c Pleiades Publishing, Ltd., 2009. ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2009, Vol. 265, pp. 154–164. c M.D. Missarov, R.G. Stepanov, 2009, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Original Russian Text Vol. 265, pp. 165–176.
On Vertex Parts of p-adic Feynman Amplitudes M. D. Missarov a and R. G. Stepanov a Received August 2008
Abstract—We reduce the calculation of the vertex parts of p-adic Feynman amplitudes to a recursive procedure for evaluating singular parts of certain integrals. We propose an algorithm for calculating these integrals in the general form. As an example, we consider vertex parts of amplitudes in the ϕ4 theory. DOI: 10.1134/S0081543809020151
On the d-dimensional p-adic space Qdp , consider the p-adic ϕ4 theory determined by the Hamiltonian H(ϕ; α; r, g) = H0 (ϕ; α) + L(ϕ(x); r, g) dx, where L(ϕ(x); r, g) = rϕ2 (x) + gϕ4 (x) and the Gaussian part is defined as 1 |x − y|−α ϕ(x)ϕ(y) dx dy. H0 (ϕ; α) = 2 A general introduction to the theory of p-adic models of mathematical physics is contained in the book [1] by Vladimirov, Volovich, and Zelenov. Here, ϕ(x) is a real- or complex-valued field, the argument x = (x1 , . . . , xd ) belongs to the d-dimensional p-adic space Qdp endowed with the norm |x|p = maxi |xi |p , and dx is the Haar measure on Qdp normalized so that Zd dx = 1, where p
Zdp = {x ∈ Qdp : |x| ≤ 1}. The parameter α is real; in what follows, we will be interested in the behavior of the model in the following cases: (i) α = 3d/2 + ε, where d is a positive integer and ε is a small parameter (for convenience, we refer to this case as the case of analytic renormalization); (ii) α = d + 2 = 6 − δ, where d = 4 − δ and δ is a small parameter (the case of dimensional renormalization). The computational formalism of perturbation theory in models of quantum field theory and statistical physics is based on the notions of Feynman graphs and amplitudes. Let G be any connected Feynman graph in the ϕ4 theory. We denote the set of its external vertices by W , the set of its internal vertices by V , and its edge set by L. The coordinate form of the Feynman amplitude corresponding to the graph G is xi(l) − xf (l) a dxv , FG (xv , v ∈ W ; a) = p l∈L
v∈V
where i(l) and f (l) are the initial and final vertices of the edge l and all variables xv belong to the space Qdp . The case of analytic renormalization corresponds to a = α − 2d = ε − d/2, and the case of dimensional renormalization corresponds to a = 2 − d = δ − 2. a Faculty of Computer Science and Cybernetics, Kazan State University, Kremlevskaya ul. 18, Kazan,
420008 Russia. E-mail addresses:
[email protected] (M.D. Missarov),
[email protected] (R.G. Stepanov).
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The “proper” singular part of a Feynman amplitude is called the vertex part of this amplitude. In [4, 2], it was proved that the p-adic ϕ4 theory is renormalizable. This means that Feynman amplitudes in the ϕ4 theory can be represented as RFG (xv , v ∈ W ) =
{V1 ,...,Vs }
FG|G1 ,...,Gs (xv , v ∈ W )
s
O(Gi ),
i=1
where RFG (xv , v ∈ W ) is the renormalized amplitude; {V1 , . . . , Vs } is an arbitrary partition of the vertex set of the graph W ∪ V ; Gi is the complete subgraph generated by the vertices from Vi for i = 1, . . . , s; G|G1 ,...,Gs is the graph obtained from G by contracting the subgraphs Gi (each to its own vertex); and O(Gi ) is the vertex part of the subgraph Gi . In the case of analytic renormalization, O(Gi ) is a polynomial in ε−1 with zero constant term, and in the case of dimensional renormalization, O(Gi ) is a polynomial in δ−1 with zero constant term, where δ = 4 − d. If |Vi | = 1, then O(Gi ) = 1. The vertex parts O(Gi ) are chosen so that the renormalized amplitude RFG , treated as a generalized d|W | function on Qp , is analytic in ε at the point ε = 0 (respectively, analytic in δ at δ = 0). To calculate the vertex part of a graph G, one must first calculate the vertex parts of its proper subgraphs. Note that the vertex parts encode important physical information, such as critical indices. Calculating the vertex parts of Feynman amplitudes is a hard computational problem in the Euclidean models of quantum field theory and statistical physics. In this paper, we discuss an algorithm for calculating the vertex parts of p-adic Feynman amplitudes of any order and present computational results for third-order diagrams. Let A(G; a) denote the action of the generalized function FG (xv , v ∈ W ; a) on the test function v∈W χ(xv ): χ(xv ) dxv , A(G; a) = FG (xv , v ∈ W ; a) v∈W
= {x ∈ Qdp : |x|p ≤ 1}. where χ(x) is the characteristic function of the ball Theorem 1. The following equality holds: s A(G|G1 ,...,Gs ; a) O(Gi ) , O(G) = −S Zdp
{V1 ,...,Vs }
i=1
where S(f ) denotes the operation of taking the principal part of the Laurent series of the function
f (ε) at the point ε = 0 and {V1 ,...,Vs } denotes summation over all partitions of W ∪ V into subsets except the partition consisting of the only element {W ∪ V }. In [4], this theorem was proved in the case where all vertices are external, but the proof can be generalized to amplitudes with nonempty set of internal vertices. Thus, the vertex parts of subgraphs can be calculated recursively, provided that the quantities A(G) are known. To calculate these quantities, we consider generalized Feynman amplitudes. Let G be an arbitrary graph (in the sense of general graph theory) with vertex set V (which is not divided into internal and external vertices) and edge set L = L(G), and let a(L) = {al ; l ∈ L} be a set of complex numbers indexed by the edges from L. We set xi(l) − xf (l) a(l) . FG (x; a(L)) = p l∈L
Consider the integral
T (G; a(L)) =
FG (x; a(L))
χ(xv ) dxv .
v∈V
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Earlier, in [3], it was is proved that T (G; a(L)) = pα(V )
A V ∈A
where α(V ) =
and
al
1 pβ(V
)
(pd − 1)! , − 1 (pd − k(V ))!
(1)
β(V ) = α(V ) + d(|V | − 1);
(2)
l∈L(G(V ))
moreover, the integral converges absolutely if Re β(V ) > 0 for all V ∈ A (for a d-dimensional version of this assertion see Lemma 3.2 in [4]). Here, the summation is over all hierarchies A on the vertex set V of the graph G, A is the set of all elements in the hierarchy A except for single vertices, and k(V ) for V ∈ A is the number of maximal elements of the hierarchy that are contained in V . In (1), G(V ) denotes the subgraph of G generated by the vertex set V , and L(G(V )) is the set of internal edges in this subgraph. Recall that a hierarchy on a set V is defined as a family A of subsets V ⊂ V such that V ∈ A, {v} ∈ A for all v ∈ V , and for any V ∈ A and V ∈ A either V ∩ V = ∅, V ⊂ V , or V ⊂ V . In the case under consideration, it is also assumed that 1 < k(V ) ≤ pd for all V ∈ A . Any set of different p-adic d-vectors x = (xv ; v ∈ V ) indexed by the elements of a set V generates a hierarchy Ax on this set: |xv − xv | < min |xv − xv | . Ax = V ⊆ V : max v, v ∈V
v ∈V , v∈V \V
The hierarchy Ax is indexed. By an indexed hierarchy on V we mean a pair (A, m), where A is a hierarchy on V and m is a positive function on A satisfying the following conditions: (i) m(V ) = 0 if and only if |V | = 1; (ii) V ⊂ V implies m(V ) < m(V ). In our case, the function mx is defined by |xv − xv |. mx (V ) = max v,v ∈V
In addition to conditions (i) and (ii), we require the function m to admit the representation m(V ) = pn(V ) ,
n(V ) ∈ Z,
for any V ∈ A. It is easy to see that for any indexed hierarchy (A, m) on V , there exists a set of vectors x such that (Ax , mx ) = (A, m). The compactification of a graph G is the graph G obtained by adding one more vertex v0 to the vertex set V of G and by adding all edges joining vertices from V with the vertex v0 to the edge set. We assume that xv0 ≡ 0. Let xi(l) − xf (l) al |xv |bv (1 − χ(xv )), FG (x; a(L), b(V )) = l∈L
v∈V
where a(L) = {al ; l ∈ L} is a set of complex numbers indexed by the elements of the edge set L = L(G) and b(V ) = {bv ; v ∈ V } is a set of complex numbers indexed by the elements of the vertex set V = V (G). Consider the integral dxv . C(G; a(L), b(V )) = FG (x; a(L), b(V )) v∈V
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Lemma 1. The following equality holds:
C(G; a(L), b(V )) = p
l∈L
al +
v∈V
bv +d|V |
T (G; a(L)),
where G is the compactification of the graph G, L is the edge set of the compactified graph G, and the exponent al is the same for all l ∈ L(G); if an edge l joins vertices v and v0 , then al = −bv − 2d − l ∈L(v) al , where L(v) denotes the set of edges in L incident to the vertex v. Proof. We supplement the vertex set V with an additional vertex v0 , set xv0 ≡ 0, and denote x = (x; xv0 ), x = (xv ; v ∈ V ). Let (A, m) be an indexed hierarchy on V ∪ {v0 }, and let B(A, m) denote the subset
B(A, m) = (xv ; v ∈ V ) : (Ax , mx ) = (A, m) . We can write C(G; a(L), b(V )) =
FG (x; a(L), b(V ))
dxv ,
v∈V
(A,m)B(A,m)
where the summation is over all indexed hierarchies on V ∪ {v0 } such that m(V ) > 1 for all sets V ∈ A containing the vertex v0 . Note that the function FG (x; a(L), b(V )) is constant on B(A, m). Let us introduce a new variable y = y(x) with yv = xv |xv |2 for all v ∈ V , where the factor |xv |2 is treated as a p-adic number. It is easy to see that this gives a homeomorphism between the domains 2 −1 {x : 1 < |xv |, v ∈ V } and {y : 0 < |yv | < 1, v ∈ V }, because xv |xv |p p = |xv |p . Take a number r = pn such that r < m(V ) for all V ∈ A and all negative n ∈ Z. The set B(A, m) is a finite union of balls of radius r of the form D(z; r) = {x : |z − x| ≤ r}, where z is a center of the ball and |x| = max{|xv |, v ∈ V }. Any point in D(z; r) can be represented as x = z + h, where |h| ≤ r, and since |hv | < 1 < |zv | for all v ∈ V , it follows that the mapping v ∈ V }, where yv = zv |zv |2 + hv |zv |2 . This means that the y = y(x) takes this point to y = {yv , ball D(z; r) is mapped to the domain v∈V Dv (zv |zv |2 ; r|zv |−2 ), where Dv (zv |zv |2 ; r|zv |−2 ) is a ball in the space of the variable yv . In other words, the Jacobian of the mapping under consideration equals v∈V |zv |−2d . It is also clear that |xv − xv | = |yv − yv | · |yv |−1 |yv |−1 . This implies
yi(l) − yf (l) al |yv |−bv −2d− l ∈L(v) al χ(p−1 yv ) dyv C(G; a(L), b(V )) = l∈L
v∈V
yi(l) − yf (l) al |yv |−bv −2d− l ∈L(v) al χ(yv ) dyv ,
= pN (V )
v∈V
l∈L(G)
where N (V ) = −
l∈L
al +
v∈V
bv +
l ∈L(v)
al
+ d|V | =
l∈L
al +
bv + d|V |.
v∈V
Finally, note that, introducing an artificial variable y(v0 ), we can rewrite the last integral as
yi(l) − yf (l) al yv − y(v0 )−bv −2d− l ∈L(v) al χ(yv ) dyv χ(yv0 ) dyv0 . l∈L(G)
v∈V
This is an integral of the form (1) for the compactified graph G, which completes the proof of the lemma. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
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In what follows, we consider graphs in the ϕ4 theory. Let G be a connected graph generated by 4-tails. This means that the degree of each vertex is greater than 0 but no greater than 4. The vertices of degree 4 are said to be internal, and the remaining vertices are external. The edges joining vertices are called internal edges. We can assume that the external vertices are incident to some additional external edges, so that the full degree of each vertex (counting both internal and external edges) equals 4. Any subset V of vertices generates a subgraph G(V ). The edges joining vertices from this subset are the internal edges of the subgraph G(V ), and all of the remaining edges incident to vertices from V are its external edges. A subgraph is said to be separable if all its external edges are incident to the same vertex not belonging to this subgraph. Consider the following generalization of the integral A(G; a(L)): xi(l) − xf (l) al dx χ(xv ) dxv , A(G; a(L)) = v p v∈V
l∈L
v∈W
where W and V are, respectively, the sets of external and internal vertices in the graph G and L is the set of internal edges in G. Theorem 2. Let G be a connected graph in the ϕ4 theory without separable subgraphs. Then, in the complex space of exponents a(L), there is an open set such that, for all a(L) from this set, the integrals A(G; a(L)) converge absolutely and pα(L2 )+α(E2 )+d|V2 | T (G1 ; a(L1 )) T (G2 ; a(L2 )), (3) A(G; a(L)) = V1 ⊂V
where the summation is over all subsets V1 of V (including the empty set ), G1 is the graph generated by the vertex set W ∪ V1 , G2 is the graph generated by the vertex set V2 = V \ V1 , G2 is the compactification of the graph G2 , L1 is the set of internal edges of the subgraph G1 , L2 is the set of internal edges of the subgraph G2 , E2 is the set of external edges of G2 , L2 is the edge set of the al is the same for all compactified graph G2 , and the set a(L2 ) is defined as follows: the exponent
l ∈ L(G2 ), and if l joins a vertex v with the vertex v0 , then al = − l ∈L(v) al − 2d, where L(v) denotes the set of edges from L = L(G) incident to v. Finally, al and α(E2 ) = al . α(L2 ) = l∈L2
l∈E2
Proof. Suppose that the integral A(G; a(L)) converges. The ultrametricity of the p-adic norm implies xi(l) − xf (l) al dxv χ(xv ) dxv = B(G1 )C(G2 ), A(G; a(L)) = p v∈V
l∈L
where the summation is over all subsets V1 of V , xi(l) − xf (l) al B(G1 ) = C(G2 ) =
l∈L(G1 )
V1 ⊂V
v∈W
χ(xv ) dxv ,
v∈W ∪V1
a xi(l) − xf (l) al |xv | l∈E2 (v) l (1 − χ(xv )) dxv , l∈L(G2 )
v∈V2
where E2 (v) is the set of external edges in the subgraph G2 that are incident to the vertex v ∈ V2 . The calculation of the integral B(G1 ) reduces directly to an application of formula (1). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
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The calculation of the integral C(G2 ) reduces to an application of Lemma 1. This implies (2). Clearly, the integral A(G; a(L)) converges if so do the integrals T (G1 ; a(L1 )) and T (G2 ; a(L2 )) for all subgraphs G1 and G2 involved in decomposition (2). The integral T (G1 ; a(L1 )) converges if Re β(V1 ) > 0 for all subsets V1 ⊂ W ∪ V1 , where al + d(|V1 | − 1) (4) β(V1 ) = l∈L1
and L1 is the set of internal edges of the subgraph generated by V1 . The integral T (G2 ; a(L2 )) converges if Re β(V2 ) > 0 for all subsets V2 ⊂ V2 ∪ {v0 }, where V2 = V \ V1 . Note that if V2 ⊂ V2 , then β(V2 ) has the form (4), where L2 is the set of internal edges of the subgraph generated by V2 . If V2 = V2 ∪ {v0 }, where V2 ⊂ V2 , then al − al + 2d + d|V2 | = − al − al − d|V2 |, β(V2 ) = l∈L 2
v∈V2
l∈L 2
l∈L(v)
l∈E2
where L2 and E2 are the sets of internal and external edges, respectively, of the subgraph generated by the vertex set V2 . In what follows, without loss of generality, we assume that the exponents al are real, because the convergence of integrals depends only on their real parts. Thus, the convergence condition for the integral A(G; a(L)) reduces to the conditions α(L ) + d(|V | − 1) > 0
(5)
α(L ) + α(E ) + d|V | < 0
(6)
for all subsets V ⊂ V ∪ W and for all subsets V ⊂ V , where α(L ) =
al ,
l∈L
α(E ) =
al ,
l∈E
and L and E are the sets of internal and external edges of the subgraph G(V ). Relations (5) and (6) are, respectively, the “ultraviolet” and the “infrared” convergence conditions for the integral A(G; a(L)). Now, let us prove the existence of a domain in the space of parameters {al , l ∈ L} on which the integral A(G; a(L)) converges. We use the ideas of [5] adapted to the case under consideration. Note at once that if the graph G contains separable subgraphs, then conditions (5) and (6) cannot be satisfied. Indeed, the existence of a separable subgraph means that there is a subset V of vertices such that the set of external edges of the subgraph G(V ) is nonempty and all these edges / V . Then condition (5) for the subgraph G(V ∪ {v0 }) and are incident to the same vertex v0 ∈ condition (6) for the subgraph G(V ) are incompatible. For each l ∈ L, we set d|D(l)| + εl , al = − |D| where D(l) is the family of trees in the graph G that contain the edge l, D is the family of all trees in G, and εl > 0. Let V ⊂ V ⊂ W . Then d d |D(l)| + εl + d(|V | − 1) = − |T ∩ L | + εl + d(|V | − 1). α(L ) = − |D| |D| l∈L
Since |T ∩
L |
|
T ∈D
l∈L
≤ |V − 1, it follows that
α(L )
l∈L
> 0 for all positive sets of εl , l ∈ L .
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Now, let V ⊂ V , L be the set of internal edges of the subgraph G(V ), and E be its set of external edges. The external edges from E connect the subgraph G(V ) with the subgraph G(V ), where V = (V \ V ) ∪ W . Suppose that the restriction of a tree T in the graph G to the subgraph G(V ) has several connected components T1 , . . . , Tk . By the restriction of T to G(V ) we mean the subgraph with vertex set V and the set of internal edges L(T ) ∩ L(G(V )). Let us identify the vertices of each subgraph Ti with one vertex vi for i = 1, . . . , k and consider the graph G determined by the vertex set V ∪ {v1 , . . . , vk } and the set of internal edges L ∪ E , in which an edge l ∈ E is incident to a vertex vi if l is incident to one of the vertices from Ti in the graph G. The tree T induces a tree T in the graph G , and |T ∩(L ∪E )| = |T |. Since |T | = |V |+k −1, it follows that |T ∩(L ∪E )| ≤ |V |. The inequality becomes strict if k > 1. Let us show that there exists a tree T0 whose restriction to the subgraph G(V ) has more than one connected component. Suppose that the tree T has only one connected component T1 in the subgraph G(V ). Take any connected component G(V1 ) of the subgraph G(V ). There are two edges l1 and l2 that connect this component with the subgraph G(V ) and are incident to vertices v1 and v2 from V (v1 = v2 ). This follows from the connectedness of the graph G, the absence of separable subgraphs in this graph, and the fact that the degree of each vertex is even (recall that G is a graph in the ϕ4 theory). Let l3 ∈ T1 be an edge belonging to the path between v1 and v2 along the subtree T1 . Let T2 be a tree in the component G(V1 ). Consider the new tree T0 in G obtained from T by removing the edge l3 and replacing all edges of T contained in the set of (internal and external) edges of the subgraph G(V1 ) with edges from T2 and the edges l1 and l2 . Note that the restriction of the tree T0 to the subgraph G(V ) has two connected components. This implies that 1 |T ∩ (L ∪ E )| < |V |, |D| T ∈D
and for sufficiently small positive εl , l ∈ L , we obtain α(L ) + α(E ) + d|V | < 0. This completes the proof of the theorem. Thus, we have proved that the integral A(G; a(L)) converges on some domain of values of the parameters al with l ∈ L and is determined by the explicit expression (3). This formula gives an explicit expression for the analytic continuation of A(G; a(L)) to the entire complex space of values of al , l ∈ L, including the diagonal al = a for all l ∈ L. Below, we give the results of calculations of the vertex parts of Feynman amplitudes in the 4 ϕ theory up to the third order of perturbation. For this purpose, we need to calculate the vertex parts of the following Feynman integrals (we omit obvious graphic representations of these amplitudes): I1 = dx1 dx2 |x1 − x2 |2a , Ω
Ω
I2 =
dx1
Ω
Ω
I3 =
dx1
Ω
dx3 |x1 − x3 |2a |x2 − x3 |2a ,
dx2
dx3 |x1 − x2 |2a |x1 − x3 |a |x2 − x3 |a ,
dx2 Ω
Ω
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I4 =
dx1
Ω
dx2 Ω
I5 =
dx1
Ω
I6 = Ω
Ω
Ω
I9 = Ω
I10 =
I11 = Ω
Ω
dx4 |x1 − x2 |2a |x3 − x4 |2a |x1 − x4 |a |x2 − x3 |a ,
dx3 Ω
Ω
dx2
Ω
dx4 |x1 − x2 |2a |x2 − x3 |2a |x3 − x4 |a |x1 − x4 |a , Ω
dx4 |x1 − x2 |a |x1 − x4 |a |x2 − x3 |a |x2 − x4 |a |x3 − x4 |2a ,
dx3
dx2
dx1
Ω
dx4 |x1 − x2 |2a |x2 − x3 |a |x2 − x4 |a |x3 − x4 |2a , Ω
Ω
Ω
dx3
dx2
dx1
Ω
dx4 |x1 − x2 |a |x1 − x4 |a |x2 − x3 |3a |x3 − x4 |a ,
dx3
dx2
dx1
Ω
Ω
Ω
dx4 |x1 − x2 |2a |x2 − x3 |2a |x3 − x4 |2a ,
dx3
dx2
dx1
dx4 |x1 − x3 |a |x1 − x4 |a |x3 − x4 |2a |x2 − x3 |a |x2 − x4 |a ,
dx3
dx2
dx1
I8 =
dx4 |x1 − x2 |a |x1 − x3 |a |x1 − x4 |a |x2 − x3 |a |x2 − x4 |a |x3 − x4 |a .
dx3 Ω
161
dx3
dx2
dx1
I7 =
Ω
Here, Ω denotes the integration over the unit p-adic ball Zdp and denotes the integration over the whole space Qdp . The Feynman integrals I1 , I3 , I10 , and I11 can be calculated by formula (1), because in these integrals the integration with respect to all variables is over the ball Ω. We set 1 1 . and g(x) = −ψ(−x) = x ψ(x) = −x 1−p p −1
The results are ψ(2a + d) , I1 = ψ(d) I3 = I10 =
ψ(4a + 2d) ψ(d)2
pd − 2 + 2g(a + d) + g(2a + d) , pd − 1
ψ(6a + 3d) (pd − 2)(pd − 3) pd − 2 4g(3a + 2d) + 2g(2a + d) + 2g(a + d) + 2g(d) + ψ(d)3 (pd − 1)2 pd − 1 + 4g(3a + 2d)g(2a + d) + 4g(3a + 2d)g(a + d) + 4g(3a + 2d)g(d) + g(2a + d)2 + g(d)2 + g(a + d)2 ,
I11 =
ψ(6a + 3d) (pd − 2)(pd − 3) pd − 2 4g(3a + 2d) + 4g(a + d) + 2g(a + d) + ψ(d)3 (pd − 1)2 pd − 1 2 + 12g(3a + 2d)g(a + d) + 3g(a + d) .
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To calculate the remaining integrals, we use the following trick. We represent integrals in the form Ω + Ω , where Ω is the complement of the ball Ω. The integrals over the ball Ω are calculated explicitly by using the formulas ψ(a + d) ψ(a + d) a , dx |x| = − dx |x|a = ψ(d) ψ(d) Ω
Ω
and the fact that in the p-adic space the relation |x1 − x2 | = |x2 | holds for any x1 and x2 such that x1 ∈ Ω and x2 ∈ Ω. The expressions for the remaining integrals are ψ(2a + d)2 ψ(−4a − d) , ψ(d)ψ(−2a)2 ψ(6a + 3d) (pd − 2)(pd − 3) pd − 2 2g(4a + 2d) + 2g(2a + 2d) + 4g(a + d) + d I4 = ψ(d)3 (pd − 1)2 p −1 + g(2a + d) + g(d) + 4g(4a + 2d)g(a + d) + 2g(4a + 2d)g(2a + d) + 4g(2a + 2d)g(a + d) + 2g(2a + 2d)g(d) + 2g(a + d)2 + g(2a + d)g(d)
I2 =
+
ψ(4a + d)ψ(2a + d) ψ(4a + d)ψ(a + d)2 ψ(2a + d)2 ψ(−4a − d)ψ(6a + 2d) , − − ψ(d)2 ψ(d)3 ψ(−2a)2 ψ(4a + 2d)ψ(d)
ψ(2a + d)3 ψ(−6a − 2d) , ψ(d)ψ(−2a)3 ψ(6a + 3d) (pd − 2)(pd − 3) pd − 2 2g(4a + 2d) + 2g(2a + 2d) + g(3a + d) + d I6 = ψ(d)3 (pd − 1)2 p −1 + 3g(a + d) + 2g(d) + 2g(4a + 2d)g(3a + d) + 2g(4a + 2d)g(a + d) + 2g(4a + 2d)g(d) 2 2 + 4g(2a + 2d)g(a + d) + 2g(2a + 2d)g(d) + g(3a + d)g(a + d) + g(a + d) + g(d) I5 =
ψ(4a + d)ψ(a + d)3 ψ(a + d)2 ψ(3a + d)ψ(−4a − d)ψ(5a + 2d) , − ψ(d)3 ψ(d)2 ψ(−a)ψ(−3a)ψ(4a + 2d) ψ(6a + 3d) (pd − 2)(pd − 3) pd − 2 g(4a + 2d) + 2g(3a + 2d) + g(2a + 2d) + d I7 = ψ(d)3 (pd − 1)2 p −1 + 2g(2a + d) + 2g(a + d) + 2g(d) + 2g(4a + 2d)g(a + d) + g(4a + 2d)g(2a + d) −
+ g(2a + d)2 + 2g(a + d)g(d) + 2g(3a + 2d)g(2a + d) + 2g(3a + 2d)g(a + d) ψ(2a + d)ψ(4a + d) , + 2g(3a + 2d)g(d) + g(2a + 2d)g(2a + d) + 2g(2a + 2d)g(d) − ψ(d)2 ψ(6a + 3d) (pd − 2)(pd − 3) pd − 2 2g(3a + 2d) + g(4a + 2d) + g(2a + 2d) + d I8 = ψ(d)3 (pd − 1)2 p −1 + 4g(a + d) + g(d) + g(2a + d) + g(3a + 2d)g(2a + d) + 4g(3a + 2d)g(a + d) PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
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+ g(3a + 2d)g(d) + 2g(4a + 2d)g(a + d) + g(4a + 2d)g(2a + d) + 2g(2a + 2d)g(a + d) ψ(a + d)2 ψ(4a + d) 2 , + g(2a + 2d)g(d) + g(a + d)g(2a + d) + g(a + d) + g(a + d)g(d) − ψ(d)3 ψ(6a + 3d) (pd − 2)(pd − 3) pd − 2 + d g(4a + 2d) + 2g(3a + 2d) + g(2a + 2d) I9 = ψ(d)3 (pd − 1)2 p −1 + 2g(2a + d) + 2g(a + d) + 2g(d) + 2g(4a + 2d)g(2a + d) + g(4a + 2d)g(d) + g(3a + 2d)g(a + d) + 2g(3a + 2d)g(2a + d) + 2g(3a + 2d)g(d) + g(3a + 2d)g(a + d) ψ(a + d)2 ψ(4a + d) 2 . + 2g(2a + 2d)g(a + d) + g(2a + 2d)g(d) + 2g(a + d)g(2a + d) + g(d) − ψ(d)3 In the case of α − 32 d -expansion, we set a = α − 2d = ε − d2 , where ε = α − 32 d. The results for the vertex parts O(Ik ) are as follows (with Sd = (ψ(d) ln p)−1 ): 1 , O(I1 ) = Sd − 2ε 1 2 , O(I2 ) = Sd 4ε2 1 + 4pd/2 + pd 1 2 ln p , − O(I3 ) = Sd 8ε2 8ε(pd − 1) 1 + 4pd/2 + pd 1 + 6pd/2 + 16pd + 6p3d/2 + p2d 1 3 2 ln p − + (ln p) , O(I4 ) = Sd − 24ε3 12ε2 (pd − 1) 36ε(pd − 1)2 1 3 O(I5 ) = Sd − 3 , 8ε d/2 + pd + p3d/2 3 p 2 (ln p) , O(I6 ) = Sd 6ε(pd − 1)2 1 + 4pd/2 + pd 1 ln p , + O(I7 ) = Sd3 − 16ε3 16ε2 (pd − 1) 1 + 4pd/2 + pd 1 + 9pd/2 + 19pd + 9p3d/2 + p2d 1 2 ln p − + (ln p) , O(I8 ) = Sd3 − 48ε3 16ε2 (pd − 1) 18ε(pd − 1)2 1 + 4pd/2 + pd 1 + 6pd/2 + 16pd + 6p3d/2 + p2d 1 2 ln p − + (ln p) , O(I9 ) = Sd3 − 24ε3 24ε2 (pd − 1) 36ε(pd − 1)2 1 + 4pd/2 + pd 1 + 6pd/2 + 16pd + 6p3d/2 + p2d 1 2 ln p − + (ln p) , O(I10 ) = Sd3 − 24ε3 24ε2 (pd − 1) 36ε(pd − 1)2 1 + 10pd/2 + 20pd + 10p3d/2 + p2d 3 2 (ln p) . O(I11 ) = Sd − 6ε(pd − 1)2 In the framework of (4 − d)-expansion, we set d = 4 − δ and a = 2 + d − 2d = δ − 2. The results for the vertex parts in this case are 1 O(I1 ) = Sd − , δ PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
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M.D. MISSAROV, R.G. STEPANOV
O(I2 ) = Sd2 O(I3 ) =
Sd2
1 , δ2
1 + 4p2 + p4 1 ln p , − 2δ2 4δ(p4 − 1)
1 + 4p2 + p4 1 + 10p2 + p4 1 2 O(I4 ) = ln p − (ln p) , − 3+ 3δ 3δ2 (p4 − 1) 18δ(p2 − 1)2 1 O(I5 ) = Sd3 − 3 , δ 2 4 6 3 p +p +p 2 (ln p) , O(I6 ) = Sd 3δ(p4 − 1)2 1 + 4p2 + p4 1 3 ln p , O(I7 ) = Sd − 3 + 2δ 4δ2 (p4 − 1) 1 + 4p2 + p4 1 + 9p2 + 19p4 + 9p6 + p8 1 3 2 ln p − (ln p) , O(I8 ) = Sd − 3 + 6δ 4δ2 (p4 − 1) 9δ(p4 − 1)2 1 + 4p2 + p4 1 + 10p4 + p8 1 3 2 ln p − (ln p) , O(I9 ) = Sd − 3 + 3δ 6δ2 (p4 − 1) 18δ(p4 − 1)2 1 + 4p2 + p4 1 + 10p4 + p8 1 3 2 ln p − (ln p) , O(I10 ) = Sd − 3 + 3δ 6δ2 (p4 − 1) 18δ(p4 − 1)2 1 + 10p2 + 20p4 + 10p6 + p8 2 (ln p) . O(I11 ) = Sd3 − 3δ(p4 − 1)2 Sd3
Comparing the above-presented vertex parts in p-adic theory with those in Euclidean theory, we see that the leading coefficients of δ−1 in these polynomials coincide. The only exceptional case is I6 ; the point is that the corresponding Feynman integral contains the subintegral dx2 dx3 |x1 − x2 |a |x2 − x3 |3a |x3 − x4 |a , which diverges as δ → 0 in Euclidean theory but does not diverge in p-adic theory. REFERENCES 1. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (Nauka, Moscow, 1994; World Sci., Singapore, 1994). 2. E. Yu. Lerner, “Feynman Integrals with a p-adic Argument in Momentum Space. III: Renormalization,” Teor. Mat. Fiz. 106 (2), 233–249 (1996) [Theor. Math. Phys. 106, 195–208 (1996)]. 3. E. Yu. Lerner and M. D. Missarov, “p-Adic Feynman and String Amplitudes,” Commun. Math. Phys. 121 (1), 35–48 (1989). 4. M. D. Missarov, “Renormalization Group and Renormalization Theory in p-adic and Adelic Scalar Models,” in Dynamical Systems and Statistical Mechanics, Ed. by Ya. G. Sinai (Am. Math. Soc., Providence, RI, 1991), Adv. Sov. Math. 3, pp. 143–164. 5. E. Speer, “Ultraviolet and Infrared Singularity Structure of Generic Feynman Amplitudes,” Ann. Inst. Henri Poincare A 23 (1), 1–21 (1975).
Translated by O. Sipacheva
PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
Vol. 265
2009
