APPENDIX

2.A Calculation Overview

In this appendix we derive the residues bλa(~λ^R(µ, ))in Eq. (2.47), λ^B_aµ^−∆a() =λ^R_a(µ, ) + b_λa(~λ^R(µ, ))

, (2.93)

that determine the beta functions at = 0 via Eq. (2.52):

β_λa(~λ^R)≡ −µ∂λ^R_a

∂µ = ¯∆_λaλ^R_a +ρ_λab_λa(~λ^R)−X

c

ρ_λcλ^R_c ∂bλa(~λ^R)

∂λ^R_c . (2.94) After establishing notation, we'll list the main results used in the main text.

Later sections provide algebraic details.

Setup

IdentifyS_E in Eqs. (2.42) - (2.44) with the bare actionS_Bby endowing all elds and couplings with bare (B) subscripts/superscripts. To simplify notation, we'll leave replica and avor indices implicit. Dene renormalized (R) elds and couplings,

ψ_B =Z_f^1/2ψ_R, a⁰_B =Z_a,0^1/2a⁰_R, a^j_B =Z_a,j^1/2a^j_R, λ^B_c =Z_c^−1/2λ^R_c (2.95) where the vector of couplings (either B or R)

~λ =g²

N_f, v, m, κ, w_x, σ_e, D_e, g_m, g₀, g_j,∆₀,∆_jT

. (2.96)

Separate SB into physical and counterterm actions:

S_B =S_p⁽¹⁾_hys+S_p⁽²⁾_hys+S_p⁽³⁾_hys+S_C⁽¹⁾_T +S_C⁽²⁾_T +S_C⁽³⁾_T (2.97) with

S_p⁽¹⁾_hys= Z

dτ d^Dx hψ¯R

γτ(∂^τ +igRµ^/2 pNf

a^τ_R) +vRµ^z−1γj(∂^j+igRµ^/2 pNf

a^j_R)

ψR

+µ^zm_Rψ¯_Rψ_R+ iκR

2 a_Rda_Ri ,

(2.98) S_p⁽²⁾_hys=

Z

dωd^Dk wRµ^z−1

2 a^T_R(−ω,−k) k²

|k|+f_R(ω, k)a^T_R(ω, k), (2.99) S_p⁽³⁾_hys=−1

2 Z

dτ dτ⁰d^Dxh

(g_m)_Rµ^2z−D( ¯ψ_Rψ_R)( ¯ψ_Rψ_R) + (g₀)_Rµ^2z−D( ¯ψ_Rγ⁰ψ_R)( ¯ψ_Rγ⁰ψ_R) + (g_j)_Rµ^2z−D( ¯ψ_Rγ^jψ_R)( ¯ψ_Rγ^jψ_R) + (∆₀)_Rµ^2z−2b_Rb_R+ (∆_j)_ReR·eR

i (2.100),

S_C⁽¹⁾_T = dτ d^Dxh ψ¯_R

γ_τ(∂^τ+ig_Rµ

pN_f a^τ_R)δ₁+v_Rµ^z−1γ_j(∂^j +ig_Rµ

pN_f a^j_R)δ₂ ψ_R +µ^zm_Rψ¯_Rψ_Rδ_m+ iκ_R

2 a_Rda_Rδ_κi ,

(2.101) S_C⁽²⁾_T =

Z

dωd^Dk w_Rµ^z−1δ_w

2 a^T_R(−ω,−k) k²

|k|+fR(ω, k)a^T_R(ω, k), (2.102) S_CT⁽³⁾ =−1

2 Z

dτ dτ⁰d^Dxh

(g_m)_Rµ^2z−Dδ_gm( ¯ψ_Rψ_R)( ¯ψ_Rψ_R) + (g₀)_Rµ^2z−Dδ_g0( ¯ψ_Rγ⁰ψ_R)( ¯ψ_Rγ⁰ψ_R) + (g_j)_Rµ^2z−Dδ_gj( ¯ψ_Rγ^jψ_R)( ¯ψ_Rγ^jψ_R) + (∆₀)_Rµ^2z−2δ_∆0b_Rb_R+ (∆_j)_Rδ_∆jeR·eR

i (2.103) ,

f_R = e²_∗(σ_e)_Rµ^z−1

|ω|+ (D_e)_Rµ^z−2k², (2.104) and the renormalization group scale µ enters in accord with the engineering dimensions listed in Table 2.2. The counterterms δ_X have poles in with co- ecients determined by the requirement that correlation functions of physical elds have no divergences as →0. We focus exclusively on the terms in δ_X proportional to 1/. Using Eq. (2.95) to impose Eq. (2.97), we relate the bare and renormalized couplings:

v^Bµ^1−z =v^R(1 +δ₂−δ₁), (2.105)

m^Bµ^−z =m^R(1 +δ_m−δ₁), (2.106)

κ^B =κ^R(1 +δ_κ), (2.107)

w^B_xµ^1−z =w_x^R(1 +δ_w), (2.108)

g_X^Bµ^D−2z =g_X^R(1−2δ₁+δ_gX), g_X ∈ {g_m, g₀, g_j}, (2.109)

∆^B₀µ^2−2z = ∆^R₀(1 +δ∆0), (2.110)

∆^B_j = ∆^R_j (1 +δ_∆j), (2.111)

σ^B_eµ^1−z =σ^R_e(1 +δ_σe), (2.112)

D^B_e =D_e^R(1 +δ_D). (2.113)

(2.114)

Thus, we can read o the residues:

b_v =v^R(δ₂−δ₁) (2.115)

b_m =m^R(δ_m−δ₁) (2.116)

b_κ =κ^Rδ_κ (2.117)

b_w =w_x^Rδ_w (2.118)

b_gX =−g_X^R(2δ₁−δ_gX), g_X ∈ {g_m, g₀, g_j}, (2.119) b_∆X = ∆^R_Xδ_∆X, ∆_X ∈ {∆₀,∆_j}, (2.120)

bσe =σ^R_eδσ, (2.121)

b_De =D_e^Rδ_D. (2.122)

2.B Counterterms

As discussed in the main text, we choose the dynamical critical exponent z in such a way that the fermion velocity v does not run, i.e., the velocity beta function is zero. In the expressions below, it's convenient to redene couplings to absorb the velocity dependence as follows:

g_m = g_m

2πv² , g₀ = g₀

2πv² , g_j = g_j

2πv² , ∆₀ = ∆₀ , ∆_j = ∆_jv² , w_x = w_x

v , σ_e = σ_e v ,

Z ∞

−∞

dzF(z) = Z ∞

−∞

v dy F(y = z

v), (2.123) where the function F(z) is introduced below. We also dene g₁ = g/16 and m=m/v.

Let us make a few remarks about the expressions below.

ˆ We use φ1 to parameterize the screening of the disorder described in Appendix2.C. φ₁ = 0 means the screening is ignored; φ₁ = 1 means the screening is included.

ˆ Terms proportional to ξ are divergent. However, this is an unphysical divergence due to our gauge choice: This divergence does not appear in physical quantities such as critical exponents.

ˆ The Ward identity guarantees the gauge eld corrections to δ₁ and δg₀ cancel; the ones in δ₂ and δ_gj likewise cancel. In the absence of the Coulomb interaction, the equality of the gauge corrections inδ1 andδ2 is a coincidence, which makesβ_g0, β_gj independent of the gauge corrections.

When the Coulomb interaction is included,β_g0 receives corrections from the gauge eld, while β_gj does not.

δ_κ, δ_w, δ_σ, and δ_D Counterterms

Quantization of the Chern-Simons level and niteness of the gauge eld self- energy in 3d implies

δκ =δw =δσ =δD = 0. (2.124) Consequently, renormalizations of κ, w_x, σ_e, and D_e are controlled by their engineering dimensions.

δ₁ Counterterm

The diagrams that contribute to δ₁ are given by taking the temporal compo- nent of Fig. 2.B.1:

δ₁=−(g_m+g₀+ 2g_j) + φ₁

g²₁w_x (g₁g_j−g₁g₀−g₀ w_x)

(g²₁+g₁w_x+κ²)² +g₁ (g₁g_j +g₁g₀+ 2g₀ w_x) g₁²+g₁w_x+κ²

+ Z ∞

−∞

1 4π²N_f

(1−y²)(g₁σ_ey²+w_xp

1 +y²|y|) +g₁|y|y²(1−y²ξ) (1 +y²)²hp

1 +y²(g₁²+κ²)(σ_e+|y|) +g₁w_x|y|i . (2.125) δ₂ Counterterm

The diagrams that contribute to δ₂ are given by taking the spatial component of Fig. 2.B.1:

δ2=−φ1

g₁²

g²₁g_j + 2g₁g_jw_x−(g₀−2g_j)κ² (g₁²+g₁w_x+κ²)²

+ Z ∞

−∞

dy 1 4π²N_f

g₁(1−y²−y⁴)σ_e−(w_xy²p

1 +y²)|y|+g₁(1−y²−y⁴ξ)|y|

(1 +y²)²hp

1 +y²(g₁²+κ²)(σ_e+|y|) +g₁w_x|y|i . (2.126)

Figure 2.B.1: Diagrams contributing to δ₁, δ₂.

Figure 2.B.2: 2-PI diagrams contributing to δ_gm, δ_g0, δ_gj.

δ_gm Counterterm

δ_gm is extracted from the diagrams in Figs. 2.B.2 and 2.B.3:

δ_gm=

2(g₀+g_m)(g_m−2g_j)

g_m + 2 g₁(g_j −g₀) +g_jw_x g₁²+g₁w_x+κ²

+φ₁

4g₁ g₁κ²(−g₀²−g_j²) +g_jg₀

2g₁³+ 4g²₁w_x + 2w_xκ²+g₁(w_x²+ 4κ²) g_m [g₁²+g₁w_x+κ²]²

+φ₁

g⁵₁(g₀−g_j) +g⁴₁w_x(2g₀−3g_j) +g₁³(g₀w_x²−2g_jw_x²+ 3g₀κ²−3g_jκ²) [g₁²+g₁w_x+κ²]³

+g₁²κ²(3g₀w_x) + 2g₁κ⁴(g_j −g₀) [g₁²+g₁w_x+κ²]³

+φ₁

2g₁⁴(g_j −g₀) + 4g₁³w_x(g_j−g₀) +g₁²(−2g₀w_x²−6g₀κ²+ 6g_jκ²)−4g₁g₀w_xκ² [g²₁+g₁w_x+κ²]²

+φ²₁

"

4g₁²g_j²κ²[g²₁(g²₁+ 2g₁w_x−w_x²) + 2g₁(g₁−w_x)κ²−κ⁴ ] gm[g₁²+g1wx+κ²]⁴

+4g²₁g₀²κ²[g²₁(g²₁+ 2g₁w_x+w_x²) + 2g₁(g₁+w_x)κ²−κ⁴] g_m[g²₁+g₁w_x+κ²]⁴

+4g₀ g_jg₁²

−g³₁(g₁+w_x)²(g₁+ 2w_x)−2g²₁(g₁+w_x)(2g₁+ 3w_x)κ²−g₁(5g₁+ 2w_x)κ⁴+ 2κ⁶ gm[g₁²+g1wx+κ²]⁴

#

− Z ∞

−∞

dy

g₁(y²+ 2)(1 +y²)σ_e+

w_x(1 +y²)p

1 +y²+g₁(2 + 3y²+y⁴ξ)

|y|

2π²Nf(1 +y²)² hp

1 +y²(g₁²+κ²)(σe+|y|) +g1wx|y|i

+ Z ∞

−∞

dy

(σ_e+|y|)

(1 +y²)(g²₁ −κ²) (σ_e+|y|) + (g₁w_xp

1 +y²)|y|

4π²N_f(1 +y²)³²hp

1 +y²(g²₁+κ²)(σ_e+|y|) +g₁w_x|y|i2 . (2.127)

Figure 2.B.3: The mass vertex contributions to δgm

δm Counterterm

δ_m is extracted from the diagrams in2.B.3:

m δm = 1 2gm

δgm−[2-PI boxes]

=g₀−2g_j +g_m− g₁(g₀ −g_j)−g_jw_x

2(g²₁+g₁w_x+κ²) +φ₁

− g₁(g₀−g_j) + 3g₁(g₀−g_j) + 2g₀w_x g₁²+g₁w_x+κ²

−g²₁ g_j[4g₁²+ 4w_x+g₁(7 + 2w_x) ]−g₀[4g₁²+w_x(7 + 2w_x) +g₁(7 + 6w_x)]

2(g₁²+g₁w_x+κ²)²

− −2g₁³(g1+wx)[−g1gj+g0(g1+wx) ] (g²₁ +g₁w_x+κ²)³

− Z ∞

−∞

dy

g₁(y²+ 2)(1 +y²)σ_e+

w_x(1 +y²)p

1 +y²+g₁(2 + 3y² +y⁴ξ)

|y|

4π²N_f(1 +y²)²h

p1 +y²(g²₁+κ²)(σ_e+|y|) +g₁w_x|y|i

+ Z ∞

−∞

dy

(σ_e+|y|)

(1 +y²)(g₁²−κ²) (σ_e+|y|) + (g₁w_xp

1 +y²)|y|

8π²N_f(1 +y²)³² hp

1 +y²(g²₁ +κ²)(σ_e+|y|) +g₁w_x|y|i2 , (2.128) where [2-PI boxes] refers to the contributions to δgm from the diagrams in Fig. 2.B.2.

δ_g0 Counterterm

δ_g0 is extracted from the diagrams in Figs. 2.B.2 and 2.B.4:

δ_g0= −2(g0+ 2gj) (g0+gm) g₀

+φ₁

4g₁²g_j g_m (g²₁ + 2g₁w_x+ 2κ²)

g₀ [g²₁+g₁w_x+κ²]² + 2g²₁ g₁²g_j + 2g₁g_jw_x+ (g_j −2g_m)κ² [g²₁ +g₁w_x+κ²]²

+2g₀g₁ g³₁+ 2g₁²w_x+ 2κ²w_x+g₁w_x²+g₁κ² [g²₁+g₁w_x+κ²]²

+ Z ∞

−∞

dy g1y²(1−y²)σe+g1y²(1−y²ξ)|y|+ (1−y²)wx

p1 +y²|y|

2π²N_f(1 +y²)²h

p1 +y²(g₁²+κ²)(σ_e+|y|) +g₁w_x|y|i . (2.129)

Figure 2.B.4: γ_µ=0 vertex component contributions to δ_g0

Figure 2.B.5: γ_µ=j vertex component contributions to δ_gj

δgj Counterterm

δ_gj is extracted from the diagrams in Figs. 2.B.2and 2.B.5:

δ_gj= −2g0 gm

g_j +φ₁

2g²₁(g0−gm)κ² [g₁²+g₁w_x+κ²]²

+φ₁

2g₁ g₀ g_m

g³₁+ 2g₁²w_x+ 2w_xκ²+g₁w_x²+ 2g₁κ² [g²₁+g1wx+κ²]²

+ Z ∞

−∞

dy 1 2π²N_f

g₁(1−y² −y⁴)σ_e−(w_xy²p

1 +y²)|y|+g₁(1−y²−y⁴ξ)|y|

(1 +y²)²hp

1 +y²(g²₁+κ²)(σ_e+|y|) +g₁w_x|y|i(2.130).

Figure 2.B.6: Diagrams contributing to δ_∆0, δ_∆j

δ_∆0 Counterterm

δ_∆0 is extracted from the diagram in Fig. 2.B.6:

δ∆0= −gm(g₁²∆j+ 2g1wx∆j +wx2∆j+ ∆0κ²) 64∆0 (g²₁ +g1wx+κ²)²

+h −g₀g_mN_fπ v²

32∆₀ +φ₁ g₁g_mN_fπ v²[−g₁g_jκ²+g₀(g₁+w_x)(g₁²+g₁w_x+ 2κ²) ] 32∆₀(g₁²+g₁w_x+κ²)²

i (2.131). δ_∆j Counterterm

δ_∆j is extracted from the diagram in Fig.2.B.6:

δ_∆j = −g_m(g₁²∆₀+ ∆_jκ²) 128∆_j (g₁²+g₁w_x+κ²)² +h −g_jg_mN_fπv²

64 ∆_j +φ₁ g₁²g_mN_f πv²[g²₁g_j+ 2g₁g_jw_x−(g₀−2g_j)κ² ] 64∆_j(g₁²+g₁w_x+κ²)²

i (2.132). 2.C Feynman Rules for Disorder and Screening

Feynman Rules for Disorder Vertices

From the action (2.100), we can read the Feynman rules for the various types of disorder.

4-fermion mass vertex:

1

2( ¯ψψ)( ¯ψψ) ⇒ +gm2πδ(ω= 0) . (2.133) 4-fermion density vertex:

1

2( ¯ψγ⁰ψ)( ¯ψγ⁰ψ) ⇒ (g₀)(γ⁰)(γ⁰)2πδ(ω = 0). (2.134) 4-fermion current vertex along the k-direction, k =xor y:

1

2( ¯ψiγ^kψ)( ¯ψiγ^kψ) ⇒ (+g_j)(iγ^k) (iγ^k)2πδ(ω = 0). (2.135)

b(τ)b(τ⁰)disordered 2-pt vertex rule:

1

2b_zb_z ⇒ ∆₀(δ_ijk² −k_ik_j)2πδ(ω)a_i(k)a_j(−k). (2.136) e_j(τ)e_j(τ⁰) disordered 2-pt vertex:

1

2(e_xe_x+e_ye_y) ⇒ (−∆_jc²) (k_x²+k²_y) 2πδ(ω)a₀(k)a₀(−k).(2.137) The factor of ¹₂ factor is canceled by symmetry factor equal to two. The 2π factor always cancels with the 1/2π that accompanies any frequency integral R _dω

2π.

Gauge Propagator

Vacuum Polarization Tensor

Π_µν(k₀,k) = ( −ig pN_f)²(v

c)^{2−δµ0−δν0} ×N_f ×(−1)

Z d²p_E

(2π)³T r[γ_µS_F(k+p)γ_νS_F(p)] (2.138)

= (+g²)(−i)² (v

c)^{2−δµ0−δν0} 1 v²

Z d²p¯_Edp₀

(2π)³ T r[γ_µγ_αγ_νγ_β] (¯k+ ¯p)_α(¯p)_β

[x(¯k+ ¯p)²+ (1−x)¯p²]², (2.139)

Π_µν(k₀,k) = −1 16

g² v²(v

c)^{2−δµ0−δν0} 1

|k|¯ [δ_µν¯k²−¯k_µk¯_ν] , ¯k= (ω, vk) , |¯k|=p

ω²+v²k². (2.140) The minus sign comes from the fermion loop. The ratio v/ccan be set to v in

future equations.

1-µ "Vacuum Polarization Vector

Π_µ(k₀,k) =√

g_m( −ig pN_f)(v

c)^1−δµ0 ×N_f ×(−1)

Z d²p_E

(2π)³T r[γ_µS_F(k+p)1S_F(p)] (2.141)

=√

g_m (+ig)(−i)² (v

c)^1−δµ0 1 v²

Z d²p¯_Edp₀

(2π)³ T r[γ_µγ_α γ_β] (¯k+ ¯p)_α(¯p)_β

[x(¯k+ ¯p)²+ (1−x)¯p²]² = 0. (2.142) The momentum part is proportional to δ_αβ p², while the trace is proportional

to the _µαβ tensor, and so it vanishes.

1-1"Vacuum polarization scalar Π_m(k₀,k) = (√

g_m)²N_f ×(−1)

Z d²p_E

(2π)³T r[1S_F(k+p)1S_F(p)](2.143)

=g_m 1 v²

Z d²p¯_Edp₀

(2π)³ T r[γ_α γ_β] (¯k+ ¯p)_α(¯p)_β

[x(¯k+ ¯p)²+ (1−x)¯p²]² = −|k|g_m 8v²(2.144). Although nonzero, when connecting external fermion lines, the resulting dia- gram would be proportional to the number of replicas n_R and vanish in the n_r →0 limit.

Eective Gauge Propagator

In Coulomb gauge (longitundal component a_L = 0), the kinetic term for the gauge eld is

S_gauge= 1 2

Z

dk²dω a₀ a_T

^g

2

16 k²

√ω²+v²k² iκ|k|

iκ|k| wx k²

|k|+f(ω,k) +^g₁₆²√

ω²+v²k²

! a₀ aT

!

, (2.145) where (k² ≡ |k|²). Recall that g₁ ≡ ^g₁₆² = ₁₆¹, the eective coulomb coupling

w_x ≡ ^+e_4π2², and f(k, ω) ≡ _|ω|+D^σek2

ek². The transverse component of the gauge eld is aT(k, ω)≡iˆkxay(k, ω)−iˆkyax(k, ω), whereˆkj =kj/|k|.

When dealing with the gamma matrix contraction in Feynman diagram calcu- lations, we have to write the eective gauge propagator obtained from S_gauge in the a₀, a_x, a_y basis (i, j =x, y):

G₀₀= 1 k²

g₁ p

ω²+v²k² + F(k, ω) g²₁+κ²+ √^g1F(k,ω)

ω²+v²k²

(2.146)

G_0i = κ k²

−_ijk_j g₁²+κ²+√^g1F(k,ω)

ω²+v²k²

, G_i0 =−G_0i = κ k²

+_ijk_j g²₁+κ²+ √^g1F(k,ω)

ω²+v²k²

(2.147)

G_ij = (δ_ij −k_ik_j k² )

g1

√

ω²+v²k²

g²₁+κ²+ √^g1F(k,ω)

ω²+v²k²

(2.148)

where F(k, ω)≡ +e² 4π²

|k|²

|k|+f(k, ω) = +e² 4π²

|k|²

|k|+ _|ω|+D^σek2

ek²

critical limit,De=0

−−−−−−−−−−→ +e² 4π²

|k|²

|k|+ ^σe_|ω|^k2 (2.149) Screened Disorder W_µν

The disorders g₀, g_j are screened by the fermion polarization. The Feynman rules in (2.134)-(2.135) have to be adjusted to account for this screening:

Wµν =W_µν⁽⁰⁾+φ1W_µν^(sc), (2.150)

whereWµν⁽⁰⁾ =Diag(g₀, i²g_j, i²g_j)is the bare part in (2.134),(2.135) andWµν^(sc)

is the screening part from the summation of fermion bubbles.

The prefactor φ₁ isolates the screened and un-screened contributions: φ₁ = 0 means that disorder screening is ignored; φ₁ = 1means that disorder screening is included. When disorder connects with the gauge propagator, we should set σ_e = 0 before setting ω = 0 (due to the presence of the δ(ω)) factor. Other- wise, there is no disorder screening. Note that the vertex factors are included inW_µν, so when applying the Feynman rules, we only need to multiply by γ_µ without any constant or velocity factor.

We separate the screening part into symmetric and antisymmetric components:

W^(sc) =W^sym+W^as, (2.151)

W₀₀^sym =g₁ −g₁g_jκ²+g₀(g₁+w_x) (g₁²+g₁w_x+ 2κ²)

(g²₁+g₁w_x+κ²)² , (2.152) W_ij^sym = g₁²(g₁²g_j + 2g₁g_jw_x−(g₀−2g_j)κ²)

(g₁²+g₁w_x+κ²)²

1

k²(k²δ_ij −k_ik_j),(2.153) W_0i^as = g₁κ(g₁g_jw_x+ (−g₀+g_j)κ²)

[g₁²+g₁w_x+κ²]²

−_ijk_j

k . (2.154)

Other components ofW^(sc) not included above vanish.

Eective Gauge Disorder D^dis_µν

The expressions in (2.136) and (2.137) 2-point vertex rules: each side of the vertex connects with dressed propagator found in (2.146)-(2.148). The eective gauge disorder is dened by

D^dis_µν =G_µαD^0,dis_αβ G_βν, (2.155) where D₀₀^0,dis = −∆_jk² dened in (2.137), D^0,dis_ij = ∆₀(δ_ijk² −k_ik_j) dened in (2.136), and D_0i^0,dis = D_i0^0,dis = 0. We decompose D_µν^dis into symmetric and

antisymmetric components:

D^dis_µν =D^S_µν+D^AS_µν, (2.156)

D^S₀₀=−g₁²v²∆_j+ 2g₁v²w_x∆_j +v²w_x²∆_j + ∆₀κ²

(g₁²+g₁w_x+κ²)² , (2.157) D^S_ij = (g1²∆₀+v²∆_jκ²)

v²(g₁²+g₁w_x+κ²)²

k²δ_ij −k_ik_j

k² , (2.158)

D^AS_0i = κ (−g₁∆₀+g₁v²∆_j+v²w_x∆_j) v (g₁²+g1wx+κ²)²

_ijk_j

k . (2.159)

Components of D^dis not listed above are zero.

SinceG_µν is constructed by the RPA sum of fermion loops, G_µν can no longer connect any more fermion loop. Consequently, D_µν^dis does not include any fermion loops. Note that D_µν^dis generates ^∆_N^0,j_f . This disorder renormalizes

∆0,∆j at 3-loop order.

2.D Fermion Self-energy

Self-energyScreened Disorder Correction Wµν

Σ_d(p₀,p) =

Z d²k

(2π)² (1_2×2)S_F(p₀−0,p−k) (1_2×2)g_m+

Z d²k

(2π)²γ_νS_F(p₀−0,p−k)γ_µW_µν(k)

=

Z d²k

(2π)² (1_2×2)S_F(p₀−0,p−k) (1_2×2)g_m+

Z d²k (2π)²γ_ν

(+i)γ₀p₀+v(p−k)_iγ_i p²₀+v²(p−k)²

γ_µW_µν(k)

= +i v²

p₀γ₀

2π [g_m] +

Z d²k (2π)²γ_ν

(+i)γ₀p₀+v(p−k)_iγ_i p²₀+v²(p−k)²

γ_µ W_µν(k) (2.160)

= +ip0γ0

2πv² g_m+ +ip0γ0

2πv² (g₀ + 2g_j) +φ₁

g²₁w_x (g₁g_j−g₁g₀−g₀ w_x)

(g²₁+g₁w_x+κ²)² +g₁ (g₁g_j +g₁g₀+ 2g₀ w_x) g₁²+g₁w_x+κ²

−i p₀γ₀ +φ₁ (−1)g²₁

g₁²gj + 2g1gjwx−(g0−2gj)κ² (g₁²+g₁w_x+κ²)²

−i

vp_jγ_j. (2.161)

Self-energyGauge Correction

Only the symmetric part of the gauge propagator produces a divergence at O(_N¹

f).

Σ_g(p₀,p) = ( −ig pN_f)² v

c

2−δµ0−δν0

Z d³k

(2π)³ γ_νS_F(p−k)γ_µG_µν(k₀,k) (2.162)

= −g²

N_f (+i) v c

2−δ_µ0−δ_ν0 1 v²

Z d²¯k dk₀ (2π)³

γ_ν(p₀−k₀)γ₀+ (¯p−¯k)_aγ_a (p₀−k₀)²+ (¯p−¯k)² γ_µ

G_µν(2.163)(k₀,¯k).

Carrying out the momentum integral and setting c= 1:

−Σ_g(p₀,p)

= Z ∞

−∞

dz ig²

(v² −z²)g₁ z²σ_e+

g₁z²v²−g₁z⁴×ξ+w_x√

v²+z²(v²−z²) z

(p₀γ₀) 4 N_f π²(v² +z²)²

√

v²+z² g₁²+κ²

σ_e+|z|

(g₁²+κ²)√

v² +z²+ g₁w_x

+ Z ∞

−∞

dz ig²

g₁(v⁴−v²z²−z⁴)σ_e−

w_xz²√

v²+z²+ g₁(−v⁴+v²z²+z⁴×ξ) z

(vp_jγ_j) 4 N_f π²(v²+z²)²

√

v² +z² g₁²+κ²

σ_e+|z|

(g²₁+κ²)√

v²+z²+ g₁w_x (2.164)

To obtain the above expression, we rst perform a gradient expansion of Σ(p0,p) around p0 = p = 0. Next, focus on the linear term in p0,p and replace the frequency integralk₀ →z|k|. When the 2d spatial momentum in- tegral is done, the result is the expression shown above. The above expression is integrable only at σ_e = 0. The ξ term is a divergent integral that arises from the choice of Coulomb gauge. Physical observables are free from any ξ dependence.

Self-EnergyEective Gauge Disorder D_µν^dis, O(^∆_N^X

f ) Σ_b(p₀,p) = (−i g

pN_f)² v c

2−δµ0−δν0

Z d²k

(2π)² γ_νS_F(p₀−0,p−k)γ_µD(2.165)_µν^dis(k)

= +i(∆₀+v²∆_j)

2N_fπv²(g₁²+κ²) γ₀p₀ +−i(g₁²∆₀+κ²v²∆_j)

2N_fπv²(g₁²+κ²)² (v γ_jp_j). (2.166) 2.E 3-point Vertex u(q)¯ δΓ^µ u(p)

Γ^µGauge Correction

Γ^µ₁ = −ig pN_f

3 v c

3−δα0−δ_β0−δµ0

Z d²kdω

(2π)³ γ_αS_F(q−k)γ_µS_F(p−k)γ_βG_βα(k, ω) (2.167)

= −ig pN_f

3 v c

3−δ_α0−δ_β0−δ_µ0 1 v²

×

Z d²¯kdω

(2π)³ γ_α(+i)γ0(q0−k0) +γc(¯q−¯k)c

(q0−k0)²+ (¯q−¯k)² γ_µ(+i)γ0(p0−k0) +γd(¯p−k)¯ d

(p0−k0)²+ (¯p−¯k)² γ_βG_βα(k, ω) (2.168) To isolate the divergent part, one can set the external momentum p =q= 0.

Following the same steps we used in the self energy diagram evaluation, we obtain

Γ^µ₁ = −ig pN_f

(−g²)

(v²−z²)g₁ z²σ_e+

g₁z²v²−g₁z⁴×ξ+w_x√

v²+z²(v²−z²) z

4 N_f π²(v²+z²)² √

v²+z² g²₁+κ²

σ_e+|z|

(g²₁+κ²)√

v²+z²+ g₁w_x(γ₀)

+ −ig pN_f

v 1

(−g²)

g₁(v⁴−v²z²−z⁴)σ_e−

w_xz²√

v²+z²+g₁(−v⁴+v²z²+z⁴×ξ) z

4 N_f π²(v² +z²)² √

v²+z² g₁²+κ²

σ_e+|z|

(g₁²+κ²)√

v² +z²+ g₁w_x (γ_j).

(2.169) As before, ξ labels the divergent part. Gauge invariance is easy to check by

comparing with Eq. (2.164): Γ^t₁ = √^−g

Nf

∂Σg

∂p0 , Γ^j₁ = √^−g

Nf

∂Σg

∂pj. Γ^µEective Gauge Disorder Correction

Γ^µ₂ = −ig pN_f

3 v c

3−δα0−δ_β0−δµ0

Z d²k

(2π)² γ_αS_F(q−k)γ_µS_F(p−k)γ_βD(2.170)_βα^dis(k)

= −i(∆₀+v²∆_j)

2N_f^3/2πv²(g₁²+κ²) γ⁰ + i(g²₁∆₀+v²κ²∆_j)

2N_f^3/2πv(g₁²+κ²)² γ^j. (2.171) Γ^µScreened Disorder Correction W_µν

Γ^µ₃ = ( −ig pN_f) v

c 1−δµ0

×

Z d²k

(2π)²γ_αS_F(q−k)γ_µS_F(p−k)γ_βW_βα (2.172)

= −ig pN_f

1

g₀+ 2g_j 2πv² +φ1

g₁²w_x (−g₁g_j +g₁g₀+g₀ w_x)

(g²₁+g₁w_x+κ²)² − g₁ (g₁g_j +g₁g₀ + 2g₀ w_x) g₁²+g₁w_x+κ²

γ⁰

+ −ig pN_f φ₁ 1

g₁²

g²₁g_j+ 2g₁g_jw_x−(g₀−2g_j)κ² (g₁²+g1wx+κ²)²

γ^j. (2.173)

Γ^µ-Random Mass Correction g_m

Γ^µ₄ = ( −ig pNf

) v c

1−δµ0

×

Z d²k

(2π)²1S_F(q−k)γ_µS_F(p−k)1g_m = −ig pNf

gm

2πv²γ⁰+ 0γ^j. (2.174) 2.F 3-point Vertex u(q) 1¯ 2×2 u(p)

12×2Gauge Correction

Γ^m₁ = ( −ig pN_f)²(v

c)^{2−δα0−δβ0} 1 v²(+i)²

×

Z d²¯kdω

(2π)³ γ_αγ₀(q₀−ω) +γ_c(¯q−k)¯ _c

(q₀ −ω)²+ (¯q−¯k)² 1 γ₀(p₀−ω) +γ_d(¯p−¯k)_d

(p₀−ω)²+ (¯p−k)¯ ² γ_β G_αβ(2.175).

Γ^m₁ =

(v²+z²) g²

g₁(2v²+z²)σ_e+w_x√

v² +z²+g₁(2v²+z²)|z|

4 N_f π²(v²+z²)² √

v²+z² g²₁+κ²

σ_e+|z|

(g₁²+κ²)√

v²+z²+ g₁w_x 12×2

= g²

g₁(2v²+z²)(v²+z²)σ_e+w_x(v²+z²)√

v²+z²+g₁(2v⁴+ 3z²v²+ξ z⁴)|z|

4 N_f π²(v² +z²)² √

v²+z² g₁²+κ²

σ_e+|z|

(g₁²+κ²)√

v² +z²+ g₁w_x 12×2(2.176). 12×2Eective Gauge Disorder Correction

Γ^m₂ = −ig pN_f

2 v c

2−δα0−δ_β0Z d²k

(2π)² γ_αS_F(q−k)1S_F(p−k)γ_βD^dis_βα(k)

= (∆₀−v²∆_j)(g₁²−c⁴κ²)

2πN_fv²(g₁²+c⁴κ²)² 12×2. (2.177)

1_2×2Screened Disorder Correction W_µν

Γ^m₃ =

Z d²k

(2π)²γ_αS_F(q−k)1S_F(p−k)γ_βW_βα

=

−g₀+ 2g_j

2πv² +φ₁1

−g²₁(2g₁ +w_x)(−g₁g_j +g₀g₁+g₀w_x

(g²₁+g₁w_x+κ²)² +g₁(3g₀g₁−3g₁g_j+ 2g₀w_x) g²₁+g₁w_x+κ²

12×2. (2.178) 12×2Random Mass Correction g_m

Γ^m₄ = ( −ig pN_f)×

Z d²k

(2π)²1S_F(q−k)1S_F(p−k)1g_m = −g_m

2πv² 12×2(2.179). 2.G 4-point Fermion-Fermion Interaction

Dene

H_µν ≡D^dis_µν( −ig pN_f)²(v

c)^{2−δµ0−δν0}. (2.180) Note that W_µν ∼ O(_N¹0

f

)and H_µν ∼ O(_N¹

f). Take the external three momenta to be p₁, p₂, p₃, p₄, where p_i = (ω_i,p_i). Schematically, the interaction has the form, [ψ(p₃)...ψ(p₁)] [ψ(p₄)..ψ(p₂)]. Dene:

Γ_A≡(γ₇, γ₀,+γ_x,+γ_y) , γ₇ ≡12×2 , T^A1 = (1g_m, W_µν, H_µν) , T˜^A1 = (1g_m, W_µν⁽⁰⁾, H_µν(2.181))

W_µν⁽⁰⁾ =Diag(g₀, i²g_j, i²g_j) (2.182)

We use A, B, C, D ={1,2,3,4} indices to label 1, γ₀, γ_x, γ_y and number sub- scripts, e.g., A₁, A₂, to label which interaction we choose: A₁ = 1 for the g_m interaction; A₁ = 2 for the W_µν interaction; A₁ = 3 for the H_µν interaction.

The diagrams below correspond to the following expressions:

B₁ =

Z d²k

(2π)²ψ(p₃)Γ_BS_F(p₁+k)Γ_Aψ(p₁) ψ(p₄)Γ_DS_F(p₂ −k)Γ_Cψ(p₂)

×T_CA^A1(k, ω= 0)T_BD^A2(p₁+k−p₃, ω = 0), (2.183) B₂ =

Z d²k

(2π)²ψ(p₃)Γ_BS_F(p₁+k)Γ_Aψ(p₁) ψ(p₄)Γ_CS_F(p₄+k)Γ_Dψ(p₂)

×T_CA^A1(k, ω= 0)T_BD^A2(p₁−p₃+k, ω = 0), (2.184) B₃ =

Z d²k

(2π)²ψ(p₃)Γ_DS_F(p₃−k)Γ_AS_F(p₁−k)Γ_Bψ(p₁)

×T_BD^A2(k, ω = 0), ψ(p₄)Γ_Cψ(p₂) ˜T_AC^A1(p₃−p₁, ω = 0), (2.185) B₄ =

Z d²k

(2π)²ψ(p₃)Γ_Aψ(p₁) ˜T_AC^A1(p₃−p₁, ω = 0)

×ψ(p₄)Γ_DS_F(p₄−k) Γ_CS_F(p₂−k)Γ_Bψ(p₂)T_BD^A2(k, ω= 0).

(2.186)

For diagrams B₃, B₄, the T_A1 vertex is un-dressed, i.e., W_µν⁰ , which is directly related to the random coupling being renormalized.

4-point InteractionBoxes B3, B4

Diagrams of type B₃, B₄ can be directly obtained from the 3-point vertex corrections in Appendices (2.E) and (2.F) with symmetry factor 2 (counting upper or lower vertices), so we don't have to recompute them at here. The terms in Γ^µ renormalizeg₀, g_j and the terms in Γ^m renormalizeg_m.

4-point Fermion InteractionBoxes B₁, B₂ Diagrams for boxes B1, B2 are presented below.

The W-H diagrams areO(^∆X_N^gX

f ). The H-H diagrams are O(^∆_N^2X2 f

).

For each interaction, ( ¯ψψ)( ¯ψψ), ( ¯ψγ0ψ)( ¯ψγ0ψ), ( ¯ψiγjψ)( ¯ψiγjψ), we sum all these diagrams with the help of computer software to O(g_X²,^g_N^X

f). The contri- bution from diagrams B₁, B₂ are the following.

BOX11=

"

2g₀g_j

πv² g_m +φ₁(−4g₁)g₁g₀ g_j(2g²₁ + 4g₁w_x+w_x²)−κ²[g₀²g₁+g₁g_j²−2g₀ g_j(2g₁ +w_x)]

(g²₁+g₁w_x+κ²)²

−φ²₁

4g²₁g_j²κ²[g₁²(g₁²+ 2g₁w_x−w_x²) + 2g₁(g₁ −w_x)κ²−κ⁴ ] g_m[g₁²+g₁w_x+κ²]⁴

+4g²₁g₀²κ²[g²₁(g²₁+ 2g₁w_x+w_x²) + 2g₁(g₁+w_x)κ²−κ⁴] g_m[g²₁+g₁w_x+κ²]⁴

+4g₀ g_jg₁²

−g³₁(g₁+w_x)²(g₁+ 2w_x)−2g²₁(g₁+w_x)(2g₁+ 3w_x)κ²−g₁(5g₁+ 2w_x)κ⁴+ 2κ⁶ gm[g₁²+g1wx+κ²]⁴

#

×( ¯ψ1ψ)( ¯ψ1ψ). (2.187)

BOXγ0γ0 =

"

2g_jg_m πv² g₀ −φ1

4g₁²g_m[g²₁g_j+ 2g₁g_j w_x−(g₀−2g_j)κ²] g₀[g²₁ +g₁w_x+κ²]²

#

( ¯ψγ0ψ)( ¯ψγ0ψ). (2.188)

BOXγjγj =

"

2g₀g_m πv²g_j −φ1

g₁g_m[−g₁g_jκ²+g₀(g³₁+ 2g₁²w_x+ 2w_xκ²) +g₁(w_x²+ 2κ²) ] g_j[g²₁+g₁w_x+κ²]²

#

( ¯ψiγjψ)( ¯ψiγjψ).

(2.189) As mentioned before, the index j =x or y; there is no index sum here. And

we assume the random current disorder variance gx=gy ≡gj(isotropic).

2.H 2-loop Vertex Corrections

At leading order, the generic two-loop diagram has the form pictured below.

The interaction legsX₁ and X₂ can be chosen to be the gauge propagator G_µν or disorder E_µν ∈ {W_µν, g_m}. In principal there are four possibible choice:

(X₁, X₂) = (G, G),(E, G),(G, E), or (E, E). In the replica limit n_r → 0, the (E, E) diagram vanishes because the fermion bubble is proportional to n_R. Also, (E, G) and (G, E) are the same diagrams so we only need to compute one of them. The top vertex can be eitherγ^µor12×2. However, we'll see below that diagrams using theγ^µ vertex are zero.

Mass Vertex u12×2uone leg gauge, one leg disorder

Γ_X1 = 1 v²

Z d²qdq¯ ₀ (2π)³

1 v²

Z d²k¯

(2π)² u(p₃) 1 γβ

!

S_F(p₁−k)[ −ig pN_f(v

c)^1−δµ0γ_µ]u(p₁)G_µν(k) (−1)×N_f

×T r

[ −ig pN_f(v

c)^1−δν0γ_ν]S_F(q−p₁+p₃)1S_F(q) 1 γ_α

!

S_F(k+q−p₁+p₃)

g_m

W_αβ(k−p₁+p₃)

! . (2.190)

Γ˜_X1 = 1 v²

Z d²qdq¯ ₀ (2π)³

1 v²

Z d²k¯

(2π)² u(p₃) 1 γ_β

!

S_F(p₁−k)[ −ig pN_f(v

c)^1−δµ0γ_µ]u(p₁)G_µν(k) (−1)×N_f

×T r

[ −ig pN_f(v

c)^1−δν0γ_ν]S_F(q−k) 1 γ_α

!

S_F(q−p₁+p₃)1S_F(q)

g_m

W_αβ(k−p₁+p₃)

! . (2.191)

The direction of the fermionic loop momenta is dierent in Γ and Γ˜. We use the upper/lower components to distinguish the diagrams that arise from either g_m/W_µν.

To extract the UV divergence, we can setp₁ =p₃ = 0. Forg_m, the divergences inΓ_X1 andΓ˜_X1 cancel (upon changing variablesq→ −qinΓ˜_X1 and using basic properties of the trace). For W_αβ, Γ_X1 and Γ˜_X1 have identical divergences.

Γ_X1 = −g² 1 (v

c)^{2−δµ0−δν0}

Z d³k (2π)³u(p¯ ₃)

γ_β k₀γ₀+v k_cγ_c k₀²+v²k² γ_µ

u(p₁)

× G_µν(k) ×W_αβ(k)×

Z d³q (2π)³ T r

γ_ν q₀γ₀+v q_dγ_d

q₀²+v²q² 1 q₀γ₀+v q_eγ_e

q²₀+v²q² γ_α (k₀+q₀)γ₀+v(k_f +q_f)γ_f (k0+q0)²+v²(k+q)²

.(2.192) Refer to the calculations in (2.204) to compute

Z d³q (2π)³T r

γ_ν q₀γ₀+v q_dγ_d q₀²+v²q²

q₀γ₀+v q_eγ_e

q²₀+v²q² γ_α (k₀+q₀)γ₀+v(k_f +q_f)γ_f (k0+q0)²+v²(k+q)²

= i _νασ (k₀, vk)_σ 8v²p

k₀²+v²k². (2.193) After settingg =c= 1,

ΓX1 = −g² 1 (v

c)^{2−δµ−δν0}

Z d³k (2π)³u(p¯ 3)

γβ

k₀γ₀+v k_cγ_c k²₀+v²k² γµ

u(p1)×Gµν(k)×Wαβ(k)× i _νασ (k₀, vk)_σ 8v²p

k₀²+v²k²

= (g₀g₁−g_jg₁−g_jw_x) 4πv²(g₁²+g₁w_x+κ²) +φ1

"

2g₁³(g₁ +w_x)(−g₁g_j +g₁g₀+g₀w_x)

(g₁²+g₁w_x+κ²)³ −g²₁[7g₀(g₁+w_x)−g_j(7g₁+ 4w_x)]

2(g²₁ +g₁w_x+κ²)² + g₁(g₀−g_j) (g₁²+g₁w_x+κ²)

# . (2.194) In total, we need to multiply by a factor of4to count the clockwise/counterclockwise

fermion loops and the exchange of W ↔Gin the diagrams.

Γ_X1 + ˜Γ_X1 +

Γ_X1 + ˜Γ_X1

W↔G = 4Γ_X1 (2.195) Vector Vertex u γ^ρuone leg gauge, one leg disorder

Replace the mass-vertex expressions 1by γ_ρ

Γ_Xρ = 1 v²

Z d²qdq¯ 0

(2π)³ 1 v²

Z d²k¯

(2π)² u(p₃) 1 γ_β

!

S_F(p₁−k)[ −ig pN_f(v

c)^1−δµ0γ_µ]u(p₁)G_µν(k) (−1)×N_f

×T r

[ −ig pN_f(v

c)^1−δν0γν]SF(q−p1+p3)γρ SF(q) 1 γ_α

!

SF(k+q−p1+p3)

g_m

W_αβ(k−p₁+p₃)

! . (2.196)

Γ˜_Xρ = 1 v²

Z d²qdq¯ ₀ (2π)³

1 v²

Z d²k¯

(2π)² u(p₃) 1 γ_β

!

S_F(p₁−k)[ −ig pN_f(v

c)^1−δµ0γ_µ]u(p₁)G_µν(k) (−1)×N_f

×T r

[ −ig pN_f(v

c)^1−δν0γ_ν]S_F(q−k) 1 γ_α

!

S_F(q−p₁+p₃)γ_ρS_F(q)

g_m

W_αβ(k−p₁+p₃)

! . (2.197)

By similar argument, the term with an even number of γ's in the trace would cancel between Γ and Γ˜, so in this case we only need to compute upper com- ponent (g_m). Setp₁ =p₃ = 0, straightforward calculation gives

Γ_Xρ = (−N_f)(−i)⁴( −ig pN_f)²(v

c)^{2−δµ0−δν0} 1 v²

Z d³q (2π)³

1 v²

Z d²k (2π)²

×

¯

u(p₃) (−k)^c

k² γ_cγ_µu(p₁)

T r[γ_ν/qγ_ρ/q(/q+/k)] 1

q²q²(k+q)²G_µν(k, ω= 0)g_m

= 0. (2.198)

So there is no contribution from Γ_Xρ,Γ˜_Xρ

Mass Vertex u1_2×2uboth legs are gauge propagators

ΓZ1 = ( −ig pNf

)⁴(v

c)^{4−δµ0−δν0−δα0−δβ0} ×(−1)×(Nf)

Z d³kd³q

(2π)³(2π)³u(p3)γβ SF(p1−k)γµu(p1)

×G_µν(k)T r

γ_νS_F(q−p₁ +p₃)1S_F(q)γ_αS_F(k+q−p₁+p₃)

G_αβ(k−p₁+p₃).

(2.199)

Γ˜_Z1 = ( −ig pN_f)⁴(v

c)^{4−δµ0−δν0−δα0−δβ0} ×(−1)×(N_f)

Z d³kd³q

(2π)³(2π)³u(p₃)γ_β S_F(p₁−k)γ_µu(p₁)

×G_µν(k)T r

γ_νS_F(−k+q+p₁−p₃)γ_α S_F(q)1S_F(q+p₁−p₃)

G_αβ(k−p₁+p₃).

(2.200) Upon taking the external momenta to zero,

ΓZ1 = g⁴ N_f(v

c)^{4−δµ0−δν0−δα0−δβ0}

Z d³k (2π)³u(p¯ 3)

γβ

k0γ0 +v kcγc

k₀²+v²k² γµ

u(p1)

×G_µν(k)G_αβ(k)×

Z d³q (2π)³ T r

γ_ν q₀γ₀+v q_dγ_d

q²₀+v²q² 1 q₀γ₀+v q_eγ_e

q₀²+v²q² γ_α (k₀+q₀)γ₀+v(k_f +q_f)γ_f (k0+q0)²+v²(k+q)²

. (2.201) Perform the q integral rst,

F_Γz(k)≡

Z d³q (2π)³T r

γ_ν q₀γ₀+v q_dγ_d q₀²+v²q²

q₀γ₀ +v q_eγ_e

q₀²+v²q² γ_α (k₀+q₀)γ₀ +v(k_f +q_f)γ_f (k₀+q₀)²+v²(k+q)²

(2.202)

= 1 v²

Z d³Q (2π)³T r

γ_ν Q_λγ_λ Q²

Q_ργ_ρ

Q² γ_α (K+Q)_σ γ_σ (K+Q)²

. (2.203)

Here we dene Q≡(q₀, vq), d³Q≡dq₀d²(vq) , K ≡(k₀, vk) Standard Feynman tricks give

FΓz(k) = i _νασ (k₀, vk)_σ 8v²p

k₀²+v²k². (2.204)

So we have (k₀ =ω) Γ_Z1 = g⁴

N_f(v

c)^{4−δµ0−δν0−δα0−δβ0}

Z d³k (2π)³u(p¯ ₃)

γ_β k₀γ₀ +v k_cγ_c k₀²+v²k² γ_µ

u(p₁)

×G_µν(k)G_αβ(k)

i _νασ (k₀, vk)_σ 8v²p

k²₀+v²k²

= ¯u(p₃)12×2 u(p₁)

× Z ∞

−∞

dz

−g⁴v²(σe+|z|)

(v² +z²)(g₁²−κ²)σe+|z|

g1wx

√v²+z²+ (g²₁−κ²) (v²+z²)

16 Nfπ²(v² +z²)³² √

v² +z²(g₁²+κ²)σe+|z|

g₁²√

v²+z²+ (g1wx+√

v²+z²κ²) 2. (2.205) The same manipulations are used in the computations of δ₁, δ₂. Note that

unlike the case of δ₁, δ₂, this term renormalizes g_m without any divergent in- tegration, labeled by ξ.

Taking the limit σ_e=w_x = 0, the expression reduces to ΓZ1

σe=wx=0 = ¯u(p3)12×2 u(p1) × −g⁴ (g₁²−κ²)

8 N_fπ²(g₁²+κ²)², (2.206) which agrees with the result in [13] Unlike the case of Γ_X1, the two legs are identical so the symmetry factor is2:

Γ_Z1 + ˜Γ_Z1 = 2Γ_Z1. (2.207) Vector Vertex u γ^ρu both legs are gauge propagators

Replace 1 byγ_ρ to obtain the vector counterparts

Γ_Zρ = ( −ig pN_f)⁴(v

c)^{4−δµ0−δν0−δα0−δβ0} ×(−1)×(N_f)

Z d³kd³q

(2π)³(2π)³u(p₃)γ_β S_F(p₁−k)γ_µu(p₁)

×G_µν(k)T r

γ_νS_F(q−p₁ +p₃)γ_ρS_F(q)γ_α S_F(k+q−p₁+p₃)

G_αβ(k−p₁+p₃).

(2.208)

Γ˜_Zρ = ( −ig pN_f)⁴(v

c)^{4−δµ0−δν0−δα0−δβ0} ×(−1)×(N_f) d kd q

(2π)³(2π)³u(p₃)γ_β S_F(p₁−k)γ_µu(p₁)

×G_µν(k)T r

γ_νS_F(−k+q+p₁−p₃)γ_α S_F(q)γ_ρS_F(q+p₁−p₃)

G_αβ(k−p₁+p₃).

(2.209) By the same argument as before, there are sixγ's in the trace, soΓ_Zρ andΓ˜_Zρ cancel one another:

Γ_Zρ + ˜Γ_Zρ = 0. (2.210)

2.I 3-loop Corrections of Disorders ∆0,∆j

With D^dis_αβ Propagator

π₁^µν =

Z d³k (2π)³

Z d³q (2π)³

Z d²QdQ₀ (2π)²

−ig pN_f

4 v c

4−δµ0−δν0−δα0−δ_β0

(−1)²(N_f)²

×T r

γ_µS_F(k−p)1S_F(k−Q)γ_αS_F(k)

× g_mδ(p₀−Q₀)δ(Q₀)D^dis_αβ(Q, Q₀ = 0) ×T r

γ_νS_F(q)γ_βS_F(q−Q)1S_F(q−p)

(2.211) (ipping the signs fork and q variables )

=

Z d²Q (2π)²

Z d³k (2π)³

v c

2−δµ0−δα0

Tr

γ_µS_F(k+p)1S_F(k+Q)γ_αS_F(k)

×

Z d³q (2π)³

v c

2−δ_ν0−δ_β0

Tr

γ_νS_F(q)γ_βS_F(q+Q)1S_F(q+p)

×

g_m −ig pN_f

4

N_f²D^dis_αβ(Q, Q₀ = 0)δ(p₀ = 0)

. (2.212)

Naively evaluating this diagram is problematic because the Feynman parame- ter integrals are not doable. To extract the divergence, we Taylor expand the

expression to second order in p. First, we dene T^µα(Q, p) =

Z d³k (2π)³

v c

^2−δµ0−δ_α0

Tr

γ_µS_F(k+p)1S_F(k+Q)γ_αS_F(k) (2.213). By reversing the trace order , we have

Z d³q (2π)³

v c

^2−δν0−δ_β0

Tr

γνSF(q)γβSF(q+Q)1SF(q+p)

(2.214)

= (−1)⁵

Z d³q (2π)³

v c

2−δν0−δ_β0

Tr

γ_νS_F(q+p)S(q+Q)γ_βS_F(q)

=−T^νβ(2.215)(Q, p).

Let π₁^µν =

Z d²Q

(2π)² [T^µα(Q, p)] [−T^νβ(Q, p)]×

gm

−ig pNf

4

N_f²D_αβ^dis(Q, Q0 = 0)δ(p0 = 0)

(2.216) and

T₂(Q, p)≡T^µαT^νβ (2.217)

T₂(Q, p) =T₂(Q,0) + ∂T₂

∂p_xp_x+ ∂T₂

∂p_yp_y+ 1 2

∂²T₂

∂p²_x p²_x+∂²T₂

∂p²_y p²_y+ 2 ∂²T₂

∂p_xp_yp_xp_y

+O(p³). (2.218)

=

∂T^µα

∂p_x

∂T^νβ

∂p_x

p=0

p²_x+

∂T^µα

∂p_y

∂T^νβ

∂p_y

p=0

p²_y +

∂T^µα

∂p_x

∂T^νβ

∂p_y +∂T^µα

∂p_y

∂T^νβ

∂p_x

p=0

p_xp_y+O(p³) (2.219) Straightforward calculation gives

T^µα(Q, p= 0) = 0. (2.220) For rst order derivatives, we can also obtain (after lengthy algebra)

∂T^µα

∂pj

= v c

^2−δµ0−δ_α0 1 v²

i(−i)³ 32|Q|¯

^µαj+ 1 Q¯²

^µατQ¯_τQ¯_j +^αjτQ¯_τQ¯_µ−^jµτQ¯_τQ¯_α (2.221). Notice that this result is true in3d with general temporal component Q₀

Plugging into Eq. (2.212) and taking the four diagrams into consideration (each triangle has either clockwise or counterclockwise owing momenta), the total result is

π_µν^tot = 4π_µν, (2.222)

where

π₁₁= (−1) g_m(g₁²∆₀+v²∆_jκ²) 1024πv⁴(g₁²+g1 wx

v +κ²)² (¯p²_y+ ¯p²_x), (2.223) π_ij = (+1)g_m(g²₁v²∆_j + 2g₁v w_x∆_j+w²_x∆_j + ∆₀κ²)

512πv⁴(g₁²+g₁ ^w_v^x +κ²)² (δ_ijp¯²−p¯_ip¯(2.224)_j),

and p¯_i = v p_i. π₁₁ renormalizes ∆_j and π_ij renormalizes ∆₀. This diagram scales as 1/N_f² if g_m,∆₀,∆_j scale as 1/N_f.

With W_αβ Propagator

Replace the internal propagator withW_αβ, the remaining calculations are the same:

˜

π_µν^tot = 4˜πµν, (2.225) where

˜

π₁₁= (−1) g_m g_jN_f

1024πv⁴(¯p²_y+ ¯p²_x) +φ₁ g₁²g_mN_f(g₁²g_j + 2g₁g_j^w_v^x + (2g_j −g₀)κ² )

1024πv⁴(g₁²+g₁ ^w_v^x +κ²)² (¯p²_y + ¯p²_x(2.226))

˜

π_ij = (+1) g₀g_mN_f

512πv⁴ (δ_ijp¯²−p¯_ip¯_j) +φ₁

g₁g_mN_f h

g₁g_jκ²−g₀(g₁+^w_v^x) (g₁²+g₁^w_v^x + 2κ²)i

512πv⁴(g²₁+g₁ ^w_v^x +κ²)² (δ_ijp¯²−p¯_ip¯_j)

(2.227) and p¯_i = v p_i. π˜₁₁ renormalizes ∆_j, and π˜_ij renormalizes ∆₀. This diagram

scales as 1/N_f if g_m, g_j scale as 1/N_f. With Gauge Propagator Gαβ

By dimensional analysis, this term should be UV nite,

∼ Z

d²Q 1 Q

1 Qp²1

Q Q0=p0

. 2.J Summary

δ¯₁p₀γ⁰+ ¯δ₂v p_jγ^j = Σ_d+ Σ_g + Σ_b (2.228)

δ_gm ( ¯ψ1ψ) ( ¯ψ1ψ) = BOX11+ 2

Γ^m₁ + Γ^m₃ + Γ^m₄ + 4Γ_X1 + 2Γ_Z1

+ 2Γ(2.229)^m₂

δ_g0 ( ¯ψγ⁰ψ) ( ¯ψγ⁰ψ) =BOXγ0γ0 + 2

Γ^µ₁ + Γ^µ₃ + Γ^µ₄

µ=1

( −ig

pN_f)⁻¹+ 2Γ^µ=1₂ ( −ig

pN_f)(2.230)⁻¹

δ_gj ( ¯ψiγ^jψ) ( ¯ψiγ^jψ) = BOXγjγj + 2

Γ^µ₁ + Γ^µ₃ + Γ^µ₄

µ=j ( −ig

pN_f v)⁻¹ + 2Γ^µ=j₂ ( −ig

pN_f v)⁻¹(2.231) whereΣ_b, Γ^m₂ , Γ^µ=1₂ , Γ^µ=j₂ are the subleading order terms in the above expres-

sions.

δ¯∆0(δijp²−pipj) = 4πij+ 4˜πij (2.232)

¯δ_∆j(p²_x+p²_y) = 4π₁₁+ 4˜π₁₁ (2.233)

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