME FORMALISM

For the calculations of thermal Feynman diagrams we adopt the real time formalism throughout this paper. We brie?y review the formalism in this section. In the real time formalism propagators are given by 2 × 2 matrices. The 4-components of the free propagator of a scalar particle with mass m are given by [14] i + n(k0 )2πδ (k 2 ? m2 ), ? m2 + iη σk0 i?F [n(k0 ) + θ(?k0 )]2πδ (k 2 ? m2 ), 12 (k ; m) = e i?F 11 (k ; m) = k2

?σk0 i?F [n(k0 ) + θ(k0 )]2πδ (k 2 ? m2 ), 21 (k ; m) = e ?i + n(k0 )2πδ (k 2 ? m2 ), i?F 22 (k ; m) = k 2 ? m2 ? iη

(II.1) (II.2) (II.3) (II.4)

where n(k0 ) is the Bose-Einstein distribution function at temperature T ≡ 1/β : n(k0 ) = 1 . exp(β |k0 |) ? 1 4 (II.5)

O?-diagonal elements depend on a free parameter σ . In this paper we make the symmetrical choice σ = β/2, leading to

F β |k0 |/2 i?F n(k0 )2πδ (k 2 ? m2 ). 12 (k ; m) = i?21 (k ; m) = e

(II.6)

Each component is the solution of the Schwinger-Dyson equation:

F ?ab (k ; m) = ?F ab (k ; m) + ?ac (k ; m)Πcd (k )?db (k ; m),

We denote the matrix of the full propagator by ? ? ?11 (k ; m) ?12 (k ; m) ? k ; m) = ? ?. ?( ?21 (k ; m) ?22 (k ; m)

(II.7)

(II.8)

where Πcd (k ) is the self-energy. The following matrix U (k ) = ? ? 1 + n(k0 ) n(k0 ) n(k0 ) 1 + n(k0 ) ? ?, (II.9)

‘diagonalizes’ the free propagator: ? ? ? ? F F F ?11 (k ; m) ?12 (k ; m) ? (k ; m) 0 ? U (k )? 1 = ? 0 ?, U (k )? 1 ? F F F ? ?21 (k ; m) ?22 (k ; m) 0 ??0 (k ; m) where ?F 0 (k ; m) is the Feynman propagator at T = 0 ?F 0 (k ; m) = k2 1 . ? m2 + iη

(II.10)

(II.11)

It is known that U (k ) also diagonalizes the full propagator [14] as well as Πab (k ) ? ? ? Π(k ) 0 ? (U (k )?1 )db . Πab (k ) = (U (k )?1 )ac ? ? ? 0 ?Π(k )

cd

(II.12)

This matrix equation gives relations between matrix elements: ? k ) = ReΠ11 (k ), ReΠ(

(II.13) (II.14) (II.15)

? k ) = tanh(β |k0|/2)ImΠ11 (k ), ImΠ( ? k ) = i sinh(β |k0|/2)Π12 (k ). ImΠ( 5

Let us next ?nd the expression of the spectral function. ?11 has the following spectral representation: ?11 (k ; m) = dω 2 ρ(ω, k)?F 11 (k0 , ω ), √ (II.16)

where we denote the argument of ?F 11 by k0 and ω ≡ i (2π )4

k2 + m2 instead of k and m. One

can prove that, if ?11 is given in Eq.(II.16), the Green function ?(t, x) = d4 keik·x ?11 (k ; m) (II.17)

obeys the Kubo-Martin-Schwinger (KMS) condition [15]. From Eq.(II.16) we obtain 1 ρ(k ) = ? tanh(β |k0 |/2)Im?11 (k ; m). π (II.18)

Our next task is thus to ?nd the expression of Im?11 (k ; m). Denoting the diagonalized full propagator by ? k ; m)U (k )?1 = ? U (k )?1 ?( ? ?(k ; m) 0 0 ??(k ; m)

?

?

?(k ; m) 0

0 ??(k ; m)?

?

we obtain ? = U (k ) ? ? = ? ?

?,

(II.19)

?

[1 + n(k0 )]?(k ; m) ? n(k0 )?(k ; m)

? U (k )

? ?

n(k0 )[1 + n(k0 )][?(k ; m) ? ?(k ; m) ]

?

?

n(k0 )[1 + n(k0 )][?(k ; m) ? ?(k ; m) ] ?[1 + n(k0 )]?(k ; m) + n(k0 )?(k ; m) (II.20)

?

?.

Taking the (1, 1)-components in the both sides of the above equation yields Im?11 (k ; m) = coth(β |k0 |/2)Im?(k ; m). (II.21)

The expression for ?(k ; m) can be found in the following manner. In Eq.(II.8), iteratively using Eq.(II.8) itself in the right hand side and diagonalizing, we obtain ? ? F ? (k ; m) 0 ? k ; m)U (k )?1 = ? 0 ? U (k )?1 ?( ? 0 ??F ( k ; m ) 0 ? ?? ?? ? F F ? ?0 (k ; m) 0 Π(k ) 0 ? (k ; m) 0 ?? ?? 0 ? + ···. +? ? ? F ? ? 0 ??F ( k ; m ) 0 ? Π( k ) 0 ? ? ( k ; m ) 0 0 (II.22) 6

The (1,1) component of this equation gives ? k ; m)U (k )?1 )11 ?(k ; m) = (U (k )?1 ?( F F ? = ?F 0 (k ; m) + ?0 (k ; m)Π(k )?0 (k ; m) + · · · 1 = 2 ? k) . k ? m2 ? Π(

(II.23)

Therefore, from Eqs.(II.18), (II.21) and (II.23), we obtain the desired expression of the ? k) ImΠ( 1 (II.24) ? k )]2 + [ImΠ( ? k )]2 . π [k 2 ? m2 ? ReΠ( Thus the spectral function can be written only with one component of the self-energies while ρ(k ) = ? all the components enter ?11 (k ; m) (see Eq.(II.8)). spectral function:

III.

BUBBLE DIAGRAM

In this section we examine the “bubble” diagram, which is a one-loop diagram shown in Fig.2. The bubble diagram appears as a subdiagram of the sunset diagram and therefore we need the expression of the former in the calculation of the latter.

φ1

φ2

FIG. 2: “Bubble” diagram.

Among the 4-components of the bubble diagram we need only the (1,1) component since the sunset diagram does not have any internal vertices. The (1,1) component of the bubble diagram for scalar particles is given by Ibub (p; m1 , m2 )11 = factors, n: Ibub (p; m1 , m2 )11 = I (2) (p2 ; m1 , m2 ) + (F (2) (p; m1 , m2 ) + (1 ? 2)) + F (3) (p; m1 , m2 ). (III.26) 7 d4 k i?F (p + k ; m1 )i?F 11 (k ; m2 ). (2π )4 11 (III.25)

Eq.(III.25) can be expressed as the sum of terms with di?erent numbers of Bose-Einstein

Each term in the right hand side is respectively given by I (2) (p2 ; m1 , m2 ) = F (2) (p; m1 , m2 ) = F (3) (p; m1 , m2 ) = d4 k i i , (III.27) 2 4 2 2 (2π ) (p + k ) ? m1 + iη k ? m2 2 + iη d4 k i n(k0 )2πδ (k 2 ? m2 (III.28) 2 ), 4 2 (2π ) (p + k ) ? m2 + iη 1 d4 k 2 2 n(p0 + k0 )2πδ ((p + k )2 ? m2 1 )n(k0 )2πδ (k ? m2 ). (III.29) (2π )4

Let us carry out the integration in the above equations. The T -independent part I (2) is given in textbooks [16]. In d-dimension it is given by I (2) (p2 ; m1 , m2 ) = i 16π 2 1 m2 m2 1 2 ? + x+ ln 2 ? x? ln 2 ?2?I , ? ? κ κ

1 ? ?

(III.30)

where κ is the renormalization point and 1 1 ≡ ? γ + ln4π ? ? ?

and x± are γ : Euler constant), (III.31)

(? = (4 ? d)/2,

1 m2 ? m2 x± ≡ ± + 1 2 2 . 2 2p I is given by ?√ √ √ (x+ ? C )(x? + C ) ? √ √ C ln + i2π ? ? (x? ? C )(x+ + C ) ? √ x? x+ I = ? arctan √ ?2 D arctan √ D D ? √ √ ? √ ? ( C ? x )( C + x ) + ? ? C ln √ √ ( C ?x )( C +x )

? +

(III.32)

for p2 > (m1 + m2 )2 , for (m1 ? m2 )2 < p2 < (m1 + m2 )2 , for p2 < (m1 ? m2 )2 , (III.33)

where C and D are C ≡ (m1 + m2 )2 1 1? 4 p2 1? (m1 ? m2 )2 , p2 D ≡ ?C. (III.34)

We see from Eqs.(III.27) and (III.33) that the discontinuous part, ImiI (2) , is non-vanishing for p2 > (m1 + m2 )2 . The divergent part in Eq.(III.27) is renormalized by applying a counter term to the Lagrangian. The ?nite part is determined in such a way that the resultant self-energy satis?es a proper normalization condition. Let us next turn to the T -dependent part, F (2) and F (3) . We ?rst discuss their discontinuous parts. As shown in Appendix A the discontinuous part of F (2) (p; m1 , m2 ) is given analytically as follows: 8

1. For p2 > (m1 + m2 )2 or 0 < p2 < (m1 ? m2 )2 , ImiF (2) (p; m1 , m2 ) = 2. For (m1 ? m2 )2 < p2 < (m1 + m2 )2 , ImiF (2) (p; m1 , m2 ) = 0. 3. For p2 < 0, ImiF (2) (p; m1 , m2 ) = Here, 1 ω± = 2

2 m2 2 ? m1 1+ p2 2

1 1 1 ? e?βω+ . ln 16π |p| β 1 ? e?βω?

(III.35)

(III.36)

1 ?1 ln (1 ? e?βω+ )(1 ? e?βω? ) . 16π |p| β

(III.37)

| p0 | ±

1?

(m2 + m1 )2 p2

1?

(m1 ? m2 )2 |p| . p2

(III.38)

F (3) has only discontinuous part. Its calculation can be done in a way similar to that for ImiF (2) (p; m1 , m2 ). We show the ?nal results: 1. For p2 > (m1 + m2 )2 , F (3) (p; m1 , m2 ) = 1 1 1 ? e?βω+ eβ (|p0 |?ω? ) ? 1 ln . 8π |p|β eβ |p0 | ? 1 1 ? e?βω? eβ (|p0 |?ω+ ) ? 1 (III.39)

2. For (m1 ? m2 )2 < p2 < (m1 + m2 )2 , F (3) (p; m1 , m2 ) = 0. 3. For 0 < p2 < (m1 ? m2 )2 , F (3) (p; m1 , m2 ) = 1 1 · β |p0 | 8π |p|β e ?1 ?βω+ 1 ? e?β (|p0 |+ω+ ) 1?e β |p0 | ? e ln × ln 1 ? e?βω? 1 ? e?β (|p0 |+ω? ) (III.40)

.

(III.41)

4. For p2 < 0, F (3) (p; m1 , m2 ) = 1 1 ?ln|1 ? e?βω+ | + e?β |p0 | ln|1 ? e?β (ω+ ?|p0 |) | ? β | p 0| ? 1 8π |p|β e 1 ?ln|1 ? e?βω? | + eβ |p0 | ln|1 ? e?β (ω? +|p0 |) | . + β |p0 | e ?1 (III.42) 9

In the above equations ω± are given by Eq.(III.38) with m1 and m2 replaced by max{m1 , m2 } and min{m1 , m2 }, respectively, since F (3) (p; m1 , m2 ) is symmetric with respect to m1 and m2 by de?nition. It should be noted that the T -dependent part has two cuts in the complex p2 plane: one starts from p2 = (m1 + m2 )2 to the right along the real axis and the other from p2 = (m1 ? m2 )2 to the left. Let us next discuss the continuous part of F (2) , which is ReiF (2) (p; m1 , m2 ) = ? 1 d4 k n(k0 )2πδ (k 2 ? m2 P 2 ), 4 (2π ) (p + k )2 ? m2 1 (III.43)

where P stands for the prescription of Cauchy’s principal value. After integration over k0 and angle, we can express Eq.(III.43) as a one-dimensional integral: ReiF (2) (p; m1 , m2 ) ∞ 1 = dωn(ω ) 16π 2 |p| m2 2 2 2 2 2 2 2 (p2 + 2p0 ω ? 2|p| ω 2 ? m2 2 + m2 ? m1 )(p ? 2p0 ω ? 2|p| ω ? m2 + m2 ? m1 ) × P ln , 2 2 2 2 2 2 2 (p2 + 2p0 ω + 2|p| ω 2 ? m2 2 + m2 ? m1 )(p ? 2p0 ω + 2|p| ω ? m2 + m2 ? m1 ) (III.44) whose integration will be carried out numerically.

IV.

STRUCTURE OF THE DISCONTINUOUS PART OF THE SUNSET DIA-

GRAM

Before proceeding to the calculation of the thermal sunset diagram, we study the structure of the discontinuous part of the diagram. In ref.[18], Weldon analyzed the discontinuous part of the bubble diagram. We will generalize it for the sunset diagram. For this purpose it is convenient to use Eq.(II.15) [17], namely ?sun (k ; m1 , m2 , m3 ) = i sinh(β |k0|/2)iIsun (k ; m1 , m2 , m3 )12 . ImiI (IV.45)

?sun (k ; m1 , m2 , m3 ) is the (1, 1) component of the diagonalized self-energy matrix for Here I the sunset diagram. We note that the self-energy which actually enters spectral functions is diagonalized one (see Eq.(II.24)). Isun (k ; m1 , m2 , m3 )12 is the (1, 2) component of the sunset diagram, which is given by the following integral: iIsun (k ; m1 , m2 , m3 )12 = d4 p i?F (p; m1 )iIbub (k ? p; m2 , m3 )12 . (2π )4 12 10 (IV.46)

For i?F 12 (p; m1 ) it is convenient to use another form:

βp0 /2 i?F f (p0 )?(p0 )2πδ (p2 ? m2 12 (p; m1 ) = e 1 ), 1 f (p0 ) = βp0 . e ?1

(IV.47)

iIbub (p; m1 , m2 )12 denotes the (1, 2) component of the bubble diagram: iIbub (k ? p; m2 , m3 )12 = i One can reduce this equation to iIbub (p; m1 , m2 )12 d4 q i?F (q ; m2 )i?F 12 (k ? p ? q ; m3 ). (2π )4 12 (IV.48)

= ieβ (k0 ?p0 )/2 f (k0 ? p0 ) d4 q 1 × {(1 + n2 + n3 ) [δ (k0 ? p0 ? E2 ? E3 ) ? δ (k0 ? p0 ? E2 + E3 )] 2 (2π ) 4E2 E3 ?(n2 ? n3 ) [δ (k0 ? p0 ? E2 + E3 ) ? δ (k0 ? p0 + E2 ? E3 )]} , (IV.49) where E2 = |q|2 + m2 2 and E3 = |k ? p ? q|2 + m2 3 . We also de?ne E1 = |p|2 + m2 1

for later use. ni is the Bose-Einstein factor de?ned by ni = n(Ei ). Using Eq.(IV.45) and Eq.(IV.46) in which Eq.(IV.49) is substituted yields ?sun (k ; m1 , m2 , m3 ) ImiI d3 p d3 q 1 = ?π?(k0 ) (2π )3 (2π )3 8E1 E2 E3 × {((1 + n1 )(1 + n2 )(1 + n3 ) ? n1 n2 n3 )δ (k0 ? E1 ? E2 ? E3 ) +(n1 n2 n3 ? (1 + n1 )(1 + n2 )(1 + n3 ))δ (k0 + E1 + E2 + E3 ) +[(n2 n3 (1 + n1 ) ? n1 (1 + n2 )(1 + n3 ))δ (k0 ? E1 + E2 + E3 ) +(n1 (1 + n2 )(1 + n3 ) ? n2 n3 (1 + n1 ))δ (k0 + E1 ? E2 ? E3 ) + (permutations)]} . (IV.50) In this equation “permutations” stands for the terms obtained by permuting the particle labels of the third and the forth terms. Let us now consider the physical content of Eq.(IV.50). The ?rst term in Eq.(IV.50) may be interpreted as the probability for the decay Φ → φ1 φ2 φ3 with the statistical weight (1 + n1 )(1 + n2 )(1 + n3 ) for stimulated emission minus the probability for the creation φ1 φ2 φ3 → Φ with the weight n1 n2 n3 for absorption. The second term is the anti-particle ?2 φ ?3 → φ1 with counter part of the ?rst term. The third term represents the probability for Φφ ?2 φ ?3 with the weight n1 (1 + n2 )(1 + n3 ). Here the weight n2 n3 (1 + n1 ) minus that for φ1 → Φφ 11

?1 with the weight n2 n3 (1 + n1 ). All processes are shown in Fig.3. minus that for φ2 φ3 → Φφ

?i stands for the anti-particle of φi . The forth term is the anti-particle counter part of the φ ?1 → φ2 φ3 with the weight n1 (1 + n2 )(1 + n3 ) third term. It represents the probability for Φφ We next ?nd the region of k 2 where the physical processes contained in Eq.(IV.50) are

possible, which is equivalent to looking for the condition under which the integral over q in Eq.(IV.50) survives. For the ?rst and the second terms in Eq.(IV.50) to be non-vanishing, k 2 must satis?es the condition k 2 > (m1 + M23 )2 , where M23 is the invariant mass of φ2 and φ3 . Therefore, the processes in Fig.3 (a), (b), (c) and (d) are possible for k 2 > (m1 + m2 + m3 )2 since M23 > m2 + m3 . The third and the forth terms survive when k 2 < (m1 ? M23 )2 . The processes in Fig.3 (e), (f), (g) and (h), therefore, take place at arbitrary k 2 . This is

reasonable since the processes in Fig.3 (g) and (h) are scattering ones and since those in Fig.3 (e) and (f) can be also regarded as scattering by interpreting the incoming Φ in (e) as an outgoing anti-Φ and by doing the outgoing Φ in (f) as an incoming anti-Φ. Accordingly, the discontinuous part of the sunset diagram is non-vanishing for arbitrary k 2 , which is a remarkable feature of the thermal self-energy at and beyond two-loop order.

V.

CALCULATION OF THE THERMAL SUNSET DIAGRAM

In this section, we explain how to calculate the thermal sunset diagram by reducing it to an expression written in terms of the bubble diagram previously obtained. The (1,1) component of the sunset diagram shown in Fig.1 is given by Isun (k ; m1 , m2 , m3 )11 = d4 p i?F 11 (p; m1 ) 4 (2π ) d4 q F i?F 11 (q ; m2 )i?11 (k ? p ? q ; m3 ). (V.51) 4 (2π )

We decompose Eq.(V.51) into terms without and with Bose-Einstein factors, which we

vac 2 th denote by Isun (k ; m1 , m2 , m3 )11 and Isun (k ; m1 , m2 , m3 )11 , respectively: vac 2 th (k ; m1 , m2 , m3 )11 + Isun (k ; m1 , m2 , m3 )11 . Isun (k ; m1 , m2 , m3 )11 = Isun

(V.52)

They are given by

vac 2 Isun (k ; m1 , m2 , m3 )11 = th Isun (k ; m1 , m2 , m3 )11

i d4 p I (2) ((k ? p)2 ; m2 , m3 ), (V.53) 4 2 (2π ) p ? m2 + iη 1 i d4 p = (2π )4 p2 ? m2 1 + iη (2) × F (k ? p; m2 , m3 ) + (2 ? 3) + F (3) (k ? p; m2 , m3 ) 12

+

d4 p (2) n(p0 )2πδ (p2 ? m2 ((k ? p)2 ; m2 , m3 ) 1) I (2π )4 + F (2) (k ? p; m2 , m3 ) + (2 ? 3) + F (3) (k ? p; m2 , m3 ) . (V.54)

Here we have expressed the second integral in Eq.(V.51) in terms of I (2) , F (2) and F (3) . The T -independent part, Eq.(V.53), has a subdivergence coming from the nested bubble diagram and a two-loop overall divergence. On the other hand, T -dependent part, Eq.(V.54), has only a subdivergence, which can be removed by carrying out renormalization at one-loop level. Hereafter, I (2) expresses that with the divergence removed. We calculate Eqs.(V.53) and (V.54) separately.

A.

T -independent part

The T -independent part of sunset type diagrams has been calculated by several authors [19-22] so far. In this paper we calculate its ?nite part using a dispersion relation.

vac 2 We ?rst ?nd the discontinuous part of Isun (k ; m1 , m2 , m3 )11 and then compute the ?nite

continuous part using the obtained discontinuous part, via the twice-subtracted dispersion relation. dispersion relation for iI (2) ((k ? p)2 ; m2 , m3 ): iI (2) ((k ? p)2 ; m2 , m3 ) = into Eq.(V.53). Then, we obtain

vac 2 Isun (k ; m1 , m2 , m3 )11 = vac 2 In order to calculate the discontinuous part of Isun (k ; m1 , m2 , m3 )11 , we substitute a

1 π

∞ (m2 +m3 )2

dM 2

ImiI (2) (M 2 ; m2 , m3 ) , M 2 ? (k ? p)2 ? iη

(V.55)

1 π ×

∞ (m2 +m3 4 )2

dM 2 ImiI (2) (M 2 ; m2 , m3 )

=

1 π

∞

dp i i 2 4 2 2 (2π ) p ? m1 + iη (k ? p) ? M 2 + iη

)2

dM 2 ImiI (2) (M 2 ; m2 , m3 )I (2) (k 2 ; m1 , M ). (V.56)

(m2 +m3

We take the discontinuous part of this equation:

vac 2 ImiIsun (k ; m1 , m2 , m3 )11

1 = π

∞ (m2 +m3 )2

dM 2 ImiI (2) (M 2 ; m2 , m3 )ImiI (2) (k 2 ; m1 , M ). (V.57) 13

Using the expression for ImiI (2) (p2 ; m, m2 ) we can easily evaluate Eq.(V.57) numerically. Next we turn to the continuous part. The continuous part of Eq.(V.53) is divergent. Therefore one needs corresponding counter terms in the Lagrangian. The second and third diagrams of Fig.4 appear at two-loop order, which cancel the divergences in the bare sunset diagram, the ?rst diagram of Fig.4. In this paper, we do not explicitly go through the renormalization procedure but concentrate on the ?nite part. In order to calculate the ?nite part, we use the twice-subtracted dispersion relation:

vac 2 ?sun ReiI (k ; m1 , m2 , m3 )11

vac 2 vac ≡ Re iIsun (k ; m1 , m2 , m3 )11 ? iIsun (0; m1 , m2 , m3 )11 ? k 2

k4 = π

vac ImiIsunset (M 2 ; m1 , m2 , m3 ) dM 2 P . M 4 (M 2 ? k 2 ) (m1 +m2 +m3 )2

∞

? iI vac (k 2 ; m1 , m2 , m3 )11 ?k 2 sun

k 2 =0

(V.58)

The second and the third terms in the left hand side are divergent and have to be renormalized but the right hand side is ?nite and remain unchanged by renormalization. Using

vac the results for ImiIsun (M 2 ; m1 , m2 , m3 ) we can easily evaluate Eq.(V.58) numerically. Af? vac (m2 ; m1 , m2 , m3 )11 is subtracted, Eq.(V.58) coincides with that in modi?ed ter ReiI sun phys

minimal subtraction scheme up to an irrelevant overall factor.

B.

T -dependent part

We rewrite the integrand of the ?rst term in the T -dependent part Eq.(V.54) so that it This can be done by recombining the two factors in the integrand of F (2) (k ?p; m2 , m3 )+(2 ? becomes the same form as that in the second term, i.e. (a delta function)×(some functions).

3) and F (3) (k ? p; m2 , m3 ), T -independent part and T -dependent part of i?F 11 . The result is as follows:

th Isun (k ; m1 , m2 , m3 )11 d4 p (2) n(p0 )2πδ (p2 ? m2 (k ? p; m2 , m3 ) + F (2) (k ? p; m2 , m3 ) + (2 ? 3) = 1) I (2π )4 +F (3) (k ? p; m2 , m3 ) (2) +n(p0 )2πδ (p2 ? m2 (k ? p; m1 , m3 ) + F (2) (k ? p; m1 , m3 ) 2) I (2) +n(p0 )2πδ (p2 ? m2 (k ? p; m1 , m2 ) . 3 )I

(V.59)

When we put k = (k0 , 0), Eq.(V.59) is reduced to

th Isun (k0 , 0; m1 , m2 , m3 )11

14

=

1 4π 2

∞ τ =± m1

(2) dωn(ω ) ω 2 ? m2 (k0 + τ ω, 1 I

ω 2 ? m2 1 ; m2 , m3 )

+ F (2) (k0 + τ ω, +F (3) (k0 + τ ω, +

∞ m2

ω 2 ? m2 1 ; m2 , m3 ) + (2 ? 3) ω 2 ? m2 1 ; m2 , m3 ) ω 2 ? m2 2 ; m1 , m3 )

(2) (k0 + τ ω, dωn(ω ) ω 2 ? m2 2 I

+F (2) (k0 + τ ω, +

∞ m3

ω 2 ? m2 2 ; m1 , m3 ) ω 2 ? m2 3 ; m1 , m2 ) , (V.60)

(2) (k0 + τ ω, dωn(ω ) ω 2 ? m2 3 ·I

Using the expressions of I (2) , F (2) and F (3) obtained in the previous section, we can evaluate the continuous and discontinuous parts of Eq.(V.60) numerically.

VI.

CONTRIBUTION OF THE THERMAL SUNSET DIAGRAM TO THE SPEC-

TRAL FUNCTION OF THE SIGMA MESON AT FINITE TEMPERATURE

The purpose of this section is to see how two-loop diagrams a?ect observables by evaluating the contribution of the thermal sunset diagram to σ spectral function at ?nite temperature in the O(4) linear sigma model. It is known that naive perturbation theory breaks down at T = 0 and that resummation of higher orders is necessary [23,24]. We adopt here a resummation technique called optimized perturbation theory (OPT) [1]. We ?rst brie?y review the procedure of OPT applied to O(4) linear sigma model. The original linear sigma model Lagrangian is as follows: L= 1 λ 2 2 (?? φi )2 ? ?2 φ2 (φ ) + hφ0 + counter terms, i ? 2 4! i (VI.61)

where φi = (σ, π ) and hφ0 being the explicitly symmetry breaking term. For the renormalized couplings ?2 , λ and h and the renormalization point κ we use the values determined in [1]: ?2 = ?(283 MeV)2 , λ = 73.0, h = (123 MeV)3 , κ = 255 MeV. In OPT one adds and subtracts a new mass term with the mass m to the Lagrangian. Thus, we have L = 1 1 2 λ 2 2 [(?? φi )2 ? m2 φ2 (φ ) + hφ0 + (counter term), i ] + χφi ? 2 2 4! i (VI.62)

where χ ≡ m2 ? ?2 . The idea of OPT is reorganization of perturbation theory: one treats the added one as a tree-level mass term while the subtracted one as perturbation. 15

When the spontaneous symmetry breaking takes place, tree level masses of π and σ read, respectively, λ 2 2 m2 0π = m + ξ , 6 for the thermal e?ective action V (ξ, T, m2 ) [1]: ?V (ξ, T, m2) = 0. ?ξ Note that the derivative with respect to ξ does not act on m2 . If Green’s functions are calculated in all orders in OPT, they should not depend on the arbitrary mass, m. However, if one truncates perturbation series at a certain order they depend on it. One can determine this arbitrary parameter so that the correction terms are as small as possible. We adopt the following condition [1]: Ππ (k 2 = m2 0π ) + Ππ (k = 0; T ) = 0, (VI.65) (VI.64) λ 2 2 m2 0σ = m + ξ , 2 (VI.63)

where ξ is the vacuum expectation value of σ and determined by the stationary condition

where the ?rst and second terms are respectively T -independent part and T -dependent part of the one-loop self-energy of π . Let us now turn to the discussion on the spectral function of σ de?ned by Eq.(II.24): ρσ (k0 , k) = ? ? σ (k0 , k) ImΠ 1 . 2 2 ? ? π (k 2 ? m2 0σ ? ReΠσ (k0 , k)) + (ImΠσ (k0 , k)) (VI.66)

? σ (k0 , k) is the self-energy of σ . Its real and imaginary parts are related with (1,1) Here Π component, Π11 σ , via Eqs.(II.13) and (II.14). As was already mentioned, the spectral function at one-loop order was studied by Chiku and Hatsuda [1]. We want to see how the spectral function at one-loop order is modi?ed by adding the thermal sunset diagram. Thus, we take

11 11 Π11 σ (k0 , k) = Πσ (k0 , k)1?loop + Πσ (k0 , k)sun .

(VI.67)

The ?rst term is the renormalized one-loop self-energy calculated in [1]. The second term is the renormalized sunset diagram depicted in Fig.5 and given by Π11 σ (k0 , k)sun = ? λ2 iIsun (k0 , k; m0π , m0π , m0σ )11 . 6 (VI.68)

We show ρσ (k0 , k = 0) at T = 200 MeV and T = 145 MeV in Fig.6. At T = 200 MeV, 16

the spectral function at one-loop order consists of a δ -function peak for σ and a continuum. By including the sunset diagram σ acquires a width of the order of 10 MeV. At lower temperature, T = 145 MeV, an enhancement of the spectrum near the threshold is observed at one-loop order. When we include the sunset diagram, this feature is not lost. Let us discuss the above results. At high temperature (T = 200 MeV), the mass of σ is smaller than twice the pion mass and the decay σ → ππ is forbidden. As a result, within the one-loop calculation σ has zero width. However, at ?nite temperature σ can interact with thermal particles in heat bath and change into other states. Among such processes, those which are taken into account by including the sunset diagram are represented by Fig.3 with (Φ, φ1 , φ2 , φ3 ) assigned to, for example, (σ, σ, π, π ). The processes which correspond to (a) and (b) in Fig.3 are possible for k0 > 2m0π + m0σ . This a?ect the spectrum at high energy. (c) and (d) in Fig.3 drop o? for positive k0 . (e) and (f) a?ect the spectrum at low energy since they are possible for k0 < m0σ ? 2m0π . The processes which correspond to (g) and (h) in Fig.3 are shown in Fig.7. They are the most important and give a ?nite width to σ since they are allowed at arbitrary positive k0 . However, we observe that their e?ects at lower temperature (T = 145 MeV) are small. The reason is traced back to Eq.(IV.50). The term representing the probabilities for (g) and (h) in Fig.3 is the fourth term. That integral at lower temperature is suppressed due to the statistical weight since at lower temperature the masses of σ and π are large. Finally we note that, as a consequence of the non-vanishing discontinuous part of the sunset diagram, the spectral function is also non-vanishing in the all range of k0 .

VII.

SUMMARY AND PERSPECTIVE

We have studied the sunset diagram for scalar ?eld theories at ?nite temperature in the real time formalism. We have explained how we can reduce it to an expression written in terms of one-loop self-energy integrals, which can be easily evaluated numerically. We have also discussed what physical processes are contained in the discontinuous part of the diagram. We have found that there exist processes which occur only at ?nite temperature and some of them are allowed at arbitrary energy. As a result, the discontinuous part of the sunset diagram is non-vanishing in all the energy region, which is a remarkable feature at ?nite temperature and manifests itself at two-loop order. 17

As an application of the result, we have demonstrated how the spectral function of σ at ?nite T at one-loop order is modi?ed when we include the thermal sunset diagram. At high temperature, where σ → ππ is forbidden, σ acquires a ?nite width of the order of 10 MeV due to collisions with thermal particles in the heat bath while σ does not have a width at one-loop order. At lower temperature the spectrum is almost unchanged. Finally we comment on the e?ect of other two-loop diagrams on the spectral function. We have seen that the threshold enhancement, which was ?rst found in the one-loop calculation, is retained if we include the sunset diagram. However, in the present calculation the e?ect of the thermal width of π in the σ → ππ decay is not included. This is taken into account by including the diagrams such as shown in Fig.8, in which an internal π changes into σ absorbing a thermal π in the heat bath. In the one-loop calculation, π has a width, Γπ = 50 MeV, at the temperature, T = 145 MeV, at which the threshold enhancement is observed for σ . If we include this e?ect as a constant complex mass shift for π in the one-loop self-energy for σ , we expect σ to acquire a width twice as that for π , i.e. Γσ = 2Γπ [25]. This implies that when the two-loop diagrams such as Fig.8 are included the spectral function for σ would be signi?cantly modi?ed with the width of about 100 MeV. The calculations of those diagrams are now in progress [26].

Acknowledgments

The authors would like to thank M. Ohtani for useful discussion. This work was partially supported by Grants-in-Aid of the Japanese Ministry of Education, Science, Sports, Culture and Technology (No.06572).

18

1

2

3

(a)

2

3

Φ

(c) (d)

Φ

2 1

2

Φ

φ

3

1

3

(e)

(f)

2

2

1

3

3

Φ

(h)

(g)

FIG. 3: The amplitudes in Eq.(IV.50) responsible for the disappearance and reappearance of Φ. ?i stands for an anti-particle of φi . (a) minus (b) corresponds to the ?rst term in Eq.(IV.50), (c) φ minus (d) to the second, (e) minus (f) to the third and (g) minus (h) to the forth. Amplitudes obtained by permuting particle labels of (e), (f), (g) and (h) also exist.

19

φ

φ

φ

φ

Φ

1

φ

1

φ

φ

φ

φ

φ

φ

φ

φ φ φ φ φ φ

Φ

2

φ

φ

1

Φ

3

(b)

φ

1

φ φ

2

3

Φ

FIG. 4: Renormalized T -independent part of the sunset diagram. The second and the third diagrams cancel the sub- and overall divergences in the ?rst diagram (bare sunset diagram) respectively.

???

· ·

20

FIG. 5: The sunset diagram for σ . Solid and dashed lines correspond to σ and π respectively.

18 16 ρσ(k0,k=0;T=200 MeV) X 10-6 14 12 10 8 6 4 2 0 0 7 6 ρσ(k0,k=0;T=145 MeV) X 10-6 5 4 3 2 1 0 0 100 200 300 400 k0 (MeV) 500 600 700 800 100 200 300 400 k0 (MeV) 500 600 700 800

FIG. 6: Spectral function of σ ρσ (k0 , k = 0) at T = 200 MeV (upper panel) and T = 145 MeV (lower panel). Solid line corresponds to ρσ at one-loop order and dashed line to that with one-loop self-energy and the sunset diagram.

21

(a)

(b)

(c)

FIG. 7: The processes which are allowed at all positive k0 contained in the discontinuous part of the sunset diagram for σ (Fig.5).

FIG. 8: two-loop self-energy diagram in which internal π changes into σ by absorbing thermal π in the heat bath.

?

22

APPENDIX A: CALCULATION OF ImiF (2) (p; m1 , m2 ) AND F (3) (p; m1 , m2 )

In this appendix, we derive Eqs.(III.35)-(III.37) and Eqs.(III.39)-(III.41). From Eq.(III.28) we obtain ImiF (2) (p; m1 , m2 ) = 1 d4 k 2 2 2πδ ((p + k )2 ? m2 1 )n(k0 )2πδ (k ? m2 ) 2 (2π )4 ∞ ∞ 1 dk0 d|k||k|2n(k0 )δ (k 2 ? m2 = 2) 4π ?∞ 0

1

×

?1

dcosθδ (p2 + 2p0 k0 ? 2|p||k|cosθ + k 2 ? m2 1 ).

(A.1)

ImiF (2) (p; m1 , m2 ) receives contribution from k which satis?es

2 |p2 + 2p0 k0 + m2 2 ? m1 | <1 2|p||k| 2 and k0 = k2 + m2 2 . We make a square of Eq.(A.2) and obtain 2 2 2 2 2 2 2 2 2 4p0 k0 (p2 + m2 2 ? m1 ) < ?(p + m2 ? m1 ) ? 4p0 m2 ? 4p |k| .

(A.2)

(A.3)

There are three cases when Eq.(A.3) holds: A. LHS> 0 and RHS> 0, (RHS)2 ?(LHS)2 > 0. B. LHS< 0 and RHS< 0, (RHS)2 ?(LHS)2 < 0. C. LHS< 0 and RHS> 0. First, we calculate (RHS)2 ?(LHS)2 . (RHS)2 ? (LHS)2

2 2 2 2 2 2 2 2 2 2 = 16p4 |k|4 + 8p2 (p2 + m2 2 ? m1 ) + 4p0 m2 ? 16(p + m2 ? m1 ) p0 |k| 2 2 2 2 2 2 2 2 = 16p4 |k|4 + 8 (p2 ? 2p2 0 )(p + m2 ? m1 ) + 4p p0 m2 |k| 2 2 2 2 + (p2 + m2 2 ? m1 ) ? 4p0 m2 2 2 2 2 2 + (p2 + m2 2 ? m1 ) + 4p0 m2 2 2 2 2 2 ? 16(p2 + m2 2 ? m1 ) p0 m2

(A.4)

Whether there exists |k|2 which satis?es (RHS)2 ? (LHS)2 < 0 depends on the sign of the following expression:

2 2 2 2 2 2 2 D = 16 (p2 ? 2p2 0 )(p + m2 ? m1 ) + 4p p0 m2 2 2 2 2 2 ? 16p4 (p2 + m2 2 ? m1 ) ? 4p0 m2 2

2 2 2 2 2 2 2 = 64p2 0 |p| (p + m2 ? m1 ) p ? (m1 + m2 )

p2 ? (m1 ? m2 )2 .

(A.5)

23

Thus, for (m1 ? m2 )2 < p2 < (m1 + m2 )2 (RHS)2 ? (LHS)2 > 0, for p2 < (m1 ? m2 )2 or (m1 + m2 )2 < p2 |k|2 < |k|2 ? or |k|2 > |k|2 + ?? ?? (RHS)2 ? (LHS)2 > 0,

2 2 |k|2 ? < |k| < |k|+

(RHS)2 ? (LHS)2 < 0,

where |k|2 ± are given by |k|2 ± = ? 1? 1+ m2 2 ? p2 m2 1

2

?

4?

|p| ±

1?

(m2 + m1 p2

)2

1?

(m1 ? m2 p2

)2

| p0 |

?2 ? ?

.

(A.6)

2 Secondly, for p2 > m2 1 ? m2

p0 k 0 ? 0

2 for p2 < m2 1 ? m2

??

LHS ? 0,

p0 k 0 ? 0 Thirdly, for p2 > 0

?? RHS < 0

LHS ? 0.

for p2 < 0 |k|2 ?

Therefore, the conditions for the above three cases are respectively given by

2 2 2 A. p2 < 0, p0 k0 (p2 + m2 2 ? m1 ) > 0 and |k| > |k|+ . 2 2 2 2 B. p2 > (m1 + m2 )2 , p0 k0 (p2 + m2 2 ? m1 ) < 0 and |k|? < |k| < |k|+ , 2 2 2 2 0 < p2 < (m1 ? m2 )2 and p0 k0 (p2 + m2 2 ? m1 ) < 0 and |k|? < |k| < |k|+ , 2 2 2 2 p2 < 0, p0 k0 (p2 + m2 2 ? m1 ) < 0 and |k|? < |k| < |k|0 . 2 2 2 C. p2 < 0, p0 k0 (p2 + m2 2 ? m1 ) < 0 and |k| > |k|0 .

2 2 2 2 (p2 + m2 2 ? m1 ) + 4p0 m2 ≡ |k|2 0 ?4p2

??

RHS ? 0

The case C can be combined with the third of the case B as 24

2 2 2 p2 < 0, p0 k0 (p2 + m2 2 ? m1 ) < 0 and |k| > |k|? .

1. For p2 > (m1 + m2 )2 or 0 < p2 < (m1 ? m2 )2 , ImiF (2) (p; m1 , m2 ) =

| k| + 1 |k| d|k| n(ωk ) 8π |p| |k|? ωk ω+ 1 dωn(ω ) = 8π |p| ω? 1 1 1 ? e?βω+ = . ln 16π |p| β 1 ? e?βω?

(A.7)

2. For (m1 ? m2 )2 < p2 < (m1 + m2 )2 , ImiF (2) (p; m1 , m2 ) = 0. 3. For p2 < 0, ImiF (2) (p; m1 , m2 ) =

∞ ∞ |k| 1 + d|k| n(ωk ) 8π |p| ωk | k| ? | k| + ∞ ∞ 1 + dωn(ω ) = 8π |p| ω? ω+ 1 ?1 = ln (1 ? e?βω+ )(1 ? e?βω? ) . 16π |p| β

(A.8)

(A.9)

where ω± = 1 2 1+

2 m2 2 ? m1 p2 2

| p0 | ±

1?

(m2 + m1 )2 p2

1?

(m1 ? m2 )2 |p| p2

(A.10)

Similarly, 1. For p2 > (m1 + m2 )2 , ImiF (3) (p; m1 , m2 ) =

| k| + 1 |k| d|k| n(ωk )n(|p0 | ? ωk ) 8π |p| |k|? ωk ω+ 1 = dωn(ω )n(|p0| ? ω ) 8π |p| ω? 1 1 ? e?βω+ eβ (|p0 |?ω? ) ? 1 1 . ln = 8π |p|β eβ |p0 | ? 1 1 ? e?βω? eβ (|p0 |?ω+ ) ? 1

(A.11)

2. For (m1 ? m2 )2 < p2 < (m1 + m2 )2 , ImiF (3) (p; m1 , m2 ) = 0. 25 (A.12)

3. For 0 < p2 < (m1 ? m2 )2 , ImiF (3) (p; m1 , m2 ) =

| k| + 1 |k| d|k| n(ωk )n(|p0 | + ωk ) 8π |p| |k|? ωk ω+ 1 dωn(ω )n(|p0| + ω ) = 8π |p| ω? 1 1 = · β |p0 | 8π |p|β e ?1 ?βω+ 1?e 1 ? e?β (|p0 |+ω+ ) β |p0 | × ln ? e ln 1 ? e?βω? 1 ? e?β (|p0 |+ω? )

. (A.13)

4. For p2 < 0, ImiF (3) (p; m1 , m2 ) ∞ ∞ |k| |k| 1 d|k| n(ωk )n(|p0 | + ωk ) + d|k| n(ωk )n(|p0 | ? ωk ) = 8π |p| ωk ωk | k| ? | k| + ∞ ∞ 1 dωn(ω )n(|p0| + ω ) + dωn(ω )n(|p0| ? ω ) = 8π |p| ω? ω+ 1 1 = ?ln|1 ? e?βω+ | + e?β |p0 | ln|1 ? e?β (ω+ ?|p0 |) | ? β | p | 0 8π |p|β e ?1 1 ?ln|1 ? e?βω? | + eβ |p0 | ln|1 ? e?β (ω? +|p0 |) | . (A.14) + β |p0 | e ?1

APPENDIX B: PHYSICAL MEANING OF |k|±

In this appendix we explain the physical meaning of |k|± de?ned by Eq.(A.6).

Let us ?rst consider the case p2 > (m1 + m2 )2 or 0 < p2 < (m1 ? m2 )2 . For simplicity we

2

suppose p0 > 0. Since internal particles are on shell, in the center-of-mass frame we have p =

2

m2 1

+

|k|2

±

m2 2

+

|k|2

,

(B.1)

where + is for p2 > (m1 + m2 )2 and ? is for 0 < p2 < (m1 ? m2 )2 . In either case |k|2 = [p2 ? (m1 + m2 )2 ][p2 ? (m1 ? m2 )2 ] . 4 p2 (B.2)

If we boost the system to the positive z-direction, the external and internal momenta become ? ? p′ = p coshθ, 0 0 (B.3) ? p⊥ = 0, p′ = p0 sinhθ (θ > 0), z ? ? k ′ = k coshθ + k sinhθ, 0 z 0 (B.4) ′ ′ ? k ⊥ = k⊥ , k = kz coshθ + k0 sinhθ.

z

26

Since ?|k| < kz < |k|, where

′ 2 ′ 2 |k′ |2 ? ≤ |k | ≤ |k |+ ,

(B.5)

2 |k′ |2 ± = (±|k|coshθ + |k0 |sinhθ ) 2 p′ p′ = ±|k| 0 + |k0| z p0 p0

1 = 4

±

(m1 + m2 )2 1? p2

2 (m1 ? m2 )2 ′ m2 2 ? m1 1? p0 + 1 + |p′ | 2 2 p p

2

.(B.6)

This is nothing but Eq.(A.6). Therefore, for p2 > (m1 + m2 )2 or 0 < p2 < (m1 ? m2 )2 , |k|+ and |k|? are respectively the maximum and minimum values of |k| such that the internal particles are on-shell. Let us next consider the case p2 < 0. In the Breit frame, p = (0, p), with p in the positive z-direction we write the momentum of the particle with mass m2 by k = (k0 , k) where k0 can be positive or negative. Then, the momentum of the particle with mass p + k = (k0 , p + k). Since internal particles are on shell, we have

2 2 2 2 2 2 k0 = m2 2 + kz + |k⊥ | = m1 + (pz + kz ) + |k⊥ | .

(B.7)

From this equation we obtain

2 2 2 ?p2 + m2 p2 1 ? m2 z + m1 ? m2 = , 2 pz 2 ?p2 2 [?p2 + (m1 + m2 )2 ][?p2 + (m1 ? m2 )2 ] 2 + |k⊥ |2 , k0 = ?4p2

kz = ?

(B.8)

When boosting the frame to the z-direction, we suppose the momenta become p′ = (p′0 , p′ ),

′ k ′ = (k0 , k′ ), ′ p′ + k ′ = (p′0 + k0 , p′ + k′ ).

(B.9)

Then, p′ and k ′ are given by ? ? p′ = |p|sinhθ, 0 ? |p′ | = |p|coshθ, ? ? k ′ = k coshθ + k sinhθ, 0 z 0 ? k′ ⊥ = k⊥ , k ′ = kz coshθ + k0 sinhθ.

z

(B.10)

(B.11)

27

Using Eqs.(B.8), (B.10) and (B.11), we obtain |k′ |2 = (kz coshθ + k0 sinhθ)2 + |k⊥ |2 . (B.12)

|k′ |2 has a minimum when |k⊥ |2 = 0 but not a maximum. Depending on the sign of k0 , the minimum value is given by |k′ |2 ± = 1 = 4 |kz | |p′ | ±

2 kz

?p2

+

m2 2

p′0 ?p2

2

m2 ? m2 1 ? 1 2 2 |p′ | ± ?p

(m1 + m2 )2 1+ ?p2

(m1 ? m2 )2 ′ 1+ p0 ?p2

2

.(B.13)

Eq.(B.13) is nothing but Eq.(A.6). Therefore, in the case of p2 < 0, both of |k|+ and |k|? are the minimum values of |k| such that the internal particles can be on-shell.

28

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[21] J.O. Andersen and E. Rev. D51, 6990 (1995).

Braaten,

Phys.

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