Debye Mass of Massless Φ⁴-Theory to Order g⁶ at Weak Coupling

Abstract

The Debye mass is calculated to order g⁶ at weak coupling of massless ϕ⁴-theory. Debye mass gets contributions from the hard and soft momentum scale. We use dimensional reduction and the effective field theory to distinct two scales of order T and gT. Three dimensional effective theory is used to calculate contributions of soft scale whereas hard scale is evaluated by applying a perturbation theory, in power series of g. For the Debye mass to g⁶-order corrections, we calculate the hard part to g⁶ order and the soft part through mass parameter to g⁶ and calculate four loop self energy diagrams. The convergence of series and loop expansion are discussed.

Appendices

Appendix A: Three-Dimensional Integrals

The infrared and ultraviolet divergences appearing in three-dimensional integrals over momenta, can be regularized by dimensional regularization. The integral is performed in d-dimensions (real number) for which the integral is convergent and then it is analytically continued back to the physical dimension, d = 3. The divergence will be isolated in an expansion parameter which goes to zero, and can be removed by renormalization. The momentum space integration measure is,

$$ {\int}_{p} \equiv \left( \frac{e^{\gamma}\mu^{2}}{4\pi}\right)^{\epsilon} \int {d^{3-2\epsilon}p \over (2 \pi)^{3-2\epsilon}} . $$

(18)

1.1 A.1 One Loop

For general loop calculations the integral is given below. It is used to calculate I₁, I₂, I₃ and I₄,

$$ \begin{array}{@{}rcl@{}} I_n&\equiv&{\int}_p{1\over(p^{2}+m^{2})^n}\\ &=&{1\over8\pi}(e^{\gamma_E}\mu^{2})^{\epsilon} {\Gamma(n-\frac{3}{2}+\epsilon) \over{\Gamma}(\frac{1}{2}) {\Gamma}(n)}m^{3-2n-2\epsilon} . \end{array} $$

(19)

1.2 A.2 Two Loop

Two-loop integrals that are required for the calculations are given below,

$$ \begin{array}{@{}rcl@{}} I_{\text{sun}}(p=im_{D})&=& {1\over4(4\pi)^{2}} \left( {\mu\over2m}\right)^{4\epsilon}\\ && \times\left[ {1\over\epsilon}+4\log2+6 +6{m\over m_{D}}\log{3m-m_{D}\over3m+m_{D}}-2\log{9m^{2}-m_{D}^{2}\over m^{2}} +\mathcal{O}(\epsilon) \right], \end{array} $$

(20)

$$ \begin{array}{@{}rcl@{}} =\\ I_{\text{sun}}^{\prime}(p=im)&=& {\log2\over4m^{2}(4\pi)^{2}}\left( {\mu\over2m}\right)^{4\epsilon} \left[1 +\mathcal{O}(\epsilon) \right]\\ && +{{\Pi}_{1}\over32m^{4}(4\pi)^{2}} \left( {\mu\over2m}\right)^{4\epsilon} \left[ 3-4\log2 +\mathcal{O}\left( \epsilon\right) \right] , \end{array} $$

(21)

$$ \begin{array}{@{}rcl@{}} J_{\text{sun}}(p=im)&=& {1\over32m^{4}(4\pi)^{2}}\left( {\mu\over2m}\right)^{4\epsilon} \left[1+\mathcal{O}(\epsilon)\right] , \end{array} $$

(22)

$$ \begin{array}{@{}rcl@{}} K_{\text{sun}}(p=im)&=& {5\over96m^{4}(4\pi)^{2}}\left( {\mu\over2m}\right)^{4\epsilon} \left[1+\mathcal{O}(\epsilon)\right] . \end{array} $$

(23)

The sunset diagram is evaluated in Ref. [17] to order 𝜖⁰. We calculate the remaining two-loop integrals in Appendix B.

1.3 A.3 Three Loop

Three-loop integrals needed are given below,

$$ \begin{array}{@{}rcl@{}} I_{\text{ball}}^{\prime} &=&{1\over8m(4\pi)^3}\left( {\mu\over2m}\right)^{6\epsilon}\left[ {1\over\epsilon}+2-4\log2 +\mathcal{O}\left( \epsilon\right) \right], \end{array} $$

(24)

$$ \begin{array}{@{}rcl@{}} J_{\text{ball}} &=&{1\over16m^3(4\pi)^3}\left( {\mu\over2m}\right)^{6\epsilon}\left[ 1+\mathcal{O}\left( \epsilon\right) \right], \end{array} $$

(25)

$$ \begin{array}{@{}rcl@{}} K_{\text{ball}} &=&{1\over32m^3(4\pi)^3}\left( {\mu\over2m}\right)^{6\epsilon}\left[ {1\over\epsilon}+5-4\log2 +\mathcal{O}\left( \epsilon\right) \right] , \end{array} $$

(26)

$$ \begin{array}{@{}rcl@{}} I_{\text{c3rung}} &=& {7\over16m(4\pi)^3}\left( {\mu\over2m}\right)^{6\epsilon} \left[\zeta(3) +\mathcal{O}\left( \epsilon\right) \right]\\ && +{{\Pi}_{1}\over96m^3(4\pi)^3}\left( {\mu\over2m}\right)^{6\epsilon} \left[ \pi^{2}+24\log2-21\zeta(3) +\mathcal{O}\left( \epsilon\right) \right], \end{array} $$

(27)

$$ \begin{array}{@{}rcl@{}} J_{\text{c3rung}} &=&-{1\over96m^3(4\pi)^3}\left( {\mu\over2m}\right)^{6\epsilon} \left[\pi^{2}-24\log2+\mathcal{O}\left( \epsilon\right) \right], \end{array} $$

(28)

$$ \begin{array}{@{}rcl@{}} K_{\text{c3rung}} &=&{1\over192m^3(4\pi)^3}\left( {\mu\over2m}\right)^{6\epsilon} \left[\pi^{2}+\mathcal{O}\left( \epsilon\right) \right] . \end{array} $$

(29)

I_ball, massive basketball diagram is evaluated to order 𝜖⁰ by Ref. [17] , and to order 𝜖 by Ref. [27]. I_ball can be differentiated with respect to m to get $I_{\text {ball}}^{\prime }$. In Appendix B the J and K loop integral are calculated.

1.4 A.4 Four Loop

Four-loop integrals needed are given below,

$$ \begin{array}{@{}rcl@{}} I_{\text{c4rung}} &=&{1\over32m^{2}(4\pi)^{4}}\left( {\mu\over2m}\right)^{8\epsilon}\left[ -2\pi^{2}\log2+ 15\zeta(3)+\mathcal{O}\left( \epsilon\right) \right], \end{array} $$

(30)

$$ \begin{array}{@{}rcl@{}} I_{\text{triangle}}^{\prime} &=&{\pi^{2}\over48m^{2}(4\pi)^{4}}\left( {\mu\over2m}\right)^{8\epsilon}\left[ 1+\mathcal{O}\left( \epsilon\right) \right], \end{array} $$

(31)

$$ \begin{array}{@{}rcl@{}} I &=& I_{\text{cballsun}}(p=im)-{1\over4(4\pi)^{2}\epsilon}I_{\text{sun}}^{\prime}(p=im)\\ &=& {\log2\over4m^{2}(4\pi)^{4}}\left[ \log{\mu\over2m} -0.01353 +\mathcal{O}\left( \epsilon\right) \right], \end{array} $$

(32)

$$ \begin{array}{@{}rcl@{}} I_{\text{fish1}}(p=im) &=&{1\over16m^{2}(4\pi)^{4}}\left( {\mu\over2m}\right)^{8\epsilon} \left[4.6029+\mathcal{O}\left( \epsilon\right) \right], \end{array} $$

(33)

$$ \begin{array}{@{}rcl@{}} I_{\text{fish2}}(p=im) &=&{1\over16m^{2}(4\pi)^{4}}\left( {\mu\over2m}\right)^{8\epsilon} \left[1.778+\mathcal{O}\left( \epsilon\right) \right] . \end{array} $$

(34)

In Ref. [25], I_triangle, the triangle diagram was calculated, so by differentiation I_triangle with respect to m² we get $I_{\text {triangle}}^{\prime }$. The fish diagrams are calculated in Appendix Appendix.

Appendix B: Explicit Calculations

1.1 B.1 Two Loop Diagrams

The sunset diagram I_sun(p = im) is known to order 𝜖⁰ in Ref. [17]. However, we need to evaluate I_sun(p) at p = im_D. By going to coordinate space, we can write

$$ \begin{array}{@{}rcl@{}} I_{\text{sun}}(p)&=& {\int}_{qr}{1\over (q^{2}+m^{2})(r^{2}+m^{2})} {1\over(\mathbf{p}+\mathbf{q}+\mathbf{r})^{2}+m^{2}} \end{array} $$

(35)

$$ \begin{array}{@{}rcl@{}} &=& {\int}_{R}e^{i\mathbf{p}\cdot\textbf{R}}V^3(R), \end{array} $$

(36)

where the coordinate-space integral is

$$ \begin{array}{@{}rcl@{}} {\int}_{R}&=&\left( {e^{\gamma_E}\mu^{2}\over4\pi}\right)^{-\epsilon} \int d^{dR} , \end{array} $$

(37)

the Fourier transform of momentum-space propagator is represented by V (R) and given as,

$$ \begin{array}{@{}rcl@{}} V(R)&=&{\int}_q{e^{i\mathbf{q}\cdot\textbf{R}}\over q^{2}+m^{2}}\\ &=& \left( {e^{\gamma_E}\mu^{2}\over4\pi}\right)^{-\epsilon} {1\over2\pi^{3/2-\epsilon}}\left( {m\over R}\right)^{1/2-\epsilon} K_{{1\over2}-\epsilon}(mR). \end{array} $$

(38)

Here, K_ν(x) is Bessel function of the second kind. After integrating over angles, (36) reduces to

$$ \begin{array}{@{}rcl@{}} I_{\text{sun}}(p)&= &\left( {e^{\gamma_E}\mu^{2}\over2p}\right)^{-\epsilon} {(2\pi)^{3\over2}\over\sqrt{p}} {\int}_{0}^{\infty}dRR^{{3\over2}-\epsilon}J_{{1\over2}-\epsilon}(pR) V^3(R). \end{array} $$

(39)

The integral (39) is ultraviolet divergent is three dimensions for small values of R, i.e. for large values of the momenta. We therefore split the integral into an integral from R = 0 to R = r and one from R = r to $R=\infty$. The first integral is calculated in d = 3 − 2𝜖 dimensions and expanded in powers of 𝜖, while the second is calculate directly in d = 3. This yields

$$ \begin{array}{@{}rcl@{}} I_{\text{sun}}(p)&=& \left( {e^{\gamma_E}\mu^{2}\over4\pi}\right)^{-\epsilon} {(2\pi)^{3\over2}\over\sqrt{p}} {\int}_{0}^{r} dR R^{{3\over2}-\epsilon}J_{{1\over2}-\epsilon}(pR) V^3(R)\\ && +{4\pi\over p}{\int}_{R}^{\infty} dR R\sin(pR)V_{0}^3(R), \end{array} $$

(40)

where we have used that $J_{1\over 2}(x)=\sqrt {2/\pi x} \sin \limits (x)$. The subscript on V₀(R) indicates that we use the expression for V (R) with 𝜖 = 0, i.e. the Coulomb potential V₀(R) = e^−mR/4πR. In the first term, we use the small-r expansion of the Bessel functions

$$ \begin{array}{@{}rcl@{}} J_{{1\over2}-\epsilon}(pR)&=&{1\over{\Gamma}[{3\over2}-\epsilon]} \left( {{1\over2}pR}\right)^{{1\over2}-\epsilon} \left[ 1+\mathcal{O}(p^{2}R^{2}) \right], \end{array} $$

(41)

$$ \begin{array}{@{}rcl@{}} V(R) &=& \left( {e^{\gamma_E}\mu^{2}\over4}\right)^{\epsilon} {\Gamma({1\over2}-\epsilon)\over{\Gamma}{1\over2}}{1\over4\pi} R^{-1+2\epsilon}\left[ 1+{m^{2}R^{2}\over2(1+2\epsilon)}+\mathcal{O}(m^{4}R^{4}) \right] \end{array} $$

(42)

$$ \begin{array}{@{}rcl@{}} &&-\left( {e^{\gamma_E}\mu^{2}}\right)^{\epsilon} {\Gamma(-{1\over2}-\epsilon)\over{\Gamma}{-1\over2}}{1\over4\pi} m^{1-2\epsilon} \left[ 1+{m^{2}R^{2}\over2(3-2\epsilon)}+\mathcal{O}(m^{4}R^{4}) \right]. \end{array} $$

(43)

The first integral in (40) is denoted by I₁ and reduces to

$$ \begin{array}{@{}rcl@{}} I_{1}&=&{1\over4(4\pi)^{2}} \left[ {1\over\epsilon} +4\log{\mu r} +2\gamma_E +\mathcal{O}(\epsilon) \right]. \end{array} $$

(44)

The second integral in (40) is denoted by I₂ and becomes

$$ \begin{array}{@{}rcl@{}} I_{2}&=& {4\pi\over p}{\int}_{R}^{\infty}dR R \sin(pR)V_{0}^3(R) |_{p=im_{D}}\\ &=& {1\over2(4\pi)^{2}m_{D}} \left [ (3m-m_{D})\log(3m-m_{D})-(3m+m_{D})\log(3m+m_{D}) \right.\\ &&\left. -2m_{D}(-1+\gamma_E+\log r) \right] . \end{array} $$

(45)

Adding (44) and (45), the r-dependent terms cancel and we obtain (20).

1.2 B.2 Three Loop Diagrams

We can rewrite the diagram I_c3rung(p = im) in (27) as,

$$ \begin{array}{@{}rcl@{}} I_{\text{c3rung}}(p=im) &=&{\int}_q{1\over(\mathbf{p}+\mathbf{q})^{2}+m^{2}} |_{p=im}[I_{\text{bub}}(q)]^{2} , \end{array} $$

(46)

where I_bub(q) is given by

$$ \begin{array}{@{}rcl@{}} I_{\text{bubble}}(q)&=& {\int}_{R}{1\over r^{2}+m^{2}}{1\over(\mathbf{r}+\mathbf{q})^{2}+m^{2}}\\ &=& {1\over4\pi q}~\arctan(q/2m). \end{array} $$

(47)

I_bub(q) can be calculated by averaging over angles first and then integrating over r. Inserting (47) into (46), averaging over the angle between p and q, and finally integrating over q, we obtain (27).

The integral J_c3rung in (28) has an extra propagator and we can write it as,

$$ \begin{array}{@{}rcl@{}} J_{\text{c3rung}}(p=im) &=&{\int}_q{1\over[(\mathbf{p}+\mathbf{q})^{2}+m^{2}]^{2}} |_{p=im}[I_{\text{bub}}(q)]^{2} . \end{array} $$

(48)

The integral is evaluated in the same manner as I_c3rung(p = im). The last integral of this type, namely K_c3rung(p = im) is,

$$ \begin{array}{@{}rcl@{}} K_{\text{c3rung}}(p=im) &=& {\int}_q{1\over(\mathbf{p}+\mathbf{q})^{2}+m^{2}} |_{p=im}I_{\text{bub}}(q)I_{\text{bub}}^{\prime}(q), \end{array} $$

(49)

where $I_{\text {bub}}^{\prime }$ is given by the derivative of (47) with respect to m²:

$$ \begin{array}{@{}rcl@{}} I_{\text{bubble}}^{\prime}(p)&=&{1\over8\pi m}{1 \over p^{2}+4m^{2}}. \end{array} $$

(50)

Inserting (47) and (50) into (49), averaging over the angle between p and q, and finally integrating over q, we obtain (29). The integral J_ball is easy to evaluate by noting that it can be written as

$$ \begin{array}{@{}rcl@{}}J_{\text{ball}} &=&{\int}_p[I_{\text{bub}}^{\prime}(q)]^{2}\\ &=& {1\over4(4\pi)^{2}m^{2}}{\int}_p{1\over(p^{2}+4m^{2})^{2}}. \end{array} $$

(51)

The integral K_ball can be evaluated by noting the relation

$$ \begin{array}{@{}rcl@{}} K_{\text{ball}}&=&{1\over2}\left( 3J_{\text{ball}}-I^{\prime\prime}_{\text{ball}}\right). \end{array} $$

(52)

1.3 B.3 Four Loop Diagrams

We can rewrite the integral I_c4rung in (30) as,

$$ \begin{array}{@{}rcl@{}} I_{\text{c4rung}} (p=im) &=& {\int}_q{1\over(\mathbf{p}+\mathbf{q})^{2}+m^{2}} |_{p=im}[I_{\text{bub}}(q)]^3 , \end{array} $$

(53)

where I_bub(q) is given by (47). Again, the integral can be evaluated by averaging over angles first and then integrating over q.

Let us next calculate I_cballsun, which is

$$ \begin{array}{@{}rcl@{}} I&=&I_{\text{sun}}^{\prime}(p=im) \left[I_{\text{sun}}(q) -6{\Delta} m^{2} \right]\\ &=& I_{\text{sun}}^{\prime}(p=im) \left[I_{\text{sun}}(q) -I_{\text{sun}}(q=im)+I_{\text{sun}}(q=im) -6{\Delta} m^{2} \right]. \end{array} $$

(54)

The first term is

$$ \begin{array}{@{}rcl@{}}I_{1}&=&I_{\text{sun}}^{\prime}(p=im) \left[I_{\text{sun}}(q)-I_{\text{sun}}(q=im)\right]\\ &=& {\int}_q{1\over q^{2}+m^{2}}I_{\text{bub}}(|\mathbf{p}+\mathbf{q}|) \left[I_{\text{sun}}(q)-I_{\text{sun}}(q=im)\right]|_{p=im} \end{array} $$

(55)

The first two terms inside the paranthesis can be written as [29, 30]

$$ \begin{array}{@{}rcl@{}} I_{\text{sun}}(q) -I_{\text{sun}}(q=im) &=& -{1\over(4\pi)^{2}} \left[{1\over2}\log\left( {q^{2}+9m^{2}\over64m^{2}} \right)+ {3m\over q}\arctan{q\over3m} \right]. \end{array} $$

(56)

The integral (55) can be calculated by averaging over the angles between q and p first and then integrate over q. We get,

$$ \begin{array}{@{}rcl@{}} I_{1}&=&-{\log2\over4m^{2}(4\pi)^{4}}\left[ 0.127234\right]. \end{array} $$

(57)

The second term is

$$ \begin{array}{@{}rcl@{}} I_{2}&=&I_{\text{sun}}^{\prime}(p=im) \left[ I_{\text{sun}}(q=im) -6{\Delta} m^{2} \right]\\ \end{array} $$

(58)

The first factor is $I_{\text {sun}}^{\prime }(p=im)$ and can be evaluated by taking the derivative of I_sun(p = im_D) with respect to m and then setting m = m_D. This yields

$$ \begin{array}{@{}rcl@{}} I_{\text{sun}}^{\prime}(p=im) &=&{\log2\over4m^{2}(4\pi)^{2}} . \end{array} $$

(59)

The second integral is simply given by the finite part of I_sun(p = im):

$$ \begin{array}{@{}rcl@{}} I_{\text{sun}}(q=im) -6{\Delta} m^{2} &=&{1\over4m^{2}(4\pi)^{2}} \left[ \log{\mu\over2m}+{3\over2}-2\log2 \right] . \end{array} $$

(60)

Multiplying (59) and (60), and adding the result to (57), we obtain (32), where we have defined C_ballsun = − 0.01353.

Let us next discuss the first fish-like diagram in (33). We can write it as,

$$ \begin{array}{@{}rcl@{}} I_{\text{fish1}}(p=im)&=& {\int}_{qr} {1\over(\mathbf{p}+\mathbf{r})^{2}+m^{2}} |_{p=im} [I_{\text{tri}}(\mathbf{q},\mathbf{r})]^{2} , \end{array} $$

(61)

where

$$ \begin{array}{@{}rcl@{}} I_{\text{tri}}(\mathbf{q},\mathbf{r}) &=&{\int}_{s}{1\over s^{2}+m^{2}}{1\over(\mathbf{q}+\mathbf{s})^{2}+m^{2}} {1\over(\mathbf{r}+\mathbf{s})^{2}+m^{2}}. \end{array} $$

(62)

The integral (62) is finite and we can be write it as, as [28]

$$ \begin{array}{@{}rcl@{}} I_{\text{tri}}(\mathbf{q},\mathbf{r})&=&{\arctan(\sqrt{D}/C)\over8\pi\sqrt{D}}, \end{array} $$

(63)

where

$$ \begin{array}{@{}rcl@{}} C&=&{q^{2}+r^{2}+\mathbf{q}\cdot \mathbf{r}+4m^{2}\over m^{2}}, \end{array} $$

(64)

$$ \begin{array}{@{}rcl@{}} D&=&{q^{2}r^{2}(\mathbf{q}-\mathbf{r})^{2}+4m^{2}[q^{2}r^{2}-(\mathbf{q}\cdot \mathbf{r})^{2}]\over4m^6} . \end{array} $$

(65)

The integral (61) can be calculated numerically by averaging over the angles first and then integrating over p and q. We get,

$$ \begin{array}{@{}rcl@{}} I_{\text{fish1}}(p=im)&=& {1\over16m^{2}(4\pi)^{4}}\left( {\mu\over2m}\right)^{8\epsilon} \left[ 4.6029 \right]. \end{array} $$

(66)

We denote the numerical constant inside the paranthesis by C_fish1 = 4.6029. Finally, we consider the second fish-like diagram and rewriting

$$ \begin{array}{@{}rcl@{}} I_{\text{fish2}}(p=im)&=& {\int}_{qr} {1\over(\mathbf{q}+\mathbf{p})^{2}+m^{2}}{1\over(\mathbf{r}+\mathbf{p})^{2}+m^{2}} |_{p=im}I_{\text{bub}}(q)I_{\text{tri}}(\mathbf{q},\mathbf{r}) \end{array} $$

(67)

The integral is finite in three dimensions but we must average over the angles between p and q, p and r, q and r. This is simplified somewhat since the direction of the external momentum p is fixed. All the angular averages as well the integrals over q and r must be done numerically [26]. This yields

$$ \begin{array}{@{}rcl@{}} I_{\text{fish2}}(p=im)&=& {1\over 16 m^{2}(4\pi)^{4}}\left( {\mu\over2m}\right)^{8\epsilon} \left[ 1.778 \right] . \end{array} $$

(68)

We denote the numerical constant inside the paranthesis by C_fish2 = 1.778.