Multi instanton contribution

Let’s come back to our main subject, the resurgent structure of the double well. We have learned that the ground state energy should be represented by the trans-series.

Consider the ground state energy expansion around the trival vacuum.

E₀ =

∞

X

n=0

c_ngⁿ, (3.50)

where the asymptotic behavior is³

c_n∼ −3ⁿn!, n  1. (3.51)

The Borel transform of this series is,

B_E0(g) = −1

1 − 3g. (3.52)

So the resulting singularity on the Borel plane is at g = ¹₃, the ambiguity of this pole is,

Im E₀ = ∓π 1 3g exp



− 1 3g



. (3.53)

This ambiguity should be canceld by other saddles. Recall that the action of single instanton in the double well is S_c= _6g¹ . The order of the ambiguity gives us a hint that this is a two instanton configuration effect and actually, it is.

We first consider the instanton-anti-instanton pair configuration. Since we are considering the ground state energy, we need to integrate over all the periodic paths, so the two instanton or the two anti-instanton configuration do not contribute. In fact, an instanton-anti-instanton pair (IA) pair is not a classical path. It is just a approximate saddle point. Only when the distance between the instanton and anti-instanton is very large, this configuration can be seen as a saddle point (dilute instanton approximation). We are looking for a configuration which is a sum of the instanton and anti-instanton but separated by a distance θ. It is given by,

q_c^θ(t) = 1

√g( 1

1 + e^t−θ/2 + 1

1 + e^−t−θ/2 − 1), (3.54)

3We have suppressed the constant C_n, which is _π³

we want to compute the action of this path. It is convenient to introduce the

the path can be rewritten by,

q^θ_c(t) = q^θ/2₋ (t) + q^θ/2₊ (t) − 1 = q₋^θ/2(t) − q₋^−θ/2= u(t) − v(t). (3.56)

The action of this path is, S(q^θ_c) =

the first tem is just the leading contribution of two instantons. Since the path is an even function, we can change the integral to twice the integral from 0 to ∞. After integration by part, then we can expand the integral in powers of v², we stop at v² because we only want to compute the leading term of θ. We also use the equation of motion of u. We find,

S(q_c^θ) = 1

the function v decay from origin very fast, so the integral saturated when t is around zero, where u = 1+O(e^θ/2) there. Because V⁰⁰(u) ∼ V⁰⁰(1) = V⁰⁰(0), the terms inside the integral cancel. And,

v(0) ˙u(0) ∼ −e^−θ

g . (3.60)

The action at leading θ contribution is, S(q^θ_c) = 1

g[1

3 − 2e^−θ+ O(e^−2θ)]. (3.61)

Now we can consider n instanton configuration separated by distance θ_i with the

We only need to keep the interaction between the nearest neighbour instantons at the leading order. The action of n-instanton configuration is just a sum of two instanton configuration.

We want to compute the n-instanton contribution to the partition, so we need to calculate the quantum fluctuation around this action. Although this is not a classical path of the equation of motion, at the large θ_i limit, it can be seen as a approximate classical path. Expand the fluctuation to second order, we need to compute the determinant of the operator M . At large θ_i limit, the spectrum of the operator M can be seen as the same spectrum of the operator at the single instanton problem but n times degenerate. The determinant of M is just the n = 1 case but with power n. Thus, the n instanton contribution to the partition function is,

Z_⁽ⁿ⁾(β) = e^−β2 β

The e^−β/2 is the ground state energy of harmonic oscillator, overall β comes from the global time translation, the factor 1/n is because the configuration is invariant under a cylic permutation. The integral over θ_i is because we need to include all possible value of θ_i. Note for n odd, the instanton contributes to odd parity Z₋₁⁽ⁿ⁾ and for n even , the instanton contributes to even parity Z₁⁽ⁿ⁾.

Now we need to do the integral of θ_i. However, the interatcion between the instantons is attractive when g positive. For g → 0⁺, the integral is saturated when the distance θ_ibetween instantons are small. When instanton and anti-instanton are close to each other, we can not distinguish that it is an IA pair or just the fluctuations around vacuum. This breaks our assumption of dilute instanton configuration. We need to do regularization to this integral, we can first do the integral for g < 0, then we do analytic continuation to g > 0. For negative g, the interaction between instantons is repulsive and the dilute instanton approximation is preserved. The

interesting thing is, the choice of the direction of analytic continuation leads to ambiguity. This is the same phenomenon when we compute the Borel resummation of the ground state energy expansion around vacuum. If we sum all instanton contribution to the ground state energy, it would finally become ambiguity free.

This is the resurgence in quantum mechanics.

In order to do the integral, it is convenient to introuduce some notation.

λ(g) = 

√πge^−1/6g, (3.65)

µ = −2

g, (3.66)

we also use the integral representation of the delta function,

δ(

the partition can be rewritten as Z_⁽ⁿ⁾(β) ∼ βe^−β/2 To evaluate I(s), we set

µe^−θ = t, (3.70) For µ positive and large, g → 0⁻, the corrections are small. We find,

I(s) ∼ µ^sΓ(−s). (3.72)

We want to sum all instanton contribution, Z(β) = e^−β/2+

∞

X

n=1

Z_⁽ⁿ⁾(β), (3.73)

using the expression of I(s), After integrate βe^−βs by parts, we find,

Z(β) = − 1

The asymptotic behavior of Gamma function makes the integral converge. The contour can also be deformed to enclose the postive half plane Re(E) > 0. So this integral is a sum of all residues,

Z(β) =X

N ≥0

e^−βEN. (3.77)

Where E_N is the N -th state energy, they are also the solutions of the equation, φ(E) = 1 − λµ^E−1/2Γ(1/2 − E) = 0. (3.78) At the weak coupling limit, λ is very small, so the zero of φ(E) is close to the pole of Γ(1/2 − E).

E_N = N + 1

2 + O(λ). (3.79)

We can expand the N -th state energy in power of λ, E_N(g) =

∞

X

n=0

E_N⁽ⁿ⁾(g)λⁿ. (3.80)

This is the multi-instanton contribution to all the energy levels at leading order.

The coefficient of order n can be found by solving (3.78). It can be rewritten as,

−i = e^−1/6g

√2π (−2 g)^EΓ(1

2− E). (3.81)

The imaginary part comes from the square root of g. Using the Euler’s reflection formula, we obtain,

Compare LHS and RHS in order of λ, we can find E_N⁽ⁿ⁾(g) order by order. For example, E_N⁽¹⁾(g) is,

E_N⁽¹⁾(g) = −  N !(2

g)^N(1 + O(g)), (3.83)

and E_N⁽ⁿ⁾(g) is,

E_N⁽²⁾(g) = 1 (N !)²(2

g)^2N[ln



−2 g



− ψ(N + 1) + O(g ln g)], (3.84) where ψ is the logarithmic derivative of the gamma function. For n-th order contri-bution, it can in general be computed. It takes the form at leading order,

E_N⁽ⁿ⁾(g) = −(2

g)^nN{P_n^N(ln

−g 2



) + O(g(ln g)ⁿ⁻¹}, (3.85) where P_n^N(a) is a polynomial of degree n − 1. For example, for N = 0, one can find,

P₂(a) = a + γ, P₃(a) = 3

2(a + γ)²+π²

12, (3.86)

here γ is the Euler’s constant, γ = −ψ(1) = 0.57721 . . . . Remember we are comput-ing the multi instanton contribution at leadcomput-ing order to the N -th energy level. We changed the coupling constnat g to negative thus we can define the multi instanton configuration. At the same time, the Borel sums of the energy without any instanton is summable since the coupling is negative. Now we want to change the coupling from negative to positive by analytic continuation, two things happen. The Borel sums become non-summable and get an ambiguous imaginary part of order two in-stanton. At the same time, the function ln

−²_g

also gets an ambiguous imaginary part ±iπ. These imaginary parts would cancel each other and the energy is still ambiguity free. The imaginary part of the ground state energy without instanton is (3.53). The imaginary part P₂ is,

Im E₀⁽²⁾(g) = 1

πge^(−1/3g)Im[P₂(ln(−g/2))] ∼ ±1

ge^−1/3g. (3.87) The same order as the imaginary part of the ground state energy (3.53) without instanton. In fact, they cancel each other. Similar cancellation appear at all order.

The leading imaginary part of E⁽¹⁾(g) is canceled by E⁽³⁾(g). For the sub imaginary part of E⁽⁰⁾(g), they are canceled by E⁽⁴⁾(g), E⁽⁶⁾(g) and to all order for n even.

Thus we can construct a rather complicated expression to the energy of double well

potential.

E_N(g) =X

n

E_N,ngⁿ+

∞

X

n=0

∞

X

k=0 k−1

X

l=0

(e^−1/6g πg )^k(ln



−2 g



)^lc_N,n,k,lgⁿ, (3.88)

where n is the order of the perturbative expansion, k is the number of instanton and l is to sum over the polynomial P_n, it also represents the number of the quasi zero modes (IA pairs). This is the trans-series of the double well potential. The construction of the trans-series is verified in several different potential and can also be extended to supersymmetry case [7, 24, 25].

4 Resurgence in QFT

In Quantum field theory, whether it is possible to construct the trans-series or not is an interesting question. There are renormalons in asymptotically free theories⁴. The infrared renormalons appear on the positive real axis of Borel plane have less action than instanton, but we can not find such classical solutions coressponding to those singularities. We also noticed in previous section that renormalon is related to the running coupling and is a strong coupled effect, therefore it is a quantum effect and it seems very hard to realize it classically. There are some conjectures of the IR renormalons in asymptotic free theories, some claim they correspond to the OPE vev takes nonzero value, others think that one may need to do second renormalization of the theory. Although there are many different conjectures, no one really gives a concrete argument for them.

Recent years, it has been discovered by Argyres and ¨Unsal [14, 29] that in the compactified gauge theory with adjoint fermions, there are new semiclassical con-figurations, e.g., bion-anti-bion events, which may correspond to the infrared renor-malons. They conjectured that these new saddle points are the leading singularities in the Borel plane and they are the incarnation of the renormalons in the weak cou-pling limit. If this is true, it may be possible to construct the trans-series of QFT and give it a non-perturbative definition at weak coupling limit. Since renormalon comes from the infrared or ultraviolet momentum contribution, these saddles should

4It has been proved that there are no renormalons in φ⁴ theory and QED in 4d because of the asymptotic behaivor of the beta function. [26, 27, 28]

also relate to the different scales of the theory. However, the relation between the bion-anti-bion configuration and the different scales is still unclear. Futhermore, the position of the bion-anti-bion on the Borel plane is at _N^8π

c/4, where the position of the closest IR renormalon is ^8π_β

0

5, which do not coincide. There are still no enough evidence to conclude these new semiclassical configurations correspond to the IR renormalons.

After several months, Dunne and ¨Unsal published another paper [13] which dis-cusses the resurgence relation may be carried out in the compactified CP^{N −1}model.

By spatial compactification and periodic boundary condition on the fermions, there are new semiclassical configurations, the kink-instantons. Kink-instantons has ac-tion of order e^SI/N, which is the instanton action divided by a factor N . The kink-anti-kink forms a bion and the bion-anti-bion configuration has ambiguous imaginary parts. The position of the bion-anti-bion and the position of the closest IR renormalon is the same order. This is a hint that IR renormalons may be realized semiclassically.

The aim of this section is to show that when some theories are compactified in a particular way, there are new semiclassical configurations and no significantly phase transition during the compactified radius change. We want to show that these new semiclassical configurations may correspond to the IR renormalons in the non-compactified theory.

4.1 QCD(adj) on R

³

× S

¹

We are going to discuss the four dimensional gauge theory with G = SU (N ) gauge group and Nf adjoint fermions. The theory is compactified to R³ × S¹ with peri-odic boundary condition for the fermions. This is the so called QCD(adj) with time direction compactified by a spatial circle with circumference L. With this compact-ification, the theory has no center symmetry changing phase transition when the radius is varied [30]. At small circle size, the theory is weakly coupled and the semi-classical analysis is valid. The gauge holonomy around S¹ takes nozero value and behaves as a Higgs field, the gauge group abelianizes at long distances, G → U (1)^r where r is the rank of the gauge group. There are new semiclassical configurations,

5β₀ is the first coefficient of the beta function, which is ¹¹₃N_c in pure bosonic QCD

monopole instantons, appearing in the theory. Instead of the 4-d BPST instantons, monopole instantons (or 3-d instantons, twisted instantons) Mi, i = 1, . . . , r + 1 make the leading contribution to the semiclassical expansion. A monopole instan-ton anti-monopole instaninstan-ton pair is a bion B = [M ¯M ]. There are two types of bion, magnetic bion B_ij = [M_iM¯_j] and neutral bion B_ii = [M_iM¯_i], where magnetic bion carries magnetic charge and neutral bion does not. What we are interested in is the neutral bions since they do not carry any topological charge (magnetic charge, instanton number) and may be related to the renormalon on the Borel plane. The bion-anti-bion [B ¯B] pair has the same quantum number as the perturbative vac-uum, just like the IA pair in quantum mechanics. Using a generalized version of the technique when calculating the IA pair in QM double well, we can compute the imaginary part of the [B ¯B] pair. Argyres and ¨Unsal found the location of the [B ¯B] pole is qualitatively of the same order as the IR renormalons and claim they correspond to the elusive IR renormalons on the Borel plane.

4.1.1 4-d theory

The Largrangian for the 4-d theories with general gauge group G with Lie algebra G and N_f massless adjoint fermions is,

L = 1

2g²(F_µν, F_µν) + 2i

g²( ¯ψ_f, ¯σ^µD_µψ_f) + iθ

16π²(F_µν, ˜F_µν), (4.1) where (·, ·) is the gauge invariant Killing form on G, f = 1, . . . , Nf is the flavor index of the Weyl fermions and ˜F_µν := ¹₂_µνρσF_ρσ. We will focus on the gauge group G = SU (Nc) case later, but now we can do some general discussion. Without mass term insertion, we can use a chiral rotation to set the theta angle equal to zero, θ = 0. The coupling constant is the following function of the energy scale µ at one loop in perturbation theory.

exp



− 8π² g²(µ)



= (Λ

µ)^β0, (4.2)

where Λ is the strong coupling scale and β₀ is the coefficient of one loop beta function, which is given by

β₀ = h^∨11 − 2N_f

3 , (4.3)

when the fermions are in the adjoint representation, h^∨ is the dual Coxeter number of the Lie algebra. The condition for asymptotic freedom is Nf < 5, we can not add too many fermions. We want to put the theory on R³ × S¹ with the S¹ of size L in the x⁴ direction, and we impose periodic boundary condition on fermions. We also assume the inverse radius L⁻¹  Λ so the theory is weakly coupled at the scale of the compactification, g(L⁻¹)  1. We study the dynamics of the effective 3-d theory at a scale µ, Λ  g/L  µ  1/L.

4.1.2 3-d effective theory

Integrating the theory along the x⁴ direction gives us the 3-d effective Lagrangian, which can also be seen as a dimensional reduction. We also assume the fields do not depend on the compactified direction x⁴. We find,

L_3d= L g²[1

2F_mnF^mn+ |D_mA₄|²+ 2i ¯ψ_fDψ/ _f − ¯ψ_fσ¯⁴A₄ψ_f]. (4.4) We can use gauge transformations to rotate A4 to its Cartan subalgebra (CSA) with generators H_i ⊂ G (i.e, we can diagonalize A₄). We define the 3-d gauge fields by,

A4(x) := 2π

Lφⁱ(x)Hi, Hi ∈ CSA, i = 1, . . . , r, (4.5) Am(x) := aⁱ_m(x)Hi+ W_m^α(x)Eα, Eα ∈ CSA^⊥, α = 1, . . . , N_c²− 1 − r, (4.6) where H_i are the basis of the Cartan subalgebra and E_α are the roots, r is the rank of the gauge group G, φⁱ are scalar fields which are related to the gauge holonomy, the a_m photons are massless bosons since they are in the CSA, the W^α fields are charged by their roots. We denote φ := φⁱHi later. With these definition, keeping only quadratic terms, the Lagrangian is,

L_3d= L

2g²(f_mn+ d_[mW_n])² + 4π²

g²L(∂_mφ + α(φ)W_m^αE_α)²+2L

g² ψ_f[i/d − 2π

Lσ¯⁴λ(φ)]ψ_f^λ. (4.7) The field strength is defined by f_mn := ∂_[ma_n] and the covariant derivative is d_m :=

∂m+ iam. The charges of the Wm boson and ψf fermions are the roots α of G, and weights λ of fundamental representation of G.

The different vacua are parameterized by different choice of hφi, the gauge in-equivalent choices of φ corresponds to points on the affine Weyl chamber. The affine

Weyl chamber, sometimes called the gauge cell, has a simple description. We denote the affine Weyl chamber by ˆT ,

T := [φ|αˆ _i(φ) ≥ 0, i = 1, . . . , r, and α₀(φ) ≥ −1], (4.8)

where α_i are a basis of simple roots, and α₀ is the lowest root of this basis. The points of ˆT correspond to the gauge inequivalent choices of φ. At interior points of T , the gauge group is Higgsed to abelian factors,ˆ

φ : G → U (1)^r for φ ∈ interior( ˆT ). (4.9) This holds for general gauge group G. Once one compactifies the theory along one direction, φ behaves like a Higgs field and Higgses the theory⁶. So now our 3-d effective theory become a theory with gauge group U (1)^r. We only want to focus on the massless content of our theory, so we integrate out all the charged fields like the W- bosons and those fermions not in CSA. The 3-d classical effective Lagrangian for the massless modes become,

L_3d= L

2g²(f_mn, f_mn) + 4π²

g²L(∂_mφ, ∂_mφ) + i2L

g² ( ¯ψ_f, /∂ψ_f). (4.10) Where f_mn = ∂_ma_n− ∂_na_m stands for the U (1)^r field strength. This is a 3-d U (1)^r gauge theory with r real, massless, neutral scalars and Weyl fermions. This is the case when we choose the interior points of ˆT as our vacuum, while at the boundaries of ˆT , the gauge symmetry is not completely broken and leaving nonabelian factors, some of W^α-bosons and ψ^α fermions also become massless. But it is not what we are interested in now.

Electric and Magnetic charges

The electric λ and magnetic µ charges in the 4-d U (1)^r theory are defined by, λ :=

Z

S²_∞

∗f, µ := 1 2π

Z

S²_∞

f, (4.11)

6At generic points on the affine Weyl chamber, the gauge symmetry is broken down to U (1)^r while r is the rank of the gauge group. However, the effective potential of φ may preserve the gauge symmetry. Only when the point on the affine Weyl chamber is at the minimum of the effective potential, it is quantum mechanically stable. Actually, only for SU(N) gauge group, the gauge symmetry is completely broken down to U(1), while for other gauge group, there are still non-abelian symmetry survive.

where the 2-form U (1)^r field strength is defined by f := ¹₂f_µνdx^µ∧ dx^ν, while the dual field strength is ∗f := ¹₂f˜_µν^∗ dx^µ∧ dx^ν. Here

f :=˜ 1

2_µνρσf_ρσ (4.12)

Our original gauge group is G and all fields transfrom in some rep of G. So the electric charges, λ, of the fields U (1)^r⊂ G live on the gauge lattice Γ_G. But now all the dynamical fields in our theory are in the adjoint representation, thus the electric charges are in the root lattice Γ_r = Γ_adj of G. By the Dirac quantization condition, the allowed magnetic charges µ are in the co-weight lattice Γ^∨_w. But if we add new massive fields inside which are not in the adjoint representation, the massive sources can have charge in the larger weight lattice Γ_w, then by Dirac quantization, the allowed magnetic charges are in the co-root lattice Γ^∨_w, which is smaller than the co-weight lattice.

We can define the corresponding electric and magnetic point operator. The Wilson loop operator wrapping the S¹ direction at a point P ⊂ R³ represents the 3-d effective electric point operator. The Wilson loop can be represented by a sum of a set of operators with different charges.

Tr P exp

 i

Z

S¹

A₄dx⁴



=X

λ

exp[2πiλ(φ)]. (4.13)

The eletirc operator at point P with charge λ is then,

E[λ, P ] := exp[2πiλ(φ)](P ). (4.14) For the magnetic operator, the t’Hooft line operator on R³× S¹ wrapping the S¹ di-rection represents a monopole operator at some point inR³. The monopole operator at point P ⊂ R³ with charge µ is given by,

M [µ, P ] creates a gauge field singularity at P,with Z

S

f = 2πµ, (4.15) for any closed surface S, f is the U (1)^r 2-form field strength.

The 3-d dual photon

Since the U (1)^r field strengths (f_mn, f_mn) decouple from all the other fields con-tent, we can replace them by the dual photon fields σ(x) because they give us the same equation of motion [31, 32]. Consider a theory contanins the 3-d U (1)^r gauge

fields a_m, vector field b_m and a scalar field σ. The partition fucntion is given by,

where the Lagrangian L is, L := g²

4L(∂_mσ + b_m)²+ i

2_mnpb_m(f_np). (4.17) We input U (1)^r gauge invariance for a_m and additional symmetry to σ and b_m,

σ → σ + σ⁰, b_m → b_m− ∂_mσ⁰. (4.18) If we fix this symmetry by setting σ = 0 and integrating out bm fields, we get

Z = to fix the gauge symmetry, we find,

Z =

This is the dual photon expression of the original U (1)^r gauge field a_m. Finally, our 3-d effective Lagrangian becomes,

L_3d= g²

4L(∂_mσ, ∂_mσ) + 4π²

g²L(∂_mφ, ∂_mφ) + i2L

g²( ¯ψ_f, /∂ψ_f). (4.21) The σ and φ fields are dimensionless, where the ψ_f fermions have dimension ³₂. There are r real scalar bosons σ, r real real scalar bosons φ and r Weyl fermions ψ_f, all the fields are massless. Under this duality, the magnetic point operator becomes a local operator,

M [µ, P ] := exp[2πiσ(µ)](P ). (4.22) This is equivalent to inserting in an gauge invariant operator e^2πiσ(µ)(P ) · e^2πiRCb(µ) with the Dirac string C ending at P into the path integral. Integrating out b_m fields and fixing the gauge by setting σ = 0 gives us the original field strength integration

M [µ, P ] := exp[2πiσ(µ)](P ). (4.22) This is equivalent to inserting in an gauge invariant operator e^2πiσ(µ)(P ) · e^2πiRCb(µ) with the Dirac string C ending at P into the path integral. Integrating out b_m fields and fixing the gauge by setting σ = 0 gives us the original field strength integration

在文檔中 在量子力學和量子場論中的Resurgence (頁 31-45)