On conjectured local generalizations of anisotropic scale invariance and their implications

Rutkevich, S. B.; Diehl, H. W.; Shpot, M. A.

doi:10.1016/j.nuclphysb.2010.09.005

Cited by 12 publications

(23 citation statements)

References 54 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When D = 2, m = 1 we have the parameters a = 3 2 , b = 1 2 and, from (24), A 2,1 = (2 ) −1 . We have been unable to prove the equivalence between (28) and (13), but present an expansion process that supports our claim that these representations are the same. Then, from (28), we obtain…”

Section: The Case D = 2 M =mentioning

confidence: 43%

“…We notice that I 3, m ( p , q ) can also be expressed in terms of elementary functions as

I_{3, m} false(p, q false) = q^{- 4 + 2 ϵ} \frac{8 normalΓ false(1 - ϵ false)}{{false(16 π false)}^{ϵ}} \frac{1}{\sqrt{X}} frakturI {false(1 - i \sqrt{X} false)}^{- 1 + ϵ}

(another version of can be found in Rutkevich et al, (A40)), or

I_{3, m} false(p, q false) = q^{- 4 + 2 ϵ} \frac{8 normalΓ false(1 - ϵ false)}{{false(16 π false)}^{ϵ}} \frac{{false(1 + X false)}^{false(- 1 + ϵ false) false/ 2}}{\sqrt{X}} \sin ([], false(1 - ϵ false) \arcsin \sqrt{\frac{X}{1 + X}}),

cf Mergulhão and Carneiro ,. (A7) These last two representations follow from and by Prudnikov et al, , 7.3.1.107 and 7.3.1.91 respectively.…”

Section: Special Cases Corresponding To Integer Dmentioning

confidence: 99%

“…where (a) n = Γ(a + n)∕Γ(a) = a(a + 1) … (a + n − 1), (a) 0 = (0) 0 ∶= 1, is the Pochhammer symbol, or rising factorial. The integral J D (p; m 1 , m 2 ) can be expressed alternatively in terms of a linear combination of two Gauss hypergeometric functions, 34, (A7);29, (7) the Appell function F 4 , 29, (17) Horn function H 3 (see the literature 29, p20;32, (33) ), as well as different versions of the Appell function F 1 : Shpot 29, (28) and Kniehl and Tarasov 35, (7) ; see also Shpot and Pogány. 27, section 2.1 Calculations of different versions of the Feynman integral J D (p; m 1 , m 2 ) in (17) using different approaches are continuing to attract attention; see, for example, the literature.…”

Section: Evaluation Of I Dm (P Q) In the General Casementioning

confidence: 99%

“…The power expansion in (28) and functional expansion in (32) have been derived under the condition that X < 1, but analytic continuation can be applied to these sums to continue the results into the region X > 1. We shall perform this in the next subsections and also derive modified versions of these expansions suitable for numerical computation.…”

Section: Continuation Of the Expansions For I Dm (P Q) Into X > 1 Amentioning

confidence: 99%

“…To derive an analytic continuation of the power series expansion of Theorem 2, given by (28), we use the technique described in Rainville. 45, §38, pp58-59 This consists of exploiting the binomial expansion of (1 − z) − appearing on the right-hand side of Pfaff's transformation (39) to derive the simple Gauss function on the left of this relation.…”

Section: The Expansion (28) and Its Modified Formmentioning

confidence: 99%

See 4 more Smart Citations

A Feynman integral in Lifshitz‐point and Lorentz‐violating theories in

Paris

Shpot

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

We evaluate a 1‐loop, 2‐point, massless Feynman integral ID,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum RD⊕Rm. Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of ID,m(p,q) in terms of powers of the variable X:=4p2/q4, where p=|p|, q=|q|, bold-italicp∈RD, bold-italicq∈Rm, and in terms of generalised hypergeometric functions 3F2(−X), when X<1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.

show abstract

Section: The Case D = 2 M =mentioning

confidence: 43%

“…We notice that I 3, m ( p , q ) can also be expressed in terms of elementary functions as

I_{3, m} false(p, q false) = q^{- 4 + 2 ϵ} \frac{8 normalΓ false(1 - ϵ false)}{{false(16 π false)}^{ϵ}} \frac{1}{\sqrt{X}} frakturI {false(1 - i \sqrt{X} false)}^{- 1 + ϵ}

(another version of can be found in Rutkevich et al, (A40)), or

I_{3, m} false(p, q false) = q^{- 4 + 2 ϵ} \frac{8 normalΓ false(1 - ϵ false)}{{false(16 π false)}^{ϵ}} \frac{{false(1 + X false)}^{false(- 1 + ϵ false) false/ 2}}{\sqrt{X}} \sin ([], false(1 - ϵ false) \arcsin \sqrt{\frac{X}{1 + X}}),

cf Mergulhão and Carneiro ,. (A7) These last two representations follow from and by Prudnikov et al, , 7.3.1.107 and 7.3.1.91 respectively.…”

Section: Special Cases Corresponding To Integer Dmentioning

confidence: 99%

Section: Evaluation Of I Dm (P Q) In the General Casementioning

confidence: 99%

Section: Continuation Of the Expansions For I Dm (P Q) Into X > 1 Amentioning

confidence: 99%

Section: The Expansion (28) and Its Modified Formmentioning

confidence: 99%

See 3 more Smart Citations

A Feynman integral in Lifshitz‐point and Lorentz‐violating theories in

Paris

Shpot

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

Boundary conformal field theory at the extraordinary transition: The layer susceptibility to O(ε)

Shpot

2021

J. High Energ. Phys.

View full text Add to dashboard Cite

We present an analytic calculation of the layer (parallel) susceptibility at the extraordinary transition in a semi-infinite system with a flat boundary. Using the method of integral transforms put forward by McAvity and Osborn [Nucl. Phys. B455 (1995) 522] in the boundary CFT, we derive the coordinate-space representation of the mean-field propagator at the transition point. The simple algebraic structure of this function provides a practical possibility of higher-order calculations. Thus we calculate the explicit expression for the layer susceptibility at the extraordinary transition in the one-loop approximation. Our result is correct up to order O(ε) of the ε = 4 − d expansion and holds for arbitrary width of the layer and its position in the half-space. We discuss the general structure of our result and consider the limiting cases related to the boundary operator expansion and (bulk) operator product expansion. We compare our findings with previously known results and less complicated formulas in the case of the ordinary transition. We believe that analytic results for layer susceptibilities could be a good starting point for efficient calculations of two-point correlation functions. This possibility would be of great importance given the recent breakthrough in bulk and boundary conformal field theories in general dimensions.

show abstract