Zero modes, Heat kernel expansions, Zeta functions. A ...campus.usal.es/~mpg/General/ymhzmhksol1Sept.pdfZero modes, Heat kernel expansions, Zeta functions. A novel approach in QFT

Zero modes, Heat kernel expansions, Zeta functions.A novel approach in QFT

A. Alonso-Izquierdo1, Juan Mateos Guilarte2

1Departamento de Matemática Aplicada, Universidad de Salamanca, Spain

2Departamento de Física Fundamental, Universidad de Salamanca, Spain

Particle and Nuclear Physics at all scales, Astroparticle Physics and Cosmology ,Santiago de Compostela, SPAIN, September2015

A. Alonso-IzquierdoJ. Mateos Guilarte (Universities of Somewhere and Elsewhere)Zero Modes and the Heat Kernel Expansion EIGMTC 2010 1 / 30

Outline

1 One-loop fluctuations in Yang-Mills-Higgs systems

2 Heat kernels and spectral zeta functions

3 Improved GDWS L-heat kernel expansion

4 One-loop mass shifts of Kinks and BPS vortices

5 One-loop shifts of Domain Wall surface and BPS Vortex string tensions


The Yang-Mills-Higgs system• Euclidean YMH action

SE[A,ΦI ] =

∫MD

?LE(FA,∇AΦI ,ΦI) =

∫MD

{14|FA ∧ ?FA| + (1)

+12

Ns∑I=1

|∇AΦI ∧ ?∇AΦI |+ ?U(∣∣∣Φ2

1

∣∣∣ , ∣∣∣Φ31

∣∣∣ , ∣∣∣Φ41

∣∣∣ , · · · , ∣∣∣Φ2Ns

∣∣∣ , ∣∣∣Φ3NS

∣∣∣ , ∣∣∣Φ4Ns

∣∣∣)}• Ta , a = 1, 2, · · · , l = r2 − 1 are l× l-antisymmetric matrices that generate LieSU(r), [Ta, Tb] = fabcTc,x ≡ (x1, x2, · · · , xD) ∈ MD are points in a Riemannian manifold of dimension D. The fields are

A =D∑α=1

Aα(x)dxα =D∑α=1

l∑a=1

Aaα(x)Tadxα , ΦI =

l∑a=1

ΦaI (x)Ta , I = 1, · · · ,Ns

FA =∑α<β

(∂Aα∂xβ

−∂Aβ∂xα

+ g[Aα,Aβ ]

)dxα ∧ dxβ , ∇AΦI =

D∑α=1

(∂ΦI

∂xα+ g[Aα,ΦI ]

)dxα

U(Φ) = c1 + c2trΦ2 + c3trΦ3 + c4trΦ4 or U(Φ1, · · · ,ΦNs ) =g2

2

∑I<J

∣∣∣[ΦI ,ΦJ ]2∣∣∣

?(em1 ∧ · · · ∧ emp ) =1

(D− p)!εm1···mp mp+1···mD emp+1 ∧ · · · ∧ emD

gαβ(x) =D∑

m=1

D∑n=1

δmnemα(x)en

β(x) , em(x) =D∑α=1

emα(x)dxα


YMH one-loop fluctuations on classical backgrounds• YMH classical backgrounds are field configurations (A, ΦI) solving the classical field equations:

∇†A

FA +∑I<J

([∇AΦI , ΦJ ] + [ΦI ,∇AΦJ ]

)= 0

∇†A∇AΦI +

δUδΦI

(Φ1, · · · , ΦNs ) = 0

∇†A = (−1)D(p+1)+1 ?∇A? : Λp(MD,LieSU(r)) −→ Λp−1(MD,LieSU(r))

• One-loop fluctuations (A,ΦI) = (A + a, ΦI + φI) of the classical background complying with PBC on anormalizing box of length L

a(x1, · · · , xi + L, · · · , xD) = a(x1, · · · , xi, · · · , xD) , ∀i = 1, · · · ,Dφα(x1, · · · , xi + L, · · · , xD) = φα(x1, · · · , xi, · · · , xD) , ∀i = 1, · · · ,D ,

are governed by the quadratic Lagrangian:

?L(2)E (A, a, ΦI , φI) ' ?LE(FA+a,∇A+a(ΦI + φI))− ?LE(FA,∇AΦI) +O(a3, a2φI , aφ2

I , φ3I )

=

(12

∣∣(da + [A, a]) ∧ ?(da + [A, a])∣∣+ |a ∧ ?FA ∧ a| +

+

Ns∑I=1

12

(∣∣ΦIΦI∣∣ |a ∧ ?a| −

∣∣ΦIa∣∣ ∧ ? ∣∣aΦI

∣∣)− Ns∑I=1

(∣∣∇AφI ∧ ?aΦI∣∣+∣∣∇AΦI ∧ ?aφI

∣∣) +

+12

Ns∑I=1

|∇AφI ∧ ?∇AφI |+12

Ns∑I=1

Ns∑J=1

?

∣∣∣∣ δ2UδΦIδΦJ

(Φ1, · · · , ΦNs )φIφJ

∣∣∣∣)

(2)


YMH HESSIAN operator• Assemble the field fluctuations in the N = l(D + Ns) vector column and write the quadratic form:

F =

aφ1...φNs

,

∫TD?{L(2)

E + LG.F.

}= 〈F|L(A, Φα)F〉 =

∫TD?

N∑i=1

N∑j=1

FiLij(A, ΦI)Fj

encompassing the quadratic Lagrangian and the background gauge enforcing term,

∇†A

a +

Ns∑I=1

[ΦI , φI ] = 0 , SG.F. =

∫TD?LG.F. = −

12

∫TD?

∣∣∣∣∣∣(∇†

Aa +

Ns∑I=1

[ΦI , φI ]

)2∣∣∣∣∣∣

1

• The Hessian L(A, ΦI) is the N × N-matrix differential operator of Laplacian type:L(A, ΦI) = −42 +Q(x1, · · · , xD) · ∇+ M2 + V(x1, · · · , xD)·

4 =

D∑β=1

∂2

∂x2β

, Q(A(x)) · ∇ =D∑β=1

Qβ(x)∂

∂xβ, lim

x21+···+x2

D→∞Q(x1, · · · , xD)) · ∇ = 0

M = diag(m1, . . . ,mN) , V[A,FA, ΦI ,∇AΦI ](x) , limx2

1+···+x2D→∞

V(x1, · · · , xD)) = 0

Lij[Aα, ΦI ] = −4 δij +D∑β=1

[Qβ ]ij(x)∂

∂xβ+ m2

i δij + Vij(x)

1Ghosts are necessary to restore unitarityA. Alonso-IzquierdoJ. Mateos Guilarte (Universities of Somewhere and Elsewhere)Zero Modes and the Heat Kernel Expansion EIGMTC 2010 5 / 30

Spectral heat and zeta functions• Hessian spectrum

L[A, ΦI ]Ψν(x1, · · · , xD) = λνΨν(x1, · · · , xD) , λν ∈ R+ , 0 < λ1 < λ2 < λ3 < · · ·

• L-heat and L-zeta functions: Mellin’s transform

hL(t) = TrL2 exp{−t L} =∞∑ν=1

exp{−tλν} , L2 = ⊕Ni=1 L2(TD)

ζL(s) = TrL2 L−s =∞∑ν=1

λ−sν =

1Γ(s)

∫ ∞0

dt ts−1 hL(t) =1

Γ(s)

∞∑ν=1

∫ ∞0

dt ts−1 e−tλν , s ∈ C

where t ∈ R+ is Schwinger proper time.

• One-loop effective action and Casimir energy

Seff [[A], ΦI ] = SE[A, ΦI ] +12

ln DetL[A, ΦI ] , 4EC[A, ΦI ]“ = ” TrL2

(L1/2[A, ΦI ]− L1/2[0, 0]

)• Zeta function regularization of traces and determinants

DetL[A, ΦI ] =∞∏ν=1

λν“ = ” exp{−

dζLds

(0)

}ζL(−1/2)“ = ” TrL2 L1/2[A, ΦI ] −→ ζL(s)“ = ” TrL2 L−s[A, ΦI ]


The L-heat kernel from the L0-heat kernel• The L-heat equation and the integral L-heat kernel KL(xα, yα; t):

N∑k=1

(∂

∂tδik + Lik[A, ΦI ]

)[KL]kj(xα, yα; t) = 0ij ; [KL]ij(xα, yα; 0) = δij

D∏β=1

δ(xβ − yβ)

hL(t) =N∑

i=1

∫TD

dvolTD [KL]ii(xα, xα; t) , dvolTD = dx1 · · · dxD

• Short t-time asymptotics of “the free " heat kernel L0 = −4+M2,

[KL0 ]ij(xα, yα; t) =δije−m2

i t

(√

4πt)D·e−

14t

∑Dα=1 |xα−yα|2

(1 + e−

ct

), [KL0 ]ij(xα, yα; 0) = δij

D∏β=1

δ(xβ−yβ)

• Factorization ansatz and transfer equation

[KL]ij(xα, yα; t) =N∑

k=1

[AL]ik(xα, yα; t) · [KL0 ]kj(xα, yα; t) , [AL]ij(xα, yα; 0) = δij

N∑k=1

δik

(∂

∂t−4

)+

D∑β=1

[xβ − yβ

t

(∂

∂xβ−

12

[Qβ ]ik(x)

)+ [Qβ ]ik(x)

∂

∂xβ

]+

+ Vik(xα)

· [AL]kj(xα, yα; t) + (m2i − m2

j )[AL]ij(xα, yα; t) = 0ij


Gilkey-DeWitt-Schwinger L-heat kernel expansion• Expanding [AL]ij(xα, yα; t) =

∑∞n=0[an]ij(xα, yα;L)tn in powers of t one finds the recurrence relations

N∑k=1

(n + 1)δik +D∑β=1

(xβ − yβ)

(δik

∂

∂xβ−

12

[Qβ ]ik

) [an+1]kj(xα, yα;L) =

=N∑

k=1

δik 4−D∑β=1

[Qβ ]ik∂

∂xβ− Vik(xα)

[an]kj(xα, yα;L)− (m2i − m2

j ) [an]ij(xα, yα;L)

to be started at [aij]0(xα, yα;L) = δij• In the x→ y range asymptotic solutions are provided by the truncations(

∂

∂t+ L

) N0∑n=0

an(xα, xα;L)KL0 (xα, xα; t) ∼= rN0 (xα)tN0 KL0 (xα, xα; t)

• Asymptotic expansion of the L-heat function and meromorphic structure of the L-zeta function

hL(t) =

N∑i=1

“∞”∑n=0

[an]ii(L)

(4π)D· tn−

D2 e−m2

i t , [an]ii(L) =

∫[an]ii(xα, xα;L) dvolTD

ζL =N∑

i=1

“∞”∑n=0

[an]ii(L)

(4π)DΓ(s)·∫ ∞

0dt · ts+n−1− D

2 · e−m2i t


L-heat kernel long t-time asymptotics and zero modes• Mismatch between the long t-time asymptotics of L and L0 when 0 < dim KerL = Nzm <∞:

KL(xα, yα; t) =

Nzm∑l=1

Ψ0l(xα)Ψ†0l(yα) +∞∑

n=1

Ψn(xα)Ψ†n(yα)e−tλn

Long and short t-time asymptotics of KL

KL(xα, yα, t)t→∞−→

Nzm∑l=1

Ψ0l(xα)Ψ†0l(yα) , KL(xα, yα, t)t→0−→

D∏α=1

δ(xα − yα)1N

to be compared with

KL0 (xα, yα, t)t→∞−→ 0 , KL0 (xα, yα, t)

t→0−→D∏α=1

δ(xα − yα)1N

• In the TD → RD infinite volume limit, a continuous spectrum labeled by wave numbersk ≡ (k1, · · · , kD), together with a discrete set of bound states, arises

KL(xα, yα; t) =

Nzm∑l=1

Ψ0l(xα)Ψ†0l(yα) +

Nb∑n=1

Ψn(xα)Ψ†n(yα)e−tω2n +

∫dDkΨk(xα)Ψ†k (yα)e−tω2(k)

• This is the kind of spectrum found when dealing with solitonic (or instantonic) fluctuations in open spaces


Entering zero modes in the L-heat kernel factorization• Incorporation of zero modes to the L-heat kernel factorization:

KL(xα, yα, t) = CL(xα, yα, t)KL0 (xα, yα, t) +

Nzm∑l=1

e−14t

∑Dα=1 |xα−yα|2 Ψ0l(xα)Ψ†0l(yα)Gl(t)

• Asymptotic choices of the N × N matrix functions Gl(t), l = 1, . . . ,Nzm, of t and CL(xα, yα, 0):

limt→∞

Gl(t) = I , Gl(0) = 0 , CL(xα, yα, 0) = 1N

• Improved transfer equation:

N∑k=1

δik

(∂

∂t−4

)+

D∑β=1

[xβ − yβ

t

(∂

∂xβ−

12

[Qβ ]ik(x)

)+ [Qβ ]ik(x)

∂

∂xβ

]+

+ Vik(xα)

· [CL]kj(xα, yα; t) + (m2i − m2

j )[CL]ij(xα, yα; t) +

+

Nzm∑l=1

N∑k=1

(√

4πt)Detm2j

{[Ψ0l]i(xα)[Ψ0l]

∗k (yα)

(∂

∂t+

D2β

)[Gl]kj(t)+

+12

N∑s=1

D∑β=1

(xβ − yβ)

(δik

∂

∂xβ−

1t[Qβ ]ik(x)

)[Ψ0l]k(xα)[Ψ0l]

∗s (yα)[Gl]sj(t)

}= 0ij


Entering zero modes in the L-heat kernel expansion• Power series in t

CL(xα, yα, t) =∞∑

n=0

cn(xα, yα,L)tn , c0(xα, yα,L) = 1N

trades the improved transfer equation by the

• improved recurrence relations∞∑

n=0

{ N∑k=1

[(n + 1)δik +

D∑β=1

(xβ − yβ)

(δik

∂

∂xβ−

12

[Qβ ]ik

)][cn+1]kj(xα, yα,L) +

+N∑

k=1

[− δik∆ +

D∑β=1

[Qβ ]ik(x)∂

∂xβ+ Vik(xα)

][cn]kj(xα, yα,L) + (m2

i − m2j )[cn]ij(xα, yα,L)

}tn

−12t

N∑k=1

D∑β=1

(xβ − yβ)[Qβ ]ik(x)[c0]kj(xα, yα,L) +

+

Nzm∑l=1

N∑k=1

(√

4πt)Detm2j

{[Ψ0l]i(xα)[Ψ0l]

∗k (yα)

(∂

∂t+

D2t

)[Gl]kj(t)+

+12

N∑s=1

D∑β=1

(xβ − yβ)

(δik

∂

∂xβ−

1t[Qβ ]ik(x)

)[Ψ0l]k(xα)[Ψ0l]

∗s (yα)[Gl]sj(t)

}= 0ij


A real scalar field in a line• Set D = 1, N = 1, Q(x) = 0 and reinterpret the Euclidean action as the static part of the energy:

SE(φ) −→ E(φ) =

∫dx

{12

(dφdx

)2

+ U(φ)

}.

The potential energy density U is chosen with a discrete set of degenerate minima:

δUδφ|φ=va

= 0 , a = 1, 2, 3, · · · ,δ2Uδφ2|φ=va

= m2 , ∀va

• Kinks are classical backgrounds that solve the first-order ODE

dφdx

=√

2U(φ) ≡ x− x0 =

∫dφ√

2U(φ); lim

x→−∞φ(x) = va , lim

x→∞φ(x) = va+1

and have finite energy: E(φ) =∫ va+1

vadφ√

2U(φ) =∣∣∣W(φ(va+1))−W(φ(va))

∣∣∣ < +∞

• Kink and vacuum Hessian operators

δ2Uδφ2|φ=φ = m2 + V(x) , lim

x→±∞V(x) = 0

L = −d2

dx2+ m2 + V(x) , L0 = −

d2

dx2+ m2


Kink zero mode improved GDWS heat kernel factorization• Normalized kink zero mode

N =

∫ ∞−∞

dx∣∣∣ dφ

dx

∣∣∣2 , Ψ0(x) =1N·

dφdx

, LΨ0(x) = 0

• Improved factorization of the L-heat kernel :

KL(x, y; t) = CL(x, y, t)KL0 (x, y; t) + e−(x−y)2

4t Ψ∗0 (y)Ψ0(x)G(t)

= e−(x−y)2

4t

(CL(x, y, t)

e−tm2

√4πt

+ Ψ∗0 (y)Ψ0(x)G(t)

)Short and long proper t-time asymptotic behaviours

limt→0

CL(x, y; t) = 1 , limt→0

G(t) = 0 , limt→∞

G(t) = 1

• Improved transfer equation:

∞∑n=0

[(n + 1)cn+1(x, y;L)−

∂2cn(x, y,L)

∂x2+ (x− y)

∂cn+1(x, y;L)

∂x+ V(x)cn(x, y,L)

]tn +

+√

4πt etm2Ψ∗0 (y)

[dG(t)

dtΨ0(x) +

G(t)2t

Ψ0(x) + (x− y)G(t)

tdΨ0(x)

dx

]= 0


Kink zero mode and improved heat kernel expansion• Optimum choice of G(t)

√4πt · etm2

·dG(t)

dt= constant ≡ G(t) = Erf (m

√t)

• Erf z = 2√π

e−z2 ∑∞n=0

2n

(2n+1)!! z2n+1 ⇒ improved recurrence relations between the cn(x, y,L)

densities and their derivatives:

(n + 1) cn+1(x, y,L)−∂2cn(x, y,L)

∂x2+ (x− y)

∂cn+1(x, y,L)

∂x+ V(x)cn(x, y,L) +

+2mΨ∗0 (y)Ψ0(x)δ0n + Ψ∗0 (y)Ψ0(x)2n+1m2n+1

(2n + 1)!!+ (x− y)Ψ∗0 (y)

dΨ0(x)

dx2n+2m2n+1

(2n + 1)!!= 0

• Improved Seeley coefficients

〈A(x)〉 =

∫ ∞−∞

dx A(x) , c0(L) = L , a0(L) = L

c1(L) = −〈V(x)〉 − 4m , a1(L) = −〈V(x)〉 , a2(L) = −16

⟨V′′(x)

⟩+

12

⟨(V(x))2

⟩c2(L) = −

16

⟨V′′(x)

⟩+

12

⟨(V(x))2

⟩+

43

m3 + 4m⟨

V(x)Ψ20(x)

⟩• Improved heat trace expansion

hL(t)− hL0 (t) =e−tm2

√4π

∞∑n=1

cn(L) tn−12 + Erf (m

√t) (3)


One-loop kink mass shift• ~-expansion of the quantum kink energy up to one-loop order

E(φ) = E(φ) + ~{(4E(φ)−4E(va)

)+4Em2 (φ)

}+O(~2)

• Regularized Kink Casimir energy

~4 EC(φK)[s] = ~4 E(φ)[s]− ~4 E(va)[s] =~2

(µ2

m2

)s+ 12 (ζL(s)− ζL0 (s)

)• Regularized energy shift due to the one-loop mass renormalization counter-term:

~4 Em2 (φ)[s] =~2〈V(x)〉

(µ2

m2

)s+ 12

limL→∞

1L

Γ(s + 1)

Γ(s)ζL0 (s + 1)

•Kink mass one-loop correction

4E(φ) = lims→− 1

2

4EC(φ)[s] + lims→− 1

2

4Em2(φ)[s]


Kink mass shift from the heat trace asymptotics:

Mellin transform of the heat trace:

~ (4EC[s] +4Em2 [s]) =

=~2

(µ2

m2

)s+ 12 1√

4π

{1

Γ(s)

[(c1(L)−

2ms

+ 〈V(x)〉)

Γ(s +12

) +∞∑

n=2

cn(L)

m2n−1Γ(s + n−

12

)

]}

=~2

(µ2

m2

)s+ 12{

mΓ(s)√π

[−(

2 +1s

)Γ(s +

12

) +12

∞∑n=2

cn(L)

m2n−1Γ(s + n−

12

)

]}.

Evaluation at the point s = − 12 + ε leaves the finite quantity − ~m

πexactly at the pole

−~m√π

lims→− 1

2

(µ2

m2

)s+ 12 Γ(s + 1

2 )

Γ(s)=

~m2π

limε→0

(1ε

+ logµ2

m2+ ψ(1)− ψ(−

12

) +O(ε2)

)

−~m

2√π

lims→− 1

2

(µ2

m2

)s+ 12 Γ(s + 1

2 )

sΓ(s)= −

~m2π

limε→0

(1ε

+ logµ2

m2+ ψ(1)− ψ(

12

) +O(ε2)

)• Asymptotic one-loop kink mass shift formula

~4 E(φ) = −~mπ

(1 +

18

∞∑n=2

cn(L)(m2)1−nΓ[n− 1]

)' −

~mπ−

~m8π

Nt∑n=2

cn(L)(m2)1−nΓ[n− 1]


The (1 + 1)-dimensional sine-Gordon and λφ4 models• U(φ) = m4

λ(1− cos

√λ

m φ)

vn = 2πnm√λ, n ∈ Z ,

δ2Uδφ2|φ=vn

= m2 , ∀n

dφdx

= 2m2√λ

sin

√λ

2mφ ⇒ φ(x) = 4

m√λ

arctanem(x−x0)

E(φ) = 8m3

λ,δ2Uδφ2|φ=φ = m2 + V(x) = m2 −

2m2

cosh2mx, Ψ0(x) =

√m2

sech(mx)

• U(φ) = λ4

(φ2 − m2

dλ

)2

v± = ±md√λ

,δ2Uδφ2|φ=v±

= 2m2d = m2

dφdx

=

√λ

2

(φ2 −

m2d

λ

)⇒ φ(x) =

md√λ

tanh(md√

2(x− x0)) , E(φ) =

2√

23

m3d

λ

δ2Uδφ2|φ=φ = m2 + V(x) = 2m2

d −3m2

d

cosh2( md√2

x), Ψ0(x) =

√3md

4√

2sech2(

md√2

x)


sine-Gordon and λφ4 kink Seeley coefficients• Routine calculation of Seeley coefficients in a Mathematica environment

Old and new Seeley Coefficients: the sine-Gordon kinkn an(L)/m2n−1 cn(L)/m2n−1

1 4.00000 0.000002 2.66667 0.000003 1.06667 0.000004 0.304762 0.000005 0.0677249 0.000006 0.0123136 0.000007 0.0018944 0.000008 0.000252587 0.000009 0.0000297161 0.00000

10 3.12801 · 10−6 0.00000

Old and new Seeley coefficients: the λφ4 kinkn an(L)/m2n−1 cn(L)/m2n−1

1 12.00000 4.000002 24.00000 2.666673 35.20000 1.066674 39.3143 0.3047625 34.7429 0.06772496 25.2306 0.01231367 16.5208 0.00189448 8.277702 0.0002525879 3.89498 0.0000297161

10 1.63998 3.12801 · 10−6

• Exact and asymptotic, new and old, “renormalized”λφ4 Kink heat function


Semiclassical λφ4-kink mass correction

Kink Seeley Coefficientsn cn(K)1 4.000002 2.666673 1.066674 0.3047625 0.06772496 0.01231367 0.00189448 0.0002525879 0.0000297161

10 3.12801 · 10−6

Kink Mass Shift EstimationNt 4E(φK ; Nt)/~γ2

d- -2 −0.6631463 −0.6657984 −0.6661775 −0.6662406 −0.6662527 −0.6662548 −0.6662549 −0.66625510 −0.666255

• Comparison with the DHN result: ∆E(φK)

~γ2d

= − 1π

(3−√

3arccos√

32 ) = 1

2√

3− 3π

= −0.666255

∆E(φK ,Nt = 10)

~γ2d

= −0.666255


BPS vortex fluctuations I• Planar BPS (λ = e2) topological solitons in the Abelian Higgs model

Eφ,A] =

∫d2x[ 1

4FijFij +

12

(Diφ)∗Diφ+18

(φ∗φ− 1)2]

D1φ± iD2φ = 0 , F12 ±12

(φ∗φ− 1) = 0

φ∗φ|S1∞

= 1 and Diφ|S1∞

= 0 ⇒∫R2

dx1dx2F12(x1, x2) = 2πn ∈ Z

• BPS vortex fluctuations

Ai(~x; n) = Vi(~x; n) + ε ai(~x) , φi(~x; n) = ψi(~x; n) + ε ϕi(~x) , i = 1, 2

ξ(~x) =(

a1(~x) a2(~x) ϕ1(~x) ϕ2(~x))t

Background gaugeB(ak, ϕ, φ) = ∂kak − (ψ1ϕ2 − ψ2ϕ1) = 0

• Zero mode fluctuations

Dξ0l(~x) = 0 , l = 1, 2, · · · , 2n , D =

−∂2 ∂1 ψ1 ψ2−∂1 −∂2 −ψ2 ψ1ψ1 −ψ2 −∂2 + V1 −∂1 − V2ψ2 ψ1 ∂1 + V2 −∂2 + V1


BPS vortex fluctuations II• BPS vortex Hessian L+ = D†D

L+ =

−∆ + |ψ|2 0 −2D1ψ2 2D1ψ1

0 −∆ + |ψ|2 −2D2ψ2 2D2ψ1−2D1ψ2 −2D2ψ2 −∆ + 1

2 (3|ψ|2 − 1) + VkVk −2Vk∂k − ∂kVk

2D1ψ1 2D2ψ1 2Vk∂k + ∂kVk −∆ + 12 (3|ψ|2 − 1) + VkVk

• SUSY partner Hamiltonian L− = DD†

L− =

−∆ + |ψ|2 0 0 0

0 −∆ + |ψ|2 0 00 0 −∆ + 1

2 (|ψ|2 + 1) + VkVk −2Vk∂k − ∂kVk

0 0 2Vk∂k + ∂kVk −∆ + 12 (|ψ|2 + 1) + VkVk

• Spectrum of vortex fluctuations

L+ = −1 (4+ 1) + V(~x) + ~Q(~x) · ~∇ , L+ξλ(~x) = λξλ(~x) , λ ≥ 0

1 =

1 0 0 00 1 0 00 0 1 00 0 0 1

, ‖ξλ(~x)‖2 > 0

‖ξ(~x)‖2 =

∫R2

d2x[(a1(~x))2 + (a2(~x))2 + (ϕ1(~x))2 + (ϕ2(~x))2] < +∞ , L0 = −1 (4+ 1)

lim~x→+∞

V(~x) = 0 , lim~x→+∞

~Q(~x) = ~0


Vortex zero modes improved GDWS recurrence relations• Optimum choice of G`(t): G`(t) = 1− e−t·14

• Recurrence relations

c1(~x,~y)−4c0(~x,~y) + (~x−~y) · ~∇c1(~x,~y) + V(~x) c0(~x,~y) = 0 , c0(~x,~y) = 14

(n + 1) cn+1(~x,~y)−4cn(~x,~y) + (~x−~y) · ~∇cn+1(~x,~y) + V(~x) cn(~x,~y) +

+ ~Q(~x) · ~∇cn(~x,~y)−12

(~x−~y) · ~Q(~x) cn+1(~x,~y) + 4π[(δn1 +

1n!

) Nzm∑`=1

Ψ0`(~x) Ψ†0`(~y) +

+1n!

Nzm∑`=1

(~x−~y) · ~∇Ψ0`(~x) Ψ†0`(~y)−1

2 (n! )(~x−~y) · ~Q(~x)

Nzm∑`=1

Ψ0`(~x) Ψ†0`(~y)]

= 0

• Seeley densities

c0(~x,~x) = 14

c1(~x,~x) = −V(~x)

c2(~x,~x) = −164 V(~x) +

16

(~Q(~x) · ~∇) V(~x) +112~Q(~x) · ~Q(~x) V(~x)−

16

(~∇ · ~Q(~x)) V(~x) +

+12

V2(~x)− 4πNzm∑`=1

Ψ0`(~x) Ψ†0`(~x)


Vortex spectral functions• Vortex heat trace

hL+ (t)− hL0 (t) =1

4π

∞∑n=1

4∑i=1

[cn(L+)]ii e−t tn−1 +

Nzm∑`=1

4∑i=1

[f`(L+)]ii (1− e−t)

[cn(L+)]ii =

∫R2

d2x [cn(~x,~x)]ii =⟨

[cn(~x,~x)]ii⟩

[f`(L+)]ii =

∫R2

d2x [Ψ0`]i(~x)[Ψ†0`]i(~x) =⟨

[Ψ0`]i(~x)[Ψ†0`]i(~x)⟩

• Vortex spectral zeta function

ζL+ (s)− ζL0 (s) =1

Γ[s]

∫ ∞0

dt ts−1 (hL+ (t)− hL0 (t))

=1

4π Γ[s]

∞∑n=1

4∑i=1

[cn(L+)]ii Γ[s + n− 1]−Nzm∑`=1

4∑i=1

[f`(L+)]ii

• Ghost fluctuations around vortex and vacuum operators and spectral zeta function

LG = −4+|ψ|2 , LG0 = −4+1

ζLG (s)− ζLG0(s) =

1Γ[s]

∫ ∞0

dt ts−1(

hLG (t)− hLG0(t))

=1

4π Γ[s]

∞∑n=1

cGn (L+G) Γ[s + n− 1]


Vortex Casimir energy• Zero point renormalization for a BPS vortex of magnetic flux 2πN ≡ Nzm = 2N

∆EVC =

~m2

[TrL2 (L+)

12 − TrL2 (L0)

12 − [TrL2

G(LG)

12 − TrL2

G(LG

0 )12 ]]

= lims→− 1

2

~µ2

(µ2

m2

)s{ζL+ (s)− ζL0 (s)−

(ζLG (x)− ζLG

0(s))}

= lims→− 1

2

~µ2

(µ2

m2

)s[ 14πΓ(s)

{ ∞∑n=1

(4∑

i=1

[cn(L+)]ii − cn(LG)

)Γ(s + n− 1)

}− 2N

]• Zeta function regularization of self-energy and tadpole divergent graphs

∆EVm2 = 2 ~m I(1) Σ(ψ,Vk) = lim

s→− 12

~µ4π

(µ2

m2

)s Γ[s]Γ[s + 1]

Σ(ψ,Vk)

Σ(ψ,Vk) =

∫R2

d2x[

1− |ψ|2 −12

VkVk

]=⟨

1− |ψ|2 −12

VkVk

⟩I(1) =

12

∫ ∞−∞

d2k(2π)2

1√k2

1 + k22 + 1

=12

lims→− 1

2

ζ−∆+1(s + 1)

∆EVm2 “ = ”−

~m4π

Σ(ψ,Vk)


One-loop BPS vortex mass shift formula• Cancellation of the n = 1 term of4EV

C against4EVm2

4∑i=1

[c1(L+)]ii = 〈5(1− |ψ|2)− 2VkVk〉 , c1(LG) = 〈1− |ψ|2)〉

14π

(4∑

i=1

[c1(L+)]ii − c1(LG)

)=

1π

Σ(ψ,Vk) , lims→−1/2

Γ(s)Γ(s + 1)

· Σ(ψ,Vk) = −1π

Σ(ψ,Vk)

• Asymptotic one-loop BPS vortex mass shift

∆EV = lims→− 1

2

[∆EVC(s) + ∆EV

m2(s)]

∆EV = −~m

16π32

∞∑n=2

( 4∑i=1

[cn(L)]ii − cn(LG))

Γ[n− 32 ]− ~mN

' −~m

16π32

Nt∑n=2

( 4∑i=1

[cn(L)]ii − cn(LG))

Γ[n− 32 ]− ~mN


Circularly symmetric BPS N-vortex zero modes• Circularly symmetric ansatz

ψ(x1, x2) = fN(r) eiNθ ; rVNθ (r, θ) = N βN(r)

• Bogomolny equation

f ′N(r) =Nr

fN(r)[1− βN(r)] ; β′N(r) =r

2N[1− f 2

N(r)]

fN(r)→ 1 , βN(r)→ 1 when r →∞fN(r) ' dNrN , βN(r) ' eNr2 when r → 0

where dN and eN are integration constants

• Normalizable zero mode of fluctuation of BPS circularly symmetric vortices

ξ(~x,N, k)

rN−k−1=

hN(r) sin[(N − k − 1)θ]hN(r) cos[(N − k − 1)θ]

− h′N(r)fN(r) cos(kθ)

− h′N(r)fN(r) sin(kθ)

,ξ⊥(~x,N, k)

rN−k−1=

hN(r) cos[(N − k − 1)θ]−hN(r) sin[(N − k − 1)θ]

− h′N(r)fN(r) sin(kθ)

h′N(r)fN(r) cos(kθ)

where k = 0, 1, 2, . . . ,N − 1


N = 4 BPS vortex zero modes

• Radial form factors hN(r) ODE

−r h′′N(r) + [1 + 2k − 2N βn(r)] h′N(r) + r f 2N(r) hN(r) = 0

ϕ(~x)

a(~x)

k = 0 k = 1 k = 2 k = 3


N = 1 BPS vortex quantum correction• Radial Seeley densities

tr c1(~x) = −2β1(r)2

r2− 5f 2

1 (r) + 5

tr c2(~x) =1

12r4f1(r)2

[f1(r)2

(8r2β1(r)− 8

(3r2 + 2

)β2

1(r) + 4β41(r) + r4

(37− 96πh10(r)2

))−

−8r2(

(14− 9β1(r))β1(r) + 8r2 − 7)

f 41 (r) + 27r4f 6

1 (r)− 96πr4h′10(r)2]

• Seeley coefficients and one-loop corrections

n tr([cn(L+)]) cn(LG)2 5.20990655492 2.605736386083 0.60457807522 0.319104645344 0.10055209843 0.022976815445 0.02634327490 0.001226454846 0.00468414064 0.00006965772

Nt ∆EN=1V

2 −1.051813 −1.054654 −1.055815 −1.056746 −1.05734


N = 1 BPS vortex quantum correction: old calculations• Radial Seeley densities

tr c1(r,L+) = −2β1(r)2

r2− 5f 2

1 (r) + 5 , c1(r,LG) = 1− f 21 (r)

tr c2(r,L+) =1

12r4

{37r4 + 4β4

1(r) + 8(

7r2 − 8r4)

f 21 (r) + 27r4f 4

1 (r)−

−8r2β1(r)[−1 + 14f 21 (r)] + 8β2

1(r)(−2− 3r2 + 9f 21 (r))

}c2(r,LG = −

16r2

{(4 + 5r2 − 8β1(r) + 4β2

1(r))f 21 (r)− 3r2 − 2r2f 4

1 (r)}

• Seeley coefficients and one-loop corrections

n tr([cn(L+)]) cn(LG)2 30.36316 2.605736386083 12.94926 0.319104645344 4.22814 0.022976815445 1.05116 0.001226454846 0.2009414064 0.00006965772

Nt ∆EN=1V

2 −1.029513 −1.083234 −1.092705 −1.094276 −1.09449


Bibliography• General

1 B.S. De Witt, “Dynamical theory of groups and fields”, Gordon and Breach, 1965

2 E. Elizalde, S. Odintsov, A. Romeo, A. Bytsenko, S. Zerbini, “Zeta function regularizationtechniques with applications”, World Scientific, Singapore, 1994

3 D. Vassilevich, “Heat kernel expansion: user’s manual”, Physics Reports 388 (2003) 279 .

• Old approach on Kinks and BPS Vortices

1 A. Alonso Izquierdo, W. Garcia Fuertes, M. de la Torre Mayado, and J. Mateos Guilarte, “Quantumcorrections to the mass of self-dual vortices”, Physical Review D 70 (2004) 061702 (R)

2 A. Alonso Izquierdo and J. Mateos Guilarte, “One-loop kink mass shifts: a computational approach”,Nuclear Physics B 852(2011) 696

• Improved GdWS-expansion and zero modes1 A.O. Barvinsky and G.A. Vilkovysky , “Covariant perturbation, 2: Second-order in the curvature”,

Nuclear Physics 333 (1990) 71

2 A. Alonso-Izquierdo and J. Mateos Guilarte, “Kink fluctuations asymptotics and zero modes”,European Physical Journal C72(2012) 2170

3 A. Alonso Izquierdo and J. Mateos Guilarte, “Quantum induced interactions in the moduli space ofdegenerate BPS domain walls”, JHEP 01(2014) 125


Documents

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