Nonperturbative characteristics of Green functions

R. Delbourgoa∗,

aSchool of Mathematics and Physics, University of Tasmania,GPO Box 252-21, Hobart, Tasmania 7001, Australia

I will describe a method of deriving a nonperturbative Green function in a manner which captures the topologyof the corresponding Feynman diagram, via the skeleton expansion. This allows one to determine the anomalousdimensions of the field theory in question as a function of the coupling constants.

1. RENORMALIZED PERTURBATIONTHEORY

The renomalization group reveals a lot aboutthe scaling properties of Green functions in renor-malizable models [1], properties which are them-selves connected with leading log behaviours inperturbation theory, after appropriate subtrac-tions have been made. They then provide thevariation of the Green function with the massscale of measurement and, in the end, the de-pendence is encapsulated by the β and γ func-tions for the theory in question. Here I wouldlike to describe what such Green functions mightlook like after summation of the perturbation se-ries, because their nonperturbative forms wouldbe very useful as inputs in skeleton expansionsand Schwinger-Dyson (SD) equations. I shall belooking at the functions in the asymptotic regimewhere mass terms may be assumed to be unim-portant and will therefore simplify the analysisby studying massless renormalizable models likeQED or QCD; the simplest model corresponds tothe trilinear interaction σφ†φ in 6-D and I willfocus on this, although Yukawa theory in 4-D hasalso undergone a similar analysis [2].

To understand how this this asymptotic be-haviour comes about, it is instructive to look atthe nature of the renormalized (at scale µ) per-turbation series for a two-point Green functionfor the field φ as it only depends on a single mo-

∗e-mail :bob.delbourgo@utas.edu.au

mentum variable p. The series takes the form

∆−1φ (p) = p2

[1 + g2c1 log

(−p2

µ2

)

+g4

{c2 log

(−p2

µ2

)+

12c21 log2

(−p2

µ2

)}+ · · ·

]

= p2

(−p2

µ2

)c1g2+c2g4+···≡ p2

(−p2

µ2

)γφ

, (1)

with γφ being interpreted as the anomalous di-mension of the field φ. Parenthetically it is worthmentioning that since the renormalization con-stant is defined by Z−1

φ = limp2→∞ p2∆φ(p), itis quite possible for Zφ to vanish if γφ is nega-tive; that would also accord with compositenessconcepts [4].

Likewise, up to possible logarithms (see Ref [1],Eq. (20.3.10)), other Green functions have thecorresponding scaling property as λ → ∞,

Γ(λp1, . . . , λpn) � λ2γΓΓ(p1, . . . , pn). (2)

where γΓ is its associated anomalous dimension,which is tied to those of the fundamental fieldsand the number of legs n. Such ideas accord withthose of conformal field theory [3] and are hardlynew. What Eq. (2) does NOT tell us is the depen-dence of Γ on the momentum ratios; this is whereall the dynamics lies hidden and it has intimateconnections with the topology of the perturba-tive Feynman diagrams whose series add up inγΓ. I will describe a method of quickly arrivingat the structures Eq. (1) and Eq. (2) from per-turbation theory, but first let us look at mattersof self-consistency.

Nuclear Physics B (Proc. Suppl.) 141 (2005) 63–67

0920-5632/$ – see front matter © 2004 Elsevier B.V. All rights reserved.

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doi:10.1016/j.nuclphysbps.2004.12.011

2. SKELETON EXPANSION

In the simple trilinear interaction modelg(σφ†φ)6D, the SD equations can be expressed inskeletal form, involving only propagators ∆ and3-point vertex functions Γ. They read

∆−1φ (p)= Zφp2−ig2Zg

∫d6kΓp,k∆φ(k)∆σ(p + k)

= Zφp2 − ig2

∫d6kΓp,k∆φ(k)∆σ(p + k)Γp,k

−g4

∫d6kd6q Γp,p+k∆φ(k)∆σ(p+k)∆φ(p+q)

∆σ(q)Γp,p+qΓq,q−k∆φ(k−q)Γp+k,k−q · · ·(3)

plus a parallel equation for the σ propagator. Ne-glecting the renormalization constants for the mo-ment, the skeleton expansion will yield terms onthe rhs,

g2F2(p, µ, γ(g2)) + g4F4(p, µ, γ(g2)) + . . . ,

that must somehow add up to the form Eq. (1). Iam going to assume that each of these contribu-tions self-consistently produce such a result, i.e.every FN scales in the same way; otherwise itis very difficult to understand what miraculouscombinations of skeleta are occurring to producethe final scaling — rendering the whole expansionworthless. It is then quite easy to establish thatthe vertex anomalous dimension is given in termsof the fields, viz. γΓ = γφ + γσ/2. As an aside,note that the quenched σ approximation amountsto neglecting quantum corrections to ∆σ, settingγσ = 0, whereupon the the scaling of Γ becomestied to that of φ; their inverse connection is un-surprising in QED because of the Ward identity,but in (σφ†φ)6D, it is a bit more unexpected.

Later on I will use such asymptotic behaviourto determine γ in terms of g2, via a transendentalequation, such that certain classes of diagrams aresummed in one shot, but I need first to clarify myself-consistent inputs in Eq. (3). The importantthing is to guarantee that the correct singularitiesof ∆ and Γ are incorporated from the word go andfor that I will turn to perturbation theory.

3. SINGULARITY STRUCTURE

Because the Feynman diagrams are divergentwhen expanded in powers of g2 they must be

regularized. Let’s use dimensional regularizationas it substantially simplifies the argument below.Working in 2� dimensions, wth general masses,the self energy Σ and proper vertex Λ can be writ-ten as

Σ(p)=c2g2

∫ ∫ 1

0

dx dyδ(x + y − 1) Γ(2 − �)

[p2xy − m21x − m2

2y]2−�,

Λ(p1, p2, p3) = c3g2

∫ ∫ ∫ 1

0

dx dy dz ×

δ(x + y + z − 1) Γ(3 − �)[p2

1yz + p22zx + p2

3xy − m21x − m2

2y − m23z]3−�

,

with � → 3 for our model. If we drop masses andrenormalize at scale µ and zero σ momentum, onefinds that [5]

Σ(p) =g2

p4

(−p2

4π

)� Γ(2 − �)Γ(� − 1)Γ(� − 1)Γ(2� − 2)

→ − g2p2

6(4π)3log

(−p2

µ2

), (4)

Λ =g2Γ(2 − �)

(4π)�(−p21p

22p

23)2−�

×[2π∆3/2−�

E Θ123− Γ2(�−1)Γ(2�−2)

×( (p2

1p22)2−�

p21+p2

2−p23

2F1

(1,

12; �− 1

2;− ∆E

(p21+p2

2−p23)2

)

+2 perms)]

, (5)

that can be transformed into

Λ =g2Γ(2−�)∆3/2−�

E

(4π)�(−p21p

22p

23)2−�

[2πΘ123

−Γ2(�−1)Γ(2�−3)

∑∫ 1

0

(p2i p

2j)

2−�t�−5/2 dt

∆3/2−�E

√4p2

i p2j−∆Et

], (6)

where ∆E =2(p21p

22 +p2

2p23 +p2

3p21)−p4

1−p42−p4

3 andΘ123 = 1 if a Euclidean can be drawn with sides√

p2i (otherwise Θ123 = 0); after renormalization

R. Delbourgo / Nuclear Physics B (Proc. Suppl.) 141 (2005) 63–6764

Λ= − g2

(4π)3

∫ 1

0

∫ 1

0

∫ 1

0

dx dy dz δ(x + y + z − 1)

× log(

p21yz + p2

2zx + p23xy

−µ2(yz + zx)

)

= − g2

2(4π3)

∫ 1

0

σdσ

∫ 1

−1

dτ log[(−p2

1

2µ2

)(1−τ)

+(−p2

2

2µ2

)(1+τ)− p2

3σ(1−τ2)4µ2(1−σ)

]. (7)

Basically these expression (4),(7) encapsulate thesingularities to be expected in the momentumvariables. A special case arises when p2

3 → 0,when you may note that Λ reduces to pure logs:

g2

2(4π)3

[log

(p21p

22

µ4

)− 2 +

p22 + p2

1

p22 − p2

1

log(

p22

p21

)],

which bears an uncanny resemblance to the dif-ference [Σ(p2) − Σ(p1)]/(p2

2 − p21) that one meets

in QED, even though we are not dealing with agauge theory!

4. GOING NONPERTURBATIVE

Presently I shall be descending to 6-D in a spe-cific way, so let me fix mass dimensions so asto ensure that the coupling g is always dimen-sionless. To that end multiply g2 by (4π/µ2)�−3.Also recognize that the first coefficient of γφ is-g2/6(4π)3; therefore if one simply substitutes� → 3 + γφ to this order,

Σ(p) → p2

(− p2

µ2

)γφ 6Γ(1−γφ)Γ(1+γφ)Γ(2+γφ)Γ(4 + 2γφ)

.

This coincides with the behaviour of the full prop-agator to all orders, Eq. (1), and it immediatelysuggests a recipe for going nonperturbative withall the Green functions: replace the first ordercoupling coefficient in the dimensionally resultby the appropriate anomalous dimension termand replace D/2 = � by 3+γφ, extracting beta-functions as well to agree with renormaliztion pre-scription. Applying this rule to the Λ, we finishup with the dimensionally-transmuted triangular

vertex function

Γ(p1, p2, p3) = 2∫ ∫ ∫ 1

0

dx dy dz δ(x+y+z−1)

×[p21yz + p2

2zx + p23xy

−µ2(yz + zx)

]γΓ

, (8)

such that the renormalized Γ(µ, µ, 0) = 1. Alter-natively

Γ(p1,p2,p3) =p21p

22p

23

∆3/2E

(p21p

22p

23

µ2∆E

)γΓ

×[2πΘ123

Γ(3 + 2γΓ)Γ2(2 + γΓ)

−3∑

k=1

∫ 1

0

(p2i p

2j)

−1−γΓ tγΓ+1/2 dt

(∆E)−γΓ−3/2√

4p2i p

2j −∆Et

]. (9)

More interestingly, when p23 = 0, the expression,

µ2

(1 + γΓ)(p21 − p2

2)

[(− p2

2

µ2

)γΓ+1

−(− p2

1

µ2

)γΓ+1]

arises, having the form [∆−1(p2)−∆−1(p1)]/(p22−

p21). We’ll return to this point shortly but let

me remark that for p23 �= 0 and nonzero masses,

the full Γ can be reduced to a Meijer G-functionwhich is unfortunately not more illuminatingthan the original Feynman parametric form.

5. γ − g2 RELATION

Having settled upon nonperturbative expres-sions for the skeletal Γ and ∆ in the asymptoticregime, we’re in a position to relate the scalingdimensions to the coupling constant to all orders.This means working out all terms in the series∑

N g2NF2N (p, µ, γ); in principle that would yieldthe connection between γ and g2. But the task isformidable — it means calculating all the Feyn-man skeleton contributions. Nevertheless I willindicate that one can get quite far just by consid-ering the very first term in the series for already atthat level we will be summing over rainbows, self-energy chains and triangular topology vertices atone swoop. For simplicity take σ to be undressedand identify γΓ = γφ ≡ γ.

R. Delbourgo / Nuclear Physics B (Proc. Suppl.) 141 (2005) 63–67 65

Insertion of Eq. (1) and Eq. (9) extremely hardto compute analaytically and it’s not for wantof trying. As a stop-gap measure I will mangleEq. (9) and use two forms for it which have someof its characteristic properties, the most imprtantbeing scaling. In my first model I will use themockup form Γ(p,−p−k, k) = (p2(p+k)2/µ2k2)γ

which provides

p2(1+γ) = ig2

∫d6k

k2(p + k)2(1+γ)

(p2(p + k)2

k2

)2γ

,

or

1 = aΓ(−1 + γ)Γ(2 − 2γ)Γ(2 + γ)Γ(4 − γ)Γ(1 + 2γ)Γ(1 − γ)

,

a ≡ g2

(4π)3. (10)

This may be contrasted with the rainbow approxi-mation where the self-consistency relation insteadreads [6], p4∆(p) = −ig2

∫∆(p + k) d6k/k2 and

1 = a/γ(γ − 1)(γ − 2)(γ − 3). To obtain a per-turbative expansion we take a series in γ or a asneeded to arrive at

γmodel−1 = −a

6+

11a2

63− 134a3

65+ · · · ,

compared with [7]

γrainbow = −a

6+

11a2

63− 206a3

63+ · · · ,

andγchain = −a

6+

11a2

63− 170a3

63+ · · · .

The second model consists in returning to theDS equation

1 = Zφp2∆φ(p)

−ig2Zg

∫d6k

k2∆φ(p + k)Γp+k,p∆φ(p), (11)

taking the p2σ→0 limit of the vertex, whereupon

∆φ(p+k)Γp+k,p∆φ(p)� [∆φ(p+k)−∆φ(p)]/[p2 − (p + k)2],

and applying the Lehmann-Kallen spectral repre-sentation,

∆φ(p) =∫ ∞

0

ρ(w2) dw2

p2 − w2, (12)

∆φ(p+k)Γ(p+k, p)∆φ(p)

�∫ ∞

0

ρ(w2) dw2

((p+k)2−w2)(p2−w2). (13)

Recalling that Z−1φ =

∫ρ(w2)dw2, the spectral

equation reduces to∫

dw2 ρ(w2)w2ZφZ−1

g + Σ(p, w)p2 − w2

= 0, (14)

where Σ(p, w) = g2∫

d2�k/[k2((p + k)2 − w2)] isthe first order self-energy for a φ field of massw, to be taken in the limit as � → 3. This islike QED, except that it is no longer true thatZg = Zφ — and this is just what one needs! Thepoint is that the self-energy carries an infinity (inits real part) corresponding to

Σ(p, w) =a

� − 3

[16p2 − 1

2w2

]

+(p2 − w2)2

π

∫ �Σ(s, w) ds

(s − w2)2(s − p2 − iε),

while Z−1φ Zg = 1 + 2a/(3(� − 3); both combine

neatly to produce a factor (p2−w2) in the numer-ator of (14). The imaginary part of (14) yields

−πp2ρ(p2) +∫ �Σ(p, w)ρ(w2) dw2

p2 − w2= 0. (15)

Since �Σ(p, w) = g2(p2 − w2)2/6(4π)3p4,

or �Σ(p, w)/(p2 − w2) = a(1 − w2/p2)2/6,

Eq. (15) may be solved via ρ(w2) ∝ (w2)−1−γ ,whence the relation 1 = −a/3γ(1−γ)(2−γ), or

γmodel−2 = −a

6+

9a2

63− 144a3

65· · ·

only exact to order a. Evidently ∆φ(p) has thesame anomalous dimension as its spectral func-tion (or imaginary part) ρ.

I should warn you that it is easy to adopt otheransatze for the vertex which also scale appropri-ately but which yet lead to infinities, so the aboveresults need to treated with great caution andsome scepticism until the full calculation of F2

has been done with Eq. (9).

R. Delbourgo / Nuclear Physics B (Proc. Suppl.) 141 (2005) 63–6766

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R. Delbourgo / Nuclear Physics B (Proc. Suppl.) 141 (2005) 63–67 67